The A Priori In The Thought Of Descartes: Cognition, Method And Science 1443817007, 9781443817004, 1443893579, 9781443893572

Table of contents :
Table of Contents......Page 6
Acknowledgements......Page 9
Note on Quotations and Bibliography......Page 10
Introduction......Page 14
Chapter One......Page 18
1.1 Cogitatio and Its Modes......Page 21
1.1.1 Understanding, Will, and Judgments......Page 24
1.1.2 Understanding and Its Modes......Page 34
1.2 The Modes of Understanding, and Innate Ideas......Page 42
1.2.1 Innate Ideas and Dispositions......Page 48
1.2.2 Innate Ideas and Implicit Apprehension......Page 51
1.3 Clear and Distinct Perception, Certainty, and Scientia......Page 60
1.3.1 Compelled Assent......Page 66
1.3.2 Metaphysical Certainty and Normativity......Page 69
1.3.3 Clear and Distinct Perception, and Scientia......Page 73
1.4 Breaking Down the Cartesian Circle......Page 81
2.1 Understanding as the Principle of Scientia......Page 88
2.2 Intuitus, Understanding, and Experientia......Page 93
2.3 The Objects of Intuitus......Page 99
2.3.1 Simple Natures......Page 102
2.3.2 Compositiones as Objects of Intuitus......Page 112
2.4 The Root of Objective Necessity, and Scientia......Page 120
2.5 Deductio......Page 130
2.5.1 Construing Deductio in Terms of Intuitus......Page 132
2.5.2 Throwing Away the Dialecticorum Vincula......Page 138
2.6.1 Approbative and Heuristic Enumeration......Page 160
2.6.2 Enumeration, Sufficiency, and Ordering......Page 164
2.6.3 Inductio, Deductio, and Enumeratio......Page 168
3.1 Links to Mathematical Contexts......Page 174
3.1.1 Mathematics as a Paradigm of a Universal Method......Page 178
3.1.2 The A Priori in Purely Mathematical Contexts......Page 184
3.1.3 The Method of Analysis in Diophantus and Pappus......Page 188
3.1.4 The “Algebre des modernes”......Page 193
3.2 Analysis in Descartes�� Mathematics......Page 203
3.2.1 A Point of Comparison: Viète��s Logistice Speciosa......Page 204
3.2.2 Descartes�� Algebraization of Geometry......Page 208
3.2.3 Algebra as an Analytical Problem-Solving Procedure......Page 218
3.2.4 Analysis and Synthesis in Pappus......Page 226
3.2.5 Descartes�� Approach......Page 232
3.2.6 Scientia and Imagination in Descartes’ Mathematics......Page 240
Chapter Four......Page 251
4.1 Mathematicæ, Mathesis Vniversalis and a Universal Method......Page 253
4.1.1 A Textual Problem......Page 255
4.1.2 The Meaning of Mathesis vniversalis in Descartes......Page 258
4.2 A Talk of the Method......Page 273
4.3 A Reconstruction of the Universal Method......Page 278
4.3.1 The General Modus Operandi......Page 281
4.3.2 The Præparatio Comparationum......Page 292
4.4 Justification and Possibility of Method......Page 308
4.4.1 The Justification Task......Page 312
4.5 Universality of the Method and the Unity of the Scientiæ......Page 320
Chapter Five......Page 330
5.1 The A Priori and the A Posteriori in the Aristotelian Tradition......Page 332
5.2 Some Aristotelian Strata in Descartes’ Conception......Page 336
5.3 A Clash with the Aristotelians......Page 339
5.3.1 Analysis as an Approbative Tool in the Aristotelian Tradition......Page 341
5.3.2 Heuristic Analysis in Aristotle......Page 343
5.3.3 Heuristic Analysis in Renaissance Aristotelianism......Page 345
5.3.4 Some Similarities to Descartes on Analysis......Page 350
5.4.1 Culs-de-Sac......Page 352
5.4.2 A Speculative Suggestion......Page 357
5.5 Coda: Synthesis as A Posteriori in Descartes......Page 364
Appendix: Abbreviations......Page 369
Bibliography......Page 371
General Index......Page 386

Citation preview

The a priori in the Thought of Descartes

The a priori in the Thought of Descartes: Cognition, Method and Science By

Jan Palkoska

The a priori in the Thought of Descartes: Cognition, Method and Science By Jan Palkoska This book first published 2017 Cambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright © 2017 by Jan Palkoska All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN (10): 1-4438-1700-7 ISBN (13): 978-1-4438-1700-4

The work on this book was supported by research grant GAP401/11/0371 “A priori, synthetic and analytic” awarded to the Institute of Philosophy, Academy of Sciences of the Czech Republic, and to the Faculty of Philosophy, Charles University, Prague, by the Czech Science Foundation.

TABLE OF CONTENTS

Acknowledgements .................................................................................. viii Note on Quotations and Bibliography ........................................................ ix Introduction ................................................................................................ 1 Chapter One ................................................................................................ 5 Cognition and Scientia 1.1 Cogitatio and Its Modes ................................................................... 8 1.1.1 Understanding, Will, and Judgments..................................... 11 1.1.2 Understanding and Its Modes ................................................ 21 1.2 The Modes of Understanding, and Innate Ideas ............................. 29 1.2.1 Innate Ideas and Dispositions ................................................ 35 1.2.2 Innate Ideas and Implicit Apprehension ................................ 38 1.3 Clear and Distinct Perception, Certainty, and Scientia .................. 47 1.3.1 Compelled Assent ................................................................. 53 1.3.2 Metaphysical Certainty and Normativity .............................. 56 1.3.3 Clear and Distinct Perception, and Scientia .......................... 60 1.4 Breaking Down the Cartesian Circle.............................................. 68 Chapter Two ............................................................................................. 75 Understanding and Scientia 2.1 Understanding as the Principle of Scientia .................................... 75 2.2 Intuitus, Understanding, and Experientia....................................... 80 2.3 The Objects of Intuitus................................................................... 86 2.3.1 Simple Natures ...................................................................... 89 2.3.2 Compositiones as Objects of Intuitus .................................... 99 2.4 The Root of Objective Necessity, and Possibility of Scientia ...... 107 2.5 Deductio ....................................................................................... 117 2.5.1 Constructing Deductio in Terms of Intuitus ........................ 119 2.5.2 Throwing Away the Dialecticorum Vincula ....................... 125 2.6 Enumeration ................................................................................. 147 2.6.1 Approbative and Heuristic Enumeration ............................. 147 2.6.2 Enumeration, Sufficiency, and Ordering ............................. 151 2.6.3 Inductio, Deductio, and Enumeratio ................................... 155

vi

Table of Contents

Chapter Three ......................................................................................... 161 The a priori in Descartes: The Mathematical Line 3.1 Links to Mathematical Contexts .................................................. 161 3.1.1 Mathematics as a Paradigm of a Universal Method ............ 165 3.1.2 The A Priori in Purely Mathematical Contexts ................... 171 3.1.3 The Method of Analysis in Diophantus and Pappus ........... 175 3.1.4 The “Algebre des modernes”............................................... 180 3.2 Analysis in Descartes’ Mathematics ............................................ 190 3.2.1 A Point of Comparison: Viète’s Logistice Speciosa............ 191 3.2.2 Descartes’ Algebraization of Geometry .............................. 195 3.2.3 Algebra as an Analytical Problem-Solving Procedure ........ 205 3.2.4 Analysis and Synthesis in Pappus ....................................... 213 3.2.5 Descartes’ Approach ........................................................... 219 3.2.6 Scientia and Imagination in Descartes’ Mathematics .......... 227 Chapter Four ........................................................................................... 238 Towards a Universal Method of Discovery 4.1 Mathematicæ, Mathesis Vniversalis and a Universal Method ..... 240 4.1.1 A Textual Problem .............................................................. 242 4.1.2 The Meaning of Mathesis Vniversalis in Descartes ............ 245 4.2 A Talk of the Method: the Discours and the Essais ..................... 260 4.3 A Reconstruction of the Universal Method.................................. 265 4.3.1 The General Modus Operandi ............................................. 268 4.3.2 The Præparatio Comparationum ........................................ 279 4.4 Justification and Possibility of Method ........................................ 295 4.4.1 The Justification Task ......................................................... 299 4.5 Universality of the Method and the Unity of the Scientiæ .......... .307 Chapter Five ........................................................................................... 317 The a priori in Descartes: Integrating the Aristotelian Line 5.1 The A Priori and the A Posteriori in the Aristotelian Tradition ..... 319 5.2 Some Aristotelian Strata in Descartes’ Conception ..................... 323 5.3 A Clash with the Aristotelians ..................................................... 326 5.3.1 Analysis as an Approbative Tool in the Aristotelian Tradition ................................................................................. 328 5.3.2 Heuristic Analysis in Aristotle ............................................ 330 5.3.3 Heuristic Analysis in Renaissance Aristotelianism ............. 332 5.3.4 Some Similarities to Descartes on Analysis ........................ 337 5.4 Analysis as A Priori in Descartes................................................. 339 5.4.1 Culs-de-Sac ......................................................................... 339 5.4.2 A Speculative Suggestion .................................................... 344 5.5 Coda: Synthesis as A Posteriori in Descartes .............................. 351

The a priori in the Thought of Descartes

vii

Appendix: Abbreviations......................................................................... 356 Bibliography ............................................................................................ 358 General Index .......................................................................................... 373

ACKNOWLEDGEMENTS

The research which underlies this book was facilitated by Czech Science Foundation grants from 2011 to 2015. In 2013—the year crucial to the shaping of this work—I was given a sabbatical by Charles University in Prague, followed by an additional six months free from teaching duties. I owe a great debt of gratitude to both these institutions. Various portions of earlier drafts of the book were presented at meetings of the Charles University Centre for the Study of Classical and Medieval Thought; I wish to thank its participants—that is, my colleagues—for their pertinent and valuable comments, which prompted me to reassess several parts of the book. I also had more than one opportunity to present the central themes of the book to my graduate courses at Charles University; I thank my students for their penetrating questions and stimulating discussions, which helped me to remain sensitive to the complexities and intricacies of Descartes’ thought. My thanks go to Michael Pockley, who read the whole manuscript more than once and not only corrected my mediocre English but also raised some important questions concerning my argument. Finally, I am deeply grateful to my partner Mirka, first for putting up with my spending so much time with a great French philosopher instead of with her and our children; but also for the fact that being a philosopher herself, she understands better than anyone else what I have been after in my Descartes project and her deep understanding has greatly helped sustain the work. It is to her that I dedicate this book.

NOTE ON QUOTATIONS AND BIBLIOGRAPHY

The abbreviations I employ are listed in the Appendix. I cite Descartes in original French and Latin, following the canonical Œuvres de Descartes, edited by Charles Adam and Paul Tannery, 11 vols. (Paris: J. Vrin, 1897–1913), in the form: AT , ; in referring to the Regulæ ad directionem ingenii, I occasionally indicate also the line number, in the form ... .. My insertions in quotations are enclosed in square brackets. I strictly follow the wording and spelling of the standard AT edition with no efforts to amend the texts either in view of modern standards or in order to remove occasional inconsistencies concerning diacritics, accents and similar matters. This includes even the titles of Descartes‫ ތ‬works. In referring to Descartes’ correspondence, I standardly use the italicized name of Descartes’ correspondent preceded by a preposition according to the AT edition (e.g. A Regius; Ad Vœtium). The only exceptions are Descartes’ letters to Mersenne and Descartes’ famous exchange with the pseudonymous Hyperaspistes; I refer to these items with abbreviations included in the above-mentioned list. The dates of Descartes’ texts, if given, are put in square brackets at the end of the reference. In general, I rely on the AT edition as regards dating Descartes’ works. The only exception is the Regulæ ad directionem ingenii: in dealing with the dating and/or chronology of some passages from this work, I take into account several suggestions made in Descartes scholarship after the appearance of the seminal work by Jean-Paul Weber. I cite texts by authors other than Descartes in the original, with the following exceptions. One is a commentary on the Regulæ in Czech by JiĜí Fiala in René Descartes, Regulæ ad directionem ingenii / Pravidla pro vedení rozumu, transl. VojtČch Balík (Prague: Oikoymenh, 2000), which I quote in my own English translation. The other are Greek authors— Aristotle, Sextus Empiricus, Diophantus of Alexandria, Pappus of Alexandria and Proclus Diadochus. I cite Aristotle’s works in standard English translations included in The Complete Works of Aristotle: The Revised Oxford Translation edited by Jonathan Barnes, 2 vols. (Princeton: Princeton University Press, 1984). Sextus’ ȆȣȡȡȫȞİȚȠȚ ‫ބ‬ʌȠIJȣʌȫıİȚȢ is cited in an up-to-date English translation, Outlines of Scepticism translated and edited by Julia Annas and Jonathan Barnes (Cambridge: Cambridge

x

Note on Quotations and Bibliography

University Press, 2000). Diophantus’ Arithmetica is quoted in an English translation by J. Winfree Smith which occurs in a translation by the same author of Viète’s In artem analyticem Isagoge, supplemented to Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (2nd ed. New York: Dover Publications, 1992). Pappus’ ȈȣȞĮȖȦȖȒ is cited in a Latin translation Mathematicæ Collectiones by Federico Commandino (Venice: Francisco de Franciscis Senens, 1588), and Proclus’ commentary to Euclid’s ȈIJȠȚȤİ߿Į is cited in a Latin translation by Francesco Barozzi (Padua: Gratiosus Perchacinus, 1560); it was these Latin editions in which Pappus and Proclus were presumably read by Descartes. In general, I use italics for Latin single words and phrases inserted in the body of the main text and in footnotes. For all more extended Latin quotations in the main text and footnotes I use normal font. French single words, phrases and quotations are uniformly presented in normal font. I occasionally also use italics to emphasize various words, phrases or sentences in English, both in my text and in quotations. I have chosen to take certain technical terms directly from Descartes’ texts and retain them in their original Latin or French form. If such terms are started with a capital letter by Descartes, I keep his capitalization throughout the text. As for transcription of Arabic names, I simply apply the way in which these are transcribed by Chikara Sasaki, Descartes’s Mathematical Thought (Dordrecht: Kluwer Academic Publishers, 2003), which has been my principal source as regards the history of algebra in the Arabic world. Finally, given the importance of the Regulæ ad directionem ingenii to the main topic of the present book, the somewhat complicated situation concerning the text of the Regulæ must briefly be considered. Having never been published during Descartes’ lifetime, nor even mentioned by him in his extant correspondence or texts, the original manuscript of the Regulæ was lost soon after Descartes’ death and serious scholarship is left to work with a few extant versions which are based upon immediate copies of the original manuscript at best, and more probably even upon copies from copies. The Regulæ were first published in 1684 in a Dutch translation under the title Regulen van de bestieringe des verstants (Amsterdam: Jan Rieuwertsz), which is now customarily referred to as the N-version of the Regulæ; an early copy of the original manuscript of the Regulæ which served as the base for the translation is lost as well. The Latin edition of the Regulæ did not appear until 1701 when it was included in Descartes’ Opuscula posthuma, physica et mathematica (Amsterdam: P. & J. Blaev). This edition is now referred to as the A-version of the Regulæ; it is unknown which copy of the original manuscript served as the base for this edition and whether the copy was identical with the basis for

The a priori in the Thought of Descartes

xi

the N-version. Finally, there is a copy purchased by Leibniz in Amsterdam some time between 1670 and 1678. This copy was discovered among Leibniz’s papers in Hanover in the mid of nineteenth century and published in Œuvres inédites de Descartes précédées d’une introduction sur la méthode edited by Louis Foucher de Careil, vol. 1 (Paris: Auguste Durand, 1859); this copy is now referred to as the H-version of the Regulæ. It is controversial and in an obvious sense indeterminable which of these three versions stands closest to the unavailable original manuscript by Descartes’ own hand. What is important for our purposes is just that the AT editors take the A-version as the basic source, consulting occasionally the H-version whilst ignoring entirely the N-version. Although this approach has been disputed by numerous Descartes scholars and several alternative strategies have been adopted in editing critically the text of the Regulæ, for the sake of uniformity I take the AT edition of the Regulæ as the basis for this study. Whenever necessary, I deal in the footnotes with suggestions for alternative readings. The most important critical editions that supersede in various respects the pioneering AT edition of the Latin text of the Regulæ include Regulæ ad Directionem Ingenii: Texte critique établi par Giovanni Crapulli avec la version Hollandaise du XVIIème siècle edited by Giovanni Crapulli (The Hague: Martinius Nijhoff, 1966); Regulæ ad Directionem Ingenii / Regeln zur Ausrichtung der Erkenntniskraft: Kritisch revidiert, übersetzt und herausgegeben edited and translated by Heinrich Springmeyer, Lüder Gäbe and Hans Zekl (Hamburg: Felix Meiner, 1973). I also consult an annotated French translation by Jean-Luc Marion: Règles utiles et claires pour la Direction de lҲesprit et la recherche de la vérité: Traduction selon le lexique cartésien, et annotation conceptuelle par Jean-Luc Marion avec notes mathématiques de Pierre Costabel (The Hague: Martinius Nijhoff, 1977); and the most recent bilingual Latin-English edition: Regulae Ad Directionem Ingenii / Rules for the Direction of the Natural Intelligence: A Bilingual Edition of the Cartesian Treatise on Method edited and translated by George Heffernan (Amsterdam: Rodopi, 1998). For a detailed account of the situation concerning the history of the text of the Regulæ and the sources for its reconstruction, see in particular Giovanni Crapulli, “Introduction,” in Descartes, Regulæ: Texte critique, xi–xxxviii; Christian Wohlers, “Einleitung,” in Descartes, Regulæ / Regeln, xxvii– lxxxvii. For a brief and condensed up-to-date survey of the situation, see George Heffernan, “Introduction: A Contextualization of the Text,” in Descartes, Regulæ / Rules, 47–54.

INTRODUCTION

My chief aim in the present study is to determine and explicate the meaning (or meanings) Descartes associates with the terms “a priori” and “a posteriori” and explore its (or their) import for relevant aspects of Descartes‫ ތ‬overall philosophical and/or scientific stance. It has been acknowledged by several specialists in the field that while Descartes‫ތ‬ usage of the pair of terms in question is at odds with the now current Kantian meaning of the a priori–a posteriori distinction, the bulk of evidence points towards the fact that Descartes’ usage does not square well, despite superficial verbal similarities, with the standard Aristotelianscholastic notion either.1 However, there is as yet little if any agreement, among those who grant or at least consider the existence of these discrepancies, as to the exact positive meaning Descartes wished to associate with the terms in question and thus, by the same token, as to the exact nature of Descartes‫ ތ‬departure from the Aristotelian conception2 and thus to the relationship of Descartes‫ ތ‬and Kant‫ތ‬s views on this score. In view of this, I wish to offer my own suggestions on at least some of these difficult interpretative issues. It should become clear in the course of the present study that the topic is of considerable interest both to those active in the interpretation of Descartes‫ ތ‬thought and those engaged in the history of philosophy from Aristotelian scholasticism to Kant. As to the former field, we shall see that our questions bear directly, among other things, upon the nature of the method Descartes claims to have discovered and employed in developing his mature metaphysics, physics, and all the other branches of the allegedly unitary scientia; and as to the latter, the answers to our questions might help, in the long run, to shed some light upon the challenging and 1

At least Stephen Gaukroger, Cartesian Logic: An Essay on Descartes’s Conception of Inference (Oxford: Oxford University Press, 1989), 99–102 and Roger Florka, DescartesҲs Metaphysical Reasoning (New York: Routledge, 2001), 69–89 and 109–17 are crystal clear on this negative point. 2 Besides the two authors mentioned in the previous footnote, the suggestions of Benoît Timmermans, “The Originality of Descartes‫ތ‬s Conception of Analysis as Discovery,” Journal of the History of Ideas 60, no. 3 (1999), 433–47 are worth noting.

2

Introduction

strangely neglected question of why Kant decided to employ the a priori– a posteriori distinction in a way which diverges so dramatically from the meaning so well established in the long Aristotelian-scholastic tradition known to him. Yet it is, of course, one thing to ask what Descartes might have meant by the terms “a priori” and “a posteriori”, and quite another to ask what if anything about Descartes‫ ތ‬use of these terms actually moved Kant to employ them in the way that he did; and the responses to each of these questions are by no means bound to be coextensive. At any rate, it is solely the former of these queries that I intend to tackle directly in the present study. The latter question is to be understood as acting merely as the chief motivational goal of the entire enterprise: I do not pretend to be in a position to answer it positively even should I succeed in answering the former. There are fifteen occurrences of the terms “a priori” and/or “a posteriori” in Descartes‫ ތ‬extant corpus.3 Even a brief initial survey reveals that Descartes prima facie employs the terms in a considerably uniform manner: as adjectives or adverbs, respectively, the terms in question modify most frequently (manners of) demonstration,4 and occasionally also (manners of) reasoning, proof, explication, deduction, investigation, and (the process of) cognition.5 It is the last item of this cluster in terms of which Descartes‫ ތ‬usage of the a priori–a posteriori pair can be rendered unified in a certain important respect: the general context is clearly that of gaining a (presumably somehow specific sort of) cognition (connoissance, cognitio); and the other terms of the cluster denote various aspects or moments or kinds of the corresponding cognitive operations or processes. It will soon become clear that the specific sort of cognition with which Descartes is properly concerned in the contexts in which the a priori–a posteriori pair enters on stage is what he generally calls scientia, i.e. the cognition that provides for certain, evident, and true judgments, or else for

3

Viz. Mers., AT I, 250–51; 489–90; AT II, 31; 432–33; AT III, 82; a Plempius, AT I, 476; au P. Vatier, AT I, 563; a M. de Beaune, AT II, 514; Hyp., AT III, 422– 23; a Regius, AT III, 505–506; a Boswell(?), AT IV, 689; Burm., AT V, 153; Resp. 2, AT VII, 155–57; Resp. 5, 358; Le Monde, AT XI, 47. 4 AT I, 476, AT III, 422: demonstrare; AT I, 489: façon de demonstrer; AT I, 563, AT II, 31, AT XI, 47: demonstration(s); AT III, 505: rationes, siue demonstrationes; AT VII, 155–56: rationes demonstrandi. 5 AT III, 82: raison; 505: rationes, siue demonstrationes; AT V, 153: argumentum; AT IV, 689: probatio; AT I, 476: explicatio; AT II, 514: deduction; AT VII, 358: investigatio; AT I, 250–51: connoissance; AT II, 433, AT XI, 47: connoistre.

The a priori in the Thought of Descartes

3

a more or less complex system of such judgments.6 Thus it sounds a reasonable point of departure to take the a priori and the a posteriori in Descartes, as regards their general function, as modifying (either in the process sense or in the product sense) various ways of gaining scientific cognition, and by analogy the resulting product, viz. a gained scientia itself. Unfortunately, the situation is much less straightforward with regard to the question of the very meaning of the a priori–a posteriori pair in Descartes. This is above all due to two closely interconnected facts. Firstly, while Descartes’ usage is thematically and functionally unified in the general way we have just indicated, he employs the terms in question in very different cognitive fields, most importantly in mathematical, physical,7 and metaphysical contexts; and it is far from clear that enough common ground could be extracted from these diverse fields to keep the meaning of the a priori and/or the a posteriori one and the same when passing from one field to another. Secondly, at least two distinct intellectual strains seem to lie in the background of Descartes‫ ތ‬notion(s) of the a priori and the a posteriori, viz. the Aristotelian conception of scientific reasoning on the one hand, and, on the other hand, the mathematical strains involving an ancient tradition of mathematical analysis and modern conceptions of algebra. While it shall turn out clear enough that each of these strains undergoes certain significant transformations in Descartes‫ ތ‬hands and that Descartes wishes both of 6 Such a general notion of scientia comes out particularly distinctly in Reg. I–IV. See especially Reg. II, AT X, 362: “Omnis scientia est cognitio certa & evidens .... Atque ita per hanc propositionem rejicimus illas omnes probabiles tantùm cognitiones, nec nisi perfectè cognitis, & de quibus dubitari non potest, statuimus esse credendum.” 7 By physics (and its grammatical kin) I will henceforth refer to what Descartes himself normally calls Physique or Physica, i.e., roughly, (i) to Descartes‫ތ‬ fundamental investigation of material reality with respect to motion and rest to be found above all in the bulk of Princ. II and in certain portions of Le Monde, and (ii) to Descartes‫ ތ‬employment of the results of (i) in explaining various material phenomena which can be found above all in the bulk of Princ. III and IV, in La Dioptrique and in Les Meteores. Roughly speaking, an essential contrast between Descartes’ physics on the one hand and, on the other, mathematics as practiced by him is drawn by Descartes in metaphysical terms; as he puts it in Burm., AT V, 160, “differentia [inter objectum Matheseos et objectum Physices] in eo solùm est, quod Physica considerat objectum suum verum et reale ens sed tanquam actu et quâ tale existens, Mathesis autem solùm quâ possibile, et quod in spatio actu non existit, at existere tamen potest” (“non solùm tanquam” is a plausible conjecture by the AT editors).

4

Introduction

them somehow to interplay in the relevant contexts, it shall turn out far from clear, once again, how exactly he thought this could work and what consequences are to be drawn with regard to the meaning(s) of the a priori and the a posteriori. As I see it, the former worry can only be addressed appropriately if it is clarified how Descartes actually proceeds, in relevant respects, in mathematics, physics and metaphysics respectively and also if the latter complication concerning the indicated interplay between the two historical strains is addressed in some detail. In view of the overwhelming complexities which the topic as a whole eventually brings about, and for reasons of space, I deliberately limit myself to the latter of the two aforementioned tasks. This, of course, is likely to amount to the most significant limitation to the present study. Furthermore, before I set out to unravel the conundrums in which the complexities of the relevant tasks are likely to result, it is necessary to outline the essentials of Descartes‫ތ‬ general conception of those matters to which, arguably, he ascribes the characterization of a priori: that is to say, scientific knowledge (scientia), and even more generally, cognition. The structure of the study is thus roughly as follows. Assuming that for Descartes the a priori has to do with gaining a specific kind of cognition, namely the so-called scientia, the task of Chapter One is to discuss Descartes‫ ތ‬general conception of cognition, explain the sense in which scientia counts as a privileged kind of cognition and secure the possibility of scientia in view of Descartes‫ ތ‬own commitments. The aim of Chapter Two is to discuss the human cognitive faculties that to Descartes are capable of and responsible for the scientiæ in the sense specified in the previous chapter, and to consider how those faculties are put to work to bring about scientiæ. Chapter Three pursues a salient strain that is arguably at work in the constitution of the meaning Descartes associates with the term “a priori”, namely deployment of a method derived from his re-interpretation and extension of analysis as a heuristic procedure in mathematics. In Chapter Four I try to provide a general account of how the method of analysis based upon the algebraic paradigm is supposed to be put to work in Descartes. Finally, the aim of Chapter Five is to integrate into the meaning of the terms “a priori” and “a posteriori”, as it will have issued from previous chapters, the causal strata of the Aristotelian meaning of the a priori–a posteriori distinction.

CHAPTER ONE COGNITION AND SCIENTIA

Familiarly enough, Descartes‫ ތ‬concerns with epistemological issues and the ontology of cognition are motivated by his wide-ranging programmatic ambition—itself framed, however provisionally, by the practical goals of individual happiness (to be attained through “acquierir toutes les vertus, & ensemble tous les autres biens, qu‫ތ‬on puisse acquerir,” DM 3, AT VI, 28) and of the envisaged “bien general de tous les hommes” (DM 6, AT VI, 61)1—to provide the entire body of human cognition with solidity and firmness by way of ensuring that, ideally, each speculative cognitive act be marked with certainty, evidence and truth.2 According to Descartes, two closely interconnected essential moments are entailed in such a fundamental project: firstly, one is to enter into the labour of overturning all one has thus far admitted as true and rebuild one‫ތ‬s body of cognition 1

As for the individual goals, see in particular the splendid passage concerning the fourth prescript of Descartes‫ ތ‬provisional morals in DM 3, AT VI, 27–28, and Reg. I, AT X, 361: “[Q]uæramus scientias vtiles ad vitæ commoda, vel ad illam voluptatem, quæ in veri contemplatione reperitur, & quæ fere vnica est integra & nullis turbata doloribus in hac vitâ felicitas.” As for the latter, collective goals, see in particular DM 6, AT VI, 61–62, and also ibid., 68–69 and 78. However, the present study is surely not the place to get involved in any detailed discussion of the practical framing of Descartes‫ ތ‬theoretical intellectual program. 2 This point (like those immediately following) is so familiar that it scarcely needs documentation. It comes out concisely e.g. in the first two precepts of the Regulæ. See Reg. I, AT X, 359: “Studiorum finis esse debet ingenij directio ad solida & vera, de ijs omnibus quæ occurrunt, proferenda judicia.” Reg. II, AT X, 362: “Circa illa tantùm objecta oportet versari, ad quorum certam & indubitatam cognitionem nostra ingenia videntur sufficere” Cf. also DM 1–2, especially the first methodical precept in DM 2, AT VI, 18: “Le premier [precepte] estoit de ne receuoir iamais aucune chose pour vraye, que ie ne la connusse euidemment estre telle: c‫ތ‬est a dire, d‫ތ‬euiter soigneusement la Precipitation, & la Preuention; & de ne comprendre rien de plus en mes iugemens, que ce qui se presenteroit si clairement & si distinctement a mon esprit, que ie n‫ތ‬eusse aucune occasion de le mettre en doute.”

6

Chapter One

anew on solid and firm foundations;3 and secondly, one is to find a correct and unitary method the employment of which would ensure that the rebuilding in which one is engaged keeps the prescribed course.4 Furthermore, Descartes makes it clear on several occasions that while beliefs concerning the nature of human cognition as such are not, of course, spared the all-embracing meliorative demolition plan, the question of the nature and scope of human cognition is the very first to be dealt with once an attempt at re-building the fabric of our knowledge upon firm foundations is set out. Thus he writes in Reg. VIII, AT X, 397–98:5 3

See Med. I, AT VII, 17: “Animadverti jam ante aliquot annos quàm multa, ineunte ætate, falsa pro veris admiserim, & quàm dubia sint quæcunque istis postea superextruxi, ac proinde funditus omnia semel in vitâ esse evertenda, atque a primis fundamentis denuo inchoandum, si quid aliquando firmum & mansurum cupiam in scientiis stabilire.” See Harry Frankfurt, Demons, Dreamers, & Madmen (Indianapolis: Bobbs-Merrill Company, 1970), ch. 2 for a brilliant discussion of the chief interpretative problems that Descartes‫ ތ‬general overthrow of belief brings about. 4 One question that will not be considered at all in this study is what exactly motivates Descartes‫ ތ‬conviction that such a radical re-establishment of literally the entire body of human beliefs is needed semel in vitâ; Descartes‫ ތ‬intention to offer a viable alternative to Aristotelian natural science via attacks on the basically empirical commonsense epistemology, and Descartes‫ ތ‬endeavour to render at least some types of human beliefs immune to attacks of the then revived radical scepticism, appear to count as the most plausible initial responses which seem, for that matter, not to preclude one another. For the most convincing expositions that emphasize the former motivation (without eschewing the latter altogether, however), see Étienne Gilson, René DescartesҲ Discours de la méthode: texte et commentaire (Paris: J. Vrin, 1925), part 2, ch. 1; Margaret Wilson, Descartes (London: Routledge, 2005), electronic edition, ch. 1; Daniel Garber, “Semel in vita: The Scientific Background to Descartes‫ ތ‬Meditations,” in Essays on DescartesҲ Meditations, ed. Amélie Rorty (Berkeley: University of California Press, 1986), 81–116; James Hill, Descartes and the Doubting Mind (New York: Continuum International Publishing Group, 2012), ch. 3–4. For classical accounts trading mainly upon the latter motivation, see Edwin Curley, Descartes against the Skeptics (Cambridge: Harvard University Press, 1978); Richard Popkin, The History of Scepticism from Erasmus to Descartes (Assen: Van Gorcum & Comp., 1960), ch. 9–10. 5 Cf. also what Descartes writes a page or two earlier in the same Regula (the passage is in fact an earlier attempt at treating the same issue; see ch. 4, fn. 104): Si quis pro quæstione sibi proponat, examinare veritates omnes, ad quarum cognitionem humana ratio sufficiat (quod mihi videtur semel in vitâ faciendum esse ab ijs omnibus, qui seriò student ad bonam mentem pervenire), ille profectò per regulas datas inveniet nihil priùs cognosci

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[C]ùm in his initijs nonnisi incondita quædam præcepta, & quæ videntur potiùs mentibus nostris ingenita, quàm arte parata, poterimus invenire, ... ijsdem priùs vtendum ad alia, quæcumque ad veritatis examen magis necessaria sunt, summo studio perquirenda .... At verò nihil hîc vtilius quæri potest, quàm quid sit humana conditio & quousque extendatur. Ideoque nunc hoc ipsum vnicâ quæstione complectimur, quam omnium primam per regulas jam antè traditas examinandam esse censemus; idque semel in vitâ ab vnoquoque ex ijs, qui tantillùm amant veritatem, esse faciendum (my emphases).

It is, as far as I can see, precisely this insight that lies behind Descartes‫ތ‬ notorious turn to the question of quisnam sim ego ille, qui jam necessario sum in Med. II, AT VII, 25: [O]mnibus satis superque pensitatis, denique statuendum sit hoc pronuntiatum, Ego sum, ego existo, quoties a me profertur, vel mente concipitur, necessario esse verum. Nondum verò satis intelligo, quisnam sim ego ille, qui jam necessario sum; deinceps que cavendum est ne forte quid aliud imprudenter assumam in locum meî, sicque aberrem etiam in eà cognitione, quam omnium certissimam evidentissimamque esse contendo.

Let us have a closer look, therefore, at what constitute the essentials of the positive views at which Descartes arrives concerning the nature and scope of human cognition.6

posse quàm intellectum, cùm ab hoc cæterorum omnium cognitio dependeat, & non contrà; perspectis deinde illis omnibus quæ proximè sequuntur post intellectûs puri cognitionem, inter cætera enumerabit quæcumque alia habemus instrumenta cognoscendi præter intellectum .... Omnem igitur collocabit industriam in distinguendis & examinandis illis ... cognoscendi modis ...” (AT X, 395–96). Also cf. RV, AT X, 505: “Poliandre. –Dites-nous donc aussy l‫ތ‬ordre que vous tiendrés pour expliquer chasque matiere. Eudoxe. –Il faudra commencer par l‫ތ‬ame raisonnable, pour ce que c‫ތ‬est en elle que reside toute nostre connoissance; & [par considerer] sa nature & ses effets ....” 6 Anything pretending to count as a full account of the issue is, of course, far beyond the scope of the present study. What follows is just an outline of Descartes‫ތ‬ complex conception. I do nonetheless take a stand on several significant points of controversy in contemporary Descartes scholarship and try to defend my interpretative conclusions.

8

Chapter One

1.1 Cogitatio and Its Modes Descartes conceives of cognition in general as a matter of acts or operations of thinking (cogitatio).7 Cogitatio in the material sense of a faculty, or active potentiality,8 plays for Descartes the rôle of the essential attribute of the mind articulated ontologically as a res cogitans.9 Particular occurrent (and usually temporary) cogitationes are then articulated by Descartes as modes or acts or actions (modi, actus, actiones) of the

7

Of dozens of references, cf. e.g. Med. III, AT VII, 37; ibid., 40; Med.(f) III, AT IX-1, 29; Resp. 2, AT VII, 160; Princ. I, 32, AT VIII-1, 17; Mers., AT I, 366. 8 See Notæ, AT VIII-2, 358: “mens sive cogitandi facultas”; ibid., AT VIII-2, 361: “nomen facultatis nihil aliud quam potentiam designat”. The employment of the Aristotelian conceptual framework here is justified by the fact that Descartes himself invokes it whenever he wishes to seriously discuss the ontology of cognition: cf. in particular Med. III, AT VII, 40–42; Resp. 2, AT VII, 160–61; Resp. 4, AT VII, 232. 9 Cf. especially Med. II, AT VII, 27; Princ. I, 53, AT VIII-1, 25; Resp. 3, AT VII, 176; DM 4, AT VI, 32–33. At AT VII, 27, Descartes treats “mens” as synonymous with “animus, sive intellectus, sive ratio”; yet he seems eventually to settle, in the Meditationes and elsewhere, on “mens” as the most appropriate term. This should come as no surprise since at least intellectus sive ratio turns out to count as just one of two principal faculties with which mens sive res cogitans is endowed according to Descartes‫ ތ‬final verdicts in Med. IV and Princ. I, 32–34, AT VIII-1, 17–18; yet erroneous identification of understanding with the principal attribute of Descartes‫ތ‬ res cogitans keeps popping out in scholarly literature on Descartes—cf. e.g. Martial Guéroult, Descartes selon l'ordre des raisons, vol. 1 (Paris: Aubier, 1953), 63–67, 76–81; Marleen Rozemond, “The Role of the Intellect in Descartes‫ތ‬s case for the Incorporeity of the Mind,” in Essays on the Philosophy and Science of René Descartes, ed. Stephen Voss (New York: Oxford University Press, 1993), 106– 107.—As is rightly noted in Hill, Doubting Mind, 65–66, the absence of “anima” in the list of arguably synonymous terms is highly significant at any rate as “it is a sign that Descartes does not wish us to equate res cogitans with Aristotle‫ތ‬s soul, ‫ދ‬the first principle of living things‫ތ‬.... So the reference to mens at the beginning of the list [of synonyms in AT VII, 27] indicates that we are at least talking of the higher rational functions peculiar to humans.” This is confirmed almost verbatim by Resp. 7, AT VII, 491: “[Q]uæsivi [in Med. II] an aliquid in me esset ex iis, quæ animæ prius a me descriptæ tribuebam, cùmque non omnia quæ ad ipsam retuleram in me invenirem, sed solam cogitationem, ideo non dixi me esse animam, sed tantùm rem cogitantem, atque huic rei cogitanti nomen mentis, sive intellectûs, sive rationis, imposui .... [A]deo ut dubitari non possit quin præcise idem tantùm per illas ac per nomen rei cogitantis intellexerim.” Cf. also Rozemond, “Role of the Intellect,” 101.

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essential attribute Cogitatio,10 and he explains them in terms of immediate conscientia.11 As for the metaphysical relationship between particular cogitationes quâ modes and the mind or res cogitans, Descartes makes it clear that while particular cogitationes can never exist and never be clearly understood separately of the mind of which they are modes,12 the corresponding mind can clearly be understood without any of its particular cogitationes,13 and although it is essential to it that it is modified with some particular cogitatio (or cogitationes)14 at any moment of its existence, it is not the case that any of the particular cogitationes it is actually modified with is essential to it.15 Descartes‫ ތ‬basic classification of the modi cogitandi can be extrapolated by way of gathering together several passages from his writings.16 To begin with, according to him modi cogitandi are of two general kinds, viz. perceptio, sive operatio intellectûs on the one hand and volitio, sive

10 See Resp. 3, AT VII, 174: “[C]ogitatio [sumi solet] interdum pro actione, interdum pro facultate, interdum pro re in quâ est facultas.” For the occurrent cogitationes as modi, actus or actiones, see in particular Princ. I, 56; 61; 64–65, AT VIII-1, 26; 29; 31–32; Med. III, AT VII, 34–35; Pour Arnauld, AT V, 221; Resp. 3, AT VII, 175–76. 11 See Resp. 2, AT VII, 160; Princ. I, 9, AT VIII-1, 7–8. 12 See e.g. Princ. I, 64, AT VIII-1, 31: “[M]odò [plures cogitationes istas] non ut substantiæ, sive res quædam ab aliis separatæ, sed tantummodo ut modi rerum spectentur. Per hoc enim, quòd ipsas in substantiis quarum sunt modi consideramus, eas ab his substantiis dinstinguimus, & quales revera sunt agnoscimus. At è contra, si easdem absque substantii, quibus insunt, vellemus considerare, hoc ipso illas ut res subsistentes spectaremus, atque ita ideas modi & substantiæ confunderemus.” 13 See Princ. I, 61, AT VIII-1, 29: “[Distinctio modalis inter modum propriè dictum, & substantiam cujus est modus] ex eo cognoscitur, quòd possimus quidem substantiam clarè percipere absque modo quem ab illâ differre dicimus, sed non possimus, viceversâ, modum illum intelligere sine ipsâ.” 14 See Burm., AT V, 148: “Quod mens non possit nisi unam rem simul concipere, verum non est: non potest quidem simul multa concipere, sed potest tamen plura quàm unum; e.g., jam ego concipio et cogito simul me loqui et me edere.” 15 See ibid., AT V, 150: “Et mens nunquam sine cogitatione esse potest; potest quidem esse sine cogitatione hac aut illâ, sed tamen non sine omni ....” 16 I take Princ. I, 32–34, AT VIII-1, 17–18 as the main point of departure, but I take into consideration the other standard loci as well, viz. Princ. I, 9 and 65, AT VIII-1, 7–8 and 32, respectively; Med. II, III, and IV, AT VII, 28 and 34, 37, and 56–57, respectively; Resp. 2, AT VII, 160; Reg. VIII and XII, AT X, 395–96, 398; and 410–16, respectively.

10

Chapter One

operatio voluntatis on the other.17 The two kinds differ, roughly and most generally, by way of the contrast between activity and passivity of the mind: as Descartes tells Regius, “Volitio vero & intellectio ... differunt tantum vt actio & passio eiusdem substantiæ. Intellectio enim propriè mentis passio est, & volitio eius actio ....” (A Regius, AT III, 372; Descartes’ emphasis), a view which is confirmed a few years later in Les Passions de l’ame.18 The modes of thought Descartes takes as falling under the heading of intellectus (or its operations) are sensation, imagination and pure understanding (sentire, imaginari, & purè

17

The classification is clearly stated in particular in Princ. I, 32, AT VIII-1, 17: “Duos tantùm in nobis esse modos cogitandi, perceptionem scilicet intellectûs & operationem voluntatis. Quippe omnes modi cogitandi, quos in nobis experimur, ad duos generales referri possunt: quorum unus est perceptio, sive operatio intellectûs; alius verò volitio, sive operatio voluntatis. Nam sentire, imaginari, & pure intelligere, sunt tantùm diversi modi percipiendi; ut & cupere, aversari, affirmare, negare, dubitare, sunt diversi modi volendi” (Descartes’ italics). Cf. also Reg. XII, AT X, 415–16; Med. IV, AT VII, 56–57; A Elisabeth, AT III, 665. Descartes refers to the two kinds at issue with modus in Princ. I, 32; it should be clear that this sense of the term “modus” (in which it is virtually synonymous with “species” or “kind”) is not to be confused with the sense in which “modus” is employed to signify particular cogitationes as opposed to the essential attribute Cogitatio (in this latter sense, “modus” is virtually synonymous with “actus” or “operatio”). 18 Cf. ibid., I, 17, AT XI, 342: „[I]l ne reste rien en nous que nous devions attribuer à nostre âme, sinon nos pensées, lesquelles sont principalement de deux genres: à sçavoir, les unes sont les actions de l‫ތ‬ame, les autres sont ses passions. Celles que je nomme ses actions, sont toutes nos volontez .... Comme, au contraire, on peut generalement nommer ses passions, toutes les sortes de perceptions ou connoissances qui se trouvent en nous ....” As has commonly been noticed, this determination of operationes intellectûs does not contradict Descartes‫ ތ‬standard talk of the particular modes of operationes intellectûs as acts (actus) since the term “act” has two different meanings in Descartes‫ ތ‬hands: in the broader sense, “act” means as much as actuality, as opposed to potentiality; and in the narrower sense, “act” means (the product of) activity, as opposed to passivity. As is remarked by Vere Chappell, “The Theory of Ideas,” in Rorty, Essays on DescartesҲ Meditations, 196, Descartes sometimes reserves the term “actus” for the broader meaning and the term “actio” for the narrower meaning (see e.g. Resp. 1, AT VII, 103 for the former case, and the just quoted AT III, 372 for the latter case); but—as Chappell also notices (ibid.)—Descartes is not consistent in this. See Lex Newman, “Descartes on the Will in Judgment,” in A Companion to Descartes, ed. Janet Broughton and John Carriero (Oxford: Blackwell Publishing, 2008), 334–36 for a more detailed up-to-date treatment of these matters.

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intelligere)19 and perhaps also memory (memoria, recordatio);20 the modes he takes as falling under the heading of voluntas (or its operations) are desire, aversion, affirmation, negation, and doubt (cupere, aversari, affirmare, negare, dubitare).21 Of numerous substantial points implied in this compressed and simplified initial exposition, at least two call for further development with regard to the purposes of the present study: namely (i) the cognitive rôles Descartes attributes to each of both kinds of modi cogitandi, viz. operationes intellectûs and operationes voluntatis, respectively, and (ii) the nature of the types of operationes intellectûs, viz. pure intellection, imagination, sensation, and the relations holding between them according to Descartes. I deal with these issues in turn in the following two subsections.

1.1.1 Understanding, Will, and Judgments Descartes deals with the issue of the respective cognitive rôles of operationes intellectûs and operationes voluntatis in terms of the pivotal distinction between apprehensions (or perceptiones as Descartes usually calls them) of an arguably propositionally structured subject matter on the one hand, and judgments concerning the content thus apprehended on the other. The distinction is most explicitly introduced, though somewhat

19

Thus in Princ. I, 32; cf. also ibid., I, 9; Resp. 2, AT VII, 160; Resp. 3, AT VII, 176; Reg. VIII, 395–96; Mers., AT I, 366. 20 Memoria is included in the list of cognitive faculties in Reg. XII, AT X, 411: “In nobis quatuor sunt facultates tantùm, quibus ad [cognitionem] vti possimus: nempe intellectus, imaginatio, sensus, & memoria.” Cf. also the very precept of Reg. XII, AT X, 410: “Denique omnibus vtendum est intellectûs, imaginationis, sensûs, & memoriæ auxilijs ...” and Reg. VIII, AT X, 398. However, Descartes mentions memory (under the name of recordatio) in a relevant context even in Princ. I, 65 (omitting sensory perception instead). Despite these occurrences, the rank of the issue of memory in Descartes‫ ތ‬lists of the operationes intellectus is somewhat precarious and I will put memory to one side for the rest of the book since it is not needed to deal with it, as far as I can tell, with regard to the aims I will be pursuing. See Anne Davenport, “What the Soul Remembers: Intellectual Memory in Descartes,” The New Arcadia Review 3 (2005), 1–5 for a good basic survey of the issue. 21 Thus in Princ. I, 32. This last enumeration seems to be complete. Other similar lists of the modi cogitandi in question add nothing over and above the items enumerated in it; cf. especially Med. II, AT VII, 28 and 34.

12

Chapter One

tentatively and, in fact, improperly (as we shall see in a moment), in Med. III, AT VII, 37:22 Quædam ex [cogitationibus meis] tanquam rerum imagines sunt, quibus solis proprie convenit ideæ nomen .... Aliæ verò alias quasdam præterea formas habent: ut, cùm ... affirmo, cùm nego, semper quidem aliquam rem ut subjectum meæ cogitationis apprehendo, sed aliquid etiam amplius quàm istius rei similitudinem cogitatione complector; & ex his ... [quædam] judicia appellantur.

Furthermore, Descartes makes it clear in several places that while it is nothing but understanding—presumably in the generic sense of operatio intellectûs—that is responsible for the propositional cognitions of apprehension, judgments are due to a certain sort of joint operation of understanding and volition:23 Non solùm intellectum, sed etiam voluntatem requiri ad judicandum. Atque ad judicandum requiritur quidem intellectus, quia de re, quam nullo modo percipimus, nihil possumus judicare; sed requiritur etiam voluntas, ut rei aliquo modo perceptæ assensio præbatur (Princ. I, 34, AT VIII-1, 18)

The proper contribution of the faculty of voluntas to the constitution of any judgment is thus clearly that it provides for acts of affirmation or else of denial, conceived of as being what we would nowadays call (the 22

Cf. also Descartes‫ ތ‬impatient explanation to Hobbes in Resp. 3 concerning this passage: “Per se notum est ... aliud esse videre hominem currentem, quàm sibi ipsi affirmare se illum videre” (AT VII, 182–83). As in Princ. I, 32, Descartes also includes the conative attitudes of desire and aversion in the list of the quædam præterea formæ in the quoted AT VII, 37. However, Descartes himself immediately draws a distinction between voluntates sive affectus (i.e. presumably desires and aversions) and judicia (i.e. presumably the doxastic, as opposed to conative, operations) there, and I limit my discussion to this latter, doxastic class of Descartes‫ ތ‬operationes voluntatis from now on as it is only judgments in the indicated sense that are relevant to our purposes. The distinction between perceptiones and judicia is then clearly at work in Descartes‫ ތ‬account of error in Med. IV and in Princ. I, 32–36. 23 Cf. also e.g. Med. IV, AT VII, 56; Notæ, AT VIII-2, 363: “Ego enim, cùm viderem, præter perceptionem, quæ prærequiritur ut judicemus, opus esse affirmatione vel negatione ad formam judicii constituendam, nobisque sæpe esse liberum ut cohibeamus assensionem, etiamsi rem percipiamus: ipsum actum judicandi, qui non nisi in assensu, hoc est, in affirmatione vel negatione consistit, non retuli ad perceptionem intellectûs, sed ad determinationem voluntatis.”

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13

doxastic kind of) propositional mental attitudes—attitudes towards a content supplied by the understanding.24 To draw the threads of the present exposition together in the meantime, it clearly emerges that for Descartes the proper locus of cognition is judgment, treated along the lines so far explicated,25 while he reserves the term “perceptio” for cognition in the narrower sense of apprehension of a given content. Furthermore, while cognition is therefore, like judgment, subject to the process-product ambiguity, Descartes is usually prone to reserving “judicium” for acts of cognition and “cognitio” for the (more or less retained) products of such cognitive acts.26 I will follow this practice in the present study from now on. This is not the place to venture any thorough assessment of the merits of Descartes‫ ތ‬prima facie somewhat awkward claim that doxastic attitudes such as affirmation or negation (or suspension of judgment for that matter) are matters of free will.27 What is to be addressed now, in view of the main topic of the present study, are several issues concerning the apprehended content, i.e. the objects of operationes intellectûs. To begin, it has been observed and commonly agreed that in so far as Descartes identifies (as he does) judgments as the only locus of truth and falsehood (or error) properly so called,28 and in so far as the relevant 24 Cf. Hyp., AT III, 432: “[N]eque enim voluntatis est intelligere, sed tantùm velle; ac ... nihil vnquam velimus, de quo non aliquid aliquo modo intelligamus ....” Cf. also A Regius, AT III, 372: “[Q]uia nihil vnquam volumus, quin simul intelligamus, & vix etiam quicquam intelligimus, quin simul aliquid velimus, ideo non facile in ijs passionem ab actione distinguimus.” Similarly also at Resp. 5, AT VII, 377. 25 Cf. e.g. Med. II, AT VII, 35: “Nempe in hac primâ cognitione nihil aliud est, quàm clara quædam & distincta perceptio ejus quod affirmo ....” 26 This tendency of Descartes‫ ތ‬is vividly present especially in Med. IV. 27 For an excellent defence of Descartes‫“ ތ‬two-faculty theory” of judgment see David Rosenthal, “Will and the Theory of Judgment,” in Rorty, Essays on DescartesҲ Meditations, 405–34. 28 Cf. Med. III, AT VII, 36–37: “[N]unc autem ordo videtur exigere, ut ... [inquiram] in quibusnam ex [cogitationibus meis] veritas aut falsitas proprie consistat .... ... Jam quod ad ideas attinet, si solæ in se spectentur, nec ad aliud quid illas referam, falsæ proprie esse non possunt .... Nulla etiam in ipsâ voluntate, vel affectibus, falsitas est timenda .... Ac proinde sola supersunt judicia, in quibus mihi cavendum est ne fallar.” The French translation of this passage (AT IX-2, 29) has “la verité ou l‫ތ‬erreur” for “veritas aut falsitas.” This is symptomatic of Descartes‫ތ‬ peculiar (and arguably controlled) running together of falsity and error—the point I will touch upon below. Cf. also Med. III, AT VII, 43: “[F]alsitatem proprie dictam, sive formalem, nonnisi in judiciis posset reperiri paulo ante notaverim ....”

14

Chapter One

mental attitudes towards the perceived content that form judgments (which attitudes he takes as amounting to the operationes voluntatis) are, according to him, the acts of assent or dissent, he is committed to treating the perceived content as essentially propositional in nature.29 It is, of course, natural to suppose that these considerations did not elude Descartes and that he does acknowledge the commitment at issue; indeed, direct textual evidence to this effect is strong enough.30 However, two familiar complications, both peculiar to Descartes‫ ތ‬treatment, are lurking here. They must now be addressed in turn. As in the above-quoted AT VII, 37 and elsewhere,31 Descartes sometimes seems to imply that the objects of judgments are ideas while he declares, by the same token, that he wishes to reserve the term “idea”, taken properly, to denote things (and more generally, perhaps in a somewhat broader sense, any sub-propositional items)32 in so far as they are the objects of one‫ތ‬s cognitive acts (or, as he usually puts it in scholastic terminology, in so far as the things at issue have so-called 29 Bernard Williams puts the core point cogently in his Descartes: The Project of Pure Enquiry (Abindgdon: Routledge, 2005), electronic edition, 167: “I can assent only to something of the nature of a proposition: one believes, or refuses to believe, that such-and-such is the case” (Williams’ emphasis). 30 Propositiones in the relevant sense are taken as the proper objects of cognitio and judicio, respectively, quite standardly throughout the Regulæ; for explicit pronouncements to this effect see e.g. Reg. III, AT X, 370; Reg. V, AT X, 379; Reg. VI, AT X, 383; 386–87; Reg. X, AT X, 405; Reg. XI, AT X, 407; 409; Reg. XII, AT X, 410; 421–22; 428; Reg. XIII, AT X, 434; Reg. XIV, AT X, 438; 449; 452; Reg. XVII, AT X, 460. The objects of judgments are explicitly identified with propositiones at Resp. 6, AT VII, 445. Further, there are numerous places in the Principia where judicium (or its grammatic varieties) is connected with a proposition (see e.g. Princ. I, 11; 66; 68; 70; Princ. II, 17; 20; 37; 52; Princ. III, 4; Princ. IV, 198; 201) while there is none in which judicium is connected with idea, and just three places in which the object of judicium is identified with res—a term, however, which is used so loosely by Descartes (as it is indeed in the bulk of the Latin tradition and perhaps even by us) that it might signify both ideas and propositions, depending on circumstances. 31 Cf. in particularMed. IV, AT VII, 56: “[P]er solum intellectum percipio tantùm ideas de quibus judicium ferre possum” (my emphasis). 32 This extension seems to be the thrust of Descartes‫ ތ‬response to Burman‫ތ‬s question concerning the claim in Med. III, AT VII, 44 that “nullæ ideæ nisi tanquam rerum esse possunt”: “[Burman:] Sed datur etiam idea nihili, quæ non est idea rei. [Descartes:] Illa idea est solùm negativa, et vix vocari potest idea; auctor autem hîc sumit ideam proprie et stricte. Aliæ etiam dantur ideæ notionum communium, quas non sunt ideæ rerum proprie; sed tum idea latius sumitur” (Burm., AT V, 153).

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objective being [esse obiective]).33 Yet it would still be rash to attribute to Descartes the weird view that representations of things tout court could ever meaningfully be assented or dissented to in the literal sense. For one thing, his remarks concerning the so-called material falsity of ideas—ideas being prima facie taken in the above-established non-propositional sense in this context34—imply that ideas in general may (though some of them perhaps need not)35 “errandi materiam præbere” (Resp. 4, AT VII, 231– 32);36 and whenever Descartes gives an explanation of what he means by this clause, he incorporates (sub-propositionally conceived) ideas as components in manifestly propositional structures which then become the proper object of the corresponding judgments.37 Furthermore—perhaps 33

Apart from AT VII, 44 quoted in the previous footnote, cf. e.g. Med. III, AT VII, 37: “Quædam ex [cogitationes meis] tanquam rerum imagines sunt, quibus solis proprie convenit ideæ nomen: ut cùm hominem, vel Chimæram, vel Cœlum, vel Angelum, vel Deum cogito.” Also cf. Princ. I, 13–20, AT VIII-1, 9–12; Resp. 1, AT VII, 102–103. 34 Cf. Med. III, AT VII, 43–44: “Quamvis enim falsitatem proprie dictam, sive formalem, nonnisi in judiciis posset reperiri paulo ante notaverim, est tamen profecto quædam alia falsitas materialis in ideis, cùm non rem tanquam rem repræsentant: ita, exempli causâ, ideæ quas habeo caloris & frigoris, tam parum claræ & distinctæ sunt, ut ab iis discere non possim, an frigus sit tantùm privatio caloris, vel calor privatio frigoris, vel utrumque sit realis qualitas, vel neutrum. Et quia nullæ ideæ nisi tanquam rerum esse possunt, siquidem verum sit frigus nihil aliud esse quàm privationem caloris, idea quæ mihi illud tanquam reale quid & positivum repræsentat, non immerito falsa dicetur, & sic de cæteris.” 35 Descartes seems to commit himself to the view that some ideas can be materially true when he deals with veras & immutabiles naturas at Med. V, AT VII, 64–65 and elsewhere. 36 Cf. also Med. III, AT VII, 37; Burm., AT V, 152. 37 Thus he says in Med. III, AT VII, 37: “Præcipuus autem error & frequentissimus qui possit in [judiciis meis] reperiri, consistit in eo quòd ideas, quæ in me sunt, judicem rebus quibusdam extra me positis similes esse sive conformes; nam profecto, si tantùm ideas ipsas ut cogitationis meæ quosdam modos considerarem, nec ad quidquam aliud referrem, vix mihi ullam errandi materiam dare possent” (my emphasis). Clearly enough, the judgment in question is not rendered false by assenting to a given idea tout court but by assenting to the propositionally structured item idea mea rei quædam extra me positæ similis est, in which the given idea amounts to just a subject. Mutatis mutandis similar remarks could be made regarding Descartes‫ ތ‬other explanations, viz. in Resp. 4, AT VII, 234, and in Burm., AT V, 152. Williams summarizes the present point aptly: “[Descartes‫ތ‬ remark that there is a certain sense in which ideas can be said to be ‫ދ‬materially false‫ ]ތ‬is no real qualification of the doctrine that ideas are not intrinsically true or false, since for the mind to be involved in any actual falsehood on the strength of one of these ideas it must do more than merely have the idea—it must move on to

16

Chapter One

even more importantly for our purposes—Descartes makes it clear on several occasions that the (presumably sub-propositional) content of at least some (and perhaps even all) ideas is, or at least can be in principle, internally structured, to the effect that the content of a given idea allows for articulation by a propositional clause.38 In the light of this, the apparent conflict between Descartes‫ ތ‬subpropositional treatment of ideas on the one hand and, on the other, his propositional commitment regarding the objects of judgments, is likely to vanish.39 Descartes emerges—at least prima facie—as entitled to treat the content of sub-propositional items as in principle capable of propositional rearrangement in either of both ways indicated and he seems to be far from failing (as some commentators complain) to distinguish carefully enough between concepts and propositional structures.40 In the course of the an assertion or judgement that things are in fact as this idea represents them” (Williams, Descartes, 116). 38 Here are the most telling passages to this effect: “[V]ostre amy n‫ތ‬a nullement pris mon sens, lors que, pour marquer la distinction qui est entre les idées qui sont dans la fantasie, & celles qui sont dans l‫ތ‬esprit, il dit que celles-là s‫ތ‬expriment par des noms, & celles-cy par des propositions. Car, quҲelles sҲexpriment par des noms ou par des propositions, ce nҲest pas cela qui fait quҲelles appartiennent à lҲesprit ou à lҲimagination; les vnes & les autres se peuuent exprimer de ces deux manieres ...” (Mers., AT III, 395; my emphasis). “[Peto], ut [lectores] examinent ideas naturarum, in quibus multorum simul attributorum complexio continetur, qualis est natura trianguli, natura quadrati, vel alterius figuræ; itemque natura Mentis, natura Corporis, & supra omnes natura Dei, sive entis summe perfecti. Advertantque illa omnia, quæ in iis contineri percipimus, vere de ipsis posse affirmari. Ut, quia in naturâ Trianguli continetur ejus tres angulos æquales esse duobus rectis, & in naturâ Corporis, sive rei extensæ, continetur divisibilitas ..., verum est dicere omnis Trianguli tres angulos æquales esse duobus rectis, & omne Corpus esse divisibile” (Resp. 2, AT VII, 163; my emphasis). Cf. also the corresponding passage in Med. V, AT VII, 65; and Mers., AT III, 383; 417. 39 Also Descartes‫ ތ‬presumably most authoritative definition of idea in Resp. 2 is thus vindicated vis-à-vis the charge that it renders the extension of the term “idea” much wider—in that it includes propositional items as well—than Descartes is entitled throughout the bulk of the Meditationes: “Ideæ nomine intelligo cujuslibet cogitationis [my emphasis] formam illam, per cujus immediatam perceptionem ipsius ejusdem cogitationis conscius sum ...” (Resp. 2, AT VII, 160). This definition is echoed in Resp. 3, AT VII, 188: “dicendo me per ideam intelligere id omne quod forma est alicujus perceptionis.” Cf. also Resp. 3, AT VII, 181: “[O]stendo me nomen ideæ sumere pro omni eo quod immediate a mente percipitur, adeo ut, cùm volo & timeo, quia simul percipio me velle & timere, ipsa volitio & timor inter ideas a me numerentur” (my emphasis). 40 Cf. e.g. Wilson, Descartes, 124 for such a complaint.

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present study these conclusions are to be borne in mind whenever subpropositional items such as concepts, natures, etc. as employed by Descartes are treated in connection with judgments or scientific cognition (i.e. cognition amounting, as we shall see, to a disposition to make judgments of a certain specific type). The other group of aforementioned familiar complications concerns apparent inconsistencies in Descartes‫ ތ‬overall distribution of labour among will, understanding and judgment. One or another inconsistency is implied—so the challenge goes—by the following triad of Descartes‫ތ‬ claims: (i) that it is judgments that are the proper locus of (formal) truth and falsity;41 (ii) that every judgment consists of an apprehended content provided by the understanding on the one hand, and of assent or else dissent supplied by the will on the other hand;42 and—as has just been established—(iii) that the apprehended content, in so far as it is the object of either assent or dissent, is to be taken as propositionally structured. For (i) and (ii) together clearly rule out the possibility that the contribution of the understanding—the apprehended content—be in itself capable of either truth or falsity in the relevant sense; yet is it not the case that propositionally structured items—which is how Descartes is committed to treat the apprehended content by (iii)—are intrinsically true or false (at least in normal cases),43 independently of whether any mind assents or dissents to them? But if so, then either (i) or (ii) must be discarded; if (i) gives way, then the likely diagnosis is that Descartes unwisely confounds error and correctness (which are properly attributed to judgments) on the one hand, and falsity and truth (which are properly attributed to propositions) on the other;44 and if (ii) is to be abandoned, then the likely diagnosis is that contrary to what Descartes holds, voluntary operations are strictly superfluous with regard to the constitution of judgments since as soon as the understanding provides for a given content, affirmation or negation is produced ipso facto on the part of the given mind.45

41

Cf. Med. III, AT VII, 36–37; 43. Cf. in particular Princ. I, 34, AT VIII-1, 18. 43 I.e. unless, for example, failure to refer, vagueness or the like come on stage. 44 This line of criticism is suggested by Wilson, Descartes, 124. As she writes, “What could error be but the affirmation of what is false, or the denial of what is true?” (ibid.; Wilson’s emphases). 45 This charge is due to Edwin Curley, “Descartes, Spinoza and the Ethics of Belief,” in Spinoza: Essays in Interpretation, ed. Eugene Freeman and Maurice Mandelbaum (La Salle: Open Court, 1975), 159–89; see especially sec. II; Curley himself credits Spinoza with it. 42

18

Chapter One

The best response on Descartes‫ ތ‬part to this apparently formidable dilemma is, I think, to reject the premise the dilemma hinges upon, namely the claim that it is necessary for propositional structures tout court to count as intrinsically either true or false.46 For such a premise might sound plausible as long as propositional content is constructed along Fregean lines as an abstract object; yet—so the suggested response goes—it is misguided to attribute to Descartes such a Fregean notion. Firstly, at least one prima facie viable alternative seems to be ready at hand, viz. a notion to the effect that propositional content amounts to an integral constituent, or aspect, of the particular occurrent mental acts; and it is far from obvious that one is bound to take such tokens of mental acts as intrinsically true or false in the relevant sense.47 Secondly, Descartes‫ ތ‬commitments with regard to the ontology of representative operationes intellectûs seem likely to license the suggestion that we attribute to Descartes this latter alternative. Briefly and roughly, Descartes conceives of ideas (presumably in the above-established broad sense of representative modes of thought tout court), in so far as their ontological constitution is concerned, as complex entities the reality of which is made up, so to speak, of two positive ontological factors, namely the so-called realitas formalis and realitas objectiva, respectively. The realitas formalis is that reality or perfection48 which particular ideas obtain precisely due to the fact that they count as real forms that actually determine the attribute of cogitatio.49 46

I owe this suggestion, as well as the following line of reasoning, to Rosenthal, “Will and Judgment,” 419–24. 47 This is not the place to assess the systematic merits of the suggested alternative. The issue of the bearers of truth values, and the familiar problems faced by propositions quâ candidates in that rôle in view of token-reflexive items, are likely to play a pivotal part in any such assessment. 48 Descartes clearly uses “realitas” and “perfectio” interchangeably throughout the relevant passages of Med. III and elsewhere (see in particular Med. III, AT VII, 40–44, especially in combination with Med., Præfatio ad lectorem, AT VII, 8; and Princ. I, 17, AT VIII-1, 11). He also occasionally uses “realitas” interchangeably with “entitas”—cf. Resp. 2, AT VII, 161. In this, Descartes seems just to follow the common scholastic practice of his time: see David Clemenson, DescartesҲ Theory of Ideas, London: Continuum International Publishing Group, 2007, 18–20. 49 Cf. Med. III, AT VII, 41: “[P]utandum est ... talem esse naturam ... ideæ, ut nullam aliam ex se realitatem formalem exigat, præter illam quam mutuatur a cogitatione meâ, cujus est modus.” Resp. 2, AT VII, 160–61: “Ideæ nomine intelligo cujuslibet cogitationis formam illam, per cujus immediatam perceptionem ipsius ejusdem cogitationis conscius sum .... ... [Ideæ] ... mentem ipsam ... informant.” That Descartes is indeed prepared to conceive of the attribute of cogitatio as of the Aristotelian materia, in the sense of what is in itself

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The realitas objectiva, on the other hand, is that reality or perfection which the very same particular ideas obtain by virtue of the fact that they represent50 a given object,51 regardless of whether that object in fact exists beyond the given idea or not.52 Now it is crucial that the bulk of the evidence (which is, admittedly, somewhat bewildering and scarcely conclusive) points towards reading Descartes to the effect that he identifies the realitas objectiva of ideas, in the sense just established, with the being that the represented object obtains precisely quâ thus represented by the given particular act of thought (i.e. with the so-called esse objective of the represented object).53 Moreover, his standard talk in terms of containment

indeterminate yet determinable and actually always already somehow determined by some form, should be clear from his referring to ideas in so far as they inform the attribute of cogitatio with the clause “idea materialiter sumpta” in Resp. 4, AT VII, 232: “[S]i [ideæ] spectarentur, ... tantummodo prout sunt operationes intellectûs, dici quidem posset materialiter illas sumi ...,” and in Med., Præfatio, AT VII, 8: “[Vox] ideæ sumi ... potest ... materialiter, pro operatione intellectûs ....” 50 The representative function seems to be of the very essence of ideas according to Descartes. Cf. e.g. Med. III, AT VII, 42: “[M]odus essendi objectivus competit ideis ex ipsarum naturâ ....” Med.(f) III, AT IX-1, 34–35: “[L]es idées estant comme des images, il n‫ތ‬y en peut auoir aucune qui ne nous semble representer quelque chose ....” 51 Cf. Med. III, AT VII, 40: “[Q]uatenus ideæ istæ cogitandi quidam modi tantùm sunt, non agnosco ullam inter ipsas inæqualitatem, & omnes a me eodem modo procedere videntur; sed, quatenus una unam rem, alia aliam repræsentat, patet easdem esse ab invicem valde diversas. Nam proculdubio illæ quæ substantias mihi exhibent, majus aliquid sunt, atque, ut ita loquar, plus realitatis objectivæ in se continent, quàm illæ quæ tantùm modos, sive accidentia, repræsentant; & rursus illa per quam summum aliquem Deum ... intelligo, plus profecto realitatis objectivæ in se habet, quàm illæ per quas finitæ substantiæ exhibentur.” Cf. also Resp. 2, AT VII, 161. 52 Cf. e.g. Med.(f) III, AT IX-1, 35: “si [les idées] sont fausses, c‫ތ‬est à dire si elles representent des choses qui ne sont point ....” This rendering resolves the ambiguity of the corresponding place in the Latin original, which reads: “si quidem sint falsæ, hoc est nullas res repræsentent ...” (Med. III, AT VII, 44). 53 See in particular Resp. 2, AT VII, 161: “Per realitatem objectivam ideæ intelligo entitatem rei repræsentatæ per ideam, quatenus est in ideâ .... Nam quæcumque percipimus tanquam in idearum objectis, ea sunt in ipsis ideis objective” (Descartes’ emphasis). Cf. also Resp. 1, AT VII, 102: “[A]dvertendum, ... me loqui de ideâ, quæ nunquam est extra intellectum, & ratione cujus esse objective non aliud significat quàm esse in intellectu eo modo quo objecta in illo esse solent” (Descartes’ emphasis).

20

Chapter One

when determining how realitas objectiva is related to ideas54 and his crucial commitment to treating realitas objectiva as an effect of some cause55 jointly indicate that he is committed to taking both realitas formalis and realitas objectiva as pertaining to one and the same entity, namely the particular occurring representative mental event, to the effect that both the realitates at issue are (to use Descartes‫ ތ‬own classification of distinctions from the Principia)56 distinct only ratione.57 All these considerations, then, lend quite a strong support to the claim that it is indeed a mistake to ascribe to Descartes the Fregean view of abstract propositions that are inherently either true or false as the paradigm of apprehended cognitive content.58 Further, in so far as the positive 54

Cf. e.g. Med. III, AT VII, 40: “[P]roculdubio illæ [ideæ] quæ substantias mihi exhibent, majus aliquid sunt, atque, ut ita loquar, plus realitatis objectivæ in se continent, quàm illæ quæ tantùm modos, sive accidentia, repræsentant; & rursus illa per quam summum aliquem Deum ... intelligo, plus profecto realitatis objectivæ in se habet, quàm illæ per quas finitæ substantiæ exhibentur” (my emphases). Ibid., AT VII, 46: “[Idea Dei] maxime clara & distincta sit, & plus realitatis objectivæ quàm ulla alia contineat ...” (my emphasis). Cf. also ibid., AT VII, 41; Resp. 1, AT VII, 103–105; Princ. I, 17, AT VIII-1, 11. 55 Cf. in particular Med. III, AT VII, 41: “[P]utandum est ... talem esse naturam ... ideæ, ut nullam aliam ex se realitatem formalem exigat, præter illam quam mutuatur a cogitatione meâ, cujus est modus. Quòd autem hæc idea realitatem objectivam hanc vel illam contineat potius quàm aliam, hoc profectò habere debet ab aliquâ causâ in quâ tantumdem sit ad minimum realitatis formalis quantum ipsa continet objectivæ” (my emphasis); Resp. 1, AT VII, 103–104: “[Q]uòd hæc idea machinæ tale artificium objectivum contineat potious quàm aliud, hoc profecto habere debet ab aliquâ causâ ...” (my emphases); Princ. I, 17, AT VIII-1, 11: “[Q]uò plus perfectionis objectivæ [ideæ quæ in nobis habemus] in se continent, eò perfectiorem ipsarum causam esse debere” (my emphases). 56 Cf. Princ. I, 60–62, AT VIII-1, 28–31. I deal with Descartes‫ ތ‬theory of distinctions in some detail below in ch. 3. 57 Cf. Chappell, “The Theory of Ideas,” 193–94. 58 Such a conclusion gains further support from the fact that, arguably, Descartes endorsed a kind of conceptualist theory of universals: cf. in particular Princ. I, 58, AT VIII-1, 27: “Numerum & universalia omnia esse tantùm modos cogitandi. ... [N]umerus ... est modus cogitandi duntaxat; ut & alia omnia quæ universalia vocamus” (Descartes’ italics). Also Princ. I, 59, ibid.: “Fiunt hæc universalia ex eo tantùm, quòd unâ & eâdem ideâ utamur ad omnia individua, quæ inter se similia sunt, cogitanda ....” To be sure, what Descartes says concerning mathematical objects and their veras & immutabiles naturas seems to compromise this claim as Descartes thus seems to endorse a sort of Platonism with regard to mathematical entities—cf. e.g. Anthony Kenny, Descartes: A Study of His Philosophy (New York: Random House, 1968), 155–56; Wilson, Descartes, 149; Gregory Brown, “Vera Entia: The Nature of Mathematical Objects in Descartes,” Journal of the

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alternative I have suggested on Descartes‫ ތ‬part—viz. to treat apprehended content as just an “objective,” i.e. representative aspect of particular eventlike ideas—is viable, Descartes‫ ތ‬notion of respective functions of judgment, will and understanding stands up to the criticisms just dealt with and remains perfectly intelligible.

1.1.2 Understanding and Its Modes Let us proceed now to the other of the two aforementioned topics: namely, the nature of the types of operationes intellectûs in Descartes; and the relations which exist between them. Having put to one side the case of memory, we saw that according to Descartes, three types of acts fall under the heading operationes intellectûs: namely sentire, imaginari, & purè intelligere. Now it would be a mistake to interpret the relation between operatio intellectûs and these three types of acts as a relation between Aristotelian genus and Aristotelian species. Descartes makes it clear that while each of the listed types indeed amounts to a result of one and the same vis intelligendi (Med. VI, AT VII, 73)59 in operation, what really matters is whether the vis at issue operates alone—in which case “mens ... se ad seipsam quoddammodo convertat, respiciatque aliquam ex ideis quæ illi ipsi insunt” (ibid.)—or whether in addition something different from the res cogitans, namely the human body (or more precisely a certain part of it, viz. the pineal gland where the immediate objects of phantasia, i.e. corporeal imagination, and/or sensus communis are situated)60 is somehow involved; and that if that vis operates alone, then the operation is called

History of Philosophy 18, no. 1 (1980), 36–37. Yet I believe that even these instances allow for a conceptualist interpretation, and that it is such a conceptualist interpretation that one should favour at the end of the day; see Lawrence Nolan, “Descartes‫ތ‬s Theory of Universals,” Philosophical Studies 89, no. 2-3 (1998), 161–80 for a convincing case in support of a similar view. 59 Cf. Med. IV, AT VII, 60 where “vis intelligendi” is treated as synonymous with “lumen naturale”. In Reg. XII, AT X, 415–16, the power at issue is called vis cognoscens; in Resp. 4, AT VII, 220 vis cognoscendi. For the unity of the vis cognoscens see Reg. XII, AT X, 415–16. I return in ch. 2 to the cognitive faculty referred to by these virtually interchangeable terms. 60 See in particular Le Monde, AT XI, 176–77; Passions I, 32, AT XI, 352–53. For a discussion of the relation of the pineal gland and phantasia, see Dennis Sepper, Descartes’s Imagination: Proportion, Images, and the Activity of Thinking (Berkeley: University of California Press, 1996), 28–35.

22

Chapter One

simply (pura) intellectio, while if the body is involved, the operation is called sensatio or else imaginatio:61 Atque vna & eadem est vis [cognoscens], quæ, si applicet se cum imaginatione ad sensum communem, dicitur videre, tangere, &c.; si ad imaginationem solam ... vt novas [figuras] fingat, dicitur imaginari vel concipere; si denique sola agat, dicitur intelligere. ... Et eadem etiam idcirco juxta has functiones diversas vocatur vel intellectus purus, vel imaginatio, ... vel sensus ... (Reg. XII, AT X, 415–16).

As for the difference between the “special” modes62 of operationes intellectûs—sensatio63 and imaginatio—Descartes explains it to Burman, lucidly and somewhat more extensively than in the above Reg. XII, as follows:64

61

Cf. also Med. VI, AT VII, 73–74: “[F]acilè intelligo, si corpus aliquod existat cui mens sit ita conjuncta ut ad illud veluti inspiciendum pro arbitrio se applicet, fieri posse ut per hoc ipsum res corporeas imaginer; adeo ut hic modus cogitandi in eo tantùm a purâ intellectione differat, quòd mens, dum intelligit, se ad seipsam quodammodo convertat ...; dum autem imaginatur, se convertat ad corpus .... ... Soleo verò alia multa imaginari, præter illam naturam corpoream, quæ est puræ Matheseos objectum, ut colores, sonos, sapores, dolorem; & similia, sed nulla tam distincte; & ... hæc percipio meliùs sensu, a quo videntur ope memoriæ ad imaginationem pervenisse ...” (my emphases); Resp. 5, AT VII, 358: “Ostendi etiam sæpe distincte, mentem posse independenter a cerebro operari; nam sane nullus cerebri usus esse potest ad pure intelligendum, sed tantùm ad imaginandum vel sentiendum.” Also Resp. 5, AT VII, 385; 387. As the context makes clear, “imaginatio” is employed by Descartes here as synonymous with “phantasia,” i.e. as referring to corporeal imagination, as quoted below. 62 Thus Med. VI, AT VII, 78: “facultates specialibus quibusdam modis cogitandi, puta facultates imaginandi & sentiendi.” Cf. also Burm., AT V, 162. 63 It will be well to bear in mind that for Descartes, sentire and its derivatives include not only the deliverances of the external senses (“sentire” in this sense referring to sense perception of external objects as prima facie understood by the contemporary reader) but also bodily sensations such as pains or tickles, natural appetites such as hunger or thirst, and even affects or emotions like joy or anger: cf. in particular Passions I, 19; 22–25, AT XI, 343; 345–48; Princ. IV, 190–96, AT VIII-1, 316–19; also Med. VI, AT VII, 74–75. 64 It will be observed that the term “idea” is used here in the sense of a corporeal image; ideas in this sense (this usage going back as early as the Regulæ and the Traité de lҲHomme in Descartes—cf. Reg. XII, AT X, 414–16; 419; Reg. XIV, AT X, 443–45; Traité de lҲHomme, AT XI, 174–76) are not to be confused with ideas in the above-established sense of intentional modes of the attribute of thought. See

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23

Quando objecta externa agunt in sensus meos, et in iis pingunt suî ideam seu potius figuram, tum mens, quando ad eas imagines quæ in glandulâ inde pinguntur advertit, sentire dicitur; cùm verò illæ imagines in glandulâ non pinguntur ab ipsis rebus externis, sed ab ipsâ mente, quæ, rebus externis absentibus, eas in cerebro effingit et format, tum imaginatio est; adeo ut differentia imaginationis et sensûs consistat in eo tantùm, quod in hoc imagines pingantur ab objectis externis, iisque præsentibus, in illâ autem a mente sine objectis externis et tanquam clausis fenestris (Burm., AT V, 162; Burman’s emphases).

Furthermore, Descartes makes it clear that it is precisely the facultas or vis intelligendi that is essential in so far as he or she counts purely as res cogitans,65 while the facultates imaginandi & sentiendi are modally distinct from that res cogitans sive substantia intelligens;66 and that the acts of those latter facultates—at least in so far as they are in fact veridical67—depend, as to their being, essentially “ab unione & quasi Michael Costa, “What Cartesian Ideas are Not,” Journal of the History of Philosophy, 21, no. 4 (1983), 537–49 for a well-arranged treatment of the topic. 65 Cf. Med. VI, AT VII, 73: “[C]onsidero ... vim imaginandi [& per implicationem vim sentiendi—J.P.] quæ in me est, prout differt a vi intelligendi, ad mei ipsius, hoc est ad mentis meæ essentiam non requiri; nam quamvis illa a me abesset, procul dubio manerem nihilominus ille idem qui nun sum ....” Ibid., AT VII, 78: “[I]nvenio in me facultates specialibus quibusdam modis cogitandi, puta facultates imaginandi & sentiendi, sine quibus totum me possum clare & distincte intelligere ....” Marleen Rozemond usefully remarks—in the context of commenting on a discussion of the topic in Wilson, Descartes, 155–58—that two distinct claims are implied in Descartes‫ ތ‬position at issue: namely (i) “intellection, unlike sensation and imagination, is [presumably causally—J.P.] independent of the body” (Rozemond, “Role of the Intellect,” 98; Rozemond’s emphasis), and (ii) “acts of intellection are not even paralleled by any physical acts” (ibid.; Rozemond’s emphasis); and that while Descartes in fact holds both (i) and (ii), it is only (i) that matters for him in the relevant contexts. 66 Cf. Med. VI, AT VII, 78: “[I]nvenio in me ... facultates imaginandi & sentiendi, sine quibus totum me possum clare & distincte intelligere, sed non vice versâ illas sine me, hoc est sine substantiâ intelligente cui insint.” This conforms perfectly to Descartes‫ ތ‬explication of distinctio modalis in Princ. I, 61 AT VIII-1, 29: “Distinctio modalis ... inter modum propriè dictum, & substantiam cujus est modus ... ex eo cognoscitur, quòd possimus idem substantiam clarè percipere absque modo quem ab illâ differre dicimus, sed non possimus, viceversâ, modum illum intelligere sine ipsâ” (Descartes’ emphasis). 67 Cf. Med. VI, AT VII, 73: “Facilè ... intelligo imaginationem ita perfici posse, siquidem corpus existat; & quia nullus alius modus æque conveniens occurrit ad illam explicandam, probabiliter inde conjicio corpus existere; sed probabiliter tantùm, & quamvis accurate omnia investigem, nondum tamen video ex eâ naturæ

24

Chapter One

permixtione mentis cum corpore” (Med. VI, AT VII, 81).68 The essential dependence of imaginatio and sensatio on the intermingling of the mind with the body comes out distinctly in Descartes‫ ތ‬counterfactual consideration presented in A Regius, AT III, 493:69 [T]amen potes, vt ego in Metaphysicis, [verum modum vnionis mentis & corporis explicare] per hoc, quod percipiamus sensus doloris, aliosque omnes, non esse puras cogitationes mentis à corpore distinctæ, sed confusas illius realiter vnitæ perceptiones: si enim Angelus corpori humano inesset, non sentiret vt nos, sed tantum perciperet motus qui causarentur ab obiectis externis, & per hoc à vero homine distingueretur.

corporeæ ideâ distinctâ, quam in imaginatione meâ invenio, ullum sumi posse argumentum, quod necessariò concludat aliquod corpus existere.” Cf. also Descartes‫ ތ‬reservations at Reg. XII, AT X, 412. 68 Here is the entire passage: “Docet etiam natura, per istos sensus doloris, famis, sitis &c., me non tantùm adesse meo corpori ut nauta adest navigio, sed illi arctissime esse conjunctum & quasi permixtum adeo ut unum quid cum illo componam. Alioqui enim, cùm corpus læditur, ego, qui nihil aliud sum quàm res cogitans, non sentirem idcirco dolorem, sed puro intellectu læsionem istam perciperem, ut nauta visu percipit si quid in nave frangatur; & cùm corpus cibo vel potu indiget, hoc ipsum expresse intelligerem, non confusos famis & sitis sensus haberem. Nam certe isti sensus sitis, famis, doloris &c., nihil aliud sunt quàm confusi quidam cogitandi modi ab unione & quasi permixtione mentis cum corpore exorti” (Med. VI, AT VII, 81). Cf. Princ. I, 48, AT VIII-1, 23: “Sed & alia quædam in nobis experimur, quæ nec ad solam mentem, nec etiam ad solum corpus referrri debent, quæque ... ab arctâ & intimâ mentis nostræ cum corpore unione proficiscuntur: nempe appetitus famis, sitis, &c.; itemque, commotiones, sive animi pathemata, quæ non in solâ cogitatione consistunt ...; ac denique sensus omnes, ut doloris, titillationis, lucis & colorum ... aliarumque tactilium qualitatum.” Princ. II, 2, AT VIII-1, 41: “[P]erspicuè advertamus dolores aliosque sensus nobis ex improviso advenire; quos mens est conscia non à se sola proficisci, nec ad se posse pertinere ex eo solo quòd sit res cogitans, sed tantùm ex eo quòd alteri cuidam rei extensæ ac mobili adjuncta sit, quæ res humanum corpus appellatur.” Fortunately, we can safely ignore, in the present study, the formidable problems of the nature of the alleged unity in question, and of the identity of a human being as, on the one hand, pure res cogitans and, on the other hand, an (allegedly “real and substantial”) unity of the mind and the body. 69 Cf. also Descartes‫ ތ‬commentary on his own claim in Med. VI, AT VII, 73 that “quamvis [vis imaginandi] a me abesset, procul dubio manerem nihilominus ille idem qui nunc sum”: “Tunc essem sicut angeli, qui non imaginantur” (Burm., AT V, 162).

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So far so good. However, the following, much disputed and closely interconnected questions are now to be addressed: in what does the very operatio intellectûs which can occur either alone or as somehow unified with a body properly consist; and due to precisely what do even imaginatio and sentio qualify at all as modi—however “special”—of the operationes intellectûs (and by implication of cogitatio)? The latter question is motivated by a commonly recognized fact that it seems prima facie odd to take, as Descartes does, acts of imaginatio, and even more so of sentio, as modes of thinking, in so far as thinking essentially implies—as it does for you and me nowadays—dealing with propositionally structured items, perhaps imprimis in inferential and ratiocinative contexts, and in so far as it therefore brings about the issue of truth and falsity. Now to address this problem satisfactorily, the former question must be answered first and it is then to be argued either (i) that the meaning Descartes associates with the term “cogitatio” as employed by him is different from—presumably wider and less “intellectualistic” than—the established common meaning of “thinking” in contemporary English, so that ultimately imaginatio and sentio do qualify as operationes intellectûs; or alternatively (ii) that while Descartes‫“ ތ‬cogitatio” and contemporary “thinking” are virtually synonymous, upon closer inspection Descartes‫ ތ‬notions of imaginatio and sentio are quite different from the notions of imaginatio and sentio that are presumably presupposed in the worry which generates the present problem, to the effect that they eventually do qualify as kinds of thought in the contemporary, narrow, “intellectualistic” sense; and perhaps mitigated forms of (i) and (ii) could somehow be combined.70 Perhaps the most significant pure instance of strategy (i) is a recent suggestion by several respected scholars to the effect that far from its contemporary, narrowly “intellectualistic” sense, “cogitatio” (and its French equivalent “pensée”) counts as—in Williams‫ ތ‬phrase—“any sort of conscious state or activity whatsoever” for Descartes, so that “it can as well be a sensation ... or an act of will” (Williams, Descartes, 62).71 This 70

I ignore this last option in what follows, however, as I do not endorse it in the present study, neither do I know of any significant resolution of the present problem to this effect. 71 Apart from Williams, Kenny, Descartes, 76–77 and Elizabet Anscombe and Peter Geach, “Translator‫ތ‬s Note,” in Descartes. Philosophical Writings: A Selection, ed. and transl. Elizabeth Anscombe and Peter Geach (2nd ed. Prentice Hall: The Open University, 1970), xlvii–xlviii have formulated a similar view. The fact that Williams in his Descartes, 62 translates “modi cogitandi” as “forms of consciousness” in Princ. I, 32, AT VIII-1, 17, illustrates well the thrust of the

26

Chapter One

line of interpretation gains its strongest prima facie support from Descartes‫ ތ‬own definitions of cogitatio in Resp. 2 and Princ. I:72 Cogitationis nomine complector illud omne quod sic in nobis est, ut ejus immediate conscii simus. Ita omnes voluntatis, intellectûs, imaginationis & sensuum operationes sunt cogitationes (Resp. 2, AT VII, 160; Descartes’ emphasis). Cogitationis nomine, intelligo illa omnia, quæ nobis consciis in nobis fiunt, quatenùs eorum in nobis conscientia est. Atque ita non modò intelligere, velle, imaginari, sed etiam sentire, idem est hîc quod cogitare (Princ. I, 9, AT VIII-1, 7).

Its proponents have adduced some further arguments in its support.73

suggestion in question. Anscombe and Geach corroborate their view, among other things, with the contention that “cogitare and its derivatives had long been used in a very wide sense in philosophical Latin” (Anscombe and Geach, “Translator’s Note,” xlvii). Yet this claim is highly dubious and was attacked, rightly I think, by John Cottingham, Cartesian Reflections: Essays on DescartesҲs Philosophy (Oxford: Oxford University Press, 2008), 98. 72 Descartes frames both the characterizations mentioned in the quoted passages as definitions (though only implicitly in the Principia). Yet he says emphatically elsewhere, “nego nos ignorare quid sit res, quidque cogitatio, vel opus esse vt alios id doceam, quia per se tam notum est, vt nihil habeatur per quod clarius explicetur” (Hyp., AT III, 426; Descartes’ emphases; I read “quidque” for “quidue” in AT III, 426.22). The characterizations at issue are then to be taken rather as explications in the light of this passage. 73 See in particular Anscombe and Geach, “Translator’s Note,” xlvii–xlviii: “Our criticism of the traditional rendering would of course fall to the ground if Descartes were maintaining that all mental acts, in spite of their apparent differences, are ‫ދ‬really‫ ތ‬thoughts ... and are only ‫ދ‬misperceived‫ ތ‬as being anything else. But Descartes expressly denies that the ‫ދ‬evil genius‫ ތ‬could make me ‫ދ‬misperceive‫ ތ‬the contents of my own mind.” Yet Hill in his “What does ‫ދ‬To Think‫( ތ‬cogitare) mean in Descartes‫ ތ‬Second Meditation?.” Acta Comeniana 19, no. 49 (2005), 99 responds with quoting a Med. II, AT VII, 31 passage “perceptio [ceræ] non visio, non tactio, non imaginatio est, nec unquam fuit, quamvis prius ita videretur, sed solius mentis inspectio” (Hill’s emphasis), and he comments pertinently: “These words can only make us very sceptical of Anscombe and Geach’s claim that the activities of the mind must, in Descartes’ view, be transparent to an unreflective subject. It is obviously impossible to doubt that, say, I seem to be perceiving a fire next to me, a dressing-gown around me or a piece of paper in my hand. But it may be that reflection can lead to a better understanding of the faculty that is in fact being employed” (Hill, ibid., 99).

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Despite all this, however, several recent scholars have suggested, in view of the bulk of other textual evidence and/or corroborative interpretive argumentation, that such a radical intensional revision of Descartes‫ތ‬ cogitatio be abandoned, and instead recommend adopting strategy (ii): they argue that Descartes has a perhaps somewhat extraordinary, strongly intellectualistic notion of sentio and imaginatio. Maybe the most audacious version on this score has recently been proposed by James Hill.74 According to him, all types of sentio75 (and by implication of imaginatio)76 in Descartes‫ ތ‬hands, in so far as they are treated accurately, i.e. as “ab unione & quasi permixtione mentis cum corpore exorta”77 (and not just as, say, a purely physical image), essentially involve judgment— i.e. a paradigmatic case of an intellectual act—as an intrinsic constitutive element, more specifically as something that operates as the inseparable form (in the fairly literal Aristotelian sense) of a given act of sentio. The professedly decisive evidence for the above rival reading—Descartes‫ތ‬ definitions of cogitatio in terms of conscientia in Resp. 2 and Princ. I, 9— are then explained away by Hill to the effect that far from the “phenomenalist” meaning which is common currency in our times, “conscientia” was employed by Descartes precisely as a characterization of “activity that is immediately present in the mind” (Hill, Doubting Mind, 130), that is to say, intellectual activity.78 Conscientia in this narrowly intellectualist sense thus seems to be just a function of cogitatio also interpreted along the now common intellectualistic lines. John Cottingham, on the other hand, has proposed a more moderate variant of strategy (ii). While he is de facto in agreement with Hill with regard to the putative evidence of Resp.

74

See Hill, “What does ‘To Think’ mean,” and Hill, Doubting Mind, ch. 7. Hill‫ތ‬s reading trades in particular on the following passages: Med. II, AT VII, 32; Passions I, 24, AT XI, 347; Princ. I, 46, AT VIII-1, 22 together with Med. VI, AT VII, 87; and Med. VI, AT VII, 78. 75 I.e. not only sense perception but also bodily sensations and even emotions—cf. fn. 63. 76 Hill focuses almost exclusively on sentio in his texts; yet he makes it clear that he means his claims to be applicable to the case of imaginatio equally well (which is corroborated with the previously quoted AT X, 415–16 and AT V, 162; cf. also Med. II, AT VII, 28: “[N]ihil aliud est imaginari quàm rei corporeæ figuram, seu imaginem, contemplari. Jam autem certò scio me esse, simulque fieri posse ut omnes istæ imagines, & generaliter quæcunque ad corporis naturam referuntur, nihil sint præter insomnia.”) Following his practice and for the sake of brevity I omit imaginatio in what follows. 77 Med. VI, AT VII, 81. 78 For Hill‫ތ‬s extensive argument to this effect, see Hill, Doubting Mind, ch. 8.

28

Chapter One

2 and Princ. I, 9,79 he conceives of the acts of sentio and imaginatio, respectively, as breaking down into two components, namely “the reflexive awareness of the mind that it is being presented with a datum of some kind” on the one hand, and “a curious residual element, which might be called the ‘qualitative feel’” (ibid., 104) on the other. Unlike Hill, therefore, he takes the intellectual component of Descartes‫“ ތ‬special” modes of cogitatio as a kind of supplementary element rather than as an intrinsic form of those modes.80 Any attempt to decide between the two above approaches would take us too far away from our proper concerns. Be this as it may, the substantial evidence both Cottingham and Hill adduce has convinced me that some version of strategy (ii) is likely to be true.81 As a consequence, I will

79

Cf. Cottingham, Cartesian Reflections, 101. Cf. in particular ibid., 104. While Cottingham shares with Hill the general point of departure—the contention that the way Descartes arrived at the certainty of the Cogito in the Meditationes and elsewhere, viz. “the method of doubt,” establishes the pertinence of a basically intellectualistic interpretation of Descartes‫ ތ‬cogitatio (cf. Cottingham, Cartesian Reflections, 99–100; Hill, Doubting Mind, 1–4)—he corroborates his more moderate reading chiefly with Burm., AT V, 149, Princ. I, 9, and Med. II, AT VII, 29. 81 I should perhaps note that as far as I can see, Hill‫ތ‬s suggestion, even if basically correct (as I believe it is), is in need of rectification in at least one important respect. That is to say, I believe Hill should have identified not judgment but rather (a sort of) apprehended propositional content as what informs the acts of sentio and imaginatio by way of an intrinsic constitutive element according to his interpretation. This is because, as we have seen and as Hill of course acknowledges, judgment is, for Descartes, a complex structure due to a joint operation of the faculties of voluntas and intellectus; yet sentio and imaginatio are unequivocally classified as operationes intellectûs by Descartes; so if it were indeed (as Hill has it) judgment that informs sentio and imaginatio quâ operationes intellectus, one would be pressed to acknowledge—given Descartes‫“ ތ‬two-factor” analysis of judgment—a distinction between operationes intellectûs and operationes voluntatis within the original heading of operationes intellectûs, with a vicious regress under way. Anyway, the suggested rectification renders untouched the essentials of Hill‫ތ‬s reading as far as I can see. It is vital for the proposed argument from regress not to confuse judgment and one‫ތ‬s perception that one is making judgment, or acts of will and one‫ތ‬s perceptions that one is willing; as far as I can see it is the former and not the latter items, respectively, that Hill is committed to take as the form (or components of the form) of sensationes and imaginationes in his interpretation. It is true that Descartes himself expressly identifies one‫ތ‬s idea that one is willing with the act of will in one instance (Mers., AT III, 294); but this must be a slip as it contradicts the authoritative exposition in 80

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assume from now on that Descartes‫ ތ‬notion of cogitatio is basically intellectualistic in the established sense; and that sentio and imaginatio acquire quite a strong intellectualistic tinge in Descartes‫ ތ‬hands.

1.2 The Modes of Understanding, and Innate Ideas One of the significant motifs connected with Descartes‫ ތ‬treatment of the modes of understanding is his categorization of the whole class of ideas as either adventitious (adventitiæ), factitious (factæ or fictitiæ) or innate (innatæ). It is not difficult to establish the general sense in which this classification is relevant to our overall project: for one thing, innateness as understood by Descartes has commonly been associated with “independence of experience” in such a sense that Descartes‫ ތ‬views of the respective roles of the senses and the understanding in the constitution and justificatory structures of human cognition are rendered compatible with the overall epistemological conception organized around the “a priori—a posteriori” distinction as understood by and as being employed in relevant contexts since the time of Kant; and for another thing, Descartes seems to ascribe to innate ideas a certain remarkable rôle in the constitution of that privileged group of cognitions which he calls scientiæ and to which, as already indicated, his own distinction between a priori and a posteriori is properly applied. Only the latter point will be tackled directly in the present study, and it must wait until later for full elaboration and assessment. Our task in the present section is only to clarify how exactly, according to Descartes, the classification of ideas in question is supposed to work, how it is related to his classification of the modes of understanding and how exactly innateness as a characterization of certain ideas is to be interpreted. At first glance, the textual basis of the issue at hand seems somewhat bewildering. It might plausibly be held that Descartes seems to promote several virtually conflicting ways in which the classification at issue, and in particular the class of innate ideas, could or should be understood, and it might seem that his pronouncements could hardly be, as they stand, pieced together to form a unitary coherent theory. Despite all this, I believe it is possible to stretch and push the texts to derive, eventually, a relatively viable unitary notion of innateness in Descartes. What is quite clear and uncontroversial is that the feature with respect to which ideas are classified

Med. III, AT VII, 37, and plunges to vicious regress the classification of cogitationes in Princ. I, 32–34, AT VIII-1, 17–18.

30

Chapter One

as adventitious, factitious or innate is their origin or source (origo).82 Further, I take it as useful and also highly plausible to treat that feature quite generally in terms of the classified ideas‫ ތ‬causal histories. After all, so much seems to be implied by Descartes himself in a passage in which he introduces an important distinction to prevent a certain elementary misunderstanding concerning the meaning of the classification in question: Pergit tamen [Regius] affirmare, ipsam ideam Dei, quæ in nobis est, non à nostrâ cogitandi facultate, cui sit innata, sed ex divinâ revelatione, vel traditione, vel rerum observatione, esse. Cujus assertionis errorem faciliùs agnoscemus, si consideremus, aliquid dici posse ex alio esse, vel quia hoc aliud est causa ejus proxima & primaria, sine quâ esse non potest; vel quia est remota & accidentaria duntaxat, quæ nempe dat occasionem primariæ, producendi suum effectum uno tempore potiùs quàm alio. Sic artifices omnes sunt operum suorum causæ primariæ & proximæ; qui verò jubent, vel mercedem promittunt, ut illa faciant, sunt causæ accidentariæ & remotæ, quia fortassis nisi justi non facerent. Non autem dubium est, quin traditio vel rerum observatio sæpe sit causa remota, nos invitans, ut ad ideam, quam habere possumus de Deo, attendamus, illamque cogitationi nostræ præsentem exhibeamus. Quòd autem sit causa proxima istius ideæ effectrix, à nemine dici potest, nisi ab eo, qui putat nihil à nobis de Deo unquam intelligi, nisi quale sit hoc nomen, Deus, vel qualis sit figura corporea quæ nobis ad repræsentandum Deum à pictoribus exhibetur (Notæ, AT VIII-2, 359–60; Descartes’ emphases).

Several things are to be borne in mind, then, to avoid misunderstanding the doctrine in question from the very start. Firstly, the classificatory differentiae are to be explicated in terms of what is, and what is not, involved in the causal histories of ideas. Secondly, only the causes which qualify as proximæ & primariæ in the sense Descartes has clarified are acceptable as components of the causal histories at issue. Finally, the particular relevant causal factors are to be related to the previously introduced ontological constituents of ideas in Descartes, viz. the realitas formalis and the realitas objectiva (or perhaps, in some cases, to both at once). Descartes makes it clear more than once that at least in some cases the realitas objectiva of an idea counts as an ontologically positive effect of some causal action on the part of an entity that is realiter distinct from

82

This is clearly indicated in particular in Med. III, AT VII, 37–39, and Descartes‫ތ‬ vocabulary in the most extensive discussion of the topic in the Notæ, AT VIII-2, 357–61 confirms this: the verbs associated there with ideas thus classified include procedere (AT VIII-2, 358), efformare (AT VIII-2, 359) and oriri (ibid.).

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the cause of the action of the mind which always counts as at least a part of the (proximate) cause of the realitas formalis of the same idea.83 Thus equipped, let us turn to the texts. As with many other heavily discussed aspects of Descartes‫ ތ‬philosophy, it is striking how rarely Descartes himself condescends to establish the classification of ideas in question distinctly and in its entirety. The following passages are the only ones in which he performs such a task: Ex ... ideis aliæ innatæ, aliæ adventitiæ, aliæ a me ipso factæ mihi videntur: nam quòd intelligam quid sit res, quid sit veritas, quid sit cogitatio, hæc non aliunde habere videor quàm ab ipsâmet meâ naturâ; quòd autem nunc strepitum audiam, solem videam, ignem sentiam, a rebus quibusdam extra me positis procedere hactenus judicavi; ac denique Syrenes, Hippogryphes, & similia, a me ipso finguntur (Med. III, AT VII, 37–38). [C]ùm adverterem, quasdam in me esse cogitationes, quæ non ab objectis externis, nec à voluntatis meæ determinatione procedebant, sed à solâ cogitandi facultate, quæ in me est, ut ideas sive notiones, quæ sunt istarum cogitationum formæ, ab aliis adventitiis aut factis distinguerem, illas innatas vocavi (Notæ, AT VIII-2, 357–58; Descartes’ emphases). Differunt ... ideæ innatæ ab adventitiis & factis sive fictitiis, quòd ad fictitias voluntatis actio concurrat, ad adventitias sensus, ad innatas sola intellectûs perceptio (Annotationes in Principia, AT XI, 655).

In the case of innate and adventitious ideas, these passages jointly yield, at least prima facie, a fairly lucid conception. As innate count precisely those ideas with a full proximate cause or origin strictly coincident with (presumably an action of) the very nature (ipsa natura) of the mind that has them, this nature being in turn identified with the mind‫ތ‬s facultas cogitandi. The source of innate ideas can then aptly be described as the mind‫ތ‬s reflexion upon its own operations and processes of thinking, or else inferences from such reflexions.84 As far as I can tell, then, the sole proximate cause of the realitas formalis of a given innate idea is an action of the facultas cogitandi and the sole proximate cause of the realitas 83 The most vivid evidence to this effect is provided by Descartes‫ ތ‬instructive exchange with Caterus in Obj. 1 and Resp. 1; see especially AT VII, 91–95; 102– 105. Cf. also Med. III, AT VII, 40–42. 84 Such a description is suggested in Desmond Clarke, DescartesҲ Philosophy of Science (University Park: The Pennsylvania State University Press, 1982), 53 and in Murray Miles, Insight and Inference: DescartesҲs Founding Principle and Modern Philosophy (Toronto: University of Toronto Press, 2012), 297.

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objectiva is some particular operations of thinking, the subject being one and the same mind. Correspondingly, the sole mental operation engaged in (or, in Descartes‫ ތ‬phrase, “concurrent with”) the apprehension of the objects of such ideas is a perceptio of (presumably pure) understanding. Similarly, as adventitious count precisely those ideas with a full proximate cause which includes—though perhaps is not entirely coincident with—(an action by) external objects, presumably of material nature. That is to say (as I interpret it), the full proximate cause of the realitas formalis of adventitious ideas essentially includes an action by external objects;85 and Descartes seems to hold that the same external objects somehow indirectly constitute (at least a part of) the realitas objectiva of the ideas in question.86 Correspondingly, the distinctive mental operation engaged in the constitution of such ideas is sensus perceptio. The case of factitious ideas somewhat upsets the neatness of the exposition. It will be observed that factitious ideas are treated quite differently in the quoted passages: the actio (or determinatio) voluntatis, with which factitious ideas are distinctively associated in AT VIII-2, 357– 58 and AT XI, 655, fits poorly with the implicit scheme within which innate and adventitious ideas have just been situated: like perceptio intellectûs and perceptio sensûs, and unlike the ipsa natura of the mind and the external objects, actio voluntatis qualifies as a mental operation; but unlike the types of perceptiones just mentioned, actio voluntatis as employed here has nothing to do with the modes of apprehension associated with the corresponding classes of ideas; rather it connotes the entity which enters, quâ the subject, into the full cause of factitious ideas. This odd performance by Descartes seems to be due to the complexity of the situation concerning factitious ideas. One complication is associated 85

Cf. Notæ, AT VIII-2, 358–59: “[J]udicemus, has vel illas ideas, quas nunc habemus cogitationi nostræ præsentes, ad res quasdam extra nos positas referri: non quia istæ res illas ipsas nostræ menti per organa sensuum immiserunt, sed quia tamen aliquid immiserunt, quod ei dedit occasionem ad ipsas, per innatam sibi facultatem, hoc tempore potiùs quàm alio, efformandas.” 86 Cf. especially an instructive treatment in Princ. IV, 197–98, in particular here: “[S]ciamus eam esse animæ nostræ naturam, ut diversi motus locales sufficiant ad omnes sensus in eâ excitandos; experiamurque illos reipsâ varios sensus in eâ excitare, non autem deprehendamus quicquam aliud, præter ejusmodi motus, à sensuum externorum organis ad cerebrum transire: omnino concludendum est, non etiam à nobis animadverti, ea, quæ in objectis externis, luminis, coloris, odoris, saporis, soni, caloris, frigoris & aliarum tactilium qualitatum vel etiam formarum substantialium, nominibus indigitamus, quicquam aliud esse quàm istorum objectorum varias dispositiones, quæ efficiunt ut nervos nostros variis modis movere possint” (AT VIII-1, 322–23).

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with the fact that the subject of the actiones voluntatis in question is clearly the very same mind that apprehends the resultant factitious ideas:87 the point is that this mind, insofar as performing those actiones voluntatis that contribute to the production of factitious ideas, does not enter into the objective reality of the resultant ideas; in this respect, the case of factitious ideas is very different from the cases of both the innate and adventitious ideas in which the entities that enter into their causal antecedents (i.e. the nature of the mind and the external objects, respectively) also form—quâ cognized—at least a part of their realitas objectiva. Another complication is associated with the fact that, arguably, the only type of actio voluntatis that eventually turns out relevant to the production of factitious ideas are operations of composing or assembling certain ideas that are incapable of further analysis.88 It is these items quâ composed in a certain way that constitute the realitas objectiva of factitious ideas in Descartes. The complication is, however, that arguably, these items can themselves be either of sensory or of purely intellectual origin.89 As a consequence, the causal history of constructed ideas includes either a certain external object (in the case of simple items of sensory origin, the external object is then, for Descartes, the pineal gland, in which “imagines pingantur ... a mente sine objectis externis et tanquam clausis fenestris,” Burm., AT V, 162) or else the very nature of the given mind (in cases wherein the simple items are of purely intellectual origin); and the concurrent apprehensive operations are, correspondingly, either perceptiones imaginationis (the former case) or else perceptiones intellectûs (the latter case). Given all these complexities, it is understandable that Descartes opted for a somewhat elliptical manner of presentation in his classifications. The following scheme surveys what I take to be the correct view of the situation as it has crystallized so far.

87

See the above-quoted AT VII, 37–38 and AT VIII-2, 357–58. This point can plausibly be extracted in particular from Reg. XII, AT X, 420–25; see especially ibid., 420: “[R]eliqua omnia quæ cognoscemus, ex ... naturis simplicibus composita esse: vt si judicem aliquam figuram non moveri, dicam meam cogitationem esse aliquo modo compositam ex figurâ & quiete; & sic de cæteris.” And ibid., 422: “Dicimus ... naturas illas, quas compositas appellamus, à nobis cognosci, vel quia experimur quales sint, vel quia nos ipsi componimus” (my emphasis). A similar conception seems to be invoked in Med. I, AT VII, 19–20. The unanalyzable items at issue, at least some (and perhaps all) of which turn out to amount to Descartes‫ ތ‬naturæ simplices or primæ notiones, will be discussed in detail in ch. 2. 89 This point is argued in ch. 2. 88

34 Fig. 1-1

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1.2.1 Innate Ideas and Dispositions Let us focus now on how the notion of innate ideas hitherto explained is further developed in Descartes‫ ތ‬hands. So far we have arrived at a notion according to which precisely those ideas count as innate the proximate causal history or constitution of which includes no sensory or voluntary input. Yet such a notion still leaves Descartes‫ ތ‬theory of innate ideas underdetermined in at least two important respects, to be considered in the present and the next subsection, respectively. To approach the first of them, I take it for granted that lest Descartes‫ތ‬ theory of innateness be rightly charged with gratuitous abuse of the word “innate” (and exchanges with his critics as mere disputes de nomine), his usage must retain, in one way or another, the implication from “i is innate to s” to “i is necessarily present in the mind of s from s‫ތ‬s very conception”.90 It will be observed that the meaning of “innate” as explained thus far seems in itself not to secure that implication. Thus the question is how the notion of innateness is or could be further explained so that it could achieve that much. One option Descartes consistently and unambiguously opposes, and even derides, is the view that innate ideas are to be identified with acts the corresponding contents of which the mind is actually aware at every moment of its existence from its very conception onwards.91 Quite to the 90

“i” stands for ideas and “s” for subjects. See in particular Notæ, AT VIII-2, 366: “[Q]uòd autem istæ ideæ [innatæ] sint actuales ... nec unquam scripsisse nec cogitasse: ... adeò ut à risu abstinere non potuerim, cùm vidi magnam illam catervam, quam Vir, fortasse minime malus, laborio se collegit ad probandum, infantes non habere notitiam Dei actualem, quandiu sunt in utero matris, tamquam si me hoc pacto egregie impugnaret” (Descartes’ italics). Cf. also Resp. 3, AT VII, 189: “[C]ùm dicimus ideam aliquam nobis esse innatam, non intelligimus illam nobis semper obversari: sic enim nulla prorsus esset innata ....” And Annotationes in Principia, AT XI, 655: “[N]on intelligo [ideas innatas] esse semper actu in aliquâ mentis nostræ parte depictas, ut multi versus in libre Virgilii continentur ....” Hill, Doubting Mind, 84 holds that “[t]his characterization of innatism is not quite a straw man as there are one or two places where Descartes seems to be working with something like it. Most famously, perhaps, he talks in the Meditations of the idea of God as ‫ލ‬the mark of the craftsman stamped on his work‫ތ‬, and in the Replies he is also happy to talk of the idea of God being ‫ލ‬implanted‫ ތ‬in each human mind” and he intimates that it might have been only in response to “pressure on this point by his critics” that he abandoned this view and offered a more sophisticated alternative (ibid., 85). I believe there are no good reasons to ascribe to Descartes such a change of opinion on the issue. As far as I can tell, neither of the two passages referred to by Hill (nor 91

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contrary, there are several passages to be found in Descartes‫ ތ‬most concentrated attempt to clarify his notion of innateness, viz. in the Notæ, which taken separately and out of context strongly suggest some relatively strong dispositional conception of innateness. Consider the following statements: Non ... unquam scripsi vel judicavi, mentem indigere ideis innatis, quæ sint aliquid diversum ab ejus facultate cogitandi ... (Notæ, AT VIII-2, 357). [I]lla omnia quæ præter ... voces vel picturas cogitamus tanquam earum significata, nobis repræsententur per ideas non aliunde advenientes quàm à nostrâ cogitandi facultate, ac proinde cum illâ nobis innatas, hoc est, potentiâ nobis semper inexistentes: esse enim in aliquâ facultate, non est, esse actu, sed potentiâ dumtaxat, quia ipsum nomen facultatis nihil aliud quàm potentiam designat (ibid., 360–61). [P]er ideas innatas me nihil unquam intellexisse, nisi quod ... nobis à naturâ inesse potentiam, quâ Deum cognoscere possumus ... (ibid., 366; Descartes’ emphasis).

One might easily be led by such statements to ascribe to Descartes the view that innate ideas are strictly identical with a certain remarkable type of mental disposition, namely the dispositions to be actually modified with acts of apprehension whose full cause would be exhausted by the very nature of the very same mind insofar as it operates alone, and which would be a matter of exercising solely perceptiones puri intellectûs. On closer inspection, however, it turns out that this just cannot be Descartes‫ތ‬ considered view, for he repeatedly and in suitable contexts characterizes ideas as operationes intellectûs and at least once he expressly contrasts such operationes with potentiæ.92 Moreover, even the above passages are yet another, namely Mers., AT I, 145) points unambiguously, let alone plausibly, towards the “actualist” view of innateness. In particular, Descartes‫ ތ‬replies to Gassendi, who pressed on the “trademark passage” from Med. III in Obj. 5, AT VII, 305–306, seem to me sound and by no means committing Descartes to such an “actualist” view. 92 Viz. in Resp. 4, AT VII, 232: “[Q]uòd nihil in mente nostrâ esse possit cujus non simus conscii; quod de operationibus intellexi, & [Arnauld] de potentiis negat” (Descartes’ italics). The relevance of the passage to our issue is established through the combination of Descartes‫ ތ‬identification of mens proper with the facultas cogitandi (cf. e.g. Notæ, AT VIII-2, 357–58; 360–61), and of the first two definitions in the Rationes at the end of Resp. 2, AT VII, 160–61: “Cogitationis nomine complector illud omne quod sic in nobis est, ut ejus immediate conscii

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counterpointed with statements which imply it is the faculty of thinking insofar as actualized with which innate ideas are to be associated, such as the following: Egregiam ... consequentiam [Regius] deducit ... tanquam, si facultas cogitandi nihil possit per se præstare, nihilque unquam percipiat vel cogitet, nisi quod accipit à rerum observatione vel traditione, hoc est, à sensibus. Quod adeò falsum est, ut è contra, quisquis recte advertit, quousque sensus nostri se extendant, & quidnam sit præcise, quod ab illis ad nostram cogitandi facultatem potest pervenire, debeat fateri, nullarum rerum ideas, quales eas cogitatione formamus, nobis ab illis exhiberi. ... [R]es [quædam] extra nos [positæ] ... tamen aliquid immiserunt, quod ei dedit occasionem ad ipsas, per innatam sibi facultatem, hoc tempore potiùs quàm alio, efformandas” (Notæ, AT VIII-2, 358–59; my emphasis).

Gathering together the above virtually conflicting commitments, I submit that as far as one can tell, the position Descartes is bound to hold might be described as follows. It is not that innate ideas are only to be identified with the dispositions in question; what Descartes is bound to mean instead is that the causal history of any idea that rightly counts as innate consists in an inherent activation (i.e. activation for which the facultas cogitandi of the given mind is fully responsible)93 of a disposition that the mind is endowed with from its very conception—namely the disposition to apprehend the mind‫ތ‬s very faculty of thinking and/or some of its cogitative processes. Thus, what count as innate stricto sensu are the dispositions just specified; and ideas deserve the title of innateness precisely insofar as they amount to (typically but not essentially temporary) inherent activations of some such innate disposition.94 simus. Ita omnes voluntatis, intellectûs, imaginationis & sensuum operationes sunt cogitationes. ... Ideæ nomine intelligo cujuslibet cogitationis formam illam, per cujus immediatam perceptionem ipsius ejusdem cogitationis conscius sum ... Atque ita non solas imagines in phantasiâ depictas ideas voco ..., sed tantùm quatenus mentem ipsam in ... cerebri partem conversam informant” (Descartes’ emphases). The latter passage also renders more precise the sense in which ideas are to be characterized as operationes: it is cogitationes in general that count as operationes in the proper sense, and ideas count as the forms of them. 93 Cf. e.g. Hyp., AT III, 424: “[N]ec dubito quin, si [mens] vinculis corporis eximeretur, ipsas [ideas innatas] apud se esset inuentura.” 94 The submitted reading also gains some support from the following passage: “[N]otandum est eas omnes res, quarum cognitio dicitur nobis esse à naturâ indita, non ideo à nobis expresse cognosci; sed tantùm tales esse, ut ipsas, absque ullo sensuum experimento, ex proprii ingenii viribus, cognoscere possimus” (Ad Vœtium, AT VIII-2, 166). It is crucial to observe that the “res, quarum cognitio

38

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1.2.2 Innate Ideas and Implicit Apprehension We shall see presently that the picture just canvassed is still in need of certain substantial modifications. To put ourselves in a position to appreciate this need, however, another well-known complication regarding Descartes‫ ތ‬conception of innateness must first be addressed. Descartes indicates in his Notæ that there is a sense in which any idea whatsoever, i.e. even ideas of sense and of imagination, rightly counts as innate:95 [N]ihil sit in nostris ideis, quod menti, sive cogitandi facultati, non fuerit innatum, solis iis circumstantiis exceptis, quæ ad experientiam spectant: quòd nempe judicemus, has vel illas ideas, quas nunc habemus cogitationi nostræ præsentes, ad res quasdam extra nos positas referri: non quia istæ res illas ipsas nostræ menti per organa sensuum immiserunt, sed quia tamen aliquid immiserunt, quod ei dedit occasionem ad ipsas, per innatam sibi facultatem, hoc tempore potiùs quàm alio, efformandas. Quippe nihil ab objectis externis ad mentem nostram per organa sensuum accedit, præter motus quosdam corporeos .... Unde sequitur, ipsas motuum & figurarum ideas nobis esse innatas. Ac tantò magis innatæ esse debent ideæ doloris, colorum, sonorum, & similium, ut mens nostra possit, occasione quorundam motuum corporeorum, sibi eas exhibere; nullam enim similitudinem cum motibus corporeis habent (Notæ, AT VIII-2, 358–59).

How to reconcile this with the fact that Descartes usually—and even in the section immediately preceding the passage cited—wishes to contrast adventitious ideas (and ideas of imagination) with those that count as innate for him? No doubt some ambiguity present in the term “innate” is to be invoked to resolve the apparent inconsistency. The question is how exactly the ambiguity is to be explicated given (i) the root meaning of innateness established above, viz. “having to do, as to its origin, with an activation of the facultas cogitandi,” and (ii) the commitment somehow to

dicitur nobis esse à naturâ indita” are not ideas but the objects of ideas; ideas are the cogitationes themselves, so that “cognoscere” is synonymous here with “ideas habere”. 95 In fact, Descartes speaks only of adventitious ideas in the cited passage. Given the above account of factitious ideas, however, Descartes‫ ތ‬claim can easily be extended to the case of the relevant type of factitious ideas (that is to say, ideas of imagination).

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retain, even in the derived meaning, the implication from “be innate to the mind of s” to “pertain to the mind of s at every moment of s‫ތ‬s existence.”96 One suggestion that must be rejected at the very start is that all ideas indiscriminately count as innate in the sense that their pertaining to the given mind is rendered possible, in a rudimentary manner, by the bare receptivity of the mind, i.e. by the mind‫ތ‬s purely passive ability to be modified by ideas. To be sure, condition (ii) remains fulfilled through the move analogical to that through which the root sense of innateness was established above: all ideas would count as innate in the sense that they modify the mind quâ characterized with something that pertains to it at every moment of its existence, namely with bare receptivity. Yet the crucial difference is that, as Descartes makes clear in the last quoted passage, even in the derivative sense to be established, innateness implies activity on the part of the mind: the mind is said there to efformare and sibi exhibere the ideas in question.97 It is, then, in view of the activity– passivity distinction that the interpretation at issue fails to satisfy condition (i). Moreover, Descartes‫ ތ‬purpose in introducing innateness in the wider sense in the Notæ is clearly to refute the thesis that there are no ideas—not even those that count as communes notiones—the origin of which would not be drawn solely “ex rerum observatione, vel traditione” (Notæ, AT VIII-2, 358), his strategy being to show that even the types of ideas that are most likely to be of such a character, viz. sensory ideas, do in fact involve some component with an origin that is not traceable back to rerum observatio, vel traditio. However, the suggested meaning of innateness in terms of bare receptivity cannot serve such a purpose: an adherent of the thesis Descartes wishes to refute can coherently concede to Descartes that all cognitions are necessarily innate in the suggested sense, and yet insist that all cognitions do have their source in “rerum observatione, vel traditione”. Innateness thus interpreted would therefore lose any theoretical significance; for no one—not even radical empiricists—would then wish to deny that all ideas qualify as innate. Yet in characterizing even sensory ideas as innate, Descartes clearly intends to advance a substantial thesis concerning the origin and nature of our cognitions, a thesis he surely regarded as anything but trivial and surely opposed resolutely to empiricist conceptions.

96

As for the latter of the conditions just mentioned, I take it that giving it up would amount to allowing Descartes, once again, to gratuitously change the meaning of a well-established term, even in the course of a single paper. 97 Cf. also Resp. 3, AT VII, 189.

40

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Desmond Clarke came up with an instructive alternative suggestion. According to him, the sought-after derivative sense of innateness is to be explicated so that even those ideas which are provoked by and correlated with extra-mental stimuli are innate in the sense of being irreducible to the type of reality which triggers them in the mind (Clarke, Descartes’ Philosophy of Science, 50).

Far from being concerned directly with the origins of ideas, the derivative meaning is then, according to Clarke, an embodiment of a thesis regarding the ontological character of ideas. The thesis in question seems to boil down to the following claim (which is correctly diagnosed by Clarke as a corollary of Descartes‫ ތ‬substantial dualism): unlike the relevant causal factors responsible for the production of ideas, all the ideas are on a par as to their ontological status; they are all distinctly immaterial as they all amount to modes of the attribute of thought. To be sure, Clarke‫ތ‬s interpretation fares much better than the former suggestion in terms of bare receptivity. Condition (ii)—that innateness is to be related to something which always pertains to the given mind—is satisfied thanks to the reference to the attribute of thought which to Descartes, of course, amounts to the very essence of the mind. Moreover, Clarke‫ތ‬s reading is superior in that it endows innateness in the sought-after wider sense with genuine theoretical significance: far from clinging to the bare receptivity of the mind which makes attribution of innateness close to trivial, Clarke‫ތ‬s interpretation makes Descartes claim that any correct conception of sense perception is bound to provide an account of sensory ideas precisely quâ the modes of an immaterial substance. What about the crucial tenet of Descartes‫ ތ‬strategy in the relevant portions of the Notæ which turned out fatal for the first suggestion we rejected? Viz., to force its opponents to admit that even the ideas which are rightly characterized as having their source in the senses do involve a component whose source is solely the facultas cogitandi? It must be admitted that even in this respect Clarke‫ތ‬s interpretation does some justice to the texts: given Descartes‫ ތ‬well-established property dualism,98 there is 98

See Jonathan Bennett, Learning from Six Philosophers, vol. 1 (Oxford: Clarendon Press, 2001), 66–67 for a condensed presentation, and idem, A Study of SpinozaҲs Ethics (Indianapolis: Hackett Publishing Company, 1984), 41–47 for a critical discussion of property dualism in the relevant sense. For a condensed exposition of the doctrine of property dualism by Descartes see Resp. 3, AT VII, 175–76. See also ch. 4, fn. 66 and 121.

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indeed, in Clarke‫ތ‬s reading, something about any idea—quâ a mode of a res cogitans—that cannot be brought about but by the facultas cogitandi, namely the idea‫ތ‬s very form or (in Descartes‫ ތ‬preferred terminology) realitas formalis. So far so good: Clarke has succeeded in providing for such a meaning of innateness in the wider sense that both the minimal conditions of any successful interpretation are satisfied; and he has also managed to retain Descartes‫ ތ‬chief strategical tenet in the relevant commentary on sec. 13 of Regius‫ ތ‬pamphlet in the Notæ. Yet unfortunately this enhanced interpretation fails to stand up to several other texts in Descartes. Most significantly, consider the following passage: Nec etiam sine ratione affirmaui, animam humanam, vbicunque sit, etiam in matris vtero, semper cogitare .... Non autem idcirco mihi persuadeo, mentem infantis de rebus Metaphysicis in matris vtero meditari; sed contra, si quid liceat de re non perspectâ conijcere ..., nihil magis rationi consentaneum est, quàm vt putemus mentem corpori infantis recenter vnitam in solis ideis doloris, titillationis, caloris, frigoris & similibus, quæ ex illâ vnione ac quasi permistione oriuntur, confuse percipiendis siue sentiendis occupari. Nec minus tamen in se habet ideas Dei, suî & earum omnium veritatum, quæ per se notæ esse dicuntur, quam easdem habent homines adulti, cum ad ipsas non attendunt; nec enim postea, crescente ætate, illas acquirit; nec dubito quin, si vinculis corporis eximeretur, ipsas apud se esset inuentura (Hyp., AT III, 423–24).

Given that ideas Dei, sui & veritatum per se notarum count as significant examples of innate ideas and that Descartes intends to make a general point, according to Clarke‫ތ‬s reading he is clearly after something much stronger here than he should. For here he is not making a tame point that it is the facultas cogitandi alone that is responsible for the formal reality of sensory ideas (exemplified in the cited passage with ideas doloris, titillationis etc.); rather, he claims that innate ideas in the unmitigated, strong sense are actually present in the given mind as full-blooded items whenever the given mind thinks at all, and a fortiori whenever it has any sensory (and by extrapolation any non-innate) ideas whatever.99 As a 99

Cf. in particular the following instructive passage (though Descartes hints there just at the idea of one‫ތ‬s own soul): “[L‫]ތ‬Ame, ... n‫ތ‬estant, comme i‫ތ‬ay demonstré, qu‫ތ‬vne chose qui pense, il est impossible que nous puissions iamais penser à aucune chose, que nous n‫ތ‬ayons en mesme temps l‫ތ‬idée de nostre Ame, comme d‫ތ‬vne chose capable de penser à tout ce que nous pensons. Il est vray qu‫ތ‬vne chose de cette nature ne se sçauroit imaginer, c‫ތ‬est à dire, ne se sçauroit representer par vne image corporelle. Mais il ne s‫ތ‬en faut pas estonner; car nostre imagination

42

Chapter One

consequence, ideas of sense and imagination turn out innate in a much stronger sense than Clarke would have it: namely, to the effect that whenever such ideas actually occur in the mind, they are essentially accompanied with apprehensions that are innate in the basic, full-blooded sense, i.e. with ideas such that even their realitas objectiva, and not just their realitas formalis, is brought about solely due to operations of the facultas cogitandi. I submit that this substantial claim indeed expresses the essentials of Descartes‫ ތ‬real considered doctrine of innateness. In order to appreciate their implications for Descartes‫ ތ‬overall view of innateness, however, we must pause to elaborate on certain salient presuppositions and complications which the substantial claim at issue seems to entail. To begin with, Descartes needs to ward off an obvious objection that by no means everyone is in a position to report, with sincerity, apprehensions of the ideas that Descartes takes as innate. An obvious way to account for this undeniable fact is to admit that not all perceptions that are there in the mind need necessarily be reflexively accessible for that very same mind, and to insist that it is only the former and not the latter relation with which the substantial claim is concerned. In other words, the contrast between implicit and explicit (or reflexive) apprehension is to be put to work in a theoretically coherent way. Indeed, the aforementioned contrast is clearly at work in several relevant passages. Thus speaking of knowing what thought and existence are (he takes these as innate ideas, of course) he claims that100 n‫ތ‬est propre qu‫ތ‬à se representer des choses qui tombent sous les sens; et pour ce que nostre Ame n‫ތ‬a ny couleur, ny odeur, ny saueur, ny rien de tout ce qui appartient au corps, il n‫ތ‬est pas possible de se l‫ތ‬imaginer, ou d‫ތ‬en former l‫ތ‬image. Mais elle n‫ތ‬est pas pour cela moins conceuable ...” (Mers., AT III, 394). 100 It will be observed that Descartes‫ ތ‬employment of the term “cognitio” in the first quoted passage does not fit well with the above-established root meaning, to the effect that “cognitio” is meant to refer to the end-product of an act of judgment. Here in AT VII, 422, cognitio interna must signify rather the act of (pre-reflexive or implicit) apprehension; this is because given Descartes‫ ތ‬conception of judgment as voluntary assent to a (propositionally structured) apprehended content, the notion of pre-reflexive judgment sounds unintelligible. The term “implicit idea,” employed in Hyp., AT III, 430 (see below), might seem virtually misleading as well: in view of Descartes‫ ތ‬definition of idea in Resp. 2, AT VII, 160, “implicit idea” might sound almost like a contradictio in adiecto. However, as James Hill has pointed out recently (and I have endorsed his claim above), it is a mistake to read conscientia here and elsewhere (most notably in Resp. 4, AT VII, 246) anachronically along the nowadays well-established “phenomenalist” lines “by which all thoughts are fully and transparently present to the phenomenal subject”

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[n]on quòd ad hoc requiratur scientia reflexa .... Sed omnino sufficit ut id sciat cognitione illâ internâ, quæ reflexam semper antecedit, & quæ omnibus hominibus de cogitatione & existentiâ ita innata est, ut, quamvis forte præjudiciis obruti, & ad verba magis quàm ad verborum significationes attenti, fingere possimus nos illam non habere, non possimus tamen reverâ non habere (Resp. 6, AT VII, 422).

And in response to Hyperaspistes‫ ތ‬charge that mirari videris omnes Dei ideam in se non percipere. Sed obijcio, non hîc deesse Geometras & Theologos, qui, postquam pro viribus mentem a corporeis rebus abstraxerunt, asserunt, nondum in se vllam Dei ideam innatam animaduertisse; nec sperant imposterum se, post decimam tuarum Meditationum lectionem, ideam istam in se reperturos (Hyp., AT III, 407; author’s emphasis)

he retorts: Nec etiam memini, me vnquam miratum fuisse, quod omnes non sentiant in se ideam Dei; tam frequenter enim animaduerti, ea quæ homines iudicabant ab ijs quæ intelligebant dissentire, vt, quamuis non dubitem quin omnes ideam Dei, saltem implicitam, hoc est aptitudinem ad ipsam explicite percipiendam, in se habeant, non mirer tamen quod illam se habere non sentiant, siue non aduertant, nec forte etiam post millesimam mearum Meditationum lectionem sint aduersuri (ibid., 430; Descartes’ emphasis).

(Hill, Doubting Mind, 130); instead, Descartes is bound, given the bulk of the textual evidence, to mean by conscientia nothing over and above immediate presence in the mind, regardless of whether the corresponding apprehension is implicit or explicit. Apart from the evidence brought together by Hill, the case of Resp. 4 strongly supports the point in view of our present considerations concerning innateness. Descartes tells Arnauld: “Quòd autem nihil in mente, quatenus est res cogitans, esse possit, cujus non sit conscia, per se notum mihi videtur, quia nihil in illâ sic spectatâ esse intelligimus, quod non sit cogitatio, vel a cogitatione dependens; alioqui enim ad mentem, quatenus est res cogitans, non pertineret; nec ulla potest in nobis esse cogitatio, cujus eodem illo momento, quo in nobis est, conscii non simus. Quamobrem non dubito quin mens, statim atque infantis corpori infusa est, incipiat cogitare, simulque sibi suæ cogitationis conscia sit, etsi postea ejus rei non recordetur, quia species istarum cogitationum memoriæ non inhærent” (Resp. 4, AT VII, 246). If “conscientia” is read phenomenally, then the passage implies, somewhat awkwardly, either that there are no innate ideas whatever (if innateness implies actual presence), or else that innate ideas are pure dispositions. Yet as we saw, there are both textual and systematic reasons that preclude ascribing either of these implications to Descartes.

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Second, in view of the above-established constraint (ii) on any theoretically significant notion of innateness, Descartes is committed not just to the claim that innate ideas, under the present interpretation, are (be it only implicitly) present in the mind whenever the mind thinks but also that the given mind thinks, i.e. actually has (implicit or explicit) ideas at every moment of its existence. Furthermore, Descartes claims this much on several occasions. He invariably presents this as a corollary of the fact that it is of the essence of the mind to think (believing this to have been demonstrated in Med. II),101 most graphically in Hyp., AT III, 423:102 [A]ffirmaui, animam humanam, vbicunque sit, etiam in matris vtero, semper cogitare: nam quæ certior aut euidentior ratio ad hoc posset optari, quàm quod probarim eius naturam siue essentiam in eo consistere, quod cogitet, sicut essentia corporis in eo consistit, quod sit extensum? Neque enim vlla re potest vnquam propriâ essentiâ priuari; nec ideo mihi videtur ille magis audiendus, qui negat animam suam cogitasse ijs temporibus, quibus non meminit se aduertisse ipsam cogitasse, quàm si negaret etiam corpus suum fuisse extensum, quamdiu non aduertit illud habuisse extensionem.

Third, the previous two points open the road for a notion of potentiality that is quite different from both the senses in which we have dealt with potentialities so far, viz. from both (i) pure receptivity of the mind, and (ii) the mind‫ތ‬s disposition to be actually modified by apprehensions whose full cause is strictly coincident with certain acts of the nature of that very same mind. Putting aside (i) which we saw Descartes himself dismisses as irrelevant, the proper actual complement of (ii) is an idea with the sort of causal history just specified which is actually present in the mind; and it has just been shown that according to Descartes, (ii) is as a matter of fact always actualized in this way, so that the ideas at issue rightly qualify as innate given the established standards. Now it is exactly these ideas precisely in so far as they amount to actualizations of (ii) in the mind that become the subjects of the potentia–actus pair in the new sense (iii) about 101 See Med. II, AT VII, 25–27. The argument there culminates in the following dictum: “[C]ogitatio ... sola a me divelli nequit. Ego sum, ego existo; certum est. Quandiu autem? Nempe quandiu cogito; nam forte etiam fieri posset, si cessarem ab omni cogitatione, ut illico totus esse desinerem. Nihil nunc admitto nisi quod necessario sit verum; sum igitur præcise tantùm res cogitans .... Sum autem res vera, & vere existens; sed qualis res? Dixi, cogitans” (ibid., 27). 102 Cf. Burm., AT V, 150: “[M]ens nunquam sine cogitatione esse potest; potest quidem esse sine cogitatione hac aut illâ, sed tamen non sine omni, eodem modo ut corpus ne quidem per ullum momentum sine extensione esse potest.” Also cf. Resp. 4, AT VII, 246; Resp. 5, AT VII, 356.

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to be established; and the feature to be modified with the potentia–actus pair is reflexive awareness or actual presence for the given mind. Innate ideas quâ permanent actualizations of disposition (ii) are potentially brought to reflexive awareness but this potentiality is not necessarily actualized: indeed, according to Descartes it is usually not actualized at all. I believe the established distinction between (ii) and (iii) can rightly be used to dispel much of the confusion concerning Descartes‫ ތ‬employment of the potentia–actus pair in the course of discussing innateness. Potentiality in sense (ii) turns out now to be at work particularly in Resp. 4, AT VII, 232 and 246–47 and in Notæ, AT VIII-2, 357–58;103 and potentiality in sense (iii) is at work particularly in Hyp., AT III, 430, in Notæ, AT VIII-2, 360–61 and 366, and in Ad Vœtium, AT VIII-2, 166. Finally, the relevant relations between innate ideas and deliverances of the senses are to be clarified. Above all, one should never conflate the fact that for Descartes innate ideas conduce sensory ideas whenever the latter occur with a quite different and mistaken notion that for Descartes the presence of innate ideas, insofar as they are just implicit in the mind, is in any sense dependent on the occurrence of sensory ideas. The latter notion is incompatible with the above-established claim that the full cause of innate ideas in the proper sense involves nothing over and above the operation of the facultas cogitandi. Descartes rejects such a notion clearly when he considers the counterfactual situation in which operations of the

103

Here are the passages just referred to: Resp. 4, AT VII, 232: “Tertium denique est, quòd nihil in mente nostrâ esse possit cujus non simus conscii; quod de operationibus intellexi, & [Arnauld] de potentiis negat” (Descartes’ italics). Ibid., 246–47: “[N]otandum est, actuum quidem, sive operationum, nostræ mentis nos semper actu conscios esse; facultatum, sive potentiarum, non semper, nisi potentiâ; ita scilicet ut, cùm ad utendum aliquâ facultate nos accingimus, statim, si facultas illa sit in mente, fiamus ejus actu conscii; atque ideo negare possimus esse in mente, si ejus conscii fieri nequeamus.” Notæ, AT VIII-2, 357–58: “Non ... unquam scripsi vel judicavi, mentem indigere ideis innatis, quæ sint aliquid diversum ab ejus facultate cogitandi; sed cùm adverterem, quasdam in me esse cogitationes, quæ non ab objectis externis, nec à voluntatis meæ determinatione procedebant, sed à solâ cogitandi facultate, quæ in me est, ut ideas sive notiones, quæ sunt istarum cogitationum formæ, ab aliis adventitiis aut factis distinguerem, illas innatas vocavi. Eodem sensu, quo dicimus, generositatem esse quibusdam familiis innatam, aliis verò quosdam morbos, ut podagram, vel calculum: non quòd ideo istarum familiarum infantes morbis istis in utero matris laborent, sed quòd nascantur cùm quâdam dispositione sive facultate ad illos contrahendos” (Descartes’ mephases).

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senses are excluded from the embodied human mind, most straightforwardly in Resp. 5, AT VII, 375:104 [Q]uia hîc tam ingenue interrogas, quas putem mentem meam habituram fuisse Dei & sui ideas, si, ex quo infusa est in corpus, mansisset huc usque in eo clausis oculis, & absque ullo aliorum sensuum usu, ingenue & candide respondeo me non dubitare (modo ipsam in cogitando non impeditam a corpore, ut neque etiam adjutam, supponamus), quin easdem, quas nunc habet, Dei & sui ideas fuisset habitura, nisi tantùm quòd multo puriores & clariores habuisset (Descartes’ italics).

Descartes even goes on immediately to designate (the deliverances of) the senses as a positive hindrance to rendering innate ideas explicit for the mind in which they are always present:105 Sensus enim ipsam in multis impediunt, ac in nullis ad illas percipiendas juvant; & nihil obstat quominus omnes homines easdem se habere æque animadvertant, quàm quia in rerum corporearum imaginibus percipiendis nimium occupantur (ibid.).

Far from positively contributing to the occurrence of innate ideas in the mind, then, the senses are, Descartes holds, responsible for the fact that innate ideas, which are always actually there in the mind, all too often remain unreflected by the very same mind, i.e. only implicit for it. To sum up, the essentials of Descartes‫ ތ‬notion of innate ideas, as it has crystallized by now, can be put as follows. Strictly speaking, the term “idea innata” refers in Descartes precisely to such ideas as are actually present in the mind quâ actualizations of the disposition of the mind to be modified by ideas whose full cause is coincident with certain operations of the very nature, i.e. of the facultas cogitandi, of that same mind. This disposition is actualized, according to Descartes, at every moment of the existence of the given mind, and it is thanks to this that Descartes is immune from the charge of abusing the word “innate”. However, the actual presence in the mind of ideas that thus qualify as innate still does 104

Cf. also the concluding sentence of the above-quoted Hyp., AT III, 423–24: the human mind just unified with the body “[n]ec minus tamen in se habet ideas Dei, suî & earum omnium veritatum, quæ per se notæ esse dicuntur, quam easdem habent homines adulti, cum ad ipsas non attendunt; nec enim postea, crescente ætate, illas acquirit; nec dubito quin, si vinculis corporis eximeretur, ipsas apud se esset inuentura.” 105 I read “percipiendas” for “percipiendias” in AT VII, 375.23-24.

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not imply that the mind is reflexively aware of those ideas: their permanent presence in the mind counts as no more than implicit unless an additional act of attending to them renders them explicit, i.e. present for the mind in which they are always present implicitly. It is, then, above all the deliverances of the senses that are responsible for the mind‫ތ‬s failure to bring innate ideas to explicit, reflexive awareness in the first place: according to Descartes, the mind, insofar as embodied (which is essentially the case of any human mind),106 is usually (and perhaps always or even necessarily) clogged with sensory ideas, and these act as a hindrance to attentive apprehension of innate ideas.107 It has already been indicated that innate ideas are to play an important part in Descartes‫ ތ‬project of establishing the body of the scientiæ; and since, presumably, it is actual scientia, one that one actually possesses, that Descartes is after in his project, it might be expected that not a negligible part of his endeavour is aimed at providing for a systematic account of how to get rid of the sensory ideas that block the mind‫ތ‬s road to explicit apprehension of innate ideas—that is to say, for a method of “leading the mind away from the senses.”108 Much is yet to be done, however, before we are ready to deal with these issues in a controlled and sufficiently clear manner. To begin with, we must proceed from considerations concerning Descartes‫ ތ‬notion of cognition simpliciter and focus on his notion of the privileged form of cognition with which we are properly concerned in this study, namely scientia.

1.3 Clear and Distinct Perception, Certainty, and Scientia By now we are in a position to appreciate, at least in general terms, both the thrust and internal complexities of Descartes‫ ތ‬following central thesis concerning his chief theoretical project of attaining a comprehensive 106 See Notæ, AT VIII-2, 351: “[P]atet, illud subjectum, in quo solam extensionem cum variis extensionis modis intelligimus, esse ens simplex: ut etiam subjectum, in quo solam cogitationem cum variis cogitationum modis agnoscimus. Illud autem, in quo extensionem & cogitationem simul consideramus, esse compositum: hominem scilicet, constantem animâ & corpore, quem videtur author noster pro solo corpore, cujus mens sit modus, hîc sumpsisse” (Descartes’ emphasis). 107 As far as I can tell, the proposed interpretation is in accord (in the chief outlines at least) with that submitted by Miles, Insight and Inference (cf. especially ch. 18– 19) and Hill, Doubting Mind (cf. ch. 6, sec. 4). 108 Cf. e.g. DM 4, AT VI, 37; Med., Synopsis sex sequentium meditationum, AT VII, 12; Med. IV, AT VII, 52; Resp. 2, AT VII, 131; Ad Vœtium, AT VIII-2, 171; A ***, AT I, 353; A Chanut, AT IV, 609–10.

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and solid system of cognitions marked with certainty and evidence: namely his claim that the clear and distinct perception of a given content109 is at least a necessary condition (though, as we shall see shortly, not always a sufficient condition) for the corresponding judgment to yield a cognition of the sought-after privileged kind. The task of the present section and subsections is to develop and discuss this thesis in some detail. At the outset I take it for granted that an extraordinarily strong kind of certainty which Descartes occasionally calls certitudo metaphysica or summa or perfectissima certitudo is at least a necessary characterization of the sought-after privileged cognitions.110 Further, Descartes repeatedly makes it clear that (presumably metaphysical) certainty of any given cognition implies clear and distinct perception,111 most vividly perhaps here (bearing in mind that for Descartes, obscuritas and confusio are the opposites of claritas and distinctio, respectively, and dubitatio is presumably the opposite of certitudo):112 109

Indeed, clarity and distinctness are characteristically attributed not to ideas but to perceptiones by Descartes; this is Descartes‫ ތ‬usual practice throughout the Princ. I (see e.g. ibid., 30; 33; 43–47; 50; AT VIII-1, 16; 17; 21–22; 24) and in the bulk of the Meditationes (cf. e.g. Med. III, AT VII, 35–36; 43; Med. IV, AT VII, 59; 61–62), although there are admittedly also several places in the Meditationes and elsewhere where Descartes applies clarity and distinctness to ideas (see Med. III, AT VII, 43–44; 46; Med. IV, AT VII, 53; Med. V, AT VII, 65; Med. VI, AT VII, 78; also cf. e.g. Princ. I, 54, AT VIII-1, 25–26). The relations between ideas and perceptiones in Descartes can easily be extrapolated upon the basis of the above-presented discussion of the objects of judgments in sec. 1.1.1. 110 Of dozens of references, see e.g. Reg. II, AT X, 362: “Circa illa tantùm objecta oportet versari, ad quorum certam & indubitatam cognitionem nostra ingenia videntur sufficere. Atque ita ... rejicimus illas omnes probabiles tantùm cognitiones, nec nisi perfectè cognitis, & de quibus dubitari non potest, statuimus esse credendum.” The term “certitudo metaphysica” occurs verbatim in Resp. 5, AT VII, 356 and Resp. 7, AT VII, 477; the term “certitude Metaphisique [sic]” in Princ.(f) I, 12, AT IX-2, 30; the term “summa certitudo” in Resp. 7, AT VII, 548; and the term “perfectissima certitudo” in Resp. 2, AT VII, 145. 111 At least one prominent scholar, viz. Jonathan Bennett, seems to deny that certainty, i.e. indubitability implies clear and distinct perception, and recommends instead that one treat clear and distinct perception on the one hand and certainty (under a peculiar factual interpretation) on the other hand as independent terms that jointly imply the sought-after privileged kind of cognitions: cf. especially Bennett, Learning from Six Philosophers, 1:366–67. In the light of the considerations that follow in the present section, however, such a move shall turn out pointless; cf. fn. 125. 112 For some other references, see e.g. DM 2, AT VI, 18: “Le premier [precepte] estoit ... de ne comprendre rien de plus en mes iugemens, que ce qui se presenteroit

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Exponam hic ... fundamentum, cui omnis humana certitudo niti posse mihi videtur. Imprimis, statim atque aliquid a nobis recte percipi putamus, sponte nobis persuademus illud esse verum. Hæc autem persuasio si tam firma sit ut nullam unquam possimus habere causam dubitandi de eo quod nobis ita persuademus, nihil est quod ulterius inquiramus: habemus omne quod cum ratione licet optare. ... Sed dubitari potest an habeatur aliqua talis certitudo, sive firma & immutabilis persuasio. Et quidem perspicium est illam non haberi de iis quæ vel minimum obscure aut confuse percipimus: hæc enim qualiscumque obscuritas satis est causæ, ut de ipsis dubitemus (Resp. 2, AT VII, 144–45; my emphases).

Although this already sounds like a satisfactory vindication of the claim that the clear and distinct perception of a given content is at least a necessary condition for the corresponding judgment to yield a certain and evident cognition, some further explications of the crucial topics just introduced—clear and distinct perception and metaphysical certainty—and a sketchy outline of my views on some important interpretive problems of Descartes‫ ތ‬overall epistemological project seem in order. Despite the pivotal rôle the characterizations of clarity and distinctness play in Descartes‫ ތ‬epistemology, Descartes just once condescends to define them, for that matter in a notoriously unsatisfactory way: Claram voco [perceptionem] illam, quæ menti attendenti præsens & aperta est: sicut ea clarè à nobis videri dicimus, quæ, oculo intuenti præsentia, satis fortiter & apertè illum movent. Distinctam autem illam, quæ, cùm clara sit, ab omnibus aliis ita sejuncta est & præcisa, ut nihil planè aliud, quàm quod clarum est, in se contineat (Princ. I, 45, AT VIII-1, 22).

Although a vast amount of literature on the topic is available, I believe that Alan Gewirth provides a most probably accurate account of what Descartes has in mind and one that renders claritas and distinctio workable theoretical terms.113 According to Gewirth, it is a mistake to take clarity and distinctness (as well as their opposites, viz. obscurity and si clairement & si distinctement a mon esprit, que ie n‫ތ‬eusse aucune occasion de le mettre endoute.” Princ. I, 43, AT VIII-1, 21: “Certum autem est, nihil nos unquam falsum pro vero admissuros, si tantùm iis assensum præbamus quæ clarè & distinctè percipiemus.” Princ. I, 45, AT VIII-1, 22: “Etenim ad perceptionem, cui certum & indubitatum judicium possit inniti, non modò requiritur ut sit clara, sed etiam ut sit distincta.” Also cf. Reg. III, AT X, 366; 368; Reg. XII, AT X, 425; DM 4, AT VI, 33; Hyp., AT III, 431. 113 Alan Gewirth, “Clearness and Distinctness in Descartes,” in Descartes, ed. John Cottingham (Oxford: Oxford University Press, 1998), 79–100.

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confusion) of an idea as “[either] intrinsic to the idea [or] explicable in terms of a simple relation between idea and perceptive act” (Gewirth, “Clearness and Distinctness,” 85); rather, the characterizations at issue are (putting details to one side) a matter of relations of (as Gewirth calls it) logical equality or inequality114 between what Gewirth calls the direct and the interpretative content of a given perception. A perception then counts as clear if its direct/interpretative content contains all that which is contained in the corresponding interpretative/direct content; and a perception counts as distinct if its direct/interpretative content contains nothing over and above what is contained in its interpretative/direct content.115 As for the other topic to be briefly discussed—metaphysical certainty —Descartes insists that far from counting as an actual intrinsic property of propositionally structured objects of judgments, certainty tout court it is to be understood as a relation between a cognizing mind or minds on the one hand and an object or objects of judgment on the other hand (and arguably also between a time or times):116 114

Gewirth himself rightly contrasts such logical (in)equality with “a narrowly quantitative” (in)equality (see ibid., 87). 115 See ibid., 86–87. Gewirth derives his interpretation chiefly from the observation that Descartes calls sense perceptions both obscure and confused, and clear and distinct in different contexts defined in terms of how the directly perceived content at issue is “viewed” (cf. Princ. I, 68, AT VIII-1, 33; Med. III, AT VII, 35; Reg. XII, AT X, 423 for one case of Descartes‫ ތ‬regarding sense perceptions as clear and distinct; and Med. VI, AT VII, 83 for another). Indeed, I take the fact that Gewirth‫ތ‬s interpretation accounts, in a very natural way, for these prima facie baffling claims of Descartes (since sense perceptions standardly count, of course, as a paradigmatic case of obscure and distinct perceptions for him) as one of the strongest reasons for accepting Gewirth‫ތ‬s reading (I deal with this issue briefly in ch. 2). I also agree with Gewirth that in allowing the interpretative content of an idea or perception one does not commit oneself to a confusion of ideas with judgments (see ibid., 88–90). Incidentally, I believe it is exactly the interpretative content of perception that Hill improperly confuses with judgment in his version of the intellectualistic interpretation of Descartes‫ ތ‬sensationes and imaginationes; cf. fn. 81. 116 Cf. also Resp. 7, AT VII, 475: “[P]atet ... dubitationem & certitudinem [male] considerari ab illo tanquam in objectis, non tanquam in nostrâ cogitatione. Alioqui enim quomodo posset fingere aliquid mihi proponi tanquam dubium, quod non esset dubium, sed certum? cùm ex hoc solo, quòd proponatur ut dubium, dubium reddatur.” And ibid., 480. The presumption of time as the relevant relata shall be justified in a moment, in the course of discussing the guaranteeing rôle of knowledge of God‫ތ‬s existence. As is rightly noticed by Jeffrey Tlumak, “Certainty and Cartesian Method,” in Descartes: Critical and Interpretative Essays, ed.

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Notandumque [Bourdinem] ubique [false] considerare dubitationem & certitudinem, non ut relationes cognitionis nostræ ad objecta, sed ut proprietates objectorum quæ perpetuo ipsis inhærant, adeo ut ea, quæ semel dubia esse cognovimus, non possint unquam reddi certa (Resp. 7, AT VII, 473).

Next, metaphysical or perfect certainty, which has been identified as the proper mark of the sought-after kind of cognition, is contrasted by Descartes with so-called practical or moral certainty. Practical certainty is characterized by him most fully in Princ.(f) IV, 205, AT IX-2, 323:117 [Une sorte de certitudes] est apelée morale, c‫ތ‬est à dire suffisante pour regler nos mœurs, ou aussi grande que celle des choses dont nous n‫ތ‬auons point coustume de douter touchant la conduite de la vie, bien que nous sçachions qu‫ތ‬il se peut faire, absolument parlant, qu‫ތ‬elles soient fausses.

Whilst practical certainty is thus characterized as hinging upon a sort of practical credibility sufficient for handling the ordinary affairs of life, metaphysical or absolute certainty takes the stage, by way of contrast, “lors que nous pensons quҲil nҲest aucunement possible que la chose soit autre que nous la jugeons” (Princ.(f) IV, 206, AT IX-2, 324; my emphasis). The task of accounting for Descartes‫ ތ‬notion of metaphysical certainty in fact amounts to the task of explaining and interpreting this too concise yet vague statement of his. What seems to be comparatively uncontroversial, to begin with, is a claim which enjoys extraordinarily strong textual support, to the effect that the characterization of metaphysical certainty under scrutiny is to be best analyzed in terms of irrevisability, i.e., roughly, in such a way that Michael Hooker (Baltimore: The John Hopkins University Press, 1978), 69 (n. 2), several commentators seem to construe certainty as a dispositional property of objects of judgment; and he has it—rightly, as far as I can tell—that “[s]o long as it is a dispositional property, whose discovery is relativized to person and time” (ibid.), such a notion of certainty could be put in accord with Descartes‫ ތ‬relational view. 117 See also e.g. DM 4, AT VI, 37–38: “[E]ncore qu‫ތ‬on ait vne assurance morale de ces choses [sc. d‫ތ‬auoir vn cors, & qu‫ތ‬il y a des astres & vne terre, & choses semblables], qui est telle, qu‫ތ‬il semble qu‫ތ‬a moins que d‫ތ‬estre extrauagant, on n‫ތ‬en peut douter, toutefois aussy, a moins que d‫ތ‬estre déraisonnable, lorsqu‫ތ‬il est question d‫ތ‬vne certitude metaphysique, on ne peut nier que ce ne soit assés de suiet, pour n‫ތ‬en estre pas entierement assuré ....” Cf. DM 5, AT VI, 56–57; Mers., AT III, 359; A Mesland, AT IV, 114 for other occurrences of the notion of moral or practical certainty and/or error.

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metaphysical certainty of any cognition (or belief)118 c amounts to c‫ތ‬s being induced by the presently available evidence, and to c‫ތ‬s being, by the same token, immune from any possible sceptical episode which would raise a possibility that additional evidence should prompt abandonment of the belief in question;119 and the textual evidence boosted with related systematic considerations seems to ensure, moreover, that the modalities with which Descartes operates when entertaining the relevant sceptical scenarios are to be best interpreted along the temporally relativized epistemic lines.120 Even accepting this as common ground, at least two big questions remain to be addressed: (a) Is the notion of metaphysical certainty that 118

This apparently anachronistic switch to contemporary parlance is justified by the fact that Descartes himself sometimes employs the term “persuasio” in the relevant contexts: see e.g. Resp. 2, AT VII, AT VII, 144–45; A Regius, AT III, 65. 119 I owe this point, as well as the conceptual devices for its articulation, to Tlumak, “Certainty and Cartesian Method,” 45–46 (I avoid Tlumak‫ތ‬s somewhat biased normative wording, however, for reasons to be presented shortly). For the most telling textual evidence in Descartes (apart from the above-quoted Resp. 2, AT VII, 144–45), see DM 4, AT VI, 31–32; Resp. 6, AT VII, 428; RV, AT X, 513. For an extended and sophisticated discussion of the details and inherent intricacies of the irrevisability reading, as well as of the rival basic explications of Descartes‫ތ‬ notion of metaphysical certainty, see Tlumak, “Certainty and Cartesian Method,” 45–53 and 65–66. 120 The idea is, roughly, that “p is epistemically possible for S at t just in case S lacks conclusive reasons for believing not-p at t” (Tlumak, “Certainty and Cartesian Method,” 48; Tlumak’s emphasis). The issue of epistemic modalities in this sense is to be distinguished carefully from the issue of epistemic interpretation of logical modalities, which also seems to be recommended by Descartes (cf. in particular Resp. 2, AT VII, 150–51; this latter issue is taken up in ch. 2). Tlumak, “Certainty and Cartesian Method,” 48 makes a good job in showing why neither practical nor logical possibilities (presumably under any interpretation, epistemic or otherwise) qualify as accurate characterizations of the doubtmakers Descartes makes use of in his search for certainty. Perhaps the most straightforward evidence for the proposed epistemic reading provides Med., Præfatio, AT VII, 7–8: “[Prima objectio] est, ex eo quod mens humana in se conversa non percipiat aliud se esse quàm rem cogitantem, non sequi ejus naturam sive essentiam [Descartes’ emphasis] in eo tantùm consistere, quod sit res cogitans, ita ut vox tantùm [Descartes’ italics] cætera omnia excludat quæ forte etiam dici possent ad animæ naturam pertinere. Cui objectioni respondeo me etiam ibi noluisse illa excludere in ordine ad ipsam rei veritatem (de quâ scilicet tunc non agebam), sed dumtaxat in ordine ad meam perceptionem [my emphasis], adeo ut sensus esset me nihil plane cognoscere quod ad essentiam meam scirem pertinere, præterquam quod essem res cogitans, sive res habens in se facultatem cogitandi.”

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Descartes employs to be interpreted along normative or factual, descriptive lines? (b) How is it that the clear and distinct perception of a given content sometimes (and indeed—as we shall see in a moment—in such cases as are of major interest for us in the present study) counts as just a necessary but insufficient condition of the metaphysical certainty of a given cognition? In one way or another, Descartes‫ ތ‬notion of compelled assent plays a pivotal rôle in the constitution of both these problems. Thus, let us turn to this first.

1.3.1 Compelled Assent The pivotal status of compelled assent in the present context is due to the fact that Descartes obviously takes that notion as the proper intermediary between clear and distinct perception on the one hand and metaphysical certainty on the other. For he constantly claims, or at least implies, that clear and distinct perception somehow brings about one‫ތ‬s forced assent to what is thus perceived and one‫ތ‬s absolute inability to doubt the perceived (propositionally structured) content. Here is, perhaps, his most explicit statement to this effect:121 “[Q]uoties aliquid clarè percipimus, ei sponte assentiamur, & nullo modo possimus dubitare quin sit verum” (Princ. I, 43, AT VIII-1, 21). It did not escape Descartes‫ތ‬ attention that the compulsion at issue together with his voluntaristic notion of assent might seem to imply that assenting to what is perceived clearly and distinctly is by no means a free act; but he makes, particularly in Med. IV, a brilliant job of showing—in a Spinozian vein—that the implication is false once we define freedom not so much in terms of control over 121

Cf. also Burm., AT V, 148: “[Q]uamdiu [auctor ad axiomata attendit, i.e. clare & distincte illa percipit], certus est se non falli, et cogitur illis assentiri.” For some other passages that jointly yield the same result, cf. e.g. Med. V, AT VII, 69: “[E]jus sim naturæ ut, quamdiu aliquid valde clare & distincte percipio, non possim non credere verum esse ....” A Regius, AT III, 64: “[M]ens nostra est talis naturæ, vt non possit clarè intellectis non assentiri.” Med. IV, AT VII, 58: “Non potui quidem non judicare illud quod tam clare intelligebam verum esse ....” DM 2, AT VI, 18: “Le premier [precept] estoit ... de ne comprendre rien de plus en mes iugemens, que ce qui se presenteroit si clairement & si distinctement a mon esprit, que ie n‫ތ‬eusse aucune occasion de le mettre en doute.” I think the fact that Descartes speaks of just clear (and not clear and distinct) perception in the quoted passage is of little moment: as the title of Princ. I, 43 reads “Nos nunquam falli, cùm solis clarè & distinctè perceptis assentimur” (AT VIII-1, 21; my emphasis), I suppose that the omission is just shorthand on Descartes‫ ތ‬part and not an indication of a doctrinal point. Cf. the quoted A Regius, AT III, 64 for a similar case. Cf. also ch. 2, fn. 5.

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whether one performs an action or not, but rather in terms of one‫ތ‬s actions being self-determined.122 Before we get involved in the above problems (a) and (b), a word or two is to be said concerning another problem connected with Descartes‫ތ‬ doctrine of compelled assent which has disturbed several recent commentators: how could Descartes‫ ތ‬doctrine of compelled assent be rendered compatible with the fact that nothing in Descartes‫ ތ‬notion of clear and distinct perception seems to block the possibility that not only true but also false propositions are capable of being perceived clearly and distinctly? For (as Margaret Wilson rightly observes) given Descartes‫ތ‬ theory of judgment, we cannot take a clear and distinct perception that P as implying a clear and distinct perception that P is true;123 but if we limit, in 122 See Med. IV, AT VII, 57–59: “[Voluntas] tantùm in eo consistit, quòd idem vel facere vel non facere (hoc est affirmare vel negare, prosequi vel fugere) possimus, vel potiùs in eo tantùm, quòd ad id quod nobis ab intellectu proponitur affirmandum vel negandum, sive prosequendum vel fugiendum, ita feramur, ut a nullâ vi externâ nos ad id determinari sentiamus. Neque enim opus est me in utramque partem ferri posse, ut sim liber, sed contrà, quo magis in unam propendeo ..., tanto liberius illam eligo .... Indifferentia autem illa, quam experior, cùm nulla me ratio in unam partem magis quàm in alteram impellit, set infimus gradus libertatis, & nullam in eâ perfectionem, sed tantummodo in cognitione defectum ... testatur .... ... Exempli causâ, cùm examinarem hisce diebus an aliquid in mundo existeret, atque adverterem, ex hoc ipso quòd illud examinarem, evidenter sequi me existere, non potui quidem non judicare illud quod tam clare intelligebam verum esse; non quòd ab aliquâ vi externâ fuerim ad id coactus, sed quia ex magna luce in intellectu magna consequuta est propensio in voluntate, atque ita tanto magis sponte & libere illud eredidi, quanto minus fui ad istud ipsum indifferens.” Cf. Rosenthal, “Will and Judgment,” 424–26 for an excellent discussion and defence of Descartes‫ ތ‬view. 123 Cf. Wilson, Descartes, 124: “Descartes‫ ތ‬doctrine of judgment effectively rules out the possibility that to ‫ދ‬clearly and distinctly perceive p‫( ތ‬where p is a proposition) involves perceiving in some overwhelmingly lucid and evident way that p is true. For in that case there would be no logical gap between clearly and distinctly perceiving p and assenting to p. Assent would be inevitable not because of an irresistible impulsion of the will, but because of logical entailment.” Bernard Williams seems to take a slightly different course without, however, doubting the gravity of the problem: “Hence it looks as though it can be good advice that one should assent to all and only the propositions which one clearly understands, only if the notion of ‘clearly understanding a proposition’ is itself taken to imply that one sees the proposition to be true. ... But once this step has been taken—and I find it very difficult to see how Descartes can avoid taking it or another only verbally different from it—the theory of assent is in difficulty. For if in this sense I clearly understand a proposition—that is to say, I can see it is true—there is nothing else I

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view of this, the content of clear and distinct perception just to the meaning of P, it seems to follow that we are sometimes compelled to assent to false propositions, so that Descartes‫ ތ‬intent to establish clear and distinct perception as the reliable guide to metaphysically certain cognition eventually breaks down. I believe that the present problem is engendered, once again, by improper assimilation of the content of perception in Descartes with Fregean propositions. Once we repudiate this assimilation and conceive of the content in question instead as an “objective” aspect of particular eventlike ideas (as I argued above), it makes perfect sense to hold that only that which can never become material for false judgments deserves to be taken as a possible object of clear and distinct perception. Indeed, this is what Descartes implies in numerous passages in which he presents clear and distinct perception as implying truth and which otherwise either would count as unintelligible or else would destroy his account of judgment.124 Descartes‫ ތ‬notion of understanding comes therefore (as David Rosenthal usefully observes) close to the standard notion of knowing as far as its relation to truth is concerned: “Like knowing, it seems we cannot truly claim to understand something that we express by means of a false propositional clause” (Rosenthal, “Will and Judgment,” 424).125

have to do in order to believe it: I already believe it. The will has nothing to do which the understanding has not already done” (Williams, Descartes, 168). 124 See e.g. Med. III, AT VII, 35: “[V]ideor pro regulâ generali posse statuere, illud omne esse verum, quod valde clare & distincte percipio” (my emphasis). Resp. V, AT VII, 376–77: “Vis ut hîc paucis dicam ad quid se voluntas possit extendere, quod intellectum effugiat [Descartes’ italics]. Nempe ad id omne in quo contingit nos errare [my emphasis]. ... Cùm ... prave judicamus, non ideo prave volumus, sed fortè pravum quid; nec quidquam prave intelligimus, sed tantùm dicimur prave intelligere, quando judicamus nos aliquid amplius intelligere quàm revera intelligamus [my emphasis].” Cf. also DM 4, AT VI, 38–39; Resp. 2, AT VII, 144; Princ. I, 43, AT VIII-1, 21. 125 Cf. especially Med. IV, AT VII, 58: “[Q]uidquid intelligo, cùm a Deo habeam ut intelligam, procul dubio recte intelligo, nec in eo fieri potest ut fallar.” Cf. also Rosenthal‫ތ‬s helpful observation that “[i]f I understand the proposition that 2 + 2 = 5, it hardly follows that I understand that 2 + 2 = 5” (Rosenthal, “Will and Judgment,” 424). Indeed, as should be clear, I am indebted to Rosenthal for the entire treatment of the present issue. It is worth noting that once the present problem is thus resolved, Bennett‫ތ‬s aforementioned gambit of divorcing logically clear and distinct perception on the one hand and metaphysical certainty on the other, a gambit devised imprimis in order to resolve the present difficulties, turns out to be unnecessary.

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Having thus adumbrated the nature of compelled assent in Descartes and having rendered it intelligible in view of several apparent problems it brings about, let us have a closer look at how the doctrine is put to work by Descartes when dealing with both the above problems.

1.3.2 Metaphysical Certainty and Normativity As for problem (a), let us begin with the observation that, complications apart, Descartes‫ ތ‬doctrine of compelled assent implies a certain minimal normative status of the notion of metaphysical certainty. Putting to one side, for the moment, the issue of whether there is something about the notion of metaphysical certainty that Descartes is committed to over and above the overall psychological inability to doubt a given content, states of such psychological inability to doubt might in general still be causally induced by significantly different factors. Most of them—such as, for example, apathy, dullness, ignorance, or pathological defiance—would bring about certainties of no interest to Descartes in the epistemological contexts set by him. The only state of certainty in which Descartes is interested—even if it were to remain purely psychological in nature126—is indeed the state induced, in the last analysis, by perceiving a given content with genuine clarity and distinctness;127 and Descartes is 126

It should perhaps be emphasized that whenever I attribute “psychological” notions to Descartes in this study, I never mean to attribute to him a radical thesis (which can be found in some, especially nineteenth-century authors) that some psychological states (be it the state of clear and distinct perception, of inability to doubt, or whatever) literally make something true, but only that (in Gaukroger‫ތ‬s pertinent phrase) “the grasp of truth is manifested in some sort of psychological clarity experienced by the knowing subject” (Gaukroger, Cartesian Logic, 54; my emphasis); in other words, not that the certainty is due solely to the psychological events but also, in part at least, due to the nature of the perceived content. 127 See Resp. 2, AT VII, 146: “Nec obstat, quòd sæpe simus experti alios deceptos fuisse in iis quæ sole clarius se scire credebant [Descartes’ italics]. Neque enim unquam advertimus, vel ab ullo adverti potest, id contigisse iis qui claritatem suæ perceptionis a solo intellectu petierunt, sed iis tantùm qui vel a sensibus, vel a falso aliquo præjudicio, ipsam desumpserunt.” Resp. 3, AT VII, 192: “Nemo ... nescit per lucem in intellectu intelligi perspicuitatem cognitionis, quam forte non habent omnes qui putant se habere; sed hoc non impedit quominus valde diversa sit ab obstinatâ opinione absque evidenti perceptione conceptâ” (my emphases). Notæ, AT VIII-2, 351–52: “[N]otandum est, hanc regulam, quicquid possumus concipere, id potest esse, quamvis mea sit, & vera, quoties agitur de claro & distincto conceptu, in quo rei possibilitas continetur, quia Deus potest omnia efficere, quæ nos possibilia esse clare percipimus; non esse tamen temere usurpandam, quia

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prepared to treat the situation in distinctively normative terms: one ought to regulate one‫ތ‬s mind towards perceiving clearly and distinctly as extensive a range of items as possible in order to achieve the right kind of cognition; one ought not to stay complacent with certainties of any other kind—these are collectively denounced as associated with mere præjudicia once the above epistemic demands raised by Descartes are applied;128 and one ought to refrain from assenting to contents perceived with less than perfect clarity and distinctness:129 facile sit, ut quis putet se aliquam rem recte intelligere, quam tamen præjudicio aliquo excæcatus non intelligit” (Descartes’ italics). Also, the “iugemens [telles] que ie n‫ތ‬eusse aucune occasion de [les] mettre en doute” are contrasted to “la Precipitation, & la Preuention” in DM 2, AT VI, 18. 128 See Princ. I, 47, AT VIII-1, 22: “Et quidem in primâ ætate mens ita corpori fuit immersa, ut quamvis multa clarè, nihil tamen unquam distinctè perceperit; cùmque tunc nihilominus de multis judicârit, hinc multa hausimus præjudicia, quæ à plerisque nunquam postea deponuntur.” Cf. also ibid., 50, AT VIII-1, 24; and Med., Synopsis, AT VII, 12. 129 Cf. also e.g. Princ. I, 44, AT VIII-1, 21: “Nos semper malè judicare, cùm assentimur non clare perceptis, etsi casu incidamus in veritatem ....” Hyp., AT III, 430–31: “[S]æpe ... multis in rebus hominum judicia ab ipsorum perceptione dissentiunt. At quicunque nullum vnquam iudicium ferunt, nisi de rebus quas clare & distincte percipiunt (quod, quantum in me est, semper obseruo), non possunt vno tempore aliter quàm alio de eâdem re iudicare.” DM 2, AT VI, 18. The tradition of criticizing Descartes for having failed to provide suitable criteria for distinguishing safely perceptions that really are clear and distinct from those that only seem to be such began as early as Gassendi‫ތ‬s challenge in Obj. 5, AT VII, 318: “Attende tamen ... difficultatem non videri an, ut non fallamur, debeamus clare atque distincte intelligere aliquid, sed quâ arte aut methodo discernere liceat, ita nos habere claram distinctamque intelligentiam, ut ea vera sit, nec fieri possit ut fallamur. Quippe initio objecimus, nos non rarò falli, tametsi nobis videamur aliquid adeò clare distincteque cognoscere, ut nihil possit clarius & distinctius. ... [E]xpectamus tamen adhuc istam artem, seu methodum, cui præcipue sit incumendum.” Recent attempts at responding to the challenge on Descartes‫ ތ‬side usually take his response to Gassendi as their starting point: “Quod denique addis, non tam de veritate regulæ [quòd ea quæ valde clare & distincte percipimus sint vera] esse laborandum, quàm de Methodo ad dignoscendum an fallamur necne, cùm existimamus nos aliquid clare percipere, non inficior; sed hoc ipsum accurate a me præstitum fuisse contendo suis in locis; ubi primùm abstuli omnia præjudicia, & postea enumeravi omnes præcipuas ideas, ac distinxi claras ab obscuris aut confusis” (Resp. 5, AT VII, 361–62; Descartes’ italics). For some such recent attempts cf. e.g. Gewirth, “Clearness and Distinctness;” James Humber, “Recognizing Clear and Distinct Perceptions,” Philosophy and Phenomenological Research 41, no. 4 (1981), 487–507; and Sarah Patterson, “Clear and Distinct Perception,” in Broughton and Carriero, Companion to Descartes, 216–34. Indeed,

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Chapter One Cùm autem quid verum sit non satis clare & distincte percipio, si quidem a judicio ferendo abstineam, clarum est me recte agere, & non falli. Sed si vel affirmem vel negem, tunc libertate arbitrii non recte utor ... (Med. IV, AT VII, 59–60; my emphases).

Yet all this still leaves open the question of whether the end-product of the compelled assent—the (relationally analyzed) state of metaphysical certainty—involves, on the part of the mind, something over and above mere psychological inability to doubt a given (propositional) content, P; more specifically, whether the clear and distinct perception that induces compelled assent might bring about a normative entitlement to be certain that P. As Jonathan Bennett aptly puts it:130 Granted that Descartes has no problem about P at [the time he clearly and distinctly perceives that P], it does not follow that he is entitled to be sure of P .... He cannot raise a problem about P, but there may be one for all that. (Bennett, Learning from Six Philosophers, 1:374)

In yet other words, the question is whether the Cartesian clear and distinct perception of P has, or could have, any justificatory, i.e. truth-conducive import with regard to the belief (persuasio) that P: does S‫ތ‬s clearly and distinctly perceiving that P provide S with a guarantee that P is true? To be sure, Descartes sometimes (and, for good or bad, quite often in places that have enjoyed the preeminent attention of modern Descartes scholars) adopts a normative language of reasons that bring about justification.131 Yet Louis Loeb and Jonathan Bennett132 convinced me that the entire machinery of the notorious Cartesian “methodical” or “hyperbolic” doubt is designed imprimis to break the alignment between præjudicia and spurious complacent certainties of wrong origins; see in particular Med., Synopsis, AT VII, 12: “In primâ [Meditatione], causæ exponuntur propter quas de rebus omnibus, præsertim matierialibus, possumus dubitare .... Etsi autem istius tantæ dubitationis utilitas primâ fronte non appareat, est tamen in eo maxima quòd ab omnibus præjudiciis nos liberet, viamque facillimam sternat ad mentem a sensibus abducendam; ac denique efficiat, ut de iis, quæ posteà vera esse comperiemus, non amplius dubitare possumus.” 130 Cf. also Williams, Descartes, 173. 131 Here are some notorious instances. Med. I, AT VII, 21–22: “Quibus sane argumentis non habeo quod respondeam, sed tandem cogor fateri nihil esse ex iis quæ olim vera putabam, de quo non liceat dubitare, idque non per inconsiderantiam vel levitatem, sed propter validas & meditatas rationes; ideoque etiam ab iisdem, non minùs quàm ab aperte falsis, accurate deinceps assensionem esse cohibendam, si quid certi velim invenire.” Med. V, AT VII, 69:

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Descartes is not entitled to such a normative attitude and, more importantly for our purposes, that he himself seems to settle for a weaker, purely factual and psychological treatment of certainty in the bulk of crucial passages in his writings, most graphically here:133 [S]tatim atque aliquid a nobis recte percipi putamus, sponte nobis persuademus illud esse verum. Hæc autem persuasio si tam firma sit ut nullam unquam possimus habere causam dubitandi de eo quod nobis ita persuademus, nihil est quod ulterius inquiramus: habemus omne quod cum ratione licet optare. Quid enim ad nos, si fortè quis fingat illud ipsum, de cujus veritate tam firmiter sumus persuasi, Deo vel Angelo falsum apparere, atque ideo, absolute loquendo, falsum esse? Quid curamus istam falsitatem absolutam, cùm illam nullo modo credamus, nec vel minimum suspicemur? Supponimus enim persuasionem tam firmam ut nullo modo tolli possit; quæ proinde persuasio idem plane est quod perfectissima certitudo (Resp. 2, AT VII, 144–45; my emphasis).

That is to say, clear and distinct perception is considered an item endowed with factual power to cause the sought-after (psychological) state of certainty rather than with normative power to provide justification and entitlement;134 and both the above commentators have done a good job of “[R]ecurratque sæpe memoria judicii ante facti, cùm non amplius attendo ad rationes propter quas tale quid judicavi, rationes aliæ afferri possunt quæ me ... facile ab opinione dejicerent ....” Cf. also Burm., AT V, 148; DM 4, AT VI, 38. 132 Louis Loeb, “The Priority of Reason in Descartes,” The Philosophical Review 99, no. 1 (1990), 3–43; Bennett, Learning from Six Philosophers, vol. 1, ch. 20. 133 Incidentally, I cannot endorse Williams‫ ތ‬observation that “[t]he reference to ‫ދ‬absolute falsehood‫[ ތ‬in AT VII, 144–45] ... is not to be taken seriously,” which Williams supports with the argument that “[w]hat appears false to God, God being omniscient, is false, so this possibility would mean that God was, radically, a deceiver” (Williams, Descartes, 185). For as far as I can see, the possibility evoked in AT VII, 144–45 is relevant also with regard to the arguments for the existence of a non-deceiving God, and therefore with regard to every inference to truth based on clear and distinct perception. Bennett, Learning from Six Philosophers, 1:376 presents a similar line of thought in a more elaborated form. 134 The contrast at issue, and Descartes‫ ތ‬attitude to it, come out vividly in Obj. 7 and Resp. 7. In AT VII, 469, Bourdin challenges Descartes with respect to what the latter has written in Med. I, AT VII, 21, as follows: “Fateris vetera omnia dubia esse, &, ut ais, coactus fateris. Quidni eam mihi vim pateris inferri, ut etiam ipse coactus fatear? Quid te, quæso, coëgit? Audivi quidem ex te modò validas fuisse & meditatas rationes. At quæ sunt illæ tandem? Si validæ, cur abdicas? quin retines? Si dubiæ & suspicionum plenæ, quâ te illæ vi coëgere?” And Descartes retorts: “Hîc ludit in verbo coactus ... inaniter. Eæ enim sunt satis validæ rationes ad

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showing that it is “intellectual stability, tranquillity, peace in the doxastic kingdom—a system of beliefs that will stay put” rather than “justification, good reasons, and truth” (Bennett, Learning from Six Philosophers, 1:377)135 with which Descartes seems to be, in fact, concerned in the first place once we look at his texts with an unbiased eye, although the latter, normative project also undeniably crops up from time to time;136 and since the discussion of problem (b)137 shall provide this interpretation with further support, I will assume it is correct in the following portions of the present study.138

1.3.3 Clear and Distinct Perception, and Scientia Let now us turn to problem (b). Why should the clear and distinct perception of a given content sometimes count as a necessary but insufficient condition of metaphysical certainty in Descartes? The best place to start is a notorious passage from Med. V, AT VII, 69–70:139 cogendum nos ut dubitemus, quæ ipsæ dubiæ sunt, nec proinde retinendæ .... Atque validæ quidem sunt, quandiu nullas alias habemus, quæ dubitationem tollendo certitudinem inducant” (Resp. 7, AT VII, 473–74; Descartes’ emphasis). As Bennett aptly comments, “[t]his roundly says that what counts is not the (normative) worth of the reason but merely its (factual) power to cause doubt” (Bennett, Learning from Six Philosophers, 1:377). Note also that Descartes speaks of causa dubitandi in the above-quoted AT VII, 144–45. 135 Cf. Loeb, “Priority of Reason,” 25–31. 136 For a brilliant and convincing discussion concerning the interrelations of the “stability” and the “normative” projects in Descartes, see Bennett, Learning from Six Philosophers, vol. 1, sec. 152–53. For various representative instances of the rival, normative approach to certainty in Descartes see Frankfurt, Demons, Dreamers, and Madmen, ch. 12; 14; Curley, Descartes against the Skeptics, ch. 5; Janet Broughton, Descartes’s Method of Doubt (Princeton: Princeton University Press, 2002), ch. 6. 137 The problem is formulated at the end of sec. 1.3. 138 This is not to say that such a psychological notion I ascribe to Descartes is unproblematic from the systematic point of view. Especially Gaukroger, Cartesian Logic, 51–56 brings to the fore various philosophical problems that such a psychological conception is likely to bring about. Yet this study is not the place to become involved in any such discussions. 139 Here are the other striking passages to the same or at least very similar effect. A Regius, AT III, 64–65: “[D]icitis: axiomatum clarè & distinctè intellectorum veritatem per se esse manifestam; quod etiam concedo, quandiu clarè & distincte intelliguntur, quia mens nostra est talis naturæ, vt non possit clarè intellectis non assentiri; sed quia sæpè recordamur conclusionum ex talibus præmissis deductarum, etiamsi ad ipsas præmissas non attendamus, dico tunc, si Deum

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Etsi enim ejus sim naturæ ut, quamdiu aliquid valde clare & distincte percipio, non possim non credere verum esse, quia tamen ejus etiam sum naturæ ut non possim obtutum mentis in eandem rem semper defigere ad illam clare percipiendam, recurratque sæpe memoria judicii ante facti, cùm non amplius attendo ad rationes propter quas tale quid judicavi, rationes aliæ afferri possunt quæ me, si Deum ignorarem, facile ab opinione dejicerent, atque ita de nullâ unquam re veram & certam scientiam, sed vagas tantùm & mutabiles opiniones, haberem. Sic, exempli causâ, cùm naturam trianguli considero, evidentissime quidem mihi, utpote Geometriæ principiis imbuto, apparet ejus tres angulos æquales esse duobus rectis, nec possum non credere id verum esse, quamdiu ad ejus demonstrationem attendo; sed statim atque mentis aciem ab illâ deflexi, quantumvis adhuc recorder me illam clarissime perspexisse, facile tamen potest accidere ut dubitem an sit vera, si quidem Deum ignorem. Possum enim mihi persuadere me talem a naturâ factum esse, ut interdum in iis fallar quæ me puto quàm evidentissime percipere, cùm præsertim meminerim me sæpe multa pro veris & certis habuisse, quæ postmodum, aliis rationibus adductus, falsa esse judicavi.

Here Descartes makes it absolutely clear, to begin with, that as long as one actually and with appropriate attention and discipline perceives something clearly and distinctly, such a state of mind counts as not just necessary but also sufficient for inducing—by way of compelled assent—the soughtafter state of metaphysical certainty;140 but that for any future time at

ignoremus, fingere nos posse illas esse incertas, quantumuis recordemur ex claris principijs esse deductas; quia nempe talis fortè sumus naturæ, vt fallamur etiam in euidentissimis; ac proindè, ne tunc quidem, cùm illas ex istis principijs deduximus, scientiam, sed tantùm persuasionem, de illis nos habuisse” (Descartes’ emphases). Princ. I, 13, AT VIII-1, 9–10: “[M]ens ... [i]nvenit etiam communes quasdam notiones, & ex his varias demonstrationes componit, ad quas quamdiu attendit, omnino sibi persuadet esse veras. Sic, exempli causâ, numerorum & figurarum ideas in se habet, habetque etiam inter communes notiones, quòd si æqualibus æqualia addas, quæ inde exsurgent erunt æqualia, & similes; ex quibus facilè demonstratur, tres angulos trianguli æquales esse duobus rectis, &c.; ac proinde hæc & talia sibi persuadet vera esse, quamdiu ad præmissas, ex quibus ea deduxit, attendit. Sed quia non potest semper ad illas attendere, cùm postea recordatur se nondum scire, an fortè talis naturæ creata sit, ut fallatur etiam in iis quæ ipsi evidentissima apparent, videt se meritò de talibus dubitare, nec ullam habere posse certam scientiam, priusquam suæ authorem originis agnoverit” (Descartes’ italics). Cf. also Resp. 4, AT VII, 246; Resp. 6, AT VII, 428; Hyp., AT III, 434. 140 Besides the passages quoted in the preceding footnote, see also Burm., AT V, 148: “[Burman:] [I]n tertiâ Meditatione probat auctor Deum esse per axiomata, cùm sibi necdum constet, se in iis non falli. [Descartes:] Probat, et scit se in iis non

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which such epistemically privileged attentive state of mind is lost (and Descartes is clear he regards the occurrence of such an event inevitable for any finite mind from time to time), something more is needed to preserve metaphysical certainty: namely clear and distinct cognition that the veracious God necessarily exists. This is because, Descartes explains, as long as one fails at present to perceive clearly and distinctly what one remembers to have thus perceived and one is ignorant of God‫ތ‬s existence at the very same moment, one is at present susceptible to all the familiar sceptical worries with respect to what he remembers to have perceived clearly and distinctly,141 and therefore one falls back, in that respect, to the undesired state of incredulity which is incompatible with the state of metaphysical certainty; for then “[p]ossum ... mihi persuadere me talem a naturâ factum esse, ut interdum in iis fallar quæ me puto quàm evidentissime percipere,” and nothing but certain cognition of the existence of the veracious God—Descartes claims—can ever dispel such fundamental worry. Thus even if the existence of God is cognized in the appropriate way, a cognition cannot count as metaphysically certain unless there is at least one moment at which a given mind perceives a given content clearly and distinctly, for otherwise it would count instead as mere probabilis cognitio;142 yet clear and distinct perception definitely ceases to count as a sufficient condition under the circumstances just described, and as long as they obtain. So there is, at any rate, a basic contrast between cognitions induced by compelled assent due to clear and distinct perception of a given content at some moment on the one hand, and on the other hand cognitions that have never enjoyed such a backing; and putting to one side the latter kind as mere probabiles opiniones, the cognitions of the former kind indiscriminately

falli, quoniam ad ea attendit; quamdiu autem id facit, certus est se non falli, et cogitur illis assentiri” (my emphasis). 141 It is important at this point to appreciate the fact that it is the very clarity and distinctness of past perceptions, and not just one‫ތ‬s remembering that one had clear and distinct perceptions in the past, with which the sceptical worry entertained by Descartes is concerned. The proponents of the so-called memory response (once influential but nowadays rightly discarded) to the charge of circularity on Descartes‫ ތ‬part with regard to his validation of clear and distinct perception failed to draw this important distinction. I deal with the circularity issue below. 142 Cf. also Reg. II, AT X, 362: “[N]unquam studere melius est, quàm circa objecta adeò difficilia versari, vt, vera à falsis distinguere non valentes, dubia pro certis cogamur admittere .... Atque ita ... rejicimus illas omnes probabiles tantùm cognitiones, nec nisi perfectè cognitis, & de quibus dubitari non potest, statuimus esse credendum.”

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deserve the title of metaphysical certainties as long as they are actually perceived attentively, clearly and distinctly.143 The crucial distinction to be drawn now within the domain of metaphysical certainties thus defined is, according to Descartes, the distinction between, on the one hand, those with respect to which there is a logical possibility that the certainty of a given cognition be dislodged by entertaining the relevant sceptical worries at any future time (this may happen, Descartes claims, anytime one ceases to attend to a given content properly enough to perceive it clearly and distinctly even if one remembers one has thus previously perceived it); and, on the other hand, those that are not threatened with such a possibility. Descartes standardly associates the distinctive features of the cognitions at issue with the characterizations of mutability vs. immutability, instability vs. solidity or firmness etc.; he calls the cognitions of the former kind opiniones or persuasiones; and he clearly indicates, in an important passage, that it is precisely the cognitions of the latter kind that deserve the title scientia:144 143

An interesting attempt at refuting this claim is presented in Peter Markie, “Clear and Distinct Perception and Metaphysical Certainty,” in René Descartes: Critical Assessments, ed. Georges Moyal, 4 vols. (London: Routledge, 1991), 1:177–84. As far as I can tell, however, Markie’s strategy is infected (putting to one side several minor points) with a fatal mistake: he improperly ties metaphysical certainty so close to scientia that the domains of what counts as metaphysically certain and of what counts as scientia are co-extensive (see in particular ibid., 179–80), which contradicts several of Descartes‫ ތ‬express claims to the contrary—cf. in particular A Regius, AT III, 64–65; Resp. 2, AT VII, 140; 145. (In AT VII, 140, Descartes complicates matters as he might seem somewhat to strengthen the concept of scientia there; even if it were so, however, it would not affect my present point as what I object to in Markie is that it is his conception of metaphysical certainty and not of scientia in Descartes that is too strong; I deal with this complication later in this section). On this issue, my reading is essentially in accord with that of Kenny, Descartes, 192. 144 See also Resp. 2, AT VII, 141: “Quod autem Atheus possit clare cognoscere trianguli tres angulos æquales esse duobus rectis [Descartes’ italics], non nego; sed tantùm istam ejus cognitionem non esse veram scientiam affirmo, quia nulla cognitio, quæ dubia reddi potest, videtur scientia appellanda [my emphasis]; cùmque ille supponatur esse atheus, non potest esse certus se non decipi in iis ipsis quæ illi evidentissima videntur ...; & quamvis fortè dubium istud ipsi non occurrat, potest tamen occurrere, si examinet, vel ab alio proponatur ....” Resp. 6, AT VII, 428: “Quantum ad scientiam Athei, facile est demonstrare illam non esse immutabilem & certam. Ut enim jam ante dixi, quo minus potentem originis suæ authorem assignabit, tanto majorem habebit occasionem dubitandi, an fortè tam imperfectæ sit naturæ, ut fallatur etiam in iis quæ sibi quàm evidentissima apparebunt ....” (Here Descartes obviously had better written “ad cognitionem

64

Chapter One [Scientiam & persuasionem tantùm] ita distinguo, vt persuasio sit, cùm superest aliqua ratio quæ nos possit ad dubitandum impellere; scientia verò sit persuasio à ratione tam forti, vt nullâ vnquam fortiore concuti possit ... (A Regius, AT III, 65).

Now it is clear enough from the passages considered thus far that according to Descartes the only way to turn persuasiones in the aboveestablished sense into pieces of scientia is to get at clear and distinct cognition of God‫ތ‬s existence. What is prima facie less clear is whether such cognition of God‫ތ‬s existence is strictly necessary for each and every piece of scientia for Descartes, i.e. whether the domains of scientia (in the above sense) on the one hand, and of cognitions induced by clear and distinct perception and backed up by cognition of God‫ތ‬s existence on the other hand, are at the end of the day strictly co-extensive. Sometimes Descartes writes to this effect, most unambiguously perhaps here:145 [P]lane video omnis scientiæ certitudinem & veritatem ab unâ veri Dei cognitione pendere, adeo ut, priusquam illum nossem, nihil de ullâ aliâ re perfecte scire potuerim. (Med. V, AT VII, 71; my emphasis)

Yet when overtly challenged on these claims, Descartes‫ ތ‬responses are best interpreted, I submit, to the effect that he is prepared to weaken his position considerably: namely to the effect that the divine guarantee is needed precisely in the cases of clear and distinct cognitions that Athei” in place of “ad scientiam Athei.”) RV, AT X, 513: “... une doctrine qui fust assés solide & assurée pour meriter le nom de science ....” Resp. 2, AT VII, 145: “... certitudo, sive firma & immutabilis persuasio ....” Resp. 7, AT VII, 473: “Est enim contrarietas inter verba scivi & dubia sunt ...” (Descartes’ emphases). Also e.g. Med. I, AT VII, 17; Med. V, AT VII, 69; Reg. II, AT X, 363; 366–67; DM 1, AT VI, 8–9; Princ. Pref., AT IX-2, 12. The doctrine embodied in all these passages provides the context for a correct reading of Descartes‫ ތ‬intimation to Mersenne: “ie repute presque pour faux tout ce qui n‫ތ‬est que vraysemblable ...” (Mers., AT I, 450). 145 See also DM 4, AT VI, 39: “[S]i nous ne sçauions point que tout ce qui est en nous de reel & de vray, vient d‫ތ‬vn estre parfait & infini, pour claires & distinctes que fussent nos idées, nous n‫ތ‬aurions aucune raison qui nous assurait, qu‫ތ‬elles eussent la perfection d‫ތ‬estre vrayes.” And Med. III, AT VII, 36: “[E]xaminare debeo an sit Deus, &, si sit, an possit esse deceptor; hac enim re ignoratâ, non videor de ullâ aliâ plane certus esse unquam posse.” For a recent account that takes—mistakenly I believe—these passages at face value, see Lawrence Nolan and John Whipple, “Self-Knowledge in Descartes and Malebranche,” Journal of the History of Philosophy 43, no. 1 (2005), 63.

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essentially imply a discursive mental operation Descartes usually calls deductio (and sometimes demonstratio); and that such a guarantee is not needed in the cases of clear and distinct cognitions that are obtained by way of an absolutely simple, non-discursive mental grasp Descartes usually calls intuitus both in the Regulæ and in his later writings.146 The root of self-sufficiency of the latter type of cognitions is then, according to him, that “nunquam possimus de iis cogitare, quin vera esse credamus” (Resp. 2, AT VII, 145).147 Indeed, granted—plausibly, given Descartes‫ތ‬ assumptions—that at the end of the day acts of remembering are instances of acts of Cartesian cogitatio, it surely follows that the sceptical worries at issue cannot arise in the cases specified; and—crucially—this is enough for such cases to qualify as instances of scientia given the criteria derived from AT III, 65. The most telling complex evidence in support of the suggested reading is gathered together from the passages in Resp. 2, AT VII, 140–41 and 145–46:148 [U]bi dixi nihil nos certo posse scire, nisi prius Deum existere cognoscamus, expressis verbis testatus sum me non loqui nisi de scientiâ earum conclusionum, quarum memoria potest recurrere, cùm non ampliùs attendimus ad rationes ex quibus ipsas deduximus. Principiorum enim notitia non solet a dialecticis scientia appellari. Cùm autem advertimus nos esse res cogitantes, prima quædam notio est ...; [atque] cùm quis dicit, ego cogito, ergo sum, sive existo, ... tanquam rem per se notam simplici mentis intuitu agnoscit .... ... [S]i quæ [certitudo, sive firma & immutabilis persuasio] habeatur, sit tantùm de iis quæ clare ab intellectu percipiuntur. Ex his autem quædam sunt tam perspicua, simulque tam simplicia, ut nunquam possimus de iis cogitare, quin vera esse credamus: ut quòd ego, 146

I discuss Descartes‫ ތ‬notions of intuitus and deductio (and their equivalents) in detail in ch. 2 where the references are given in extenso. For the time being, we need only consider the notorious explication of both operations in question that Descartes provides in Reg. XI, AT X, 407: “[A]d mentis intuitum duo requirimus: nempe vt propositio clarè & distinctè, deinde etiam vt tota simul & non successivè intelligatur. Deductio verò, si de illâ faciendâ cogitemus ..., non tota simul fieri videtur, sed motum quemdam ingenij nostri vnum ex alio inferentis involvit ....” 147 Cf. Reg. III, AT X, 368: “Per intuitum [Descartes’ emphasis] intelligo ... mentis puræ & attentæ tam facilem distinctumque conceptum, vt de eo, quod intelligimus, nulla prorsus dubitatio relinquatur [my emphasis] ....” 148 Cf. Descartes‫ ތ‬commentary on DM 4, AT VI, 39 at Burm., AT V, 178: “Si enim ignoraremus veritatem omnem oriri a Deo, quamvis tam claræ essent ideæ nostræ, non sciremus eas esse veras, nec nos non falli, scilicet cùm ad eas non adverteremus, et quando solùm recordaremur nos illas clare et distincte percepisse. Aliàs enim, etiamsi nesciamus esse Deum, quando ad ipsas veritates advertimus, non possumus de iis dubitare ....”

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Chapter One dum cogito, existam; quòd ea, quæ semel facta sunt, infecta esse non possint, & talia, de quibus manifestum est hanc certitudinem haberi. Non possumus enim de iis dubitare, nisi de ipsis cogitemus; sed non possumus de iisdem cogitare, quin simul credamus vera esse, ut assumptum est; ergo non possumus de iis dubitare, quin simul credamus vera esse, hoc est, non possumus unquam dubitare. ... Alia sunt, quæ quidem etiam clarissime ab intellectu nostro percipiuntur, cùm ad rationes ex quibus pendet ipsorum cognitio satis attendimus, atque ideo tunc temporis non possumus de iis dubitare; sed quia istarum rationum possumus oblivisci, & interim recordari conclusionum ex ipsis deductarum, quæritur an de his conclusionibus habeatur etiam firma & immutabilis persuasio, quandiu recordamur ipsas ab evidentibus principiis fuisse deductas; hæc enim recordatio supponi debet, ut dici possint conclusiones. Et respondeo haberi quidem ab iis qui Deum sic norunt ut intelligant fieri non posse quin facultas intelligendi ab eo ipsis data tendat in verum; non autem haberi ab aliis (Descartes’ emphases).

As already noted, it might seem as if the first article of the quoted passage witnessed Descartes‫ ތ‬endorsement of the view that cognitions gained through pure intuitus should not, strictly speaking, be called (pieces of) scientia. If this really were the case then the passage, far from supporting the proposed interpretation, would instead run straight against the wider notion of scientia in AT III, 64–65. Yet I submit that such a reading is incorrect. For the complaint Descartes tries to address reads that in view of the above-quoted AT VII, 71, Descartes is not entitled to treat the cognitions of his own existence and his being a res cogitans (attained at Med. II, AT VII, 27–28) as pieces of scientia;149 and significantly, Descartes in his response stops short of renouncing his talk in terms of scientia (and metaphysical certainty) in Med. II and tries instead to neutralize the troublesome AT VII, 71 through setting it within the context of the above-quoted AT VII, 69–70. In view of this, the second sentence of the passage under scrutiny is best interpreted to the effect that Descartes invokes the usage of “the dialecticians” in order to explain why he failed, in AT VII, 69–71, to highlight in due fashion the distinct areas within the domain of scientia as understood by him: he intends to tell the author(s) of Obj. 2, I submit, that he adopts the usage of “the dialecticians” in the troublesome passages just mentioned, which usage nonetheless is not his 149

See Obj. 2, AT VII, 124–25: “[C]ùm nondum certus sis de illâ Dei existentiâ, neque tamen te de ullâ re certum esse, vel clare & distincte aliquid te cognoscere, posse dicas, nisi prius certo & clare Deum noveris existere, sequitur te nondum clare & distincte scire quòd sis res cogitans, cùm ex te illa cognitio pendeat a clarâ Dei existentis cognitione, quam nondum probasti locis illis, ubi concludis te clare nosse quòd sis.”

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own. Moreover, the first sentence of the quoted AT VII, 140 would amount, upon closer inspection, to a very poor start indeed if Descartes were about to say just that the cognitions treated in AT VII, 27–28 ultimately do not deserve the title of scientia according to him. I conclude, therefore, that Descartes‫ ތ‬use of the term “scientia” in the present contexts and in the passages considered thus far is perfectly univocal and that his considered meaning is captured most cogently and authoritatively in AT III, 64–65.150 To sum up the present discussion, I suggest that Descartes‫ ތ‬views on how cognitiones are to be classified with regard to our present concerns are best captured in the following scheme: Fig. 1-2

150

Nicholas Jolley, “Scientia and Self-knowledge in Descartes,” in Scientia in Early Modern Philosophy: Seventeenth-Century Thinkers on Demonstrative Knowledge from First Principles, ed. Tom Sorell, G. A. J. Rogers, and Jill Kraye (Dordrecht: Springer, 2010), 84–97 seems to oppose this suggestion: whilst the criteria of scientia in AT III, 64–65 are clearly indifferent as to whether the cognitions at issue are considered in isolation or as more or less complex or systematically interconnected, Jolley claims that “[for Descartes,] scientia is a body of knowledge as opposed, say, to a set of isolated intuitions” (Jolley, “Scientia and Self-knowledge,” 86). While I surely agree with Jolley that scientia in the sense of a body of knowledge is what Descartes is anxious to establish in the first place in his overall philosophical project, I am considerably less confident about his suggestion that Descartes‫ ތ‬notion of scientia be restricted to that domain only. Jolley supports his case with but one passage from RV, AT X, 513 while in the above exposition I have adduced several authoritative passages to the contrary.

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1.4 Breaking Down the Cartesian Circle The proposed interpretation of Descartes‫ ތ‬notion of scientia gains further support from the manner in which it allows one to resolve the notorious problem of the so-called Cartesian Circle. More precisely, it enables one to exonerate Descartes from the familiar charge of circularity in such a way that—unlike any other proposed solution of which I am aware—all of the following basic and presumably undisputed desiderata are really satisfied: (i) the solution should not compromise the professedly universal scope of Descartes‫ ތ‬meliorative epistemological project; (ii) the solution should accord with all the relevant passages in Descartes; and (iii) the solution should not jeopardize the pretension that scientia, in the established strict sense established, is possible to a reasonable extent for human beings. I shall explain. The charge of circularity in question is brought forward perspicuously, and in the standard form, by Arnauld in Obj. 4 and by Gassendi in his Disquisitio metaphysica:151 Unicus mihi restat scrupulus, quomodo circulus ab eo non committatur, dum ait, non aliter nobis constare, quæ a nobis clare & distincte percipiuntur, vera esse, quàm quia Deus est. At nobis constare non potest Deum esse, nisi quia id a nobis clare & evidenter percipitur; ergo, priusquam nobis constet Deum esse, nobis constare debet, verum esse quodcunque a nobis clare & evidenter percipitur (Obj. 4, AT VII, 214; Arnauld’s emphasis). [V]ideatur ab hoc usque loco inchoari circulus, quatenus es certus futurus Deum esse, & non esse deceptorem, quoniam habeas claram, & distinctam ejus rei notitiam; & certus sis futurus claram, & distinctam notitiam esse veram, quia noveris esse Deum, qui non possit esse deceptor (Gassendi, Disquisitio Metaphysica, 99).

Both thinkers thus read Descartes as claiming, on the one hand, that one cannot be certain of the veracity of anything that is clearly and distinctly perceived unless one is certain that the veracious God exists; and as being committed, on the other hand, to claiming that one cannot be certain of the 151

Pierre Gassendi, Disquisitio Metaphysica: seu Dubitationes et Instantiæ: Adversus Renati Cartesii Metaphysicam, & Responsa (Amsterdam: Iohannes Blaev, 1644). Cf. also Burm., AT V, 148: “Videtur [quòd auctor circulum commiserit]; nam in tertiâ Meditatione probat auctor Deum esse per axiomata, cùm sibi necdum constet, se in iis non falli.” And Obj. 2, AT VII, 124–25.

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existence of the veracious God unless one perceives clearly and distinctly that such a God exists. The problem thus stated might be expected to prove considerably intractable, if not outright insoluble, as long as the certainty at issue is treated along full-blooded normative lines, for the veracious God is then to be called in to protect against the Demon Hypothesis even clear and distinct perceptions arising from pure intuitus; and I do not believe that any satisfactory solution is ever forthcoming under such circumstances.152 However, even several alternative solutions that are not essentially tied to the strong normative stance—since they accommodate, in one way or another, Descartes‫ ތ‬claim that judgments on the basis of clear and distinct perception need no additional guarantee as long as a given content is actually perceived clearly and distinctly—have recently been commonly recognized as failing in the face of the above specified desiderata.153 152

Wilson‫ތ‬s treatment illustrates the present point well. On the one hand, Wilson dismisses most resolutely the psychological inability to doubt what one actually perceives clearly and distinctly as being relevant to the problem of circularity: [Descartes, in Resp. 2,] tries to exempt from the scope of the [Deceiver] Hypothesis ... every distinct perception that does not rely on memory, at the time we are having it [Wilson’s emphasis]. This might mean that our inability to doubt [Wilson’s emphasis] while in the grips of a distinct perception somehow makes the Hypothesis irrelevant to what we‫ތ‬re perceiving then. ... But this line is unpromising. The crucial issue is whether we can know certain propositions prior to proving God; the observation that there are certain moments when we cannot for the moment doubt is epistemically irrelevant [my emphasis]” (Wilson, Descartes, 117).

On the other hand, the dismissal leaves her with just one line of defence against the charge of circularity: namely the one that trades upon Descartes‫ ތ‬claim that “the idea of God is the most clear and distinct of all” (ibid., 116), for “[t]hen we could ascribe to Descartes the claim that ‫ދ‬doubts‫ ތ‬of distinct perceptions of lesser distinctness ... are removed by the absolutely distinct perception of God‫ތ‬s perfection” (ibid.; Wilson’s emphasis). Yet Wilson herself recognizes that such a strategy is unsatisfactory (though she still recommends it faute de mieux) since “[i]f I am going to suppose that my understanding may be systematically deceptive/defective, there is simply no reason to feel assured as the degree of ‫ދ‬distinctness‫ ތ‬increases ...” (ibid.). 153 I have in mind in particular (1) the so-called memory defence—promoted prominently in Alan Stout, “The Basis of Knowledge in Descartes,” Mind 38, no. 151–52 (1929): 330–342; 458–72 and in Willis Doney, “The Cartesian Circle,” Journal of the History of Ideas 16, no. 3 (1955): 324–38—according to which the

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Yet a promising line of addressing the charge of circularity has been taken up and developed in various ways by several recent scholars.154 Their common starting point is that they credit Descartes—correctly as I argued—with the view that divine guarantee is definitely not needed for attaining metaphysical certainty155 with regard to a given cognition as long as one actually perceives a given content clearly and distinctly at a given moment. The crucial gambit is that Descartes seems to hold both (i) that there is no reason in principle not to apply that view to the cognition that God exists—provided, of course, that one is able to perceive clearly and distinctly at a single moment that God exists; and (ii) that the above proper objective of the divine guarantee is not so much immunity of past clear and distinct perceptions to subsequent dislodgment by sceptical worries but rather the reliability of our recollecting that we did perceive a given content clearly and distinctly; and (2) what Bennett has recently called “the Bréhier view” (see Bennett, Learning from Six Philosophers, 1:368; the solution is recommended prominently in Émile Bréhier, Histoire de la philosophie, vol. 2, La philosophie moderne (Paris: Félix Alcan, 1929–1932), 81–83 and in John Etchemendy, “The Cartesian Circle: Circulus ex tempore,” Studia Cartesiana 2 (1981): 5–42), according to whom the proper object of the divine guarantee is the subsequent immutability of the truth of a judgment once made on the basis of an actual clear and distinct perception. The memory defence has been refuted convincingly in Harry Frankfurt, “Memory and the Cartesian Circle,” Philosophical Review 71, no. 4 (1962): 504–11 (and most commentators have followed suit since then), and the Bréhier view was definitively discredited in Bennett, Learning from Six Philosophers, 1:369. I shall not venture any extended discussion of these issues, however, as I have nothing new to say and it would take us too far away from our proper concerns. 154 See in particular Frankfurt, “Memory and the Cartesian Circle.” And Bennett, Learning from Six Philosophers, vol. 1, ch. 19. 155 To prevent confusion, it is to be borne in mind that metaphysical certainty was interpreted by me along strongly psychological lines, and this is of course how I understand the notion in the present discussion. In this respect, I differ significantly from the bulk of commentators who otherwise have adopted a similar approach: while they acknowledge that the certainty associated with actual clear and distinct perceptions cannot be but psychological, they still hold that Descartes was after a stronger, normative certainty (and they usually reserve only for this stronger category the term “metaphysical certainty”): cf. in particular Alan Gewirth, “The Cartesian Circle,” The Philosophical Review 50, no. 4 (1941): 368–95; Anthony Kenny, “The Cartesian Circle and the Eternal Truths,” The Journal of Philosophy 67, no. 19 (1970): 685–700; and James Van Cleve, “Foundationalism, Epistemic Principles, and the Cartesian Circle,” The Philosophical Review 88, no. 1 (1979): 55–91. I can therefore steer clear of the project of crossing the gap between psychological and normative certainties, which I take to amount to an impossible task.

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requirement can in fact be met by human beings, at least in so far as one is able to perceive clearly and distinctly each step of the ex hypothesi sound proofs of God‫ތ‬s existence Descartes provides,156 and to grasp a given proof in its entirety at a single moment. It is not difficult to see that given all this, the alleged circle breaks down: as long as one actually, and in a single intuitus, perceives clearly and distinctly both the premises and the conclusion of a proof that the veracious God exists, one undoubtedly has all that is required to be metaphysically certain that the veracious God exists; and as long as one is certain of this, one is metaphysically certain of everything one remembers having perceived clearly and distinctly at any time in the past since the certainty of the existence of the veracious God precludes the arising of any sceptical worries concerning any past clear and distinct perceptions. Indeed, there is no circle here. There might be some worries as to whether Descartes himself envisaged a solution along the suggested lines;157 but the following passage sounds perfectly conclusive: [Burman:] Videtur [quòd auctor circulum commiserit]; nam in tertiâ Meditatione probat auctor Deum esse per axiomata, cùm sibi necdum constet, se in iis non falli. [Descartes:] Probat, et scit se in iis non falli, quoniam ad ea attendit; quamdiu autem id facit, certus est se non falli, et cogitur illis assentiri. [Burman:] Sed mens nostra non potest simul nisi unam rem concipere. Demonstratio autem illa longior est, et constat ex pluribus axiomatis. Tum omnis cogitatio fit in instanti, et menti in illâ demonstratione multæ subordiuntur cogitationes. Et sic ad illa axiomata attendere non poterit, cùm una cogitatio impediat aliam. [Descartes:] 1o Quod mens non possit nisi unam rem simul concipere, verum non est: non potest quidem simul multa concipere, sed potest tamen plura quàm unum .... Tum 2o quod cogitatio etiam in instanti, falsum est, cùm omnis actio mea fiat in tempore, et ego possim dici in eâdem cogitatione continuare et perseverare per aliquod tempus. ... Cùm igitur cogitatio nostra ita plura quàm unum complecti queat, et in instanti non fiat, manifestum est nos demonstrationem de Deo integram complecti posse, quod dum facimus certi 156 I shall not discuss Descartes‫ ތ‬proofs of God‫ތ‬s existence in the present study. Since my ambition is chiefly historical, I feel entitled simply to go along with Descartes‫ ތ‬obvious belief that his proofs were sound. 157 Cf. especially Williams, Descartes, 183: “Frankfurt claims that for this answer [i.e. for the interpretation whose essentials I have just promoted—J.P.], it would at the very least be necessary when recollecting some previous demonstration that I should carry in my mind an actual intuition of the proof that God exists and is no deceiver, and there is no reason to suppose that Descartes regarded this feat as necessary, or perhaps, even possible.” The following quotations show manifestly that Williams is wrong.

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Chapter One sumus non nos falli, et sic omnis difficultas tollitur (Burm., AT V, 148–49; my emphasis).

Furthermore, Descartes‫ ތ‬entitlement to such a stance is boosted with his earlier account of the way one can attain a single intuitive grasp of comparatively long chains of reasoning in Reg. VII, AT X, 387–88: [A]d illas veritates inter certas admittendas, quas suprà diximus à primis & per se notis principijs non immediatè deduci [quod] sit interdum per tam longum consequentiarum contextum, vt, cùm ad illas devenimus, non facilè recordemur totius itineris, quod nos eò vsque perduxit; ideoque memoriæ infirmitati continuo quodam cogitationis motu succurrendum esse dicimus. ... Quamobrem illas continuo quodam imaginationis motu singula intuentis simul & ad alia transeuntis aliquoties percurram, donec à primâ ad vltimam tam celeriter transire didecerim, vt ferè nullas memoriæ partes reliquendo, rem totam simul videar intueri ... (my emphasis).

Putting details and internal complexities to one side, I believe that the submitted response is the best that can be given on Descartes‫ ތ‬part, and also the one that is in accord with almost all the relevant texts.158 Yet a formidable complication, unnoticed by commentators as far as I can discover, is lurking here. To begin with, the proponents of the present interpretation have commonly recognized a certain embarrassing peculiarity that the solution brings about. As Bennett so elegantly put it, [e]ven if one could get the proof of God‫ތ‬s existence and veracity into a single intuition, and become assured of the truth-rule on that basis, this assurance would last only as long as one was [perceiving clearly and distinctly] the proof of God .... That puts our philosophy and physics on an intermittently shaky basis, unless we have a one-step or intuitive grasp of God‫ތ‬s existence which we can bring to mind at will, or perhaps one that is always present in our thinking. The entire story is becoming fanciful, is it not? Ought not Descartes to be embarrassed by being committed to such a burdensome demand as that a satisfactory body of knowledge requires that a veracious God be there in one‫ތ‬s thinking—complete with a proof of his existence—all the time? (Learning from Six Philosophers, 1:369–70).

Bennett is undoubtedly right that the commitment at issue sounds somewhat embarrassing. Yet, of course, there is no problem of principle 158

The only significant exceptions are a few passages in which Descartes commits himself to a strong normative approach towards the issue of the validation of human cognitive operations. The passages are listed in fn. 131.

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here: fanciful as it perhaps is, Descartes seems to dispose of resources to render the required state of mind achievable in principle, even by human beings. For we have just seen him envisaging a technique of compressing discursive argumentative structures into a single intuitive grasp in Reg. VII; and attaining the required constancy seems to be, at root, just a matter of drill and discipline. While it is thus certainly possible in principle to devote such exercise to any discursive cognition, it remains advisable to focus one‫ތ‬s energies on the cognition of God‫ތ‬s existence as clearly it shall spare one from the same labour with all the rest. So far so good. Even if everything up to the present point is conceded, however, I suspect—contrary to what sympathetic commentators seem to suppose—that the matter must not be left at this stage, lest at the end of the day Descartes‫ ތ‬project of providing the foundations for a considerably large body of scientia collapses in the face of the circularity charge. For recall that scientia as understood by Descartes in the present contexts was defined, among other things, in terms of logical impossibility that the certainty of a given cognition be dislodged, cæteris paribus, by entertaining the relevant sceptical worries at any future time. If so, however, then eventually the cognition of God thus far established fails to carry out the job it is, according to Descartes, designed to do, viz. to render judgments based on past clear and distinct perceptions units of scientia: for even if one were to attain, presumably by way of the disciplined exercises prescribed in the Regulæ, a state of mind such that one would, throughout some period, constantly be blessed with actual clear and distinct cognition of God‫ތ‬s existence, it still remains logically possible that one might lose that wonderful state of mind some time in the future; and, as a consequence, one would never really get over the position Descartes himself ascribes to atheists at Resp. 2, AT VII, 140. That is to say, the domain of one‫ތ‬s scientia would eventually shrink back to comprehending nothing over and above mere temporary cognitions on the basis of actual clear and distinct perceptions. However, this would run directly contrary to the third of the above stated desiderata; consequently, the line of defence against the circularity charge we have considered would collapse. The considerations up to now seem to warrant the conclusion that if we wish to retain the present solution to the circularity problem (as I believe we should), Descartes is left with just one way out: namely to the effect that the cognition of God‫ތ‬s existence can in principle be rendered, even by human beings, to count as a full-blooded simple intuitus, i.e. as an item which satisfies Descartes‫ ތ‬description of what are “tam perspicua, simulque tam simplicia, ut nunquam possimus de iis cogitare, quin vera esse credamus ...” (Resp. 2, AT VII, 145; my emphasis); and fortunately

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there is at least one very authoritative passage in which Descartes claims almost exactly as much, namely Resp. 2, AT VII, 163–64: [Peto] ... ut [lectores] diu multumque in naturâ entis summe perfecti contemplandâ immorentur; & inter cætera considerent, in aliarum quidem omnium naturarum ideis existentiam possibilem, in Dei autem ideâ non possibilem tantùm, sed omninò necessariam contineri. Ex hoc enim solo, & absque ullo discursu, cognoscent Deum existere; eritque ipsis non minus per se notum, quàm numerum binarium esse parem, vel ternarium imparem, & similia. Nonnulla enim quibusdam per se nota sunt, quæ ab aliis non nisi per discursum intelligantur (my emphases).

Further, given that the passage comes from the very same Resp. 2 in which the above description is expressly introduced, and that Descartes‫ތ‬ examples of even and odd numbers seem to be of the same type as the example given in AT VII, 145 (“quæ semel facta sunt, infecta esse non possint”),159 I conclude not only that Descartes does envisage the possibility of a strictly intuitive grasp of the simple fact that God exists, besides an intuitive grasp of the proof that God exists, but even that it is, eventually, precisely this cognitio per se nota of God‫ތ‬s existence that is needed for Descartes‫ ތ‬meliorative epistemological program to emerge unscathed, and for scientia as conceived by him to be possible.160

159

The status of these cognitions is discussed in some detail in ch. 2. Some portions of ch. 2, in which the institution of reducing immediate deductiones to intuitus is discussed, will provide further, more developed support for this crucial interpretative claim.

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After we have thus elaborated Descartes‫ ތ‬notion of scientia as a (body of) cognition(s) upon which Descartes wishes to build in carrying out his project of firmly founding the human cognition tout court, and having secured, in certain respects at least, the possibility of scientia as conceived by him, the next task is to consider in some detail how Descartes treats those of the human cognitive faculties that are in this specific sense capable of and responsible for the scientia. We shall thus accomplish the preparatory work needed to be appropriately equipped for tackling our central issue, viz. the meaning or meanings of the term “a priori” in Descartes.

2.1 Understanding as the Principle of Scientia On the face of it, Descartes‫ ތ‬position looks crystal clear: it is exclusively the understanding that is responsible for those cognitions that are based upon clear and distinct perception and that also amount to (discrete units of) scientia; and the operations of the understanding through which these pieces of scientia could ever be attained are precisely two in number—namely intuitus and deductio. Thus he claims expressly in Reg. VIII, AT X, 398 that “solum intellectum esse scientiæ capacem,”1 and he confirms the claim when he says in Reg. IV, AT X, 372 that “nullam scientiam haberi posse, nisi per mentis intuitum vel deductionem”2 whilst 1

Cf. Reg. XII, AT X, 411: “Solus intellectus equidem percipiendæ veritatis est capax ....” Resp. 2, AT VII, 145: “Superest ... ut, si quæ [talis certitudo, sive firma & immutabilis persuasio] habeatur, sit tantùm de iis quæ clare ab intellectu percipiuntur.” 2 Cf. Reg. XII, AT X, 425: “[C]olligitur ... nullas vias hominibus patere ad cognitionem certam veritatis, præter evidentem intuitum, & necessariam deductionem ....” Reg. III, AT X, 370: “Atque hæ duæ viæ [sc. intuitus & deductio] sunt ad scientiam certissimæ, neque plures ex parte ingenij debent admitti, sed aliæ omnes vt suspectæ erroribusque obnoxiæ rejiciendæ sunt ....” Ibid., 366: “Circa

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making it clear that intuitus and deductio are actions or operations of the understanding:3 [H]îc recensentur omnes intellectûs nostri actiones, per quas ad rerum cognitionem absque vllo deceptionis metu possimus pervenire: admittunturque tantùm duæ, intuitus scilicet & deductio (Reg. III, AT X 368; my emphasis). Expositis duabus intellectûs nostri operationibus, intuitu & deductione, quibus solis ad scientias addiscendas vtendum esse diximus ... (Reg. IX, AT X, 400; my emphasis).

Yet a critical complication is lurking here which must be treated with diligent care lest we miss, from the very beginning, the whole point and complexity of Descartes‫ ތ‬overall notion of scientific enterprise, the proper understanding of which is pivotal for the project of this book. Recall that in the canonical exposition in Princ. I, 32, the understanding is treated in objecta proposita, ... quid clarè & evidenter possimus intueri, vel certò deducere, quærendum est; non aliter enim scientia acquiritur.” 3 The last word of the first quotation reads “inductio” instead of “deductio” in AT X, 368 as well as in both A- and H-versions of the Regulæ. There is a respectable tradition of amending “inductio” for “deductio”, in particular in view of Descartes‫ތ‬ back references to AT X, 368 in Reg. IV, AT X, 372 and Reg. IX, AT X, 400—cf. in particular Leslie Beck, Method of Descartes: A Study of the Regulae (Oxford: Clarendon Press, 1952), 51, fn. 2 and 84, fn. 3; Beck refers to the separate editions of the Latin text of the Regulæ by Artur Buchenau (Leipzig: Dürr, 1907), Henri Gouhier (Paris: J. Vrin, 1931) and Georges Le Roy (Paris: Boivin, 1933) in which the emendation is adopted; the now standard English translation of the Regulæ also follows suit—cf. René Descartes, The Philosophical Writings of Descartes, vol. 1, ed. and transl. by John Cottingham, Robert Stoothoff, and Dugald Murdoch (Cambridge: Cambridge University Press, 1985), 14 (by way of contrast, the once standard English translation by Elizabeth Haldane and G. R. T. Ross in René Descartes, The Philosophical Works of Descartes, ed. and transl. by Elizabeth Haldane and G. R. T. Ross, vol. 1 [Cambridge: Cambridge University Press, 1911], 7 sticks to “induction”). Following the editions by Giovanni Crapulli (The Hague: Martinius Nijhoff, 1966), Jean-Luc Marion (The Hague: Martinius Nijhoff, 1977), and Clarke, Descartes’ Philosophy of Science, 75–76, fn. 30, I am ultimately inclined to stick to “inductio” in view of the entire situation of the Regulæ since in the end a specific type of deductio turns out to be tantamount to what Descartes calls inductio sive enumeratio (particularly in Reg. VII and XI, to be discussed in detail towards the end of this chapter). However, I wish to ignore these complications for the present and to bear with “deductio”, with the observation that as a matter of fact (to be argued later) “deductio” is synonymous with “inductio” in the quoted AT X, 368.

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two senses: in the wider sense as the passio mentis, i.e. as perceptio simpliciter, in which sense it is contrasted with volitio as the actio mentis; and in the narrower sense as just one of several modi of that passio mentis, viz. as intellectio pura, in which sense it is contrasted with imaginatio and sentio. So the question is now in which of these two specified senses the understanding is taken by Descartes as responsible for (the pieces of) scientia in the passages just quoted and elsewhere. Contrary to what one would perhaps expect (due to stubborn interpretative prejudices or bad textbooks), both the textual and the systematic evidence seems to point strongly towards opting for the former, wider sense of intellectus quâ passio mentis. The systematic evidence must be put to one side for the moment since it cannot be appreciated unless Descartes‫ ތ‬general conception of mathematics and of the rôle of imagination in it is elaborated (ch. 3); and part of the textual evidence will be presented in the next section. For the time being, let us consider just some portions of textual evidence in order to endow the professed option with some initial plausibility as our interim interpretative hypothesis. The evidence is provided by Descartes‫ ތ‬occasional but quite authoritative claims that it is not only pure understanding but also the senses (and presumably imagination)4 that sometimes deliver clear and distinct perceptions, most notably in the following passages:5 Supersunt sensus, affectus, & appetitus, qui quidem etiam clarè percipi possunt, si accuratè caveamus, ne quid ampliùs de iis judicemus, quàm id præcisè, quod in perceptione nostrâ continetur, & cujus intimè conscii sumus. Sed perdifficile est id observare, saltem circa sensus: quia nemo nostrûm est, qui non ab ineunte ætate judicârit, ea omnia quæ sentiebat, esse res quasdam extra mentem suam existentes, & sensibus suis, hoc est, 4

I will put the case of imagination to one side in the following discussion, both for the sake of brevity and because Descartes himself does not treat it expressly in the passages under consideration. 5 Again, I do not put much weight on the fact that Descartes speaks only of clear and not of distinct perception in the first passage. He is somewhat careless in employing his all-important distinction between clear and distinct perceptions in the course of Princ. I, 63–70; witness, for example, that the title of Princ. I, 63 reads “Quomodo cogitatio & extensio distinctè concipi possint ut constituentes naturam mentis & corporis” (AT VIII-1, 30; my emphasis), while Descartes goes on smoothly to speak of clear and distinct cognitions concerning the very same issue in the body of the article. Moreover, in the second quoted passage (Princ. I, 68) Descartes designates as clear and distinct what I take to be the same type of perceptions as is the type which is designated solely as clear in the first passage. See also ch. 1, fn. 121.

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Chapter Two perceptionibus quas de illis habebat, planè similes (Princ. I, 66, AT VIII-1, 32). [D]iligentissimè est advertendum, dolorem quidem & colorem, & reliqua ejusmodi, clarè & distinctè percipi, cùm tantummodo ut sensus, sive cogitationes, spectantur (ibid., 68, AT VIII-1, 33). [S]ensuum perceptionibus, quæ proprie tantùm a naturâ datæ sunt ad menti significandum quænam composito, cujus pars est, commoda sint vel incommoda, & eatenus sunt satis claræ & distinctæ, utor tanquam regulis certis ad immediate dignoscendum quænam sit corporum extra nos positorum essentia, de quâ tamen nihil nisi valde obscure & confuse significant (Med. VI, AT VII, 83; my emphases).

Indeed, once these statements are read in the light of Gewirth‫ތ‬s interpretation (endorsed above by me) of what clarity and distinctness of perceptions amount to in Descartes, the claims come out as perfectly intelligible: generally speaking, it is—in principle at least—possible to establish the relations of what Gewirth calls logical equality between Gewirthian direct and interpretative contents in the domain of sensations and sense perceptions no less than in the domain of pure intellections. Further, once it is appreciated that the dichotomy of perceptions which do and which do not count as clear and distinct goes thus orthogonally through the domain of the particular modi intelligendi (pure intellection, imagination, sensation, and perhaps memory),6 I can see no principal obstacle that would prevent one treating judgments constituted through assent to clear and distinct perceptions in the domain of the senses as capable of conforming to the strictest standards Descartes requires of cognitions that deserve the title of scientia—provided one respects the critical caveat that “ne quid ampliùs de iis judicemus, quàm id præcisè, quod in perceptione nostrâ continetur, & cujus intimè conscii sumus” 6

It is important to bear in mind here that to Descartes even the deliverances of pure understanding may fail to count as clear and distinct. See e.g. Princ. I, 54, AT VIII-1, 26: “[H]abere possumus ideam claram & distinctam substantiæ cogitantis increatæ & independentis, id est Dei: modò ne ... quidquam etiam in eâ esse fingamus, sed ea tantùm advertamus, quæ revera in ipsâ continentur, quæque evidenter percipimus ad naturam entis summè perfecti pertinere.” Hyp., AT III, 430: “[F]requenter ... animaduerti, ea quæ homines iudicabant ab ijs quæ intelligebant dissentire .... Sic, cum iudicant spatium, quod inane appellant, nihil esse, illud nihilominus vt rem positiuam intelligunt. Sic, cum accidentia putant esse realia, repræsentant sibi ipsa tanquam substantias, etsi substantias esse non iudicent ....”

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(Princ. I, 66, AT VIII-1, 32). I know of only one passage (which, unfortunately, comes from the highly authoritative Resp. 2) that prima facie goes straight against the proposed interpretation: [C]ùm hydropicus sitit, vel cùm ictericus nivem videt ut flavam: non ... minus clare & distincte illam sic videt, quàm nos ut albam. ... Nec obstat, quòd sæpe simus experti alios deceptos fuisse in iis quæ sole clarius se scire credebant. Neque enim unquam advertimus, vel ab ullo adverti potest, id contigisse iis qui claritatem suæ perceptionis a solo intellectu petierunt, sed iis tantùm qui vel a sensibus, vel a falso aliquo præjudicio, ipsam desumpserunt (Resp. 2, AT VII, 145–46; Descartes’ italics).

Here clear and distinct perceptions ab intellectu are unambiguously contrasted, in a relevant context, with clear and distinct perceptions a sensu, a position barely compatible with the proposed interpretation while it squares well with equating the understanding quâ responsible for scientiæ with pura intellectio. I am not able to explain away this odd passage in an entirely unproblematic way. The best I can do is to employ here Alan Gewirth‫ތ‬s suggestion that for Descartes, “[i]n the context of science, ... the concern is with the essences of things, [so that] the connection between the two contents [viz. the direct and the interpretative one] must be necessary” (“Clearness and Distinctness,” 87). For reasons that will emerge in ch. 4, I reject this claim in its generality as being by far too strong; yet I suggest it is precisely such a strong notion of scientia that Descartes might have had in mind in the quoted troublesome passage; and thus he seems to stand on firm ground, given his general metaphysical commitments, in claiming that in so far as the direct content is delivered solely by the senses (and presumably by the imagination as well) the resultant perceptions can never be employed regarding the essences of any thing; or, to put it from another perspective which will be explicated below in this chapter, it can never enter into any necessary compositions and thus remains incapable of entering into any deductive chains that are constitutive for many scientiæ. For the time being, it is enough to observe that, if correct,7 the suggested conjecture manages to resolve the discrepancy at issue by way of pointing out an equivocation regarding the term “scientia”: while in the bulk of his texts Descartes assigns responsibility for a given cognition‫ތ‬s counting as scientia to understanding 7

In view of considerations advanced in the closing section of ch. 3, it shall become clear that even if this conjecture is correct, Descartes still must have misstated his point in an important respect.

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in the wider sense of passio mentis,8 he switches to a much stronger conception of scientia in AT VII, 145–46, such a shift committing him to tie scientia exclusively to pure understanding as opposed to the other modes of operationes intellectûs. As a result, I adopt as a hypothesis the interpretation according to which it is understanding in a general sense of passio mentis that Descartes means when he designates understanding as the faculty which is—in the form of intuitus or deductiones—properly and exclusively responsible for any cognition to count as scientia in the Regulæ. The significance of some salient implications of this hypothesis can barely be overestimated: in particular, the hypothesis has it—contrary to what many readers have assumed—that there are no obstacles of principle to treating not just pure understanding but also imagination and even the senses as capable of taking an essential part in delivering cognitions that deserve the title of scientia in the strict sense specified above. This seems to open rich prospects for Descartes to integrate even various overtly empirical cognitive enterprises into his general epistemic fabric of the scientiæ and helps, in the long run, to make some sense of the seemingly incompatible standards he appears to set on various occasions with regard to what he calls scientia. Let us proceed step by step, however. The next task is to engage in a thorough account of the two operationes intellectûs that according to Descartes are, as we saw, uniquely responsible for human scientia: intuitus and deductio.

2.2 Intuitus, Understanding, and Experientia Descartes explains what he means by intuitus most straightforwardly in Reg. III, AT X, 368: Per intuitum intelligo, non fluctuantem sensuum fidem, vel malè componentis imaginationis judicium fallax; sed mentis puræ & attentæ tam facilem distinctumque conceptum, vt de eo, quod intelligimus, nulla prorsus dubitatio relinquatur; seu, quod idem est, mentis puræ & attentæ non dubium conceptum, qui à solâ rationis luce nascitur ... (Descartes’ emphasis).

On the face of it, the explication seems to confirm both chief points made above with respect to intuitus: viz. that (1) Descartes refers with intuitus to precisely those items of perception that bear the mark of absolute 8

Cf. below for numerous references.

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indubitability per se9 (i.e., one might wish to add, in particular without any recourse to cognition of a benevolent deity) and that (2) acts of intuitus are a matter of conceptions delivered by intellectiones which are contrasted with fluctuans sensuum fides as well as with imaginationis judicium fallax and are said to be conceptions of the mens pura and engendered à solâ rationis luce. As for (1), we have already encountered the most telling passage concerning the principles of the alleged absolute indubitability of intuitus, namely Resp. 2, AT VII, 145 where the self-sufficiency of intuitus is traced back to the characterization that “nunquam possimus de iis [rebus] cogitare, quin vera esse credimus.” Further, I take numerous claims in the Regulæ to the effect that the absolute indubitability per se of intuitus is due to their utmost simplicity, facility, perspicuity and evidential character, as pointing in the same direction.10 The mildly psychological character of indubitability Descartes‫ ތ‬notion of intuitus brings about, its relation to truth and the implications of its psychological nature for the status of Descartes‫ ތ‬overall epistemological enterprise were established and defended in the previous chapter. I have nothing to add at present. As for (2), it is less than clear once again—as with the issue of the faculty which is responsible for scientia—whether Descartes means in his quoted definition of intuitus that the conceptus de eo quod intelligimus is due to the understanding in the wider sense of passio mentis or rather in the narrower sense of pura intellectio. It appears natural at first glance to read Descartes‫ ތ‬definition of intuitus to the latter effect, with an allimportant implication that at the end of the day the entire Cartesian scientia is precisely a matter of deliverances of pure understanding.11 To be sure, there is some evidence that might be invoked in support of such an intellectualist reading. After all, the conceptus that the intuitus is identified with in the quoted AT X, 368 are sharply contrasted with both fluctuans sensuum fides and malè componentis imaginationis judicium fallax. Moreover, there is the following passage: [P]our moi, ie distingue deux sortes d‫ތ‬instincs: l‫ތ‬vn est en nous en tant qu‫ތ‬hommes & est purement intellectuel; c‫ތ‬est la lumiere naturelle ou intuitus mentis, auquel seul ie tiens qu‫ތ‬on doit fier; l‫ތ‬autre est en nous en tant qu‫ތ‬animaux, & est vne certaine impulsion de la nature a la conseruation 9

Cf. e.g. Reg. III, AT X, 369: “intuitûs evidentia & certitudo”. Cf. e.g. Reg. III, AT X, 368–70 and Reg. XI, AT X, 407–408 for lists of these features in various combinations. 11 Leslie Beck is a prominent proponent of this reading; cf. his The Method of Descartes, 52–54. 10

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Chapter Two de nostre cors, a la iouissance des voluptez corporelles &c., lequel ne doit pas tousiours estre suiui (Mers., AT II, 599; Descartes’ italics).

Yet I wish to argue once again that the bulk of both textual and systematic evidence points towards taking not only pure understanding but also imagination and sensation—insofar as these faculties are employed correctly—as capable of delivering intuitus in the strict sense established by Descartes; and that, as a consequence, whenever Descartes picks out intellectus as the faculty responsible for intuitus in general, “intellectus” is to be taken as denoting a passio mentis in general. To begin, there are several moments discernible in Descartes‫ތ‬ treatment which, while singularly inconclusive, jointly suggest there be something wrong with the restrictive association of intuitus with pura intellectio. Thus, the wording of AT X, 368 allows of a reading quite different from the straightforward one that indeed supports the rival interpretation in question: could it be that alongside fluctuans sensuum fides, some fixa sensuum fides is possible for human beings? Again, could it be that apart from imaginatio malè componens, it is possible to employ imagination in such a way that it would compose benè? And could it be that in so far as (and only in so far as) this is the case, the employment of imaginatio and sensus renders one entitled to designate them as modi or operationes intellectûs in the sense employed in the quoted AT X, 368?12 A similar treatment should work with regard to the quoted AT II, 599. Next, Descartes commits himself to the notion of intuitus as a type of experientia. While holding firmly that intuitus and deductio are the only operations that ever yield any scientia,13 he also claims that “nos duplici viâ ad cognitionem rerum devenire, per experientiam scilicet, vel deductionem” (Reg. II, AT X, 364–65). Given the contrast between intuitus and deductio (to be established later in this chapter), intuitus must then count as a—presumably epistemologically privileged—type of experientia14 (and experientia in its widest meaning must then amount to nothing more specific than actual reflexive presence, or explicit apprehension or actual awareness, of the mind of an idea in the mind). The 12

The intellectualist conception endorsed above of Descartes‫ ތ‬cogitare, even with respect to imaginatio and sentio, proposed by Hill and (perhaps even more readily in the present context) Cottingham, should help to render this point intelligible or perhaps even appealing. 13 Cf. Reg. III, AT X, 368; Reg. IV, AT X, 372; Reg. IX, AT X, 400; and the references in fn. 2. 14 I owe this important point to Clarke, Descartes’ Philosophy of Science, 74–75 and n. 17.

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point is confirmed (to some degree at least) in Reg. VI where Descartes writes15 [n]otandum ... paucas esse duntaxat naturas puras & simplices, quas primò & per se, non dependenter ab alijs vllis, sed vel in ipsis experimentis, vel lumine quodam in nobis insito, licet intueri (AT X, 383; my emphasis).

Next, all this accords well with Descartes‫ ތ‬propensity to clarify the meaning to be associated with intuitus by way of analogies with visual perception and of visual metaphors.16 Finally, there is his warning that especially with regard to the word “intuitus” he is prepared to ignore its established meaning(s), taking into account instead just “quid singula verba Latinè significent, vt, quoties propria desunt, illa transferam ad meum sensum, quæ mihi videntur aptissima” (Reg. III, AT X, 369); and Latin intueor properly means “to look inside”. Adherents to the rival intellectualist interpretation might dismiss these hints for considerable reasons. They might insist that all the visual and etymological material just invoked indeed amounts to nothing but metaphors whose analog is completely devoid of sensory and imaginative content. Their case might then be strengthened with a remark that while it 15 In view of Reg. III, AT X, 368 where intuitus is defined as a “conceptum, qui à solâ rationis luce nascitur” without qualification, the wording “vel in ipsis experimenta, vel lumine quodam in nobis insito” (my emphases) clearly generates an inconvenient dilemma of interpretation. Either we take the Latin vel ... vel to mean here what it normally means, viz. exclusive disjunction; but then one must hold (putting to one side the option that Descartes is just inconsistent) that “lumen in nobis insitum” of AT X, 383 and “lumen rationis” of AT X, 368 are not synonymous, the former being just a species of the latter. Or else one sticks to the synonymy of these two terms; but then one must take the Latin vel ... vel to mean synonymy (which function is normally entrusted to seu or sive in Latin) and also swallow the strict synonymy of lumen in nobis insitum and experimenta. We shall see soon that the former option is better in terms of some other texts. Yet whatever the correct solution to the dilemma, what matters right now is that AT X, 383 is somewhat disconcerting for the adherents of a strict alliance between intuitus and pura intellectio as it might suggest that according to Descartes, experientia, in a sense going over and above the objects of pure understanding, does matter with respect to at least some intuitus. 16 Cf. Reg. XI, AT X, 400–401; Reg. XII, AT X, 427; Reg. XVI, AT X, 454. Descartes was, of course, aware of the long tradition of treating cognition in terms of visual metaphors, going through Augustine and Plotinus back to Plato. For references see John Cottingham, A Descartes Dictionary (Oxford: Blackwell Publishers, 1993), 94–95.

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is undeniable Descartes sometimes associates experientia with empirical observation and more generally with the operations of imagination and/or sense perception (especially in his essays concerning physics),17 the bulk of relevant occurrences witness him referring experientia unambiguously to reflexive cognitive operations whose objects count as the objects of pure understanding, viz. to cognitions of acts of thinking and even more frequently to acts of will.18 In view of this, it might seem possible and even plausible to interpret Descartes‫ ތ‬commitment to intuitus as a type of experientia to the effect that it is certain objects of pure understanding (such as acts of thinking, volitions, etc.) that form direct objects of the experientiæ involved in the case of intuitus.19 This sounds like a stand-off. However, as far as I can see the following evidence from Reg. XII definitely outweighs the case against the intellectualist interpretation:20 17

See e.g. Reg. VIII, AT X, 393–95; Reg. XII, AT X, 410–11; 427; La Dioptrique III, AT VI, 108; ibid., IX, AT VI, 202; Les Meteores VIII, AT VI, 334; Geom. II, AT VI, 422. 18 For references of experientia to thinking, see e.g. Resp. 5, AT VII, 358; Resp. VI, AT VII, 427; RV, AT X, 523. For references to volitions and the freedom of the will, see e.g. A Elisabeth, AT IV, 332–33; Med. III, AT VII, 38; Med. IV, AT VII, 56–59; Resp. 3, AT VII, 191; Resp. V, 377–78; Princ. I, 6, AT VIII-1, 6; Princ. I, 20, AT VIII-1, 20. In Burm., AT V, 146 and in Resp. 2, AT VII, 140 Descartes associates experientia even to notiones communes. 19 This suggestion gains further support from the following passage in Reg. VIII, AT X, 394: “Quia [aliquis] [proportionis, quam servant anguli refractionis ad angulos incidentiæ] indagandæ non erit capax, cùm non ad Mathesim pertineat, sed ad Physicam, hîc sistere cogetur in limine, neque aliquid aget, si hanc cognitionem vel à Philosophis audire, vel ab experientiâ velit mutuari .... Ac præterea hæc propositio composita adhuc est & respectiva; atqui de rebus tantùm purè simplicibus & absolutis experientiam certam haberi posse ...” (my emphases). It sounds plausible to have it that according to the present passage, experimenta through which one arrives at the “physical” cognition concerning matters which are compositæ & respectivæ are sensory in nature while it still might hold that experimenta de rebus purè simplicibus & absolutis—which surely amount to intuitus—have only purely intellective items as their possible objects. 20 Cf. also Reg. XII, AT X, 419–20: “Et quidem hæ [naturæ quæ communes dicendæ sunt] possunt vel ab intellectu puro cognosci, vel ab eodem imagines rerum materialium intuente” (my emphases). I imagine at least one influential recent commentator, namely John Schuster, might object to such a synchronic employment of the passages from Reg. XII to establish a similar reading of the passages in the earlier parts of the Regulæ. For, according to Schuster, it was not until Reg. XII that corporeal faculties and their deliverances were established as legitimate and even indispensable factors thanks to their crucial rôle in what

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Dicimus ... naturas illas, quas compositas appellamus, à nobis cognosci, vel quia experimur quales sint, vel quia nos ipsi componimus. Experimur quidquid sensu percipimus, quidquid ex alijs audimus, & generaliter quæcumque ad intellectum nostrum, vel aliunde perveniunt, vel ex suî ipsius contemplatione reflexâ. Vbi notandum est, intellectum à nullo vnquam experimento decipi posse, si præcisè tantùm intueatur rem sibi objectam, prout illam habet vel in se ipso vel in phantasmate ... (AT X, 422–23; my emphasis). [P]erspicuum est, intuitum mentis, tum ad illas omnes [naturas simplices] extendi, tum ad necessarias illarum inter se connexiones cognoscendas, tum denique ad reliqua omnia quæ intellectus præcisè, vel in se ipso, vel in phantasiâ esse experitur. (AT X, 425; my emphases)

Here we get all that is needed to surmount the defence of the intellectualist view offered in the previous paragraph. For on the one hand, the experimenta at issue are unambiguously tied to intuitus in the strict sense, and some of them are, on the other hand, unambiguously associated with the deliverances of phantasia, i.e. of corporeal imagination which (as we already know) is a common base for both sentio and imaginatio. Thus, I take it as demonstrated that for Descartes intuitus quâ operatio intellectûs is by no means restricted to acts of pure understanding. On the contrary, even certain deliverances of the senses and imagination have once again turned out capable of becoming objects of intuitus in the strict sense in which the resultant cognitions are capable of forming pieces of scientia according to Descartes‫ ތ‬overall epistemological picture. It will be observed that the results just presented are in perfect accord with the interpretative hypothesis submitted at the end of sec. 2.1. Indeed, the final sentence of Schuster calls the ontological certification of the faculty of intuitus in view of Descartes’ reinforced sceptical considerations due to his stay in Paris in the 1620s. According to Schuster the opening rules of the Regulæ, including Reg. III and VI, were written earlier than Reg. XII and bear no traces of the corporeal faculties’ being involved in the procedures productive of scientiæ. See in particular John Schuster, “Descartes‫ ތ‬Mathesis Universalis: 1619–28,” in Descartes: Philosophy, Mathematics and Physics, ed. Stephen Gaukroger (Sussex: The Harvester Press, 1980), 55–64. I have some strong reservations concerning Schuster’s overall developmental construal, and I will present some aspects of my criticism in the following chapters. For the present, it is enough to note that even if Schuster were right, the main thrust of the present section—viz. that even the corporeal faculties are capable of delivering intuitus in the strict sense established by Descartes— remains intact and would even come out as easier to establish much as the intellectualist strains of the passages from Reg. III and VI would come out as irrelevant at the risk of undue anachronism.

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the quoted AT X, 422–23 amounts to but a paraphrase of Descartes‫ތ‬ treatment in Princ. I, 66 and 68 where, as we saw, a possibility of scientia due to the senses (and presumably the imagination as well) is arguably established. What we have learned from the present considerations, however, is that according to Descartes the capability of the imagination and the senses to deliver pieces of scientia in the strict sense is eventually a function of their capability of operating, at least sometimes, to the effect that they deliver intuitus.

2.3 The Objects of Intuitus We are by now in a position to appreciate the thrust and significance of Descartes‫ ތ‬overall treatment of the objects of intuitus. It will be convenient first to introduce Descartes‫ ތ‬enumerations and classifications of all the types of items that could ever fall within the range of human cognition. The classification will then be correlated, at a basic level, with Descartes‫ތ‬ most authoritative enumeration of the proper objects of intuitus. The stated general classification of the types of items falling within the range of one‫ތ‬s cognition can be extracted from the following passages of Reg. XII: [H]îc de rebus non agentes, nisi quantùm ab intellectu percipiuntur, illas tantùm simplices vocamus, quarum cognitio tam perspicua est & distincta, vt in plures magis distinctè cognitas mente dividi non possint ...; reliquas autem omnes quodam modo compositas ex his esse concipimus (AT X, 418). [Præter] res illas, quæ respectu nostri intellectûs simplices dicuntur ..., possimus ... dicere reliqua omnia quæ cognoscemus, ex istis naturis simplicibus composita esse (AT X, 419–20). Dicimus ... conjunctionem harum rerum simplicium inter se esse vel necessariam vel contingentem (AT X, 421).

Here we have Descartes‫ ތ‬presumably exhaustive, audaciously reductive treatment of literally omnia quæ cognoscimus in its upmost generality: any content that can ever be provided by the operations of the human understanding amounts either to a certain simple nature or else to a compositio or mixtura of simple natures; and the compositions always count

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either as necessary or as contingent conjunctiones (or connexiones).21 Several fundamental and challenging questions arise that must be confronted to endow this abstract and tantalizingly indeterminate sketch with content and to render it a workable philosophical position. In particular, one must ask (1) what are the distinctive features of simple natures, what are the particular simple natures, how are they classified and how one recognizes that they actually form the content of a given perceptive act; and (2) what are the principles of connecting the simple natures, i.e. how exactly the compositions of simple natures are to be or can in principle be formed; and last but not least, (3) what is the root of necessity and contingency, respectively, of the two kinds of compositiones? All these questions shall be addressed in due course. For the moment, however, let us focus on a much more sober preliminary issue: according to Descartes, which items of the present scheme do or might form the objects of intuitus? Here is a passage in which Descartes seems to present his considered view on the question: [P]erspicuum est, intuitum mentis, tum ad illas omnes [naturas simplices] extendi, tum ad necessarias illarum inter se connexiones cognoscendas, tum denique ad reliqua omnia quæ intellectus præcisè, vel in se ipso, vel in phantasiâ esse experitur. (Reg. XII, AT X, 425)

It is thus absolutely uncontroversial that one segment of the possible objects of intuitus in Descartes amounts to the complete set of simple natures. Furthermore, Descartes makes it sufficiently clear that he also means that simple natures as such can be the objects of no operation other than intuitus. The situation is less clear with regard to another item in the quoted list, viz. the necessary connections of simple natures. While it surely holds that some such connections are objects of intuitus for Descartes, the issue is complicated by the fact that according to Descartes (as we shall see in more detail below) at least some necessary connections of simple natures are due to deductio,22 and it is not a trivial task to determine whether, or in which sense, such necessary connections due to deductio ultimately count 21

Thus Reg. XII, AT X, 425.16 (quoted below). See in particular Reg. XII, AT X, 424–25: “[I]n [deductione] tamen etiam plurimi defectus esse possunt: ... atque ita fit, quoties ex re particulari vel contingenti aliquid generale & necessarium deduci posse judicamus. Sed hunc errorem vitare in nostrâ potestate situm est, si nulla vnquam inter se conjungamus, nisi vnius cum altero conjunctionem omnino necessariam esse intueamur.” Also cf. Reg. III, AT X, 369. 22

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(or can count) as objects of intuitus as well. I shall postpone dealing with this complication, however, and limit myself for the moment only to the well-documented claim that for Descartes, only those necessary connections of simple natures which are grasped “vt tota simul & non successivè” (Reg. XI, AT X, 407) count as another type of object of intuitus. Finally, there is the prima facie somewhat confusing sort of reliqua omnia quæ intellectus præcisè, vel in se ipso, vel in phantasiâ esse experitur. So much seems clear that given both the quoted AT X, 418–21 and AT X, 425, one cannot but conclude that the set of items thus described falls under the heading of contingent connections of simple natures. Yet what is it about the members of this species of (nonsuccessively and totally) grasped contingent connections of simple natures that renders this species suitable to count as another type of object of intuitus? It must be something specific about certain simple natures that accounts both (i) for the contingency of the envisaged connections (lest all the connections that could ever become objects of intuitus boil down to necessary ones) and (ii) for the peculiar character of precisely those contingent connections that qualify as objects of intuitus (lest—given Descartes‫ ތ‬reductionist treatment of omnia quæ cognoscimus—literally everything that is ever grasped simultaneously and as a whole, qualifies as an object of intuitus and thus as a piece of the Cartesian scientia in the wider sense). As for requirement (ii), a likely feature is, of course, that the appropriate simple natures be apprehended explicitly in a given whole or (which comes to the same thing) that a given whole is perceived clearly and distinctly. As for crucial requirement (i), I submit that once the conditions thus far employed are combined with the wording of Descartes‫ތ‬ phrase under scrutiny, it turns out that the envisaged (clearly and distinctly perceived) simple natures are precisely those that are not employed regarding the essences of things, that is to say (given the results of sec. 2.2), precisely those simple natures that are perceived by the senses or by the imagination: for one thing, in Descartes phantasia usually refers to corporeal imagination (though sometimes to a physical organ responsible for the acts of corporeal imagination),23 i.e. to that part of human nature in which the senses and the imagination get in touch; moreover, we saw Descartes having charged the adverb “præcisè” with the peculiar rôle of indicating the way in which the deliverances of the senses (and presumably of the imagination as well) are to be interpreted (i.e. employed as interpretative content in the Gewirthian sense) in order to count as clear

23

Cf. Sepper, Descartes’s Imagination, 31, fn. 31.

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and distinct perceptions;24 and last but not least, I know of no other way in which to make good sense of the quoted AT X, 425 once it is combined with Descartes‫ ތ‬considered classification of omnia quæ cognoscimus. Having thus established the basic rationale of Descartes‫ ތ‬list of the types of objects of intuitus, the acuteness of the above introduced (clusters of) questions concerning Descartes‫ ތ‬reductive treatment of omnia quæ cognoscimus, in particular of (1) and (2), emerges distinctly. Clusters (1) and (2) shall be taken up immediately, preparing the way for (3).

2.3.1 Simple Natures Most of Descartes scholars agree on two things. First, that the doctrine of what Descartes calls naturæ simplices in the Regulæ and notiones simplices or notions primitiues in his mature texts25 is of upmost significance to his philosophy; and second, that the doctrine is obscure, vague and difficult to handle in a unitary way. As regards this alleged intractability of the doctrine, this is usually regarded as being due to several factors: (i) even in the Regulæ—the only text in which simple natures are brought to the fore and discussed at some length—Descartes fails to determine unambiguously and with sufficient clarity what he means by simple natures; (ii) the samples of simple natures he lists, somewhat casually, in Reg. VI and XII (AT X, 381–82 and AT X, 419–20) jointly form a prima facie bewilderingly heterogeneous jumble of items that seem to differ dramatically both in ontological and in logical respects; finally, (iii) the indications Descartes provides in Reg. VI and XII, respectively, on what simple natures are and how are they are to be employed, seem to be so divergent that the task of determining in which sense Descartes‫ ތ‬discussions of simple natures in Reg. VI and XII respectively are concerned with one and the same doctrine has usually been considered tricky or worse. Although most of what has just been said in (i)–(iii) is by and large 24 Cf. Princ. I, 66, AT VIII-1, 32. Also Med. II, AT VII, 29: “Sed verò etiam ego idem sum qui imaginor; nam quamvis forte, ut supposui, nulla prorsus res imaginata vera sit, vis tamen ipsa imaginandi revera existit, & cogitationis meæ partem facit. Idem denique ego sum qui sentio, sive qui res corporeas tanquam per sensus animadverto: videlicet jam lucem video, strepitum audio, calorem sentio. Falsa hæc sunt, dormio enim. At certe videre videor, audire, calescere. Hoc falsum esse non potest; hoc est proprie quod in me sentire appellatur; atque hoc præcise sic sumptum nihil aliud est quàm cogitare.” 25 See, respectively, Princ. I, 47, AT VIII-1, 22; A Elisabeth, AT III, 665 for the mature terms.

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true as it stands, I do not think that (i)–(iii) jointly warrant the complaints of abundant obscurity, vagueness and fragmentation regarding Descartes‫ތ‬ treatment of simple natures. For I believe that although (i) is perfectly correct, still Descartes tells the reader enough on simple natures to perform the tricky task mentioned in (iii) satisfactorily; and that in view of this solution, the logical and ontological heterogeneity mentioned in (ii) turns out innocuous and indeed unsurprising. These general claims will now be clarified in the course of my positive treatment of simple natures as objects of intuitus. The only passage that might be read as containing at least rudiments of a substantial definition of simple natures is the following: Dicimus igitur primò, aliter spectandas esse res singulas in ordine ad cognitionem nostram, quàm si de ijsdem loquamur prout revera existunt. Nam si, ver. gr., consideremus aliquod corpus extensum & figuratum, fatebimur quidem illud, à parte rei, esse quid vnum & simplex: neque enim, hoc sensu, compositum dici posset ex naturâ corporis, extensione, & figura, quoniam hæ partes nunquam vnæ ab alijs distinctæ exstiterunt; respectu verò intellectûs nostri, compositum quid ex illis tribus naturis appellamus, quia priùs singulas separatim intelleximus, quàm potuimus judicare illas tres in vno & eodem subjecto simul inveniri. Quamobrem hîc de rebus non agentes, nisi quantum ab intellectu percipiuntur, illas tantùm simplices vocamus, quarum cognitio tam perspicua est & distincta, vt in plures magis distinctè cognitas mente dividi non possint: tales sunt figura, extensio, motus, &c.; reliquas autem omnes quodam modo compositas ex his esse concipimus. Quod adeò generaliter est sumendum, vt nequidem excipiantur illæ, quas interdum ex simplicibus ipsis abstrahimus ... (Reg. XII, AT X, 418).

Here Descartes indicates that simple natures are precisely those entities that count as cognitive primitives or unanalyzables, i.e. those entities that exist as the fundamental elements of which each and every item26 is composed insofar as it is considered in ordine ad cognitionem nostram. Such a general characterization calls for explanation in at least three respects: (a) the way(s) in which simple natures are cognized; (b) the cognitive function(s) simple natures are expected to perform; and (c) the ontological status of simple natures. These issues shall now be addressed in turn; as shall (d), the question of the unity of Descartes‫ ތ‬doctrine of 26

Descartes speaks of res but it is clear that the term should be taken here in the loosest way, in particular as not implying that the doctrine of simple natures is to be restricted to the domain of real entities as opposed to mere fictions (i.e. a subset of entia rationis).

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simple natures. (a) Cognition of simple natures. We have just encountered the passage in which the simple natures‫ ތ‬way of being cognized is described in terms of the utmost perspicuity and distinctness: [I]llas tantùm [res] simplices vocamus, quarum cognitio tam perspicua est & distincta, vt in plures magis distinctè cognitas mente dividi non possint ... (Reg. XII, AT X, 418).

This is clearly an explication of what Descartes means with his usual claim that simple natures are per se notæ27 (the cognitive operations involved having to do either with experimenta or with lumen in nobis insitum). I submit that in view of what was argued in ch. 1, the mind‫ތ‬s inability to “divide” some contents in plures magis distinctè cognitas is to be treated in strictly psychological terms. Under this supposition, the following passage provides the best clue for grasping the way in which the doctrine of simple natures is to be integrated into the epistemology worked out so far: Dicimus ... naturas illas simplices esse omnes per se notas, & nunquam vllam falsitatem continere. Quod facile ostendetur, si distinguamus illam facultatem intellectûs, per quam res intuetur & cognoscit, ab eâ quâ judicat affirmando vel negando; fieri enim potest vt illa quæ revera cognoscimus, putemus nos ignorare, nempe si in illis præter id ipsum quod intuemur, sive quod attingimus cogitando, aliquid aliud nobis occultum inesse suspicemur, atque hæc nostra cogitatio sit falsa. Quâ ratione evidens est nos falli, si, quando aliquam ex naturis istis simplicibus à nobis totam non cognosci judicemus; nam si de illâ vel minimum quid mente attingamus, quod profectò necessarium est, cùm de eâdem nos aliquid judicare supponatur, ex hoc ipso concludendum est, nos totam illam cognoscere; neque enim aliter simplex dici posset, sed composita ex hoc quod in illâ percipimus, & ex eo quod judicamus nos ignorare (Reg. XII, AT X, 420–21).

In ch. 1 the sense was clarified in which (psychologically interpreted) metaphysical certainty—an inability to doubt—of the deliverances of intuitus (i.e. non-discursive clear and distinct cognitions) is at root due to the fact that “nunquam possimus de iis cogitare, quin vera esse credamus” (Resp. 2, AT VII, 145); and we can now grasp the sense in which this very fact is taken by Descartes as a function of the cognitive unanalyzability that is essential to simple natures (which have already been identified as 27

Cf. Reg. VI, AT X, 383; Reg. XII, AT X, 420; 425.

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objects of intuitus): in view of the Gewirthian notion of clarity and distinctness, a sine quâ non of falsity is a discrepancy between the direct and the interpretative contents of a given perception (i.e. idea or proposition); but such a discrepancy necessarily implies plurality within the content in question: as a consequence, wherever plurality within the content is ruled out by definition, the possibility of falsity is also ruled out. Such is the case, of course, with simple natures as treated by Descartes. Consequently, insofar as a simple nature is apprehended per se, it cannot provide matter except for a true cognition.28 Whenever we think there still remains some room for error (or ignorance), we simply mistake simple natures for something complex: all that can be legitimately concluded from such a situation is that we have not thus far apprehended any simple nature as such. (b) Cognitive functions of simple natures. The topic of the cognitive functions of simple natures is addressed by Descartes most straightforwardly in Reg. VI. The content of this pivotal rule will be discussed in more detail in ch. 4. For the time being, let it be enough to say that “natura simplex” is an honorific ascribed to any item that plays a certain indispensable and extremely momentous rôle as an item that amounts to the formal limit of a specific procedure which Descartes recommends as the proper way “vt aliquam veritatem inveniamus” (Reg. V, AT X, 379), that is to say, the procedure of ordering “res omnes ... inquantum vnæ ex alijs cognosci possunt” (Reg. VI, AT X, 381). The success of the procedure depends, among other things, upon finding an item that Descartes designates as the maximè absolutum (see ibid., 382) relative to the given set of items to be ordered, i.e. an item to which any complex content must be related in a controlled and explicit, discursive manner if it is to count as cognized by 28

Brian O‫ތ‬Neil offers an instructive alternative reading of what Descartes means by the per se cognition of simple natures: “The [cognitive] relation [between the intellect and simple natures] is, for Descartes, direct; it is the immediate union of intuition. Nothing comes between the mind and the simple nature which is its object. The simple nature which is known is known per se and not ‫ދ‬by means of‫ތ‬ an idea which represents it. The mind‫ތ‬s act is one of ‫ދ‬immediate vision‫ތ‬, of a ‫ދ‬direct seeing‫( ”ތ‬idem, “Cartesian Simple Natures,” in Moyal, Critical Assessments, 1:118–27). I find this untenable in view of what I take as a firmly established interpretation according to which a perception or idea, be it in the sense of a sub-propositional or of a propositionally structured content, is an essential constituent of any cognition whatever in Descartes. Given the interpretation I promote, the point is not (as O‫ތ‬Neil wishes) that “nothing [and a fortiori no perception or idea–J.P.] comes between the mind and the simple nature which is its object” but rather that the perception or idea by means of which the mind perceives a given simple nature can never count as materially false.

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way of scientia; and the important point for the present is that Descartes wishes to reserve the term natura simplex solely for any item that necessarily constitutes at least one of the possible maximè absoluta in any methodical ordering of the cogitabilia in the course of pursuit of scientia. It is against the background of this functional treatment that Descartes‫ތ‬ following pivotal dictum is to be approached:29 Colligitur ... omnem humanam scientiam in hoc vno consistere, vt distinctè videamus, quomodo naturæ istæ simplices ad compositionem aliarum rerum simul concurrant (Reg. XII, AT X, 427).

Further, it is in view of this treatment that Descartes provides, in Reg. VI, the following sample list of absoluta (i.e. of what is to count as simple natures) and relativa: Absolutum voco ... omne id quod consideratur quasi independens, causa, simplex, vniversale, vnum, æquale, simile, rectum, vel alia hujusmodi .... Respectivum verò est, ... quidquid dicitur dependens, effectus, compositum, particulare, multa, inæquale, dissimile, obliquum, &c. (AT X, 381–82).

(c) The ontological status of simple natures. It will be observed that the items from this list amount, in any case, to entities that seem to be of a merely conceptual nature, i.e. to have no independent reality over and above that of the contents of the mind that apprehends them. Is it correct, then, to extrapolate to the effect that the ontological status of all simple natures is like this? Several additional reasons seem to support this interpretation. To begin, consider once again the following passage: [N]aturas ... simplices ... sunt ... eædem, quas in vnâquâque serie maximè simplices appellamus. Cæteræ autem omnes non aliter percipi possunt, quàm si ex istis deducantur, idque vel immediatè & proximè, vel non nisi per duas aut tres aut plures conclusiones diversas; quarum numerus etiam est notandus, vt agnoscamus vtrùm illæ à primâ & maximè simplici propositione pluribus vel paucioribus gradibus removeantur. Atque talis est vbique consequentiarum contextus, ex quo nascuntur illæ rerum quærendarum series, ad quas omnis quæstio est reducenda, vt certâ methodo possit examinari (Reg. VI, AT X, 383; my emphases).

29

The dictum is discussed in more detail below. Cf. also Reg. XII, AT X, 422: “Dicimus ... nihil nos vnquam intelligere posse, præter istas naturas simplices, & quamdam illarum inter se mixturam sive compositionem ....”

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The text suggests that the relations between absolute terms, i.e. between those involving simple natures and relative terms are, fundamentally, inferential in nature; and it might seem hard—given our contemporary wisdom—to imagine anything other than items of a propositional, and by implication of a conceptual nature as capable of counting as the terms of inferential relations: after all, does not Descartes himself talk of propositiones in this particular context? Another reason worthy of consideration30 draws upon Descartes‫ ތ‬repeated insistence that in his treatment of simple natures he refuses to consider things nisi quantùm ab intellectu percipiuntur, that is to say, in ordine ad cognitionem nostram or respectu intellectûs nostri rather than prout revera existunt or à parte rei (see Reg. XII, AT X, 418).31 Thus it might be tempting to endorse or at least to consider, once again, a sort of essentially idealist interpretation to the effect that, for Descartes, no simple nature ever denotes the elements of the mind-independent world since simple natures amount only to the essentials of the way the human understanding constructs representations of the mind-independent world.32 Yet a strong case can be made against such an idealist reading. To begin with, the way in which simple natures (or simple or primitive notions, as Descartes later calls them) are classified in Descartes‫ ތ‬texts seems to imply quite clearly that at least some simple natures are to count as constituents of a mind-independent reality. He says in Reg. VIII that 30

It might be tempting to employ here yet another snippet of Descartes‫ ތ‬treatment of simple natures, namely his important remark that “hîc loci ... quædam assumenda sunt quæ fortasse non apud omnes sunt in confesso; sed parùm refert, etsi non magis vera esse credantur, quàm circuli illi imaginabiles, quibus Astronomi phænomena sua describunt, modo illorum ope, qualis de quâlibet re cognitio vera esse possit aut falsa, distinguatis” (Reg. XII, AT X, 417). However, this would be a mistake due to the confusion of formal and material modes of speech: to be sure, quâ hypothesis any item whatever will count as something merely imaginabile, i.e. mind-dependent in the presently applicable sense. However, what matters in the present reasoning is the status of the content of hypotheses provided they turn out and as long as they count as correct. It is of course an open question whether all the simple natures quâ contents of the relevant hypotheses are mind-dependent or not. 31 Cf. also Reg. VI, AT X, 381: “[R]es omnes per quasdam series posse disponi, non quidem in quantum ad aliquod genus entis referuntur, sicut illas Philosophi in categorias suas deviserunt, sed in quantum vnæ ex alijs cognosci possunt ....” The dictum is discussed in more detail in ch. 4. 32 Jean LeBlond, “Les natures simples chez Descartes,” Archives de Philosophie 12, no. 2 (1937), 243–60 has come up with the most straightforward reading to this effect.

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“[e]x simplicibus nullæ esse possunt, nisi vel spirituales, vel corporeæ, vel ad vtrumque pertinentes” (AT X, 399) and promises to extend this in Reg. XII, which he indeed does in AT X, 419–20. Here he refers to the three above genera, respectively, as purè intellectuales, purè materiales and communes and he goes on to say: Purè intellectuales illæ sunt, quæ per lumen quoddam ingenitum, & absque vllius imaginis corporeæ adjumento ab intellectu cognoscuntur: tales enim nonnullas esse certum est, nec vlla fingi potest idea corporea quæ nobis repræsentet, quid sit cognitio, quid dubium, quid ignorantia, item quid sit voluntatis actio, quam volitionem liceat appellare, & similia; quæ tamen omnia revera cognoscimus, atque tam facile, vt ad hoc sufficiat, nos rationis esse participes. Purè materiales illæ sunt, quæ non nisi in corporibus esse cognoscuntur: vt sunt figura, extensio, motus, &c. Denique communes dicendæ sunt, quæ modò rebus corporeis, modò spiritibus sine discrimine tribuuntur, vt existentia, vnitas, duratio, & similia. Huc etiam referendæ sunt communes illæ notiones, quæ sunt veluti vincula quædam ad alias naturas simplices inter se conjungendas, & quarum evidentiâ nititur quidquid ratiocinando concludimus. Hæ scilicet: quæ sunt eadem vni tertio, sunt eadem inter se; item, quæ ad idem tertium eodem modo referri non possunt, aliquid etiam inter se habent diversum, &c. ... Cæterùm, inter has naturas simplices, placet etiam numerare earumdem privationes & negationes, quatenus à nobis intelliguntur: quia non minus vera cognitio est, per quam intueor, quid sit nihil, vel instans, vel quies, quàm illa per quam intelligo, quid sit existentia, vel duratio, vel motus (AT X, 419–20).

Further, a similar classification, though with some minor shifts both in terminology and in content,33 appears much later in Princ. I, 47–49, AT VIII-1, 22–24: [S]ummatim hîc enumerabo simplices omnes notiones, ex quibus cogitationes nostræ componuntur .... Quæcunque sub perceptionem nostram cadunt, vel tanquam res, rerumve affectiones quasdam, consideramus; vel tanquam æternas veritates, nullam existentiam extra cogitationem nostram habentes. Ex iis quæ tanquam res consideramus, maximè generalia sunt substantia, duratio, ordo, numerus, & si quæ alia sunt ejusmodi, quæ ad omnia genera rerum se extendunt. Non autem plura quàm duo summa 33

The most remarkable difference between the classifications in Reg. XII and in Princ. I is clearly the inclusion of “[quæ] ab arctâ & intima mentis nostræ cum corpore unione proficiscuntur” within, and the omission of privations and negations of simple natures from the communes genus in the latter classification. Yet these differences are of little significance to our present issue, viz. an overall determination of the ontological status of simple natures.

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What matters now is that at least as regards the simple items that are purè materiales and purè intellectuales, it is scarcely possible, without quite unnatural stretches, not to read the texts to the effect that the items at issue, taken in themselves, do count as real affectiones of things that are constituents of mind-independent reality: it is not that the cognizing mind ascribes to bodies (if there are any) figure, extension etc. without the bodies being really affected by these items; rather, the mind—insofar as its cognition is true—forms the ideas of figure, extension etc., if and only if the bodies are really affected by these items (amounting to the simple natures the given mind cognizes); and similarly with the case of the affections of the mind insofar as it really exists as a constituent of mindindependent reality. Moreover, Descartes in Princ. I, 48–49 characterizes precisely notiones communes as “nullam existentiam extra cogitationem nostram habentes” (Princ. I, 48, AT VIII-1, 22) or as “quæ in mente nostrâ sedem habe[n]t” (Princ. I, 49, AT VIII-1, 23), and he expressly contrasts them with (the cognitions of) “res ... existens [vel] rei modus” (ibid.). I cannot see how one could avoid the conclusion that at least the modes of extension and thought that count as simplices notiones with which Descartes deals in Princ. I, 48 are to be considered as real affectiones of res existentes.

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Anyone who adopts the stance for which I have just argued—that at least some simple natures amount to the elements of mind-independent reality according to Descartes—is bound to deal with the reasons adduced above in support of the contention to the contrary. As for the reasoning to the effect that no elements of mind-independent reality can literally enter into inferential relations of any sort, my discussion must be postponed until Descartes‫ ތ‬notion of inference is accounted for later in this chapter. As for the reasoning turning upon Descartes‫ ތ‬remarks that he will discuss simple natures only in ordine ad cognitionem nostram, Brian O‫ތ‬Neil convinced me that the alleged, strongly idealist conclusion is a non sequitur that draws upon an undue overburdening of the innocent observation that the understanding in its cognitive operations at least sometimes (and perhaps always or even essentially) “somewhat rearrange[s] the elements of the world in the process of grasping them” (O‫ތ‬Neil “Cartesian Simple Natures,” 133).34 (d) The unity of the doctrine of simple natures. The class of simple natures has thus turned out to be extremely heterogeneous from the ontological point of view. For one thing, the domains into which simple natures pertain range from that of mental operations (res intellectuales) and affections of bodies (res materiales), over “trans-categorial properties”35 (quæ modò rebus corporeis, modò spiritibus sine discrimine tribuuntur) to mathematical axioms, logical principles and rules of inference (communes notiones). And for another thing, according to Descartes while some types of simple natures indeed seem to be of a purely conceptual nature, other seem to enjoy mind-independent existence. 34 O‫ތ‬Neil‫ތ‬s apt diagnosis is worth quoting at some length: “Whenever he talks about analysing the components of the world, Descartes reminds his readers that we have to pay attention to the world as we understand it. To me this says nothing more than the well-known Scholastic axiom: ‫ދ‬The thing known is in the mind of the knower after the fashion of the knower.‫ ތ‬This does not interpose a veil, or deny directness and accuracy of understanding. It merely says that the process of understanding is sui generis and will, in consequence, somewhat rearrange the elements of the world in the process of grasping them. ... It amazes me that some distinguished commentators on Descartes can fail to notice what is to me an obvious parallel between the doctrines of abstraction in Aquinas and Descartes. Instead they frequently seize on a few of those words of Descartes [in Reg. XII, AT X, 418], and proceed to base a whole theory on a misunderstanding of them. They conclude that simple natures are just ‫ދ‬notions‫ ތ‬because Descartes has said he is considering things only in reference to our mind‫ތ‬s understanding of them.” (O‫ތ‬Neil, ibid., 133–34; O’Neil’s emphases). 35 The term is borrowed from the discussion of dualisms in Bennett, A Study of Spinoza’s Ethics, 41–47.

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The ontological heterogeneity of the class of simple natures detailed above might, of course, sound somewhat embarrassing; and this brings us back to the general issues presented above. It will be remembered that there I claimed the ontological heterogeneity in question was innocuous and even predictable, and that as such it by no means implies that Descartes‫ ތ‬doctrine of simple natures is worthless due to its disunity and vagueness; and by now we have gathered together all the threads required to defend such a claim.36 The chief point is to distinguish two closely related yet different ways in which simple natures are employed by Descartes in his account of the foundations of human cognition. On the one hand, “natura simplex” is the term employed by Descartes—particularly in Reg. VI—to denote items in terms of which a certain unique rôle or function, namely that of constituting an absolutum, can essentially be described in the course of capturing the way in which particular contents are to be ordered by the human mind to attain at scientific cognition of anything that is in principle capable of being cognized in this privileged way. On the other hand, Descartes also undertakes—in particular in Reg. XII—a survey of various ontological domains and tries to determine in concreto at least some salient samples of what could and should instantiate those essentials of the above unique rôle as picked out in Reg. VI. Further, there seems to be nothing strange about Descartes‫ ތ‬readiness to call these simple items “naturæ simplices” once again. Since Reg. VI recommends (in continuation of Reg. V) finding the absolute and inventing an ordered series of all related relative items, it provides a sample list of some general functional terms the presence of which will suffice to designate a certain item as an absolutum—such as “independens, causa, simplex, vniversale, vnum, æquale, simile, rectum” (Reg. VI, AT X, 381). By way of contrast, Descartes surveys in Reg. XII the available ontological domains and determines in concreto some of the items by which these general functional terms can be instantiated.37 This interpretation, I submit, accounts satisfactorily for the apparent incompatibility of the sample lists of simple natures in Reg.VI and Reg. XII (plus Princ. I), respectively, and also shows that the ontological heterogeneity of the types of simple natures as classified in Reg. XII and Princ. I in fact does not jeopardize the essential unity or Descartes‫ތ‬ doctrine of simple natures. This is because it is exclusively the functional account in Reg. VI that provides for the basic and unitary meaning of the term “natura simplex”; the heterogeneity and apparent disunity of the class of simple natures has turned out to be but a function of the diversity of the 36

I am indebted to O‫ތ‬Neil‫ތ‬s treatment in his “Cartesian Simple Natures,” 119–23 in the rest of the present paragraph. 37 Cf. O‫ތ‬Neil, “Cartesian Simple Natures,” 122 for a similar apt treatment.

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ontological domains from which the items to instantiate the roles picked out in Reg. VI might come.

2.3.2 Compositiones as Objects of Intuitus Let us proceed now to the remaining types of objects of intuitus in Descartes, other than simple natures. We saw that these remaining types share the feature of counting as ex naturis simplicibus mixturas sive compositiones and that there are two species of these compositiones that rightly count as objects of intuitus: namely (i) those necessary compositions of simple natures that are grasped “vt tota simul & non succesivè” (Reg. XI, AT X, 407) by the human understanding, and (ii) those contingent compositions of simple natures which are such that the involved simple natures are perceived explicitly but are not employed regarding the essences of things. This exposition will be developed in two respects in the present section: (1) the two species of compositions just mentioned will be located within Descartes‫ ތ‬general theory of how the compositiones could be cognized by the human understanding; and (2) the nature of necessary connections of the relevant type will be examined in more detail. In Reg. XII Descartes provides a presumably complete classification of the ways in which the compositiones can be cognized:38 Dicimus ... naturas illas, quas compositas appellamus, à nobis cognosci, vel quia experimur quales sint, vel quia nos ipsi componimus. Experimur quidquid sensu percipimus, quidquid ex alijs audimus, & generaliter quæcumque ad intellectum nostrum, vel aliunde perveniunt, vel ex suî ipsius contemplatione reflexâ. Vbi notandum est, intellectum à nullo vnquam experimento decipi posse, si præcisè tantùm intueatur rem sibi objectam, prout illam habet vel in se ipso vel in phantasmate .... ... Componimus autem nos ipsi res quas intelligimus, quoties in illis aliquid inesse credimus, quod nullo experimento à mente nostrâ perceptum est .... ... Dicimus [deinde], hanc compositionem tribus modis fieri posse: nempe per impulsum, per conjecturam, vel per deductionem. Per impulsum sua de rebus judicia componunt illi, qui ad aliquid credendum suo ingenio feruntur, nullâ ratione persuasi, sed tantùm determinati, vel à potentiâ aliquâ superiori, vel à propriâ libertate, vel à phantasiæ dispositione .... ... Quidquid autem hac ratione componimus, non quidem nos fallit, si tantùm 38

The treatment is announced in Reg. VIII: “[E]x compositis [naturis] alias quidem intellectus tales esse experitur, antequam de ijsdem aliquid determinare judicet; alias autem ipse componit. Quæ omnia fusiùs exponentur in duodecimâ propositione ....”

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Chapter Two probabile esse judicemus atque nunquam verum esse affirmemus, sed etiam doctiores non39 facit. Superest igitur deductio, per quam res ita componere possimus, vt certi simus de illarum veritate; in quâ tamen etiam plurimi defectus esse possunt: vt si [naturas quasdam malè conjugamus]; atque ita fit, quoties ex re particulari vel contingenti aliquid generale & necessarium deduci posse judicamus. Sed hunc errorem vitare in nostrâ potestate situm est, si nulla vnquam inter se conjugamus, nisi vnius cum altero conjunctionem omnino necessariam esse intueamur ... (AT X, 422–25).

It should be clear upon closer inspection that according to Descartes both specific types of compositiones with which we are dealing now fall within the class of what experimur and not of what we nos ipsi componimus.40 It has already been established that intuitus is a (privileged) sort of experientia in Descartes and we know that Descartes does include both types of compositiones among the objects of intuitus.41 Moreover, cognitions from intuitus are, by definition, free from error;42 yet the only species within the class of quæ nos ipsi componimus that might aspire to securing such safety from error are, according to the above passage, the deliverances of correctly performed deductiones; but—we learn from the same passage—contingent compositions can never enter into safe deductions, and necessary compositions that are the objects of intuitus, quâ objects of intuitus,43 count immediately always as the terms, and never as the deliverances, of deductions. We can therefore conclude that according to Descartes the human mind is basically passive even with respect to certain privileged compositions of simple natures, namely those that amount to objects of intuitus. Even such intuitively cognized compositions are thus numbered amongst what Descartes invokes as “spontè obvias veritates” (Reg. VI, AT X, 384) or 39

I read “non” for “nos” in AT X, 424.19. This is not to say, of course, that the class of quæ experimur is exhausted with the aforementioned types of compositiones. Presumably a great number of contingent compositions that are not experienced præcisè by the intellect fall within that class. Yet we can safely put these to one side as the cognition of them can clearly never aspire to the title of Cartesian scientia. 41 See in particular Reg. XII, AT X, 425. 42 Notice in this connection Descartes‫ ތ‬remark in Reg. XII, AT X, 423: “[N]otandum est, intellectum à nullo vnquam experimento decipi posse, si præcisè tantum intueatur rem sibi objectam, prout illam habet vel in se ipso vel in phantasmate ...”: the description of the items that never deceive the intellect is strikingly similar to that of (arguably contingent) connexiones in Reg. XII, AT X, 425.17–18. 43 The significance of this qualification will be clarified below. 40

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“quod sponte obvium est” (Reg. XII, AT X, 411). This conclusion is of utmost significance, particularly with regard to the case of necessary compositions as objects of intuitus, for it amounts at the very least to a claim that the synchronic necessary connections between simple natures that the understanding apprehends while operating correctly are not to be gauged as mere deliverances of productive acts of the understanding somehow applied to reality which in itself is devoid of such modal features: quite to the contrary, they are somehow encountered by the human understanding as integral parts of mind-independent reality. The negative claim just formulated is, however, in need of further disambiguation if we are to grasp the essentials of the positive sense in which Descartes seems to hold that necessities in reality may be cognized by the human understanding. This need brings us to the latter of the two tasks mentioned at the beginning of the present section. Even if we accept—as I believe we should by force of the argument just submitted—that Descartes‫ ތ‬epistemology of modalities is realistic in the sense that at least some necessary connections (namely those whose terms involve simple natures) are integral parts of mind-independent reality, the task of determining with some precision the nature of such real necessary connections in Descartes turns out quite challenging. The challenge might take the form of an apparently intractable aporia in which Descartes seems to get stuck and which can be put conveniently (if anachronically) in terms of the following distinction between analytic and non-analytic connections between real items. Let us stipulate that a necessary connection between items a and b is analytic if and only if the simpler concept of a/b is included in the more complex concept of b/a; and that a necessary connection between a and b is non-analytic if and only if a and b are necessarily connected and their connection is not analytic in the sense just stipulated.44 44 Never mind, for the moment, the strikingly negative and uninformative character of the latter stipulation. The question of how Descartes‫ ތ‬positive pronouncements concerning the nature of non-analytic necessary connections and the manner in which they are cognized and are to be interpreted is a difficult one and I will confront it in sec. 2.4. In any case, it is due to the openness of that question that I abstain from referring to non-analytic connection by use of the term “synthetic” in my stipulation (unlike several distinguished commentators, most notably O‫ތ‬Neil, “Cartesian Simple Natures,” 129–35 and Leonard Miller, “Descartes, Mathematics, and God,” The Philosophical Review 66, no. 4 [1957: 451–65): as long as “synthetic” is read along the well-established Kantian lines, synthetic connection is just one of several options of how the nature of non-analytic necessary connections might be resolved.

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Given these stipulations, let us approach the announced aporia. On the one hand, Descartes commits himself to the claim that simple natures can never be conceived as divisible let alone actually divided by the mind into simpler items that are cognized magis distinctè: [I]llas tantùm [res quantùm ab intellectu percipiuntur] simplices vocamus, quarum cognitio tam perspicua est & distincta, vt in plures magis distinctè cognitas mente dividi non possint ... (Reg. XII, AT X, 418).

Furthermore, he intimates later to Princess Elizabeth that lors que nous voulons expliquer quelque difficulté par le moyen d‫ތ‬vne notion [primitiue] qui ne luy appartient pas, nous ne pouuons manquer de nous mesprendre; comme aussi lors que nous voulons expliquer vne de ces notions par vne autre; car, estant primitiues, chacune dҲelles ne peut estre entenduë que par elle mesme (AT III, 665–66; my emphasis).

I take this as implying, in the final analysis, that it is, strictly speaking, impossible for simple natures ever to become connected in some analytic manner.45 This implication is less straightforward than it might appear, however, and it will be well to consider it in more detail before we proceed further. For is it not—to take Descartes‫ ތ‬own example—that figura, which unambiguously counts as a simple nature for Descartes, is conceptually divisible into, say, extensio and terminus, of which at least extensio also counts as a simple nature?46 Well, consider the immediate continuation of the last cited AT X, 418: [T]ales [res simplices] sunt figura, extensio, motus, &c.; reliquas autem omnes quodam modo compositas ex his esse concipimus. Quod adeò generaliter est sumendum, vt nequidem excipiantur illæ, quas interdum ex simplicibus ipsis abstrahimus: vt sit, si dicamus figuram esse terminum rei extensæ, concipientes per terminum aliquid magis generale quàm per figuram, quia scilicet dici potest etiam terminus durationis, terminus motûs, &c. Tunc enim, etiamsi termini significatio à figurâ abstrahatur, non tamen idcirco magis simplex videri debet quàm sit figura; sed potiùs, cùm alijs 45 Cf. a persuasive discussion to the same effect in Miller, “Descartes, Mathematics, and God,” 456–57. However, neither Miller nor any other commentator I know of addresses the complication with which I deal in the rest of the present paragraph. 46 See in particular Reg. XII, AT X, 419: “[P]urè materiales [res respectu nostri intellectûs simplices] ... sunt figura, extensio, motus, &c.” Cf. also the opening words of the following quotation.

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etiam rebus tribuatur, vt extremitati durationis vel motûs &c., quæ res à figurâ toto genere differunt, ab his etiam debuit abstrahi, ac proinde est quid compositum ex pluribus naturis planè diversis, & quibus non nisi æquivocè applicatur (Reg. XII, AT X, 418–19).

As I read it, Descartes‫ ތ‬proper point here is exactly to warn against the feeling that just any complex description of a term implies a division of that term in the sense relevant to his criterion of simplicity regarding the res quantùm ab intellectu percipiuntur. This feeling is misleading— Descartes argues—since some components of a given complex description may in fact count not as simpler but rather as more complex than the term described in view of the criterion at issue;47 and he indicates that this is the case at least whenever—as with terminus—the spuriously simpler component is the product of an act of abstraction from the term described. In which sense does such an end-product of abstraction count as complex? In view of the latter half of the last quoted passage, I submit Descartes‫ ތ‬meaning is that in so far as the end-product of a given abstraction (e.g. terminus) is considered simpliciter, it remains—to employ Descartes‫ ތ‬later technical terminology—non-distinct (more exactly, nondistinctly perceived) in the sense specified in ch. 1, this also being how I interpret Descartes‫ ތ‬charge of equivocity in the above passage. The only way to render it distinct is to determine the ontological domain within which it is to apply, e.g. terminus durationis, terminus motûs etc. However, such a determination renders it ipso facto complex; and the chief point is that the content of this resultant complex does not amount to the content of the (ex hypothesi simple) term from which the abstract in question was derived in the first place. Thus, in Descartes, as far as necessary connections between simple natures are concerned, to show that some term can be isolated from the content of a given term T is not 47

It is important not to confuse the present suggestion with the case of switching between the orders ad cognitionem nostram and à parte rei. Such a switch may render one and the same item once as simple and then as composite according to Descartes: “[S]i, ver. gr., consideremus aliquod corpus extensum & figuratum, fatebimur quidem illud, à parte rei, esse quid vnum & simplex: neque enim, hoc sensu, compositum dici posset ex naturâ corporis, extensione, & figura, quoniam hæ partes nunquam vnæ ab alijs distinctæ exstiterunt; respectu verò intellectûs nostri, compositum quid ex illis tribus naturis appellamus, quia priùs singulas separatim intelleximus, quàm potuimus judicare illas tres in vno & eodem subjecto simul inveniri” (Reg. XII, AT X, 418). What we are considering now, however, is the situation in which a component of a complex description of a term is more complex than the term described, without leaving the order respectu intellectûs nostri. This order counts as default throughout the current considerations.

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enough to entitle us to take T as complex. More is needed for such a conclusion—to wit, that the isolated term must be capable of being perceived distinctly as it stands. In this way, then, the apparent counterexamples are eliminated and the above highlighted implication from the indivisibility of simple natures to the non-analytic nature of their connections remains secure. On the other hand, however, proceeding to the other strain in constructing the aforementioned aporia, Descartes seems to imply, in a difficult and oft-quoted passage, that even simple natures, insofar as their connection is necessary, are related in some analytic manner:48 Necessaria [conjunctio rerum simplicium inter se] est, cùm vna in alterius conceptu confusâ quâdam ratione ita implicatur, vt non possimus alterutram distinctè concipere, si ab invicem sejunctas esse judicemus: hoc pacto figura extensioni conjuncta est, motus durationi, sive tempori, &c., quia nec figuram omni extensione carentem, nec motum omni duratione, concipere licet. Ita etiam si dico, quatuor & tria sunt septem, hæc compositio necessaria est; neque enim septenarium distinctè concipimus, nisi in illo ternarium & quaternarium confusâ quâdam ratione includamus. Atque eodem modo, quidquid circa figuras vel numeros, demonstratur, necessariò continuum est cum eo de quo affirmatur. Neque tantùm in sensibilibus hæc necessitas reperitur, sed etiam, ex. gr., si Socrates dicit se dubitare de omnibus, hinc necessariò sequitur: ergo hoc saltem intelligit, quòd dubitat; item, ergo cognoscit aliquid posse esse verum vel falsum, &c., ista enim naturæ dubitationis necessario annexa sunt (Reg. XII, AT X, 421).

Two points in this passage seem to support the disconcerting implication now in question: (1) Descartes says that even simple things are connected necessarily if (presumably the concept of) one “in alterius [rei] conceptu confusâ quâdam ratione implicatur”, which suggests a conceptual inclusion of the type essential to analytic connections; and (2) at least one of the examples (or perhaps illustrations) Descartes adduces to clarify his conception, namely the one concerning numbers, might seem to be an instance of materially analytic inclusion, a reading which appears to gain support from Descartes‫ ތ‬statement that three and four is, once again,

48

Princ. I, 14, AT VIII-1, 10 might prima facie be read to the same effect: “[U]t [mens] ex eo quòd, exempli causâ, percipiat in ideâ trianguli necessariò contineri, tres ejus angulos æquales esse duobus rectis, planè sibi persuadet triangulum tres angulos habere æquales duobus rectis: ita ex eo solo quòd percipiat existentiam necessariam & æternam in entis summè perfecti ideâ contineri, planè concludere debet ens summè perfectum existere.”

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included in seven.49 How to resolve this aporia? I submit that what is to stand firm is the former claim to the effect that simple natures cannot ever be connected analytically since the argument in its support to me ultimately sounds unassailable; and that, consequently, what is to be explained away are the two above points that seem to support the contrary claim.50 Let us begin with (1). My attempt at getting to grips with it draws upon the distinction between implicit and explicit apprehension which was established in the above treatment of innate ideas. When Descartes first introduces the relation of being implied confusâ quâdam ratione in AT X, 421, his only general explication reads “vt non possimus alterutram distinctè concipere, si ab invicem sejunctas esse judicemus” (ibid.; my emphases). The act of judgment surely implies explicit apprehension of what the judgment is about. Moreover, we already know that if apprehended at all, each simple nature cannot but be apprehended distinctè. In view of this, the explication at issue seems to indicate that as long as one explicitly apprehends two simple natures of the right type, one cannot (where the “cannot” is arguably psychological) but judge them to be conjunctas, lest one mistakes one or both of them for something else— and if this is so it sounds natural, I submit, to interpret the confusa implicandi ratio in question not as referring specifically to situations in 49

I put to one side the other case, i.e. the dictum of Socrates and its implications. The case is clearly enthymematic as Descartes puts it and it is therefore particularly hard to assess its rôle in Descartes‫ ތ‬reasoning. Be this as it may, the case can be handled in a way similar to the way I am about to recommend treating the case of numbers. 50 Leslie Beck takes a very different course in his instructive interpretation of the nature of necessary connections between simple natures. He claims that “necessary connexion, in Descartes‫ ތ‬use of the term, is akin to logical implication in the sense that one concept presupposes another” (Beck, Method of Descartes, 94), and that “Descartes must ... be understood to be arguing that the simple natures, known per se [Beck’s italics], such as movement, duration, and figure, are the elements or implicates of wider conceptual wholes [my emphasis] which are clear and distinct and constitute the res per se notæ [Beck’s italics]” (ibid., 95–96), while he makes it clear that the limited cases of those “wider conceptual wholes” are cogitatio and extensio which count as the “two really simple natures” (ibid., 97). As far as I can see, such a conception involves a commitment to the notion that some simple natures can after all be analytically connected with certain other simple natures. However, such a commitment runs counter, in the final analysis, to Descartes‫ތ‬ clearly declared claim that simple natures are indivisible in the above indicated relevant sense. By contraposition, the entire interpretation of necessary connections proposed by Beck must be rejected.

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which what remains implicit is the relation of conceptual inclusion but more generally to situations in which one or both of two simple natures are not explicitly apprehended by the given mind. In other words, the ratio confusa refers to the following counterfactual conditional insofar as its antecedent is false:51 if m apprehended items A and B explicitly and distinctly, then m could not judge that A and B were sejunctas. Descartes surely had no Berkeleyan scruples concerning abstract ideas, so it is certainly possible, from his perspective, to apprehend explicitly just extension (or else just a particular shape) without considering explicitly any particular shape (or else extension). Yet what Descartes claims, according to my reading, is that once both extension and (a particular) shape are apprehended explicitly and simul and distinctly, one cannot but judge them to be conjunctas; and there is no indication whatever that the judgment be based on inspection of the meanings of extension and (a particular) shape. In view of the submitted interpretation, (2) can be dealt with as follows. The first thing to notice is that for Descartes, particular numbers obviously are not simple natures so that the case at issue is to be taken at best as an illustration by way of analogy. For our purposes, we do not need to pass judgment upon the tricky question of whether Descartes means to take the number example as an instance of an analytic or non-analytic proposition. For if he means it to be non-analytic, then this very fact provides strong support for the way in which I was treating (1): Descartes‫ތ‬ point is then not that three and four are analytically included in seven but that if one explicitly apprehends both seven and three and four, one cannot but judge both these items to be conjunctas. On the other hand, if Descartes means the relation in question to be analytic, then I wish to insist that in Descartes‫ ތ‬eyes analytic connections have something in common with non-analytic connections between simple natures, something significant enough for him to adduce the number example to highlight that common ground. The common ground I have in mind is the very same fact that is expressed with the conditional I have just formulated. For it will be observed that the conditional holds even for items that are related analytically; and Descartes‫ ތ‬point is, I submit, just that the analytical relation is not the only case by which the conditional at issue can ever be satisfied. To sum up, two important claims concerning the necessary connections between simple natures quâ objects of intuitus in Descartes have been 51

‫ދ‬m‫ ތ‬refers to a human mind. Descartes evokes this situation vividly in Resp. 1, AT VII, 117–19.

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established. First, the synchronic, necessary connections between simple natures are not to be conceived as produced by the human mind but as encountered by that mind as parts of mind-independent reality; and second, the necessary connections between simple natures are to be conceived, in spite of some prima facie evidence to the contrary, as strictly non-analytic in nature. To complete the picture canvassed thus far, Descartes characterizes the contingent connections between simple natures which are objects of intuitus simply as “quæ nullâ inseparabili ratione conjunguntur” (Reg. XII, AT X, 421), i.e. simply as non-necessary connections with reference to the treatment of those necessary connections just discussed.52

2.4 The Root of Objective Necessity, and Scientia The discussion of the nature of necessary connections between simple natures remains incomplete in at least two important respects: both (1) the way in which the necessary connections are intuited by the human mind and (2) the root of the necessity of arguably non-analytic connections between simple natures have yet to be confronted. I begin with (1), which shall bring us to (2). We saw that, according to Descartes, it is not that the human mind somehow supplements simple natures with either necessary or contingent relations but that the mind discovers those relations as holding between simple natures independently of its own productive acts. It is therefore appropriate to ask what exactly it is that the mind discovers à parte rei which makes it assent rightly to the perception of a necessary connection between the simple natures it intuits. Unfortunately, Descartes never gives any distinctive answer over and above the bare statement that the human mind does sometimes, somehow discover the necessary relations in 52 Descartes also remarks that “plurimarum propositionum, quæ necessariæ sunt, conversas esse contingentes: vt quamvis ex eo quòd sim, certò concludam Deum esse, non tamen ex eo quòd Deus sit, me etiam existere licet affirmare” (Reg. XII, AT X, 422). As I understand it, this suggests, in the present context, that it would be a mistake to conceive the connections between simple natures quâ objects of intuitus—both necessary and contingent—as a matter of bare juxtaposition of the given simple natures. Rather, Descartes’ example seems to suggest that the connections in question are to be conceived as one-way conditionals expressing hypothetical necessities (the case of necessary connections) or else hypothetical non-necessities (the case of contingent connections). However, since this complication, important as it is, is too fine-grained to affect the subsequent discussion in the present study, I set it to one side.

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question, employing visual metaphors which remain unhelpful.53 This seems to be a matter of principle rather than of casual neglect, for it seems virtually impossible to address that question in view of Descartes‫ތ‬ commitments concerning the way in which simple natures are cognized by the human mind. There seem to be just two ways in which simple natures left to themselves could ground cognitions of the relations in question: either insofar as they are taken per se, or else by means of something inherent to them. However, the former option is a non-starter since simple natures are clearly conceived by Descartes as cognized completely apart from any relations into which they might ever enter;54 and the latter option is also to be rejected since we saw Descartes insisting that simple natures are cognized in such a way “vt in plures magis distinctè cognitas mente dividi non possint” (Reg. XII, AT X, 418). However, the problem just highlighted affects not only the issue of the mind‫ތ‬s cognition of necessary connections between simple natures but even the ontological issue of the very foundations à parte rei of the connections in question. For Descartes cannot make use of the suggestion that simple natures left to themselves do, after all, ground the necessary connections that hold between them despite the fact that the human mind has no cognitive access to the manner in which they do so. Such a move would completely undermine his fundamental project of founding scientia upon cognitions which are metaphysically certain without recourse to anything else, even to a benevolent God. It is, of course, precisely the cognitions we are dealing with now, viz. the deliverances of intuitus, that are determined by Descartes to play such a founding rôle—and it is but an implication of this that the deliverances of intuitus are cognized by the human mind precisely as they are in themselves: otherwise the mind would be driven to assent to something that is not perceived clearly and distinctly by it, the Vicious Demon would prevail and Descartes‫ތ‬ meliorative epistemological project would indeed become stuck in a 53

Leonard Miller comments aptly: “[A]dopting an analogy with vision, [Descartes] simply reiterates again and again that the necessity is revealed by the ‫ދ‬eyes of the mind,‫ދ ތ‬mental vision,‫ދ ތ‬spiritual illumination,‫ ތ‬or the ‫ދ‬natural light.‫ ތ‬It is quite clear that the natural light is cast upon the simples, thus revealing the necessary relations between them, but the exact manner in which reason inspects its objects and the exact nature of the relations it observes remain obscure. ... For Descartes the lumen naturale reveals necessary connections but does not illuminate the source or nature of this necessity” (“Descartes, Mathematics, and God,” 458– 59; Miller’s italics). 54 Cf. the above-quoted AT III, 665–66: “... estant primitiues, chacune d‫ތ‬elles ne peut estre entenduë que par elle mesme.”

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vicious circle. As a consequence, in the special case of the deliverances of intuitus, it holds that if the human mind is not able to discern anything about them that could determine their relations, this is because there is simply nothing about them by which their relations could ever be determined. Thus we see Descartes confronting a real problem here. On the one hand, he can hardly renounce his claim that the human mind intuits some simple natures as connected in certain significant ways since this is clearly an essential component of his project of establishing scientia. On the other hand, his own theory of simple natures seems to leave him with no prospect of accounting for those significant connections. The nature of the connections remains mysterious. Descartes‫ ތ‬effort to escape this unsatisfactory situation seems to form a significant part of the motives for his famous belief that eternal truths have been created by God, in the sense that even those truths that count as necessary for the human mind entirely depend on the will of God as to their truth value.55 It would be preposterous to deny that Descartes‫ ތ‬chief 55

The most informative statement of the doctrine can be found in Mers., AT I, 149–50: “Pour les veritez eternelles, ie dis derechef que sunt tantum veræ aut possibiles, quia Deus illas veras aut possibiles cognoscit, non autem contra veras à Deo cognosci quasi independenter ab illo sint veræ. Et si les hommes entendoient bien le sens de leurs paroles, ils ne pourroient iamais dire sans blaspheme, que la verité de quelque chose precede la connoissance que Dieu en a, car en Dieu ce n‫ތ‬est qu‫ތ‬vn de vouloir & de connoistre; de sorte que ex hoc ipso quod aliquid velit, ideo cognoscit, & ideo tantum talis res est vera” (Descartes’ italics). For other explicit claims to this effect see in particular Mers., AT I, 145– 46; 151–53; AT II, 138; Pour Arnauld, AT V, 223–24; Resp. 5, AT VII, 380; Resp. 6, AT VII, 431–32; 435–36. It is commonly agreed that Descartes’ doctrine at issue is extremely difficult and controversial. A vast amount of secondary literature presenting various attempts at interpreting the doctrine has been produced in recent decades, and I cannot and do not pretend that I even come close to an exhaustive treatment of the topic. In fact, I will ignore almost entirely the aspect of the doctrine at issue which has enjoyed prominent attention from modern Descartes scholars, viz. the implications of the doctrine for the metaphysics of modality. I will focus almost exclusively on an aspect which seems most relevant to the chief topic of the present book, viz. the alleged devastating implications of the doctrine at issue for Descartes’ overall meliorative epistemological project. The most important contributions to the doctrine at issue which focus imprimis on the aforementioned general metaphysical issues include Harry Frankfurt, “Descartes on the Creation of the Eternal Truths,” The Philosophical Review 86, no. 1 (1977), 36–57; Alvin Plantinga, Does God have a Nature? (Milwaukee: Marquette University Press, 1980); Jacques Bouveresse, “La théorie du possible chez Descartes,” Revue International de Philosophie 146, no. 3 (1983), 293–310; Edwin

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and most explicitly articulated reasons for the doctrine are theological, being drawn from the consideration that the denial of the doctrine is in conflict with God‫ތ‬s supreme power.56 Yet putting these theological motives to one side,57 the issue of positive grounds of necessary connections between simple natures seems to point in the same direction. For if those connections cannot be determined by anything about simple natures left to themselves, it must be admitted that they are wholly determined by something external to them. Once combined with the supposition that finite minds do not invent but discover the connections in question, this consideration seems to lead to the suggestion that, at the end of the day, it cannot be but God himself who is fully responsible for the connections holding as they actually do, in the sense that God counts as their “efficiens & totalis causa” (Mers., AT I, 152).58 Given (ex hypothesi) that there is indeed nothing inherent to simple natures that could preclude a supremely powerful being from connecting them otherwise than they actually are connected, it finally follows that God has determined the

Curley, “Descartes on the Creation of the Eternal Truths,” The Philosophical Review 93, no. 4 (1984), 569–97; Hidé Ishiguro, “The Status of Necessity and Impossibility in Descartes,” in Rorty, Essays on Descartes’ Meditations, 459–72; Jonathan Bennett, “Descartes’s Theory of Modality,” Philosophical Review 103 (1994), 639–67; James Van Cleve, “Descartes and the Destruction of the Eternal Truths,” Ratio 7, no. 1 (1994), 58–62; Dan Kaufman, “Descartes’s Creation Doctrine and Modality,” Australasian Journal of Philosophy 80, no. 1 (2002), 24– 41; Lilli Alanen, “Omnipotence, Modality, and Conceivability,” in Broughton and Carriero, Companion to Descartes, 353–71. 56 Cf. in particular A Mesland, AT IV, 118: “Pour la difficulté de conceuoir, comment il a esté libre & indifferent à Dieu de faire qu‫ތ‬il ne fust pas vray, que les trois angles d‫ތ‬vn triangle fussent égaux à deux droits, ou generalement que les contradictoires ne peuuent estre ensemble, on la peut aisement oster, en considerant que la puissance de Dieu ne peut auoir aucunes bornes ....” Cf. also Mers., AT I, 145–46; 149–50; Resp. 6, AT VII, 431–32; 435–36. 57 The theological dimension of the doctrine at issue is discussed extensively and perspicuously in Jean-Luc Marion, Sur la théologie blanche de Descartes (Paris: Presses universitaires de France, 1981). See also a brief summary in Gaukroger, Cartesian Logic, 60–62. 58 The entire sentence reads: “Vous me demandez in quo genere causæ Deus disposuit æternas veritates. Ie vous répons que c‫ތ‬est in eodem genere causæ qu‫ތ‬il a creé toutes choses, c‫ތ‬est à dire vt efficiens & totalis causa” (ibid., 151–52; Descartes’ italics). Cf. also ibid., 149: “[V]eritez eternelles ... sunt tantum veræ aut possibiles, quia Deus illas veras aut possibiles cognoscit, non autem contra veras à Deo cognosci quasi independenter ab illo sint veræ” (Descartes’ italics).

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connections simply by a contingent fiat.59 While Descartes might thus seem to avoid the disconcerting conclusion that nothing can in principle account for the necessary connections between simple natures given his premises, there is a worry, putting to one side fundamental concerns regarding Descartes’ commitments in the field of the general metaphysics of modalities, that the rather extreme way in which he achieves this in fact threatens to subvert or at least to jeopardize his entire meliorative epistemological project. It is worth taking a closer look at this charge as it will sharpen those aspects of the doctrine of the creation of eternal truths that are immediately relevant to our next issue, viz. Descartes‫ ތ‬conception of deduction. To begin, it is well to take stock on the situation as it has crystallized thus far. The difference between the divine and the finite mind‫ތ‬s epistemic situation concerning necessity (and a fortiori necessary connections) seems to boil down to the fact that the criterion for any competent and attentive mind m and for any proposition p, if it is unintelligible for m that ™p, then p is necessary yields dramatically different results for the divine mind and finite minds with regard to the class of necessary p‫ތ‬s. While the class of necessities is likely to be quite numerous as far as finite minds are concerned, it is bound to remain—putting to one side the peculiar case of the proposition that God exists60—strictly empty with respect to the divine mind; for given Descartes‫ ތ‬doctrine under scrutiny, it is intelligible for the divine mind that ™p for any p whatever.61 In other words, fundamentally all truths are 59

Cf. ibid., 152: “[I]e dis que [Dieu] a esté aussi libre de faire qu‫ތ‬il ne fust pas vray que toutes les lignes tirées du centre à la circonference fussent égales, comme de ne pas creer le monde.” This statement amounts to saying that, as Miller aptly puts it in his “Descartes, Mathematics, and God,”, 462, fn. 27, “the notions ‫ދ‬circle‫ތ‬, ‫ދ‬radius‫ތ‬, and ‫ދ‬equal‫ ތ‬could have been related so that the sentence ‫ދ‬The radii of a circle are not all equal‫ ތ‬is true ....” 60 The existence of God is clearly a presupposition for creation tout court and therefore a fortiori for putting to work the entire machinery of Descartes’ theory in question. This is why given Descartes’ commitments, the proposition that God exists is best not treated on a par with the rest of (what are normally taken as) necessary propositions, and perhaps it is not to be characterized as necessary on pain of equivocation. For the sake of brevity, from now on I shall ignore this complication. 61 This seems to be the thrust of Descartes‫ ތ‬intimation in Mers., AT I, 146 that “generalement nous pouuons bien assurer que Dieu peut faire tout ce que nous

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contingent from the divine point of view. This seems to imply, somewhat embarrassingly, complete elimination of necessity from the domain of immediate objects of the divine cognition. Descartes’ hints to the effect that God has decreed that He shall not change His will regarding connections which count as necessary for finite minds, and that such necessary connections hold sempiternally as a consequence,62 are also insufficient because sempiternality can in principle be instantiated in the domain of (what counts for finite minds as) contingent truths. Indeed, under the present circumstances it seems to me that the best one can do to establish necessity as a sharp-edged category in the divine cognition is to construct it no stronger than as an extrinsic denomination. “It is necessary that p” would then be true with respect to the divine mind precisely for those (absolutely contingent) p‫ތ‬s which are rightly taken as necessary (and not just as sempiternal) by the competent finite minds. Be this as it may, I presume it is reflections on the situation constructed along the suggested lines that have arisen more than once in attempts to show that Descartes‫ ތ‬doctrine of the creation of eternal truths brings down, in one way or another, his whole epistemological project. Let us have a brief look at two such attempts which I consider as representative of the worries that are likely to arise for anyone confronted with the doctrine at issue. The stronger, quite straightforward suggestion is that this doctrine eventually implies that God cannot but be a deceiver. Leonard Miller puts the charge as follows: The crucial difference between our understandings is not that I cannot think of an alternative while He can, but rather that I see it is senseless to think of pouuons comprendre, mais non pas qu‫ތ‬il ne peust faire ce que nous ne pouuons pas comprendre; car ce seroit temerité de penser que nostre imagination a autant d‫ތ‬estendue que sa puissance.” Cf. also A Mesland, AT IV, 118: “Dieu ne peut auoir esté determiné à faire qu’il fust vray, que les contradictoires ne peuuent estre ensemble, & que, par consequent, il a pu faire le contraire ....” 62 See Mers., AT I, 145–46: “On vous dira que si Dieu auoit establi ces verités, il les pourroit changer comme vn Roy fait ses lois; a quoy il faut respondre qu‫ތ‬ouy, si sa volonté peut changer.—Mais ie les comprens comme eternelles & immuables.— Et moy ie iuge le mesme de Dieu.—Mais sa volonté est libre.—Ouy, mais sa puissance est incomprehensible ....” Cf. also Resp. 5, AT VII, 380: “[Q]uemadmodum Poëtæ fingunt a Iove quidem fata fuisse condita, sed postquàm condita fuere, ipsum se iis servandis obstrinxisse; ita ego non puto essentias rerum, mathematicasque illas veritates quæ de ipsis cognosci possunt, esse independentes a Deo; sed puto nihilominus, quia Deus sic voluit, quia sic disposuit, ipsas esse immutabiles & æternas.”

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there being one, while He does not. When I recognize that a proposition is necessarily true, I also recognize that it would be a mistake to suppose it could be false. Consequently, if I were able to admit the possibility of an alternative, the very nature of my belief would change. Since this is not so for God, we are not aware of the same thing when we each apprehend what is supposed to be the same truth. Descartes cannot escape the conclusion that God has put exceedingly misleading ideas in my mind. ... As a result, the metaphysical guarantee he has developed really destroys the significance of the criterion it is supposed to support. Descartes believes that reason is able to reveal the basic nature and structure of the universe and that it can do this by apprehending the manner in which simples are and are not necessarily related to each other. After the guarantee of the criterion of clear and distinct ideas has been elaborated, however, it turns out that the relations apprehended by reason are but misleading representatives of the true relations whose basic nature must remain a mystery to us (“Descartes, Mathematics, and God,” 463–64).

As I see it, this line of attack is profoundly mistaken. Given Descartes‫ތ‬ conceptions of judgment, compelled assent and the rôle of will in all of this, God would turn out to be a deceiver if and only if to what any finite mind is compelled to assent (i.e. what is perceived clearly and distinctly by a given mind) would count as false from God‫ތ‬s perspective. Now for any apprehended p which is necessary from our perspective, it is one thing (i) to assent to p and quite another thing (ii) to assent to the apprehension that p is necessary. In the former case (i), Descartes is surely committed to holding that the assent to p is compelled. However, no deception on the divine part is implied by the doctrine of the creation of eternal truths in this case since the doctrine in question does not imply that the p in question is false from His perspective; it only implies that if p is true to Him, p is only contingently so. The competent finite minds are metaphysically certain that p, they are compelled to assent to p, and there is nothing in the doctrine of eternal truths that precludes it being indeed true that p from God’s perspective since He has decreed that p as a matter of fact. Matters are very different in Descartes as regards the latter case, (ii). The fact that God is able to change the truth value of p at will implies that it is indeed false from His perspective that it is necessary that p. However, this is not enough, of course, to accuse Him of deception in the specific context now at issue, for as I have already pointed out, He would turn out to be a deceiver if and only if he had created us in such a way that we were compelled to assent that it is necessary that p under the present circumstances. Yet I cannot see that Descartes commits himself to anything like this. On the contrary, the fact that we are able to conceive—

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however nebulously63—the possibility that p, whilst being necessary to us is contingent from God’s perspective, implies, given Descartes‫ ތ‬premises, that we possess no metaphysical certainty (in the sense worked out in ch. 1) that it is necessary that p. As a consequence, Descartes commits himself to the claim that the right thing to do is to suspend judgment with regard to the apprehension that it is necessary that p and I cannot see anything in his doctrine that would eventually render such advice impossible or incoherent. In yet other words, for any p that is necessary to us, it is one thing to assess the perception that p and quite another thing to assess the perception that it is necessary that p. While the former perception comes out as clear and distinct, the latter does not and even cannot come out clearly and distinctly given Descartes‫ ތ‬premises—and I cannot find anything in Descartes‫ ތ‬texts or in his doctrine that would imply the impossibility of suspending judgment on the latter perception.64 Stephen Gaukroger has come up with a weaker and prima facie more plausible suggestion. According to him, it is not that the doctrine of the creation of eternal truths renders God a deceiver but rather that it eventually “undermines any intelligible connection between God and us” (Gaukroger, Cartesian Logic, 68) and thus a fortiori “poses an immense problem for the idea that our knowledge could have a divine guarantee” (ibid.). Given that divine guarantee is, strictly speaking, not required as long as the deliverances of intuitus are concerned (as I argued above), Gaukroger‫ތ‬s line of attack might seem irrelevant to our present issue, viz. 63

Cf. Mers., AT I, 152: “[I]e sçay que Dieu est Autheur de toutes choses, & que ces veritez sont quelque chose, & par consequent qu‫ތ‬il en est Autheur. Ie dis que ie le sçay, & non pas que ie le conçoy ny que ie le comprens; car on peut sçauoir que Dieu est infiny & tout-puissant, encore que nostre ame estant finie ne le puisse comprendre ny conceuoir; de mesme que nous pouuons bien toucher auec les mains vne montagne, mais non pas l‫ތ‬embrasser comme nous ferions vn arbre, ou quelqu‫ތ‬autre chose que ce soit, qui n‫ތ‬excedast point la grandeur de nos bras: car comprendre, c‫ތ‬est embrasser de la pensée; mais pour sçauoir vne chose, il suffit de la toucher de la pensée ....” 64 Miller might appear to consider, and to reject, something similar: “God would be absolved [from the charge of deception] if Descartes could suspend judgment about the nature of the necessity, but since he cannot conceive the [necessary] propositions to be false, he cannot do this” (“Descartes, Mathematics, and God,” 464). However, in fact he just misses the point. Of course Descartes cannot conceive the necessary propositions to be false and this is why he must assent to them, without, however, being deceived on this score; but he presumably can conceive the proposition that a proposition is necessary to be false, and in such a case he just can and should suspend judgment. The nature of necessity is not in question here at all.

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necessary connections between simple natures in so far as they are objects of intuitus. However, since even rightly performed deductio has to do with necessary connections between the deliverances of intuitus,65 it will be convenient to discuss the challenge here. The chief reason Gaukroger puts to use is that the situation described above brings about, in the final analysis, an equivocation on the very notion of truth which turns out fatal for the intelligibility of the notion of the divine guarantee. For, Gaukroger argues, [a] cognitively omnipotent God might well be able to divide sentences into those that we would regard as true and those that we would regard as false, but they might as well be designated ‫ދ‬T‫ ތ‬and ‫ދ‬F‫ތ‬, or ‫ދ‬1‫ ތ‬and ‫ދ‬0‫ތ‬, unless he possessed an independent understanding of truth, an understanding which took the form of a grasp of the point of the exercise. For a God who created truths by fiat ... such an independent understanding would be wholly irrelevant, and it is the very irrelevance of such an understanding that shows that it is not truth, in the sense in which we understand it, that, as far as the cognitively omnipotent God is concerned, he is creating. ... God would [therefore] be being asked to guarantee something that would surely make as little sense to him as truth-for-God does to us (Cartesian Logic, 68).

I readily admit that from the divine perspective, the designations of the propositionally structured contents as necessary and non-necessary amount just to extrinsic denominations, in a way analogical to that in which Gaukroger wants the designations true and false to count as extrinsic denominations from the divine point of view. However, I deny that the analogy just specified can actually be drawn in a way that could support Gaukroger‫ތ‬s challenge in question; for the radical equivocation on the term “truth” which Gaukroger puts to use holds precisely for the special case of propositions which ascribe a modal status to given contents: here indeed we cannot comprehend at all how what we cognize as necessary could hold as merely contingent for God, so that the gulf between the divine and our understanding of truth is insurmountable. However, it has already been established that there is nothing that needs to be guaranteed in this area: as long as we can “toucher de la pensée” (Mers., AT I, 152) that God is cognitively omnipotent (as Gaukroger aptly puts it), the 65 Cf. Reg. III, AT X, 369: “[P]er [deductionem] intelligimus, illud omne quod ex quibusdam alijs certò cognitis necessariò concluditur.” Reg. XII, AT X, 424–25: “Errorem [in deductione performanda] vitare in nostrâ potestate situm est, si nulla vnquam inter se conjungamus, nisi vnius cum altero conjunctionem omnino necessariam esse intueamur ....”

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perception that it is necessary that p is rendered metaphysically uncertain and the right doxastic attitude is that of suspending judgment, in which case no divine guarantee is needed. On the other hand, God‫ތ‬s cognitive omnipotence by no means precludes that the p‫ތ‬s which competent finite minds judge to be necessarily true count as true for God as well. To be sure, there remains an important difference in what true means for God and for us, respectively, even in this area. The difference concerns, so to speak, only the method of verification of what counts as true, viz. employing the criterion of clarity and distinctness in the case of finite minds vs. bare fiat in the case of God; and it therefore by no means jeopardizes, as far as I can see, the claim that if any finite mind m (willynilly) assents to any content p that is perceived clearly and distinctly, then p is true even from God‫ތ‬s perspective, independently of whether p is cognized as necessary or not by m and despite the established fact that “necessary” counts as an extrinsic denomination for God. As long as this claim can be sustained, however, the intelligibility of the divine guarantee in the cases to which it is relevant can also be sustained. It should be clear how much is at stake in assessing the present criticisms and the defence I have offered. I have argued that Descartes was driven to his doctrine of the creation of eternal truths, among other things, by the need to account for the root of something that is indispensable if his project of achieving scientia is to gain any significance: namely the root of the necessity of connections between simple natures. As a consequence, if the doctrine at issue really were to imply either that God were a deceiver or that the divine guarantee of deductive connections were unintelligible, the implication would amount to a fatal stroke at the very heart of Descartes‫ ތ‬epistemological project. This is why I have struggled so much to vindicate Descartes on this score. Yet be this as it may, the very fact that necessary connections between simple natures, being vital for endowing the project of building the body of human scientia with any significance, are rooted in the voluntaristic doctrine of the creation of eternal truths and consequently marked with the complications discussed above, brings about (among other things) the conclusion, rightly highlighted by Gaukroger,66 that the operations constitutive of the divine cognition can no longer serve as a paradigm for the human cognitive operations in any way, including, of course, the privileged case of the human scientia. The import of this conclusion for an adequate assessment of Descartes‫ ތ‬conceptions of inference, method and scientific cognition can hardly be overestimated. Hopefully I will be able to 66

Cf. in particular Gaukroger (1989), 68–70.

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show that it is this groundbreaking move from which the most original and audacious aspects of those conceptions of Descartes are drawn, however inconspicuous the connections might be at first sight. Descartes‫ ތ‬conception of deductio, to which I now turn, should provide, in the long run, an instructive specimen of the structures amenable to such a treatment.

2.5 Deductio To approach the complex topic of deductio in Descartes, let me begin with the observation that the term “deductio” denotes in Descartes, in its arguably core meaning67 and for the moment regardless of various functions with which the denotate could be assigned in various dialectical or pragmatic contexts,68 either the process or the end-product of inference (illatio),69 in the sense no more specific than the duration-implying passage from or succession of one content to/after another, with the crucial conditions that the passage preserve metaphysical certainty on the part of a given mind concerning the cognition reached at a given time, and that the connection between the premis(es) and the conclusion is necessary.70 This result is yielded by a gathering together of the following passages from the Regulæ along with the already established fact that deductio and intuitus 67 At least since Desmond Clarke‫ތ‬s Descartes’ Philosophy of Science, it has been a well-established fact that “deducere”, and even more its French equivalent “déduire,” is used by Descartes in several meanings concerning the logical character of the denotate, including most remarkably deduction, induction and enumeration. Indeed, Descartes occasionally employs “inductio”, “enumeratio”, along with “conclusio” and “consequentia” interchangeably with what he usually refers to as “deductio”. Cf. Clarke, Descartes’ Philosophy of Science, 63–70 and 207–10 for a standard, detailed treatment of the issue and for some references. 68 Clarke, ibid. has shown persuasively that in Descartes‫ ތ‬hands “deducere” and “déduire,” or inferences denoted with these terms, can play the rôles of demonstration, proof, explanation or justification, depending on context. 69 Cf. Reg. II, AT X, 365: “deductionem ..., sive illationem puram vnius ab altero ....” And as Clarke notes (Descartes’ Philosophy of Science, 67), Descartes has “inferendum” when referring in Reg. XIII, AT X, 431 to what he denoted with “deducere” in Reg. XII, AT X, 427. 70 Roger Florka, Descartes’s Metaphysical Reasoning, 16–34 offers a perspicuous discussion of the general intricacies of the very concept of inference, which both reflects recent achievements in the field and is sensitive to the peculiarities of Descartes‫ ތ‬thought. I shall not venture to take up the issue in the present study. I just hope that the conception that will emerge from my treatment shall imply, among other things, Descartes‫ ތ‬response to those extremely general questions concerning the nature of inference tout court.

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are the only two ways of gaining scientia: [P]er [deductionem] intelligimus, illud omne quod ex quibusdam alijs certò cognitis necessariò concluditur (Reg. III, AT X, 369). Deductio verò, si de illâ faciendâ cogitemus, vt in regulâ tertiâ, non tota simul fieri videtur, sed motum quemdam ingenij nostri vnum ex alio inferentis involvit ... (Reg. XI, AT X, 407).71 Dicimus ... compositionem [rebus quas nos ipsi componimus] tribus modis fieri posse: nempe per impulsum, per conjecturam, vel per deductionem. ... [Ex his] [s]uperest igitur sola deductio, per quam res ita componere possimus, vt certi simus de illarum veritate ..., ... si nulla vnquam inter se conjugamus, nisi vnius cum altero conjunctionem omnino necessariam esse intueamur ... (Reg. XII, AT X, 423–25).72 71 Similarly Reg. III, AT X, 370: “Hic igitur mentis intuitum à deductione certâ ita distinguimus ex eo, quòd in hac motus sive successio quædam concipiatur, in illo non item ....” 72 As a matter of fact, a part of the passage just quoted with omissions reads “[s]uperest igitur sola deductio, per quam res ita componere possimus, vt certi simus de illarum veritate; in quâ tamen etiam plurimi defectus esse possunt: vt si, ex eo, quòd in hoc spatio aëris pleno nihil, nec visu, nec tactu, nec vllo alio sensu percipimus, concludamus illud esse inane, male conjungentes naturam vacui cum illâ hujus spatij; atque ita fit, quoties ex re particulari vel contingenti aliquid generale & necessarium deduci posse judicamus. Sed hunc errorem vitare in nostrâ potestate situm est, nempe, si nulla vnquam inter se conjungamus, nisi vnius cum altero conjunctionem omnino necessariam esse intueamur ...” (Reg. XII, AT X, 424–25; my emphasis). The passage has suggested to several readers that Descartes allows for faulty deductiones (the authors of the standard translation of the Regulæ in English are in this number; they translate the crucial “in quâ tamen etiam plurimi defectus esse possunt” with “yet even with deduction there can be many drawbacks” [Descartes The Philosophical Writings, 1:48]). This plays havoc with Descartes‫ ތ‬salient and self-confident claims in Reg. II, AT X, 365 that “experientias rerum sæpe esse fallaces, deductionem verò, sive illationem puram vnius ab altero, posse quidem omitti, si non videatur, sed nunquam malè fieri ab intellectu vel minimùm rationali” (my emphases). Cf. also ibid.: “Arithmetica & Geometria ... totæ consistunt in consequentijs rationabiliter deducendis. ... [I]gitur ... in illis citra inadvertentiam falli vix humanum videatur.” The only way to reconcile the militating passages consistent with the suggested reading is to propose that Descartes has in mind the cases of inadvertence in the quoted piece of Reg. XII. However, it seems to me that in such a case the compositio would not count as a deductio at all but rather as impulsus since it would satisfy the characterization “[p]er impulsum sua de rebus judicia componunt illi, qui ad aliquid credendum suo ingenio feruntur, nullâ ratione persuasi, sed tantùm

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One thing that emerges from this preliminary exposition is that one must resist the temptation to read into Descartes‫ ތ‬deductio the rather strong notion of a very specific type of inference, viz. inference which is valid either due to form alone or due to peculiar (to wit, analytic) relations between the concepts involved in the premises and the conclusion. Descartes‫ ތ‬handling of the term is clearly much looser and wider, and we shall soon see that his conception of deductio73 eventually precludes the stronger notion just mentioned from being a possible component of its meaning. Our present task is to explicate Descartes‫ ތ‬notion of deductio in some detail and to draw consequences which shall complete our understanding of his conception of the privileged type of cognition called scientia. We shall begin with focusing on two striking and interrelated aspects of his doctrine of deductio indicated in the last quoted passages: namely Descartes‫ ތ‬assimilation of the limiting cases of deductio to intuitus; and then considering what stands behind his resolute appeal to free one‫ތ‬s practice of reasoning from what he calls the vincula of the dialecticians.

2.5.1 Construing Deductio in Terms of Intuitus74 The last quoted passages make it clear, among other things, that from Descartes’ perspective deductio is in one respect precisely a matter of connexiones (or compositiones) necessarias. However, since necessary connections of simple natures also form one of the three classes of the objects of intuitus and since, as we saw, there are eventually no compositiones other than those made up of simple natures, one must deal with the question of the delimitation of those necessary connections that are the objects of intuitus and of those that fall within the scope of deductiones.

determinati ...” (Reg. XII, AT X, 424; my emphasis). I submit therefore that the “in quâ” within “in quâ tamen etiam plurimi defectus esse possunt” should be read as referring to compositio in general as treated in the relevant portion of Reg. XII (AT X, 422–25). While this is a somewhat less natural reading of Descartes‫ ތ‬Latin, it renders his texts consistent. The professed infallibility of deductio will be taken up in a moment. 73 I will keep using the Latin word whenever referring to Descartes‫ ތ‬concept in order to remind the reader that Descartes‫ ތ‬meaning, to be determined in crucial respects, differs dramatically from the generally accepted meaning today. 74 The title is taken from Gaukroger‫ތ‬s highly illuminating phrase in his Cartesian Logic, 51: “construing deduction in terms of intuition rather than rules of inference.”

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At first sight the question is easy to address. Descartes himself confronts it early in the Regulæ and seems to handle it briefly and satisfactorily: At verò hæc intuitus evidentia & certitudo, non ad solas enuntiationes, sed etiam ad quoslibet discursus requiritur. Nam, exempli gratiâ, sit hæc consequentia: 2 & 2 efficiunt idem quod 3 & 1; non modo intuendum est 2 & 2 efficere 4, & 3 & 1 efficere quoque 4, sed insuper ex his duabus propositionibus tertiam illam necessario concludi. Hinc jam dubium esse potest, quare, præter intuitum, hîc alium adjunximus cognoscendi modum, qui fit per deductionem: per quam intelligimus, illud omne quod ex quibusdam alijs certò cognitis necessario concluditur. Sed hoc ita faciendum fuit, quia plurimæ res certò sciuntur, quamvis non ipsæ sint evidentes, modò tantùm à veris cognitisque principijs deducantur per continuum & nullibi interruptum cogitationis motum singula perspicuè intuentis: non aliter quam longæ alicujus catenæ extremum annulum cum primo connecti cognoscimus, etiamsi vno eodemque oculorum intuitu non omnes intermedios, à quibus dependet illa connexio, contemplemur, modò illos perlustraverimus successivè, & singulos proximis à primo ad vltimum adhærere recordemur. Hîc igitur mentis intuitum à deductione certâ distinguimus ex eo, quòd in hac motus sive successio quædam concipiatur, in illo non item; & præterea, quia ad hanc non necessaria est præsens evidentia, qualis ad intuitum, sed potiùs à memoriâ suam certitudinem quodammodo mutuatur (Reg. III, AT X, 369–70; Descartes’ emphasis).

Thus, the sought-after difference seems to consist simply in that unlike in the case of intuitus, establishing necessary connections in the case of deductio essentially requires succession and thus duration on the part of the apprehending mind. Even if things were left at this stage, it is worth noting that the exposition Descartes has just presented goes no more than half-way towards his mature classification of cognitiones with regard to the project of attaining scientiæ. On the one hand, the present description of the situation belies the (most likely provisional) view implied in Reg. II, AT X, 362: Omnis scientia est cognitio certa & evidens .... Atque ... rejicimus illas omnes probabiles tantùm cognitiones, nec nisi perfectè cognitis, & de quibus dubitari non potest, statuimus esse credendum (AT X, 362; my emphases).

In Reg. III the complacent alliance of certainty and evidence is broken. Whilst it is clear that in so far as intuitus is concerned, (presumably

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metaphysical) certainty becomes a function of præsens evidentia, in the case of deductio (metaphysical) certainty is a function of memory and the evidential character of each individual memory of the successive moments of the motus ingenij in which deductio consists. Assimilating somewhat to Fig. 1-2 referred to above in ch. 1, the situation of Reg. III—and indeed of the entire extant Regulæ—can be captured in the following scheme: Fig. 2-1

As with Descartes‫ ތ‬mature conception, the credibility of the deliverances of memory is not in question. What makes the conception of the Regulæ crucially different from Descartes‫ ތ‬mature position is the fact that neither the evidential character nor (in the terms Descartes preferred later in his career) clarity and distinctness of the particular steps of the succession making up a given deductio as re-presented in the corresponding recollections is ever challenged in the Regulæ. In other words, the entire scientia per deductionem insofar as literally proposed in the Regulæ would count not as scientia but as persuasio tantùm of the mature scheme. This is because, of course, the need for divine guarantee in cases of certitudo mutuata is neither felt nor met in the whole of the extant Regulæ, which is again due to the fact that the institute of the Vicious Demon is absent throughout this early and suppressed fragment. Nonetheless, having thus compared the situation of the Regulæ with Descartes‫ ތ‬mature account of scientia per deductionem, I feel free to run the relevant portions of both the early and the mature texts in a single harness since I take the mature conception as just a more restrictive development of the earlier one, with no indications that the development should preclude the mutatis mutandis

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compatibility of both schemes.75 However, things become even more problematic than this. To begin with, Descartes makes it clear in the last quoted passage and elsewhere that however “borrowed” the certainty of a given deductio, it is essential to this certainty that strictly each of the successive steps of the deductio at issue be immediately evident to a given mind, i.e. count as a piece of scientia per intuitum,76 presumably of the compositiones necessariæ naturarum simplicium sort.77 However, Descartes is after something even more ambitious than this. Consider the following claims: [Q]uæcumque vna ex alijs immediate deduximus, si illatio fuerit evidens, illa ad verum intuitum jam sunt reducta (Reg. VII, AT X, 389). [S]implicem ... deductionem vnius rei ex alterâ diximus [in Reg. VII] fieri per intuitum. ... Si ... ad [deductionem], vt jam facta est, attendamus, sicut in dictis ad regulam septimam, tunc nullum motum ampliùs designat, sed terminum motus, atque ideo illam per intuitum videri supponimus, quando est simplex & perspicua ... (Reg. XI, AT X, 407–408). [D]ici posse illas quidem propositiones, quæ ex primis principijs immediatè concluduntur, sub diversâ consideratione, modò per intuitum, modò per 75 As might be expected, this interpretative decision is not entirely uncontroversial. Several recent commentators (such as Daniel Garber and John Schuster) who claim to have determined certain deep break-off between the early and the mature Descartes would most probably insist that the mature scheme is an expression of a radically new approach to the issue of validation of cognition and, by implication, to the issues of the nature of scientia and the nature and status of method as a key to scientia. However, it is too early to get involved in polemics on these global issues of interpretation right now. I confront certain aspects of the mentioned alternative approaches later in ch. 4. 76 See Reg. III, AT X, 369: “[P]lurimæ res certò sciuntur, quamvis non ipsæ sint evidentes, modò tantùm à veris cognitisque principijs deducantur per continuum & nullibi iterruptum cogitationis motum singula perspicuè intuentis ...” (my emphasis). Reg. VII, AT X, 388: “Addimus autem, nullibi interruptum debere esse hunc motum [ingenij deductionem constituentem]; frequenter enim illi, qui nimis celeriter & ex remotis principijs aliquid deducere conantur, non omnem conclusionum intermediarum catenationem tam accuratè percurrunt, quin multa inconsideratè transiliant. At certè, vbi vel minimum quid est prætermissum, statim catena rupta est, & tota conclusionis labitur certitudo.” Reg. XI, AT X, 408: “... per motum quemdam cogitationis singula attentè intuentis simul & ad alia transeuntis.” 77 Of course, each particular step of the deductio is also to be represented as a piece of scientia per intuitum in the corresponding recollection, but this is irrelevant to my present argument.

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deductionem cognosci ... (Reg. III, AT X, 370).

Descartes clearly states here that the barrier between intuitus and deductiones is by no means rigid and impermeable: if the conditions spelled out in the quoted AT X, 389 are satisfied, i.e. if the cognizing mind limits itself just to intuited items that immediately (and presumably necessarily, for otherwise the case would not count as a deductio at all) occur to it one after another and if the transition bears the mark(s) of evidence, then the deductio is ipso facto literally reduced to a new intuitus. It is important to appreciate that such a reduction of immediate deductiones to intuitus amounts to a genuine yet by no means the only way of extending one‫ތ‬s cognitive capacities, which for Descartes is one of the necessary conditions required in order to extend the range of one‫ތ‬s scientia by way of discoveries. This is because he clearly has it that one‫ތ‬s memory capacity is one of several factors that directly determine the cognitive capacities of finite minds and we can see by now that the more one is able to reduce singular acts of deductiones to intuitus the more new deductive steps one is able to retain in one‫ތ‬s memory with certainty.78 It comes as no surprise, therefore, that Descartes overtly recommends the pursuit of such reductions in Reg. XI, AT X, 407:79 78 Cf. Reg. XI, AT X, 408–409: “[H]æ duæ operationes [sc. intuitus & enumeratio seu deductio] se mutuò juvent & perficiant, adeò vt in vnam videantur coalescere .... Cujus rei duplicem vtilitatem designamus: nempe ad conclusionem, circa quam versamur, certiùs cognoscendam, & ad ingenium alijs inveniendis aptius reddendum. Quippe memoria, à quâ pendere dictum est certitudinem conclusionum, quæ plura complectuntur quàm vno intuitu capere possimus, cùm labilis sit & infirma, revocari debet & firmari per continuum hunc & repetitum cogitationis motum ...; quamobrem mihi necesse est illas iteratâ cogitatione percurrere, donec à prima ad vltimam tam celeriter transierim, vt fere nullas memoriæ partes relinquendo rem totam simul videar intueri. Quâ quidem ratione ingenij tarditatem emendari nemo non videt, & illius etiam amplificari capacitatem.” Cf. also in particular Reg. VII, AT X, 387–88. 79 It is worth noting that the passage is most naturally read as supporting the interpretation according to which Descartes should allow for the possibility of more than two simple natures being perceived in a single act of intuitus. This is inevitable once it is allowed (i) that necessary connections of ex hypothesi just two simple natures quâ objects of original intuitus can enter as the terms into a single piece of an immediate deductio and (ii) that such an immediate deductio can (at least sometimes and by some finite minds at least) be reduced to a new intuitus. This being conceded, it does not greatly matter exactly how many simple natures can in principle fall within the scope of a single intuitus. The answer is presumably mind-relative and the indeterminacy seems to do as little harm here as the

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Chapter Two Postquam aliquot propositiones simplices sumus intuiti, si ex illis aliquid aliud concludamus, vtile est easdem continua & nullibi interrupto cogitationis motu percurrere, ad mutuos illorum respectus reflectere, & plura simul, quantum fieri potest, distinctè concipere: ita enim & cognitio nostra longè certior sit, & maximè augetur ingenij capacitas (my emphasis).

Furthermore, I believe that once adequately interpreted, the reductions at issue coincide with “proportionibus ... eò reducendis, vt æqualitas inter quæsitum, & aliquid quod sit cognitum, clarè videatur” (Reg. XIV, AT X, 440), of which Descartes says that “præcipuam partem humanæ industriæ non in alio collocari” (ibid.). Descartes provides a predictable and prima facie somewhat psychological hint at how the desired reductions are to be achieved in practice, viz. by iterative running through the particular steps of which a given deductio consists, “vt ferè nullas memoriæ partes reliquendo” (Reg. VII, AT X, 388).80 It is of much greater importance to appreciate the overall significance both of his general resolution of complex (i.e. non-immediate) deductiones in terms of series of intuitus, and even more of his appeal to reduce as many deductiones as possible to new intuitus by means of exercising a certain genuine cognitive activity. For according to him these factors eventually imply nothing less than ruling out the possibility that inferences proper—which always take the form of motus ingenij in Descartes—at the end of the day really contribute to the constitution of scientific cognition. To be sure, finite minds cannot dispense with deductiones due to their limited cognitive capacities. However, this does not mean that whatever is essential to deductiones ever really adds any cognitive content over and above what is already involved in the (set of) intuitus to which any particular deductio is reducible: Descartes implies with sufficient clarity that nothing essential is lost if one achieves a reduction of a deductio to a new intuitus, and he seems to hold this to be a matter of principle. We shall soon see that this is by no means the entire story since the ordering of series of intuitus is introduced into the picture by Descartes as another essential element and he takes this ordering (among other things) indeterminacy concerning one‫ތ‬s ability to imagine a heptagon or octagon makes in Burm., AT V, 162–63. What matters is just that the number of simple natures thus amenable to a single intuitus is very small in any case and that deductiones (and the corresponding exercises of memory, invokings of divine guarantee etc.) are de facto inevitable for posessing scientia to any extent that Descartes would find desirable and satisfactory. 80 Cf. Reg. XI, AT X, 408–409.

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as the device par excellence available to finite minds for discovering new cognitions of the right (viz. scientific) type. Yet even though certain deductiones will indeed qualify as indispensable for arriving at some cognitions—due to ordo counting as an essential constituent of those deductiones—and even though the very establishment of an ordo allowing of some discovery thus rightly counts as a genuine cognitive achievement from Descartes’ perspective, at the end of the day deductiones ordinatæ still amount to but subsidiary devices for arriving at new cognitions, devices which are perhaps causally indispensable in this sense but can be discarded salva cognitione once the sought-after cognition is achieved. Thus despite the momentous complication just introduced, the submitted implications of Descartes‫ ތ‬reductive treatment of deductio still hold forth. To appreciate their groundbreaking character and to push the exposition further, let us focus now on a salient issue that emerges against the background of the implications in question, namely on Descartes‫ތ‬ contemptuous treatment of syllogistic reasoning and (virtually more generally) of the formal devices of the “dialecticians” as something repugnant to his own conception of discursive reasoning and of scientia.

2.5.2 Throwing Away the Dialecticorum Vincula There seems to be nothing about the initial conditions presented in the opening paragraph of sec. 2.5 which precludes Descartes‫ ތ‬deductiones being amenable to treatment in terms of certain standard formal systems designed to handle deductive inferences, most notably in terms of Aristotelian syllogistic reasoning,81 which still counted as the most developed and comprehensive system of deductive logic in Descartes‫ތ‬ time.82 However, Descartes‫ ތ‬vigorous reductive treatment of deductio 81 Cf. Aristotle‫ތ‬s classical definition of syllogism in An. Pr. I, 1, 24b19–22: “A deduction [i.e. a syllogism—J.P.] is a discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so. I mean by the last phrase that it follows because of them, and by this, that no further term is required from without in order to make the consequence necessary” (Aristotle, Complete Works, 1:40). As far as I can see, nothing in this definition is at odds with the initial conditions Descartes requires for deductio. 82 This study is not the place to become involved in any detailed examination of the general conceptions of dialectics and/or logic which formed the actual context of Descartes‫ ތ‬treatment of logic and dialectics, nor of those questions which concerned those authors writing on these subjects of whom Descartes actually knew, directly or second hand, and in particular whom he might have had in mind in his mostly disapproving attacks upon the “dialecticians” and/or syllogistic reasoning. Likewise, I shall not venture to determine exactly which variant theory

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presented in sec. 2.5.1 clearly shows not only that he eventually dismisses such a possibility but even that his general conception of deductio must be incompatible with it in some sense, for the syllogistic form is commonly understood as embodying and spelling out the proper achievement of a given cognitive enterprise, as the form essentially contributes to spelling out the relation between the subject and the predicate term via the middle term upon a causal or at least a distinctively explanatory basis (in the case of what Aristotle calls scientific syllogisms), or else the way in which the middle term can be arrived at via correlations between the subject and the predicate term (in the case of what Aristotle calls inductive syllogisms). As a consequence, it is impossible to be rid of a syllogistic form which a given cognition takes without losing the essentials of what that cognition consisted in from the point of view of the standard theory of syllogistic reasoning; and this, of course, is in sharp contrast with what Descartes commits himself to in his treatment of deductio up to the present point of our exposition. It is, I submit, in view of this situation that Descartes‫ ތ‬attacks against syllogistic reasoning, and more generally against the devices of the “dialecticians,” are to be understood and assessed: far from being just somewhat casual digressions to come to terms with a rival and professedly inadequate conception, I read Descartes‫ ތ‬struggle with the vincula Dialecticorum as a considered attempt at contrastive highlighting of what he took as the crucial and most revolutionary moments and implications of of syllogistic reasoning served as received wisdom in Descartes‫ ތ‬time and whether Descartes was just towards this standard view in his criticisms. My discussion is going to be general enough to safely ignore these important historical issues. I will just assume that although in Descartes‫ ތ‬time there were strong currents driving the rejection of deductive logic in general and syllogistic reasoning in particular as useless for the only task professedly attributed to dialectics, viz. to organize and convey previously attained knowledge (I mean the Humanist tradition culminating in the contributions of Rudolph Agricola and Peter Ramus), deductive logic was still considered valuable at least in the Jesuit schools where it was still treated in a form (conceived broadly enough and perhaps transformed in many respects) of Aristotelian syllogistic reasoning. For an invaluable survey of all the above issues and for further references see Gaukroger, Cartesian Logic, especially 19–25 and 31–47. I will also assume, following Florka, Descartes’ Metaphysical Reasoning, 43, that syllogistic reasoning (including inductive syllogisms of topics) is coextensive with formal logic for Descartes and that, as a consequence (to put it in Florka‫ތ‬s phrase) “[h]ad Descartes been given the chance ... to consider that there are arguments whose formal validity cannot be captured by predicate logic, he would ... have been willing to extend the criticisms of syllogistic reasoning to cover there other formally valid arguments” (ibid.).

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his rethinking the nature and purpose of inferential reasoning. Let us therefore examine his criticisms with diligent care. Although commentators generally have not failed to notice and discuss Descartes‫ ތ‬hostility towards syllogistic reasoning and formalism, the issue all too often still seems pervaded with confusion and superficial complacence.83 Thus some preliminary tidying up is perhaps in order. To begin with, there are no less than three different modi operandi with respect to which the devices of the “dialecticians” are critically assessed by Descartes (though he usually considers more than one of those modes at once in his discussions): regulation, discovery, and exposition. His reasons for dismissing the devices of the “dialecticians” as well as the logical strength of the rebuttal implied by those reasons differ considerably depending on which of these modi operandi he has in mind. Moreover, Descartes is highly sensitive to the somewhat concealed perils of formalisms involved in the dialectical devices, more specifically in a peculiar synergy, which is inherent in syllogistic reasoning, of semantically understood inferential validity on the one hand, and certain forms (stock arguments etc.) construed syntactically on the other.84 Indeed, it is often above all due to this ineliminable formal component that Descartes finds fault with the dialectical devices. Yet it will be important to distinguish the cases in which what Descartes rejects are formalisms tout court (and only a fortiori the peculiar forms of the “dialecticians”) from the cases in which it is exactly the peculiarities of the dialectical devices that Descartes dismisses. Finally, two types of operations having to do with intuitus and deductiones are to be carefully distinguished in this context, namely carrying out single intuitus or single immediate deductiones on the one hand, and handling a number of implemented intuitus or immediate deductiones on the other; we shall see that Descartes‫ ތ‬assessments of the dialectical devices differ with respect to the former and latter types. With these preliminaries in mind, let us consider Descartes‫ ތ‬critique of the dialectical devices relative to the two types of operations with intuitus and deductiones, starting with the former case of carrying out intuitus and immediate deductiones.

83

Honourable exceptions include Gaukroger, Cartesian Logic and Florka, Descartes’s Metaphysical Reasoning. Yet although I build much on their discussions of the issue (as I happily acknowledge), I still believe even their treatments are in need of several corrections and greater precision. 84 See Florka, ibid., 41–42 for a good treatment of this issue.

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2.5.2.1 Beyond Regulation By regulation as one of the three modi operandi, with which I contended Descartes is concerned while dealing with the vincula Dialecticorum, I understand subjecting cognitive operations to certain rules (regulæ). To regulate one‫ތ‬s cognitive operations thus amounts to setting certain normative constraints upon the ways one employs or can employ one’s cognitive faculties in order to correctly carry out one‫ތ‬s ratiocinative operations; and one salient implication of regulation understood along these lines is that the human ingenium left to itself might operate either correctly or incorrectly unless regulated by adequate rules. There is a long and respectable tradition, given a good start in Aristotle‫ތ‬s Organon and culminating in the Ramist and Jesuit theories of Descartes‫ ތ‬time, according to which one important, or perhaps even the only task of dialectic is to provide for the rules that, once learned and observed, would achieve exactly that much with regard to cognitive workings of the human mind, viz., in a word, ensure that the mind‫ތ‬s cognitive operations preserved truth.85 It is beyond doubt that Descartes rejects such a dialectic agenda at least with regard to intuitus and immediate deductiones, for he makes it clear that according to him, intuitus and immediate deductiones are strictly infallible and the only rationale for regulation—viz. that the human ingenium could operate incorrectly without observing certain rules—is thus ruled out, in any case, by this supposition:86 Notandum ... deductionem ... sive illationem puram vnius ab altero, posse quidem omitti, si non videatur, sed nunquam malè fieri ab intellectu vel minimùm rationali. Et parùm ad hoc prodesse mihi videntur illa Dialecticorum vincula, quibus rationem humanam regere se putant .... Omnis quippe deceptio, quæ potest accidere hominibus ... nunquam ex malâ illatione contingit ... (Reg. II, AT X, 365 ; my emphases). 85 Once again, I cannot undertake here a detailed historical survey which would supply firm ground for such a strong claim. See Gaukroger, Cartesian Logic, especially 56–60 for a good overall survey of the field. Cf. also Wilhelm Risse, Die Logik der Neuzeit, i: 1500–1640 (Stuttgart: Friedrich Frommann, 1964); idem, “Zur Vorgeschichte der cartesischen Methodenlehre,” Archiv für Geschichte der Philosophie 45, no. 3 (1963), 269–91; Earline Ashworth, Language and Logic in the Post-Medieval Period (Dordrecht: D. Reidel, 1974), especially 787–96. 86 Admittedly, Descartes does not discuss intuitus in the quoted passage. However, he writes referring back to the quoted passage, as early as in Reg. III, AT X, 368: “Per intuitum [Descartes’ emphasis] intelligo ... mentis puræ attentæ non dubioum conceptum, qui ... ipsâmet deductione certior est, quia simplicior, quam tamen etiam ab homine malè fieri non posse suprâ notavimus” (my emphasis).

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What cannot be decided, upon the basis of this passage alone, is simply the nature of the relation between ex hypothesi infallible intuitus and (immediate) deductiones on the one hand and the professed inappropriateness of their regulation on the other. One option is that their regulation is possible in principle but superfluous or even virtually harmful since it is inappropriate to regulate that which by its very essence ex hypothesi works perfectly without regulation. The other option is that it is a matter of principle that intuitus and immediate deductiones—regardless their ex hypothesi infallibility—cannot be regulated, i.e. that they are essentially beyond regulation. The issue is decided in another closely related passage from the Regulæ. Having introduced method as something that “[n]ecessaria est ... ad rerum veritatem investigandam” (Reg. IV, AT X, 371) and having characterized it as “regulas certas & faciles, quas quicumque exactè servaverit, nihil vnquam falsum pro vero supponet, & ... perveniet ad veram cognitionem eorum omnium quorum erit capax” (ibid., 371–72), Descartes goes on to claim that [n]eque enim etiam [methodus] extendi potest ad docendum quomodo hæ ipsæ operationes [sc. intuitus & deductio] faciendæ sint, quia sunt omnium simplicissimæ & primæ, adeò vt, nisi illis vti jam antè posset intellectus noster, nulla ipsius methodi præcepta quantumcumque facilia comprehenderet (ibid., 372).

It is clear enough how the argument is supposed to work. (1) If we could be taught at all how intuitus and deductio are to be completed correctly, then it is solely the method under discussion that would have to be properly responsible for such a task; but even the method cannot be responsible for this, since (2) our ability to carry out intuitus and deductiones correctly is, Descartes maintains, a preliminary condition for the method to be capable of teaching us anything at all; therefore, nothing could ever teach us to carry out those operations. Their correct functioning is a condition of any regulation whatsoever, and they are thus beyond regulation. Due to its importance for Descartes‫ ތ‬overall view of the way in which scientia is to be attained, the argument is worth some explication. Premise (1) is plausible enough given Descartes‫ ތ‬statement that it is the proper task of the method under discussion to explicate (explicare) “quomodo mentis intuitu sit vtendum, ne in errorem vero contrarium delabamur, & quomodo deductiones inveniendæ sint, vt ad omnium cognitionem perveniamus” (ibid.), i.e. that whatever is immediately related to intuitus and deductio in the context of attaining scientia falls within the compass of the method. It is, then, premise (2) that is crucial for evaluating the argument. What might stand behind Descartes‫ ތ‬confidence that (2) is

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true? Following Gaukroger in having recourse to an analogical case Descartes argued in a letter to Mersenne, that we cannot be taught what truth is because it is a sine quâ non of learning anything at all to know what truth is,87 I submit that Descartes must have had in mind something like the following. Suppose intuitus and immediate deductiones are indeed in need of regulation according to method (cf. premise [1] above). Such a regulation implies that the operations to be regulated are cognized in the course of their methodical assessment. Now, since the context is that of attaining scientia, the cognition implied in regulation must itself meet scientific standards. However, as established above, no cognition can meet such standards unless it in itself counts as a result of operations of intuitus and/or (eventually immediate) deductio. Thus if, according to the present hypothesis, these operations are generally in need of regulation then the regulative operations of the method must themselves be regulated in the first place, with a vicious infinite regress under way. As a consequence, the present hypothesis must be dismissed, and the claim that intuitus and 87

The passage is worth quoting at length: [P]our moy, ie n‫ތ‬en [sc. ce que c‫ތ‬est que la Verité] ay iamais douté, me semblant que c‫ތ‬est vne notion si transcendentalement claire, qu‫ތ‬il est impossible de l‫ތ‬ignorer: en effect, on a bien des moyens pour examiner vne balance auant que de s‫ތ‬en seruir, mais on n‫ތ‬en auroit point pour apprendre ce que c‫ތ‬est que la verité, si on ne la connoissoit de nature. Car quelle raison aurions nous de consentir a ce qui nous l‫ތ‬apprendroit, si nous ne sçauions qu‫ތ‬il fust vray, c‫ތ‬est a dire, si nous ne connoissions la verité? Ainsy on peut bien expliquer quid nominis a ceux qui n‫ތ‬entendent pas la langue, & leur dire que ce mot verité, en sa propre signification, denote la conformité de la pensée auec l‫ތ‬obiet, mais que, lors qu‫ތ‬on l‫ތ‬attribue aux choses qui sont hors de la pensée, il signifie seulement que ces choses peuuent seruir d‫ތ‬obiets a des pensées veritables, soit aux nostres, soit a celles de Dieu; mais on ne peut donner aucune definition de Logique qui ayde a connoistre sa nature. Et ie croy le mesme de plusieurs autres choses, qui sont fort simples & se connoissent naturellement ..., en sorte que, lors qu‫ތ‬on veut definir ces choses, on les obscurcist & on s‫ތ‬embarasse (Mers., AT II, 596–97; Descartes’ italics).

Gaukroger rightly points out, in his lucid discussion (Cartesian Logic, 52–53), that while Descartes‫ ތ‬argument concerning truth turns on the issues of definition and of the difference between truth and falsity (since the professed objective of learning is what truth is), his supposed analogical argument concerning intuitus and deductio is to be understood rather as turning on the issues of justification and the difference between the correct and incorrect implementation of these operations (since the professed objective of learning is their correct implementation).

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immediate deductio are beyond regulation must be acknowledged. This is clearly intended to hold for any form of regulation whatever, and a fortiori also for Aristotelian syllogistic reasoning. Let us proceed to another of the above modi operandi, namely to discovery. This mode requires but a brief, general treatment in the context of intuitus and immediate deductiones. For although, as we shall see, discovery is not the only domain in which regulation is at work, Descartes makes it clear that discoveries worth the title generally do require (a specific sort of) regulation—which, of course, he takes as coincident with his method.88 So if, as we have established, Descartes views no regulation whatever as being possible in the domain of intuitus and immediate deductiones as such, no (real) discovery is possible there either, and consequently in that context no formalisms, dialectical or otherwise, can assume any rôle as tools of discovery. The remaining modus operandi to be discussed in the context of intuitus and immediate deductiones is exposition. It is a well-established fact that demonstrative syllogisms were intended, at least by Aristotle himself, neither to regulate cognitive operations in the strong sense pointed out by Descartes nor to facilitate discoveries but rather to provide expository and didactic tools, i.e. to impart and convey already established knowledge in an economical, neatly-arranged manner.89 Furthermore, Descartes indeed concedes on several occasions to the vulgaris Dialectica 88

This can safely be gathered together, above all from the complex introductory treatment of the envisaged method in Reg. IV. The method, as we saw, is defined there as “regulas certas & faciles, quas quicumque exactè servaverit, ... gradatim semper augendo scientiam, perveniet ad veram cognitionem eorum omnium quorum erit capax” (AT X, 371–72; my emphases), as explaining, among other things, “quomodo deductiones inveniendæ sint, vt ad omnium cognitionem perveniamus” (ibid., 372; my emphasis), and as a discipline which “ad veritates ex quovis subjecto eliciendas se extendere debet” (ibid., 374). Cf. also A Beeckman, AT I, 160: “Tria genera inuentorum tibi proponam. Primo, si quid habes alicuius momenti, quod solius ingenij vi & rationis ductu poteris excogitare, fateor te laudandum ...” (my emphasis; the other two types of inventions are dismissed by Descartes as not really deserving the name—cf. ibid., 160–62). 89 Cf. Jonathan Barnes, “Aristotle‫ތ‬s Theory of Demonstration,” in Articles on Aristotle I: Science, ed. Jonathan Barnes, Malcolm Schofield, and Richard Sorabji (London: Duckworth, 1975), 65–87. There is an important Galenic tradition in which demonstrative syllogisms were employed, in a sense, as a means of discovery. Yet this complication can be put to one side for the present since even in the Galenic tradition the expository import of demonstrative syllogism is not ruled out, and my present point is just that demonstrative syllogism was intended first of all as a tool of presentation in the Aristotelian tradition.

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that “prodesse tantummodo interdum posse ad rationes jam cognitas faciliùs alijs exponendas” (Reg. X, AT X, 406).90 Yet despite this somewhat hesitant concession,91 for Descartes the dialectical devices turn out to be not much better off with regard to exposition than they were shown to be with regard to regulation in the domain of immediate deductiones, for Descartes makes it clear many times that the aforementioned flimsy pedagogical benefit is debased due to the fact that the formal devices of the “dialecticians” effectively do (or easily can do) a degree of harm to the appropriate workings of human cognitive faculties in other, more substantial respects. As he puts it concisely to Burman,92 Dialectica, cùm doceat nos de omnibus rebus disserere, ... bonam mentem magis evertit quàm adstruit; nam dum nos divertit et digredi facit in hos locos communes et capita, quæ rei externa sunt, divertit nos ab ipsâ rei naturâ (Burm., AT V, 175).

Above all it is, then, this unwelcome effect of diverting the mind from the nature of a cognized thing, that is, from the concrete content to be apprehended, and considered, to something external that prompts 90

This is most probably the benefit that Descartes has in mind in Reg. II where he says that although the Dialecticorum vincula “parùm ad [deductiones bene faciendas] prodesse mihi videntur ..., etiamsi eadem alijs vsibus aptissima esse non negem” (AT X, 365). Cf. a similar concession in Princ. Pref., AT IX-2, 13–14: “[L]a Logique ... de l‫ތ‬eschole ... n‫ތ‬est à proprement parler, qu‫ތ‬vne Dialectique qui enseigne les moyens de faire entendre à autruy les choses qu‫ތ‬on sçait ....” 91 Descartes has, in fact, more to say on what the dialectical devices might be good for, namely (i) for “puerorum ingenia exercenda” (Reg. II, AT X, 363) and (ii) for “parler, sans iugement, de celles qu‫ތ‬on ignore” (DM 2, AT VI, 17; cf. a similar phrase in Princ. Pref., AT IX-2, 13). However, the sarcasm redolent in these remarks is too evident to take them seriously. 92 Similarly in Reg. X, AT X, 406: “[H]îc nos præcipuè caventes ne ratio nostra ferietur, dum alicujus rei veritatem examinamus, rejicimus istas formas vt adversantes nostro instituto, & omnia potiùs adjumenta perquirimus, quibus cogitatio nostra retineatur attenta ....” Princ. Pref., AT IX-2, 13–14: “[L]a Logique ... de l‫ތ‬eschole ... corrompt le bon sens plustost qu‫ތ‬elle ne l‫ތ‬augmente ....” RV, AT X, 521: “[Lumen rationis & sanus sensus] ubi solus per se agit, erroribus minùs est obnoxius, quàm cùm mille diversas regulas, quas artificium & desidia hominum, ad illum corrumpendum potiùs quàm reddendum perfectiorem, invenerunt, anxiè observare studet.” Cf. also Reg. IV, AT X, 372–73; DM 2, AT VI, 17. In the quoted AT V, 175, Dialectica is contrasted with Logica, quæ de omnibus rebus demonstrationes dat and Descartes seems to approve of the latter. To save the present interpretation, I suggest Descartes refer here by “Logica” to his own analytical procedure which is introduced and discussed in the following chapters.

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Descartes to present the dialectical devices quite generally as impediments to cognitive operations and as something which “corrompt le bon sens” (Princ. Pref., AT IX-2, 13). Descartes is not altogether clear whether the problem is just that dialectical devices distract the mind and thus render it inattentive or whether the employment of these devices could even somehow interfere with actual performance of the cognitive faculties, but I put this issue to one side.93 For be this as it may, what matters now is that in so far as the case of immediate deductiones is concerned, the possible expository virtues of the dialectics fall away and what remains is only its unwelcome obstructive impact. Given the immediacy of deductiones in question and the above-established cognitive divisibility of their terms, I suspect Descartes would hold that there could be nothing that any formal exposition whatever (and a fortiori any formalism of the dialectical cast) could ever clarify or better organize, for the only adequate formula to render each of the immediate deductiones seems to be simply A?B (where “A” and “B” stand for intuitive cognitions and “_?_” means “the latter follows immediately and necessarily from the former”);94 and such a 93

The former option surely forms at least a part of Descartes‫ ތ‬meaning (witness the well-known passage in Reg. X, AT X, 405–406, quoted below). The latter option is clearly stronger as it implies that the dialectical devices actually engage in the actual operations of the cognitive faculties. The distinction between distraction and interference as two possible interpretations of the dialectics as a hindrance to cognition is drawn and perspicuously discussed in Florka, Descartes’s Metaphysical Reasoning, 47–53. I have nothing substantial to add to Florka‫ތ‬s treatment. 94 This claim is controversial in an illuminating way. Referring to Günter Patzig, Die aristotelische Syllogistik: Logisch-philologische Untersuchungen über das Buch A der “Ersten Analytiken” (Göttingen: Vandenhoeck & Ruprecht, 1959), Stephen Gaukroger in his Cartesian Logic, 57–58 points out that in the so-called perfect syllogisms, i.e. in the categorial first-figure syllogisms of the Barbara mode, the connection between the premises and the conclusion might rightly be characterized as immediately evident in the sense coming close to, if not coincident with, the sense envisaged by Descartes in the case of immediate deductiones. If so, why should Descartes feel entitled to deny the claim of such syllogisms to count as the basic forms of immediate deductiones? I believe a part of Descartes‫ ތ‬response would be that what such syllogisms capture is, in fact, an ordered sequence of several cognitions based upon the corresponding immediate deductiones (Descartes seems bound to hold that at very least there is one immediate deductio behind the maior, another immediate deductio behind the minor and still another behind the passage from the joint intuitive grasp of both maior and minor to the conclusion), so that they must fail to fulfil the professed basic rôle. I believe it is something like this that stands behind Descartes‫ ތ‬claim

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formula not only clarifies or organizes nothing but it even threatens to obscure the denoted structure by suggesting reification of the relation of necessary sequence at issue. Moreover and most importantly, even this trivial formula (not to speak of more complex syllogistic forms were they—counterfactually for Descartes—taken as expressing immediate deductiones adequately) indeed abstracts from the actual content of the rendered deductio and thus diverts the mind from the crucial moment upon which Descartes‫ ތ‬entire conception of scientia hinges, viz. the psychological state of irresistible compulsion to proceed from A to B once both A and B are cognized intuitively. To sum up, Descartes reserves no place for the vincula Dialecticorum in so far as the operations of intuitus and immediate deductiones as such are concerned. With regard to regulation and discovery, dialectical devices are ruled out a fortiori due to Descartes‫ ތ‬sweeping denial of the possibility of regulation, and by implication of discovery, in the given domain. With regard to exposition, the situation turned out somewhat differently. While a certain sort of formalism can in principle aspire to fulfil the rôle of an expository device, the formalisms of dialectical cast—syllogisms—are definitely non-starters in this respect. Yet even if they were not, they would have been accused of impeding the workings of the human ingenium no less than the formalism that seems acceptable. The moral to be drawn so far with regard to our larger topic—viz. Descartes‫ ތ‬understanding of the rôle of inference in the constitution of scientia—thus dramatically exceeds the narrow constraints of the issue of the Dialecticorum vincula: it has turned out that no formalisms whatsoever can contribute anything positive either to the workings or the better understanding not only of intuitus but even of immediate deductiones. That is to say, however, due to the reductive treatment established above, that at the end of the day Descartes wants the entire body of what he envisages as scientia proper to remain absolutely clear of formalisms. We can see by now that this is—exposition apart—a matter of principle. Before we proceed further into the realms in which, according to Descartes, regulation does have its place, it will be well to remark that I take Descartes as referring exactly to those cognitive faculties of the that “[a]liæ autem mentis operationes, quas harum priorum [sc. intuitus & deductionis] auxilio dirigere contendit Dialectica, hîc sunt inutiles, vel potiùs inter impedimenta numerandæ, quia nihil puro rationis lumini superaddi potest, quod illud aliquo modo non obscuret” (Reg. IV, AT X, 372–73). This criticism is to be carefully distinguished from considerations presented in the sec. 2.5.2.2 concerning the epistemic dependence of the premises of syllogisms on singular deductiones.

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human mind whose employment is essentially beyond regulation in the sense just established—viz. whose employment issues in intuitus and immediate deductiones—with the virtually interchangeable terms “vis intelligendi”, “vis cognoscendi” and “vis cognoscens” introduced in ch. 1. Indeed, Descartes seems to refer to those faculties—in certain contexts at least—with a set of other virtually interchangeable terms, of which “bon sens” or “bona mens” will play a rather significant rôle later in this study.95 Setting interpretative details to one side,96 it will suffice for our purposes to corroborate my above interpretative contention with the celebrated opening section of DM 1, AT VI, 1–2: Le bon sens est la chose du monde la mieux partagée ... [L]a puissance de bien iuger, & distinguer le vray d’auec le faux, qui est proprement ce qu’on nomme le bon sens ou la raison, est naturellement esgale en tous les hommes; et ainsi ... la diuersité de nos opinions ne vient pas de ce que les 95

Apart from “bon sens” and “bona mens,” also “bon esprit,” “ingenium,” “raison,” “recta ratio,” “lumen rationis,” and “lumen naturale” fall within the mentioned set of virtually synonymous terms. The prima facie interchangeability of “bon sens”, “bon esprit”, and “raison” is established most authoritatively in DM 1, AT VI, 1–2. These words are rendered as “bona mens,” “ingenium,” and “recta ratio” in the Latin translation, most probably reviewed by Descartes, of the Discours—cf. AT VI, 540. “Bona mens” is employed in arguably the same sense most notably in Reg. I where it is used virtually interchangeably with “ingenium” and with “naturale rationis lumen”—cf. ibid., AT X, 360–61. “Vis intelligendi” is used interchangeably with “lumen naturale” in Med. IV, AT VII, 60. There are dozens of occurrences in Descartes’ texts of lumen naturale in the present sense from the Meditationes onwards—cf. e.g. Med. III, AT VII, 38–52; Med. VI, AT VII, 82; Resp. 1, AT VII, 107–108; Resp. 2, AT VII, 134–36; 148; 161; Resp. 4, AT VII, 238–41. Finally, ratio is employed in Princ. I, 1, AT VIII-1, 5 in a similar context and meaning as bon sens/bona mens is at the beginning of DM 1; and in Princ. I, 30, AT VIII-1, 16 Descartes defines lumen naturæ in a manner which renders it virtually interchangeable with the ingenium and naturale rationis lumen of Reg. I. To be sure, at least some of the above terms seem to assume narrower meanings here and there even in the very works just cited (in particular, this is almost certainly true as regards ingenium, which seems to get tied quite closely to imagination in Reg. XII, AT X, 416—see in particular Sepper, Descartes’s Imagination, 96–98). Yet I do not wish to become involved in detailed terminological debates here and will henceforth use only “bona mens” and “bon sens” to convey the declared meaning. 96 See e.g. John Morris, “Descartes’s Natural Light,” in Moyal, Critical Assessments, 1:413–32 for a typical sample of detailed treatment. One important interpretative issue concerning the meaning of bona mens will be taken up towards the end of ch. 4.

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Here Descartes makes it clear that it is one thing to possess a power “de bien iuger, & distinguer le vray d’auec le faux”, which is identical with le bon sens (also called here la raison and l’esprit bon) common to all humans, and that it is quite another thing to be able to apply that common power adequately, a capacity which is not common by nature to all humans and is somehow to be achieved by each of them individually. Furthermore, it comes as no surprise that this capacity “d’appliquer [le bon sens] bien” is conceived as a direct function of the possession of a method in the subsequent sections of DM 1.97 Thus “auoir [le bon sens]” as opposed to “d’appliquer [le bon sens] bien” clearly amounts to possession of the capacity “de bien iuger, & distinguer le vray d’auec le faux” beyond any regulation. And we have learned from the above discussion of the opening sections of Reg. IV that it is exactly intuitus and immediate deductiones that amount to actualizations of the cognitive ability in so far as it is beyond regulation. As already announced, the notion of bona mens, in the sense just introduced, will play a significant rôle later in dealing with major issues of justification and possibility of method in Descartes and of his peculiar notion of the unity of scientiæ (ch. 4). For the time being, however, let us return to our present topic of vincula Dialecticorum. 2.5.2.2 Regulation, and Advancing Knowledge Despite this terminal hostility to formalisms, there remains a large area in which, according to Descartes, certain types of formalism can and do play some important and perhaps even indispensable rôles in attaining scientiæ. For although, as we saw, Descartes holds that intuitus and 97

See AT VI, 2–4: “[P]our la raison, ou le sens, d’autant qu’elle est la seule chose qui nous rend hommes ..., ie veux croire qu’elle est toute entiere en vn chacun ... Mais ie ne craindray pas de dire que ie pense auoir eu beaucoup d’heur, de m’estre rencontré dés ma ieunesse en certains chemins, qui m’ont conduit a des considerations & des maximes, dont i’ay formé vne Methode, par laquelle il me semble que i’ay moyen d’augmenter par degrez ma connoissance ... Ainsi mon dessein n’est pas d’enseigner icy la Methode que chascun doit suiure pour bien conduire sa raison, mais seulement de faire voir en quelle sorte i’ay tasché de conduire la miene.”

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immediate deductiones are in themselves beyond regulation, he makes it clear that at least in so far as regulation counts as a tool of discovery or of advancing knowledge, certain formalisms have important auxiliary cognitive functions. Much remains to be accomplished to understand adequately Descartes‫ ތ‬own positive conception in this respect. Our present task, however, is only to explain why syllogistic reasoning is ruled out by Descartes even in those areas in which certain other types of formalisms can and are to be employed according to him, and what we can learn from his rebuttal with regard to his own positive views on the nature of inference in the contexts of advancing knowledge. Let us begin with a delineation of the areas of regulation in Descartes. To him, any regulation that is virtually relevant to the project of establishing scientiæ falls under the heading of method. This is established by his previously quoted definition of method, [p]er methodum autem intelligo regulas certas & faciles, quas quicumque exactè servaverit, nihil vnquam falsum pro vero supponet, & nullo mentis conatu inutiliter consumpto, sed gradatim semper augendo scientiam, perveniet ad veram cognitionem eorum omnium quorum erit capax (Reg. IV, AT X, 371–72; my emphasis),

in combination with his claim that this method “ad veritates ex quovis subjecto eliciendas se extendere debet” (ibid., 374). Now Descartes continues immediately: Notanda autem hîc sunt duo hæc: nihil nimirum falsum pro vero supponere, & ad omnium cognitionem pervenire. Quoniam, si quid ignoramus ex ijs omnibus quæ possumus scire, id fit tantùm, vel quia nunquam advertimus viam vllam, quæ nos duceret ad talem cognitionem, vel quia in errorem contrarium lapsi sumus. At si methodus rectè explicet quomodo mentis intuitu sit vtendum, ne in errorem vero contrarium delabamur, & quomodo deductiones inveniendæ sint, vt ad omnium cognitionem perveniamus: nihil aliud requiri mihi videtur, vt sit completa, cùm nullam scientiam haberi posse, nisi per mentis intuitum vel deductionem ... (ibid., 372).

Thus according to him there are precisely two areas which jointly exhaust the range of possible regulative operations of the envisaged method: (1) use of intuitus in order that “ne in errorem vero contrarium delabamur,” and (2) handling intuitus and immediate deductiones in order to explain “quomodo deductiones inveniendæ sint, vt ad omnium cognitionem perveniamus.”

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It is important to clarify of what exactly the envisaged regulations consist. To begin with, one might wonder how this distribution of regulation, in which intuitus and deductiones are involved, could be reconciled with Descartes‫ ތ‬claim that intuitus and immediate deductiones are beyond regulation. To dispel this air of apparent paradox, however, one only needs to distinguish (i) implementing intuitus and immediate deductiones from (ii) employing implemented intuitus and immediate deductiones in other operations. It is, of course, only sense (i) in which intuitus and immediate deductiones are beyond regulation; and it is sense (ii) within which the distribution of regulation in question is intended to be effected. By the same token, however, the ways in which intuitus are supposed to be employed in regulations of type (1) and the ways in which (presumably) both intuitus and immediate deductiones are supposed to be employed in regulations of type (2) seem to differ considerably. As for (1), the proper objects of regulation are non-discursive cognitions, and intuitus are employed as standards of certainty required for each cognition to qualify as (a piece of) scientia; this régime of regulation has in fact been established and discussed in numerous sections up to the present point in this study.98 As for (2), the proper objects of regulation are arguably certain specific ways in which particular implemented intuitus and immediate deductiones are related to one another in order to facilitate discoveries; far from operating as standards, the implemented intuitus and immediate deductiones thus count as essential components of the regulanda. It is this latter régime of regulation, i.e. the régime amounting to what has frequently been called ars inveniendi, that Descartes himself took as the chief modus operandi of his method and he clearly thought his contribution to this topic was particularly momentous.99 It will be one of our principal tasks in the following chapters to provide an adequate account of this important, extremely complex and controversial aspect of Descartes‫ ތ‬thought. However, as previously suggested let us for now limit ourselves to the question of why Descartes thought that syllogistic 98 It might also be said that this régime is dealt with in what precedes the crucial Reg. IV, in particular in Reg. II and III, and that régime (2) is the chief concern of what follows, i.e. the entire extant Regulæ starting with Reg. V. 99 See Reg. IV, AT X, 373: “[H]ujus methodi vtilitas sit tanta, vt sine illâ litteris operam dare nociturum esse videatur potiùs quam profuturum ....” Also ibid., 374: “Hæc [methodus] ad veritates ex quovis subjecto eliciendas se extendere debet; atque, vt libere loquar, hanc omni aliâ nobis humanitùs traditâ cognitione potiorem, vtpote aliarum omnium fontem, esse mihi persuadeo.” These are the private counterparts of Descartes‫ ތ‬presentation of his method in DM 1 which is permeated with the rhetoric of humility; see in particular AT VI, 3–4.

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reasoning was a non-starter in both areas in which regulation was to be executed. Descartes never even mentions syllogistic reasoning in connection with the use of intuitus in the course of discriminating truths from falsities, and this should come as no surprise: syllogisms, with their inherently discursive character, are of course blatantly inadequate as the means of capturing intuitus as conceived by Descartes and he may well have considered this too obvious to merit any discussion.100 The situation is very different as regards the other mode of regulation, viz. handling intuitus and immediate deductiones in the course of seeking new scientific cognitions. Although Descartes‫ ތ‬complex reasoning against the use of syllogism in these contexts is to be gathered together from hints that are scattered across his writings, his overall position turns out sufficiently developed and remarkably unitary. To begin with, Descartes is ready to apply a fortiori to the case of syllogistic reasoning in the contexts of discovery a general criticism which he raised against formalisms in the context of treating intuitus and immediate deductiones in the expository mode of operation. That is to say, he rejects the practice of purely mechanical manipulation of the formal devices in the course of reaching conclusions, the rejection of which seems applicable a fortiori in the mode of discovery: [Dialectici] quasdam formas disserendi præscribunt, quæ tam necessariò concludunt, vt illis confisa ratio, etiamsi quodammodo ferietur ab ipsius illationis evidenti & attentâ consideratione, possit tamen interim aliquid certum ex vi formæ concludere: quippe advertimus elabi sæpe veritatem ex istis vinculis, dum interim illi ipsi, qui vsi sunt, in ijsdem manent irretiti. ... Quamobrem hîc nos præcipuè caventes ne ratio nostra ferietur, dum alicujus rei veritatem examinamus, rejicimus istas formas vt adversantes nostro instituto, & omnia potiùs adjumenta perquirijmus, quibus cogitatio nostra retineatur attenta ... (Reg. X, AT X, 405–406).

There is more to Descartes‫ ތ‬attack against syllogistic reasoning in the context of discovery, however. For even if the requirement implied in the aforementioned criticism is fulfilled, i.e. even if the content associated with the formal device at issue is attended to and no conclusions purely ex 100

Professed immediate evidence of a certain type of syllogism, namely the categorial first-figure syllogisms in the mode Barbara, might perhaps suggest with some initial plausibility that syllogisms could count for Descartes at most as the basic forms of immediate deductiones, i.e. of the minimal discursive structure (though, as we saw, he has theoretical means to reject this claim).

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vi formæ are allowed, then according to Descartes syllogistic reasoning encounters systematic problems. Let us begin with his famous dictum against syllogistic reasoning in Reg. X, AT X, 406:101 [A]dvertendum est, nullum posse Dialecticos syllogismum arte formare, qui verum concludat, nisi priùs ejusdem materiam habuerint, id est, nisi eamdem veritatem, quæ in illo deducitur, jam antè cognoverint. Vnde patet illos ipsos ex tali formâ nihil novi percipere, ideoque vulgarem Dialecticam omnino esse inutilem rerum veritatem investigare cupientibus ....

Several scholars have suggested that what Descartes has in mind here is one line of Sextus‫ ތ‬classical two-pronged refutation of demonstrative reasoning, namely the charge of cognitive circularity.102 Indeed, Sextus considers an inference such as Everything human is an animal. But Socrates is human. Therefore, Socrates is an animal103

and he argues that the inference is cognitively circular if its universal maior is interpreted as established by way of inductive generalization, in the sense of establishing a universal statement upon the basis of a finite number of particular instances: the assent to the maior cannot then be justified unless the truth of the conclusion is established in the first place, which surely renders the argument cognitively circular.104 101

See also Reg. XIII, AT X, 430: “[Dialectici] ad syllogismorum formas tradendas, eorumdem terminos, sive materiam cognitam esse supponunt ....” 102 See e.g. Gaukroger, Cartesian Logic, 11–18; Danielle Macbeth, “Viète, Descartes, and the Emergence of Modern Mathematics,” Graduate Faculty Philosophy Journal 25, no. 2 (2004), 94. 103 Sextus Empiricus, Outlines of Scepticism, ed. and transl. by Julia Annas and Jonathan Barnes (Cambridge: Cambridge University Press), II, 195. 104 Sextus’ own account of the situation reads as follows: “[T]his proposition— Everything human is an animal—is confirmed inductively from the particulars; for from the fact that Socrates, being human, is also an animal, and similarly with Plato and Dio and each of the particulars, it is thought possible to affirm that everything human is animal. ... Thus when [the Peripatetics] say: Everything human is an animal. But Socrates is human. Therefore, Socrates is an animal, and wish to conclude from the universal proposition ‫ދ‬Everything human is an animal‫ތ‬ to the particular proposition ‫ދ‬Therefore Socrates is an animal‫ ތ‬..., they fall into the reciprocal argument, confirming the universal proposition inductively by way of each of the particulars and the particular deductively from the universal ...” (Sextus

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I do not intend to doubt the well-established fact that Descartes was acquainted with, and profoundly influenced by, Sextus‫ ތ‬thought.105 Yet I believe that the present suggestion is a red herring. For one thing, recall that the context of Descartes‫ ތ‬consideration in question concerning syllogistic reasoning is that of the conditions of scientia taken in the above restrictive sense. Since Descartes would be, of course, by no means the only one to point out that inductive generalization can never establish anything over and above what he has called probabilis opinio, the interpretation in terms of cognitive circularity would render his rebuttal curiously inappropriate in view of his own standards of significance. Moreover, several other passages from Descartes jointly suggest quite a different reading of his chief complaint against syllogistic reasoning, a reading which is both consistent with the dictum in Reg. X and much more adequate in view of what Descartes seeks in the present context. Consider first Descartes‫ ތ‬comment on the Disquisitio Metaphysica by Gassendi: [L]‫ތ‬erreur qui est icy la plus considerable, est que cet Auteur suppose que la connoissance des proportions particulieres doit tousiours estre deduite des vniuerselles, suiuant l‫ތ‬ordre des syllogismes de la Dialectique: en quoy il montre sçauoir bien peu de quelle façon la verité se doit chercher; car il est certain que, pour la trouuer, on doit toûjours commencer par les notions particulieres, pour venir après aux generales, bien qu‫ތ‬on puisse aussi reciproquement, ayant trouué les generales, en deduire d‫ތ‬autres particulieres (A Clerselier, AT IX-1, 205–206; my emphasis).

I take this to be a generalized rendering of the point famously made by Descartes in Resp. 2, AT VII, 140–41: Cùm ... advertimus nos esse res cogitantes, prima quædam notio est, quæ ex nullo syllogismo concluditur; neque etiam cùm quis dicit, ego cogito, ergo sum, sive existo [Descartes’ italics], existentiam ex cogitatione per syllogismum deducit, sed tanquam rem per se notam simplici mentis intuitu agnoscit, ut patet ex eo quòd, si eam per syllogismum deduceret, novisse prius debuisset istam majorem, illud omne, quod cogitat, est sive existit [Descartes’ italics]; atqui profecto ipsam potius discit, ex eo quòd apud se experiatur, fieri non posse ut cogitet, nisi existat. Ea enim est natura nostræ mentis, ut generales propositiones ex particularium cognitione efformet [my emphasis]. Empiricus, Outlines of Scepticism, II, 195–96). That Sextus means by reciprocal argument what I have called cognitively circular inference is confirmed also ibid., I, 169. 105 See Popkin, History of Skepticism, ch. 9 for convincing evidence to this effect.

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At first glance, it might seem that nothing precludes the interpretation of Descartes‫ ތ‬invocation in these passages of forming general propositions ex particularium cognitione in terms of inductive generalizations which are relevant to the charge of cognitive circularity. However, Descartes explains to Burman how he wishes his words to be understood: [S]cilicet ad id tantùm attendo quod in me experior, ut, cogito, ergo sum [Burman’s italics] non autem ita attendo ad generalem illam notionem, quicquid cogitat, est [Bruman’s italics]; nam ... non separamus illas propositiones a singularibus, sed eas in illis consideramus [my emphasis]; et hoc sensu verba hæc ... hîc citata [sc. “ex nullo syllogismo” from AT VII, 140—J.P.] intelligi debent (Burm., AT V, 147).

Thus, the crucial difference is that Descartes is interested now in such cases as a general proposition, far from being derived by inductive generalization from a number of particular instances, is “considered” within a single particular instance which is actually intuited by the mind. The point is that general propositions of this type are not disqualified in advance as mere probabiles opiniones but can count as scientiæ (as we shall see more distinctly in a moment); and that, as a consequence, Descartes is bound to offer an independent argument to discredit the pretension of syllogisms, the universal maior of which is a general propositions of this type, to count as possible tools for genuine scientific discoveries.106 It is here that his apparently vague charge in the abovequoted AT X, 406 finds its proper place: once it is granted that “on doit 106

Danielle Macbeth has suggested in her “Viète, Descartes,” 94–95 that Descartes in fact draws here upon the other line of Sextus‫ ތ‬campaign against demonstrative reasoning. Sextus is concerned here with syllogisms whose universal premise is not established by way of inductive generalization and his point is, roughly, that the envisaged case of non-inductive universal premise makes no sense unless it is conceived as just an explicit statement of a necessary connection between the terms; and that if so, the premise is strictly speaking redundant: “[I]f being an animal follows being human, and for that reason the proposition ‫ދ‬Everything human is an animal‫ ތ‬is agreed to be true, then at the same time as it is said that Socrates is human, it may be concluded that he is an animal, so that it is enough to propound the argument thus: Socrates is human. Therefore, Socrates is an animal, and the proposition ‫ދ‬Everything human is an animal‫ ތ‬is redundant” (Sextus Empiricus, Outlines of Scepticism, II, 165). Admittedly, there is remarkable affinity between the cases taken up by Sextus on the one hand and on the other by Descartes in AT VII, 140–41 and AT IX-1, 205–206. Yet it seems to me that the comparison with Sextus‫ތ‬ procedure throws little light on the significance of Descartes‫ ތ‬treatment, especially in view of the complications introduced in sec. 2.5.2.3.

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toûjours commencer par les notions particulieres, pour venir après aux generales” (A Clerselier, AT IX-1, 206), and that such is the “natura nostræ mentis” (Resp. 2, AT VII, 140), it follows that every cognition of any universal maior, even if it amounts to scientia, always depends upon intuitive cognition of the corresponding particular instance, which is why at the end of the day no syllogism with such a major can ever count as a tool for genuine discovery.107 2.5.2.3 General Notions and the Advance of Knowledge However, this is far from being the end of the story. Descartes himself complicates matters when he claims, in the highly authoritative Principia, that ubi dixi hanc propositionem, ego cogito, ergo sum, esse omnium primam & certissimam, quæ cuilibet ordine philosophanti occurrat, non ideò negavi quin ante ipsam scire oporteat, quid sit cogitatio, quid existentia, quid certitudo; item, quod fieri non possit, ut id quod cogitet non existat, & talia ... (Princ. I, 10, AT VIII-1, 8; Descartes’ italics).

This prima facie directly contradicts the passage which provided the basis for the submitted interpretation of Descartes‫ ތ‬rationale against the heuristic pretensions of syllogistic reasoning where we read that si [quis existentiam ex cogitatione] per syllogismum deduceret novisse prius debuisset istam majorem, illud omne, quod cogitat, est sive existit; atqui 107

Two minor objections are perhaps to be addressed at this point to clear the stage. To begin with, one might complain that while Descartes‫ ތ‬point is easily established, it is worthless as it is the topics and not the syllogisms that are supposed to play the rôle of tools of discovery in Aristotle and his followers. However, Gaukroger has shown convincingly in his Cartesian Logic, 21–25 that the original heuristic function of the topics was dramatically obscured if not entirely lost from sight during the Middle Ages and the Humanist Renaissance. Second, someone might object that the reasoning is not hitherto concerned with valid syllogisms tout court but at most only with syllogisms involving at least one universal premise, positive or negative (the domain of relevance of the argument perhaps being even narrower, limiting itself just to the perfect or even only to the demonstrative syllogisms of Aristotle—see e.g. Gaukroger, Cartesian Logic, 15– 18; but we need not dwell at this issue now). However, Descartes‫ ތ‬reasoning can plausibly be supplied with a suppressed assumption that if any syllogisms had any chance at all to count as tools of scientific discovery then it would be syllogisms with at least one universal premise.

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Chapter Two profecto ipsam potius discit, ex eo quòd apud se experiatur, fieri non posse ut cogitet, nisi existat (Resp. 2, AT VII, 140; Descartes’ italics).

So how can Descartes hold, as he does in Princ. I, 10, that the general illud omne, quod cogitat, est sive existit108 is after all known (scire) before the particular cogito, ergo sum, and at the same time to insist that the cogito, ergo sum is not deduced syllogistically on the grounds that the general counterpart is not “noticed” (novisse) before the particular cogito, ergo sum? Descartes was directly challenged by Burman on this tricky point,109 and his response is as follows: Ante hanc conclusionem: cogito ergo sum [Burman’s italics], sciri potest illa major: quicquid cogitat, est [Bruman’s italics], quia reipsâ prior est meâ conclusione, et mea conclusio illâ nititur. Et sic in Princip. [Burman’s italics] dicit auctor eam præcedere, quia scilicet implicite [my emphasis] semper præsupponitur et præcedit; sed non ideo semper expresse et explicite [my emphasis] cognosco illam præcedere et scio ante meam conclusionem, quia scilicet ad id tantùm attendo quod in me experior, ut, cogito, ergo sum [Burman’s italics], non autem ita attendo ad generalem illam notionem, quicquid cogitat, est [Burman’s italics]; nam ... non separamus illas propositiones a singularibus, sed eas in illis consideramus; et hoc sensu verba hæc ... hîc citata [AT VII, 140] intelligi debent. (Burm., AT V, 147).

Descartes thus deploys, once again, the distinction between implicit and explicit apprehension110 to resolve the apparent inconsistency in question. General notions and propositions, quâ implicitly apprehended, precede and condition any explicit cognition of the corresponding particular instances that are reflexively intuited by a given mind; and this explicit cognition of the particulars in turn precedes and conditions any explicit cognition of the

108

It will be observed that quod fieri non possit, ut id quod cogitet non existat, which is the wording of Princ. I, 10, amounts roughly (modal context apart) to the contraposition of the positive general statement in AT VII, 140. 109 Burm., AT V, 147: “[AT VII, 140]: ex nullo syllogismo...—Sed annon contrarium ponitur, Princ. I, 10?” (Burman’s italics). 110 Descartes in fact commits himself to implicit scientia in his response to Burman. This fits poorly with his well-established usage of the term “scientia”, which is to denote a certain type of cognitiones and thus can hardly be employed univocally on the pre-reflexive level, which is the domain of the implicit, for reasons articulated in ch. 1, fn. 100. I amend Descartes‫ ތ‬usage in the following discussion.

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general items which are by default apprehended only implicitly.111 This development in terms of the “implicit–explicit” distinction allows one to understand Descartes‫ ތ‬motivation for complicating the issue in Princ. I, 10: it is his commitment to innateness (where, it will be remembered, the operation of implicit apprehension is at home in the first place) that presses him to insist on the primacy of general notions and propositions in the order of cognition. By the same token, his treatment of innate structures in terms of implicit apprehension helps him to hold without inconsistency in Resp. 2 that in so far as explicit cognitions are concerned, it is cognitions of particulars that come first. We thus arrive at a more adequate understanding of what exactly Descartes finds wrong with syllogisms in the domain of advancing knowledge: it is not simply that in (the relevant sort of) syllogisms general notions are set prior to the corresponding particular instances in the order of cognition—we have just seen Descartes himself is committed to this in a certain sense; rather, the problem has turned out to be connected with the fact that in syllogistic reasoning general notions are taken as primary in the mode of explicit cognition. Yet even this enhanced diagnosis is to be qualified as it rules out more than Descartes needs. For if things were left at this stage, the heuristic pretensions of syllogistic reasoning would be disproved only at the price of committing oneself to the claim that general notions in the explicit mode of cognition can play no rôle whatever in facilitating discoveries—and this is by no means what Descartes holds. To see this, consider first the following: [Q]uantùm ad principia communia et axiomata, exempli gratiâ, impossibile est idem esse et non esse [Burman’s italics], attinet, ea homines sensuales, ut omnes ante philosophiam sumus, non considerant, nec ad ea attendunt; sed quia tam clare sibi innata sunt, et quia ea in semetipsis experiuntur, omittunt, et non nisi confuse considerant, nunquam verò in abstracto et separata a materiâ et singularibus [my emphasis]. Si enim ita considerarent, nemo de iis dubitaret ..., quoniam ea ab eo, qui attente ad illa animadvertit, negari non possunt (Burm., AT V, 146).

Descartes clearly admits here that consideration of the general items in abstracto et separata a materiâ et singularibus is possible (and indeed that such consideration amounts to a desired end which philosophy is supposed to facilitate) and it can be safely extrapolated that such apprehension in abstracto 111

I am indebted to Murray Miles, “Descartes‫ތ‬s Method,” in Broughton and Carriero, Companion to Descartes 147–50 for perspicuous clarification of this vexed issue within the “implicit–explicit” and “general–particular” dimensions.

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amounts to explicit cognition of the principia communia et axiomata. Descartes is prone to ascribe to such general items, in so far as they are explicitly cognized, certain irreducible functions in his project of advancing knowledge. For “principia communia et axiomata” are beyond doubt synonymous with “communes notiones” in both the Principia and the Regulæ112 and Descartes states resolutely concerning the communes notiones that sunt veluti vincula quædam ad alias naturas simplices inter se conjungendas, & [earum] evidentiâ nititur quidquid ratiocinando concludimus (Reg. XII, AT X, 419; my emphasis).

However, these ratiocinando conclusiones cannot but amount to certain series of deductiones and we saw that according to Descartes at least some such series count as vehicles of discovery. Finally, the invocation of evidentia in the last quoted passage ensures that it is explicit cognition of the communes notiones that is at work in this context. Thus according to Descartes, general notions in the explicit mode of cognition, more precisely that subclass of them which he usually calls notiones communes, can and do play certain substantial rôles in facilitating discoveries. In view of all the complexities taken up in the discussion hitherto, we can finally appreciate what good is brought by the struggle with Descartes‫ތ‬ critique of syllogistic reasoning: the critique allows us to shine a bright light upon the perils of employing explicitly cognized general notions in the domain of advancing knowledge. We are in a position by now to see what, from Descartes’ perspective, is to be avoided: to wit, the employment of explicitly cognized general notions in facilitating discoveries concerning particulars such that the particulars the given discovery is professed to be about are the same as those which allowed the corresponding general notions to be explicitly cognized in the first place. Further, we are in a position by now to appreciate in which sense this is by no means the only way in which general notions can, according to Descartes, be employed in the context of facilitating discoveries.

112 Impossibile est idem simul esse & non esse is included in the sample list of communes notiones, sive axiomata in Princ. I, 49, AT VIII-1, 23–24. And the entire passage of Princ. I, 47–49 was shown above to amount to a later version of the classification of simple natures including communes notiones in Reg. XII, AT X, 419–20.

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2.6 Enumeration Although Descartes insists (as we saw) that only two operations of the understanding, viz. intuitus and deductio, can eventually ever bring the human mind to scientia, he introduces yet another operation in this and related contexts which he usually calls enumeratio and sometimes inductio.113 The status and rôles of this operation in Descartes‫ތ‬ epistemological project must now be clarified. The issue is vexed due to several factors, including Descartes‫ ތ‬carelessness and terminological shifts; stern generality of the exposition in the Regulæ (which remains by far the most informative source for reconstruing Descartes‫ ތ‬ideas in this area); and his decision to treat at least two quite different types of operations under the heading of enumeratio. Yet it must be addressed with diligent care since it has to do, in certain respects, with some challenging interpretive issues in the areas with which we are concerned. I begin with introducing the operation of enumeratio and distinguishing the aforementioned two meanings it comprises. Then I discuss the features which are to be attributed to any operation deserving the title of enumeration in the technical sense envisaged by Descartes, namely the features of being sufficient and being ordered. Finally, I make use of all this to clarify Descartes‫ ތ‬somewhat perplexing usage of the terms “enumeratio”, “inductio”, and “deductio” in this particular context.

2.6.1 Approbative and Heuristic Enumeration A general characterization of enumeratio can be found in Reg. VII, AT X, 388:114 Est igitur ... enumeratio ... eorum omnium quæ ad propositam aliquam quæstionem spectant, tam diligens & accurata perquisitio, vt ex illâ certò evidenterque concludamus, nihil à nobis perperàm fuisse prætermissum ....

Descartes makes it clear that this operation has properly to do with deductiones, more specifically that it is required “ad illas veritates inter certas admittendas, quas ... diximus à primis & per se notis principijs non immediatè deduci” (ibid., 387). His basic idea seems clear enough: as far 113 See fn. 124 for references. Descartes‫ ތ‬usage of enumeratio and inductio is discussed in sec. 2.6.3. 114 Cf. also DM 2, AT VI, 19: “Et le dernier [precepte estoit], de faire partout des denombremens si entiers, & des reueuës si generales, que ie fusse assuré de ne rien omettre.”

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as cognitive operations over and above intuitus and immediate deductiones are concerned, a specific type of structured review or survey of steps within complex deductiones is needed to ensure the certainty required for scientia.115 Enumeration in this general sense thus seems to be characterized aptly by Hannequin as an operation “qui condense en un rapport unique une somme de rapports et qui les organise.”116 However, matters are complicated due to the fact that even given this general definition, Descartes apparently speaks of enumeratio with several different meanings in mind. In a sense, this should come as no surprise once it is appreciated that rather than a determinate stage in the complex fabric of Descartes‫ ތ‬methodical procedure, in Descartes‫ ތ‬hands enumeratio is to be taken as a sort of recapitulation to which one should grow accustomed after having completed any of the determinate methodical stages in order to keep control over the epistemic constraints required by the aim at establishing a (new piece of) scientia. Of course, this interpretative claim cannot be fully vindicated until one goes through the particular stages of the methodical procedure. For the time being, however, a preliminary general classification of Descartes‫ ތ‬usage of the term “enumeratio” is in order. There are clear indications that putting to one side any further ramifications, Descartes has two quite different types of operations in mind when speaking of enumeratio. On the one hand, there is a type (call it approbative from now on) the purpose of which is to establish or to support or to approve of the (presumably metaphysical) certainty of a deductive mental motus which has already been completed, i.e. (as Descartes puts it later in Reg. XI) certainty of deductio in so far as deductio “motum nullum ampliùs designat, sed terminum motûs” (ibid., AT X, 408). Enumeratio in the approbative sense is described by Descartes as follows:117

115

Cf. Reg. VII, AT X, 389: “Nam quæcumque vna ex alijs immediatè deduximus, si illatio fuerit evidens, illa ad verum intuitum jam sunt reducta. Si autem ex multis & disjunctis vnum quid inferamus, sæpe intellectûs nostri capacitas non est tanta, vt illa omnia possit vnico intuitu complecti; quo casu illi hujus operationis certitudo debet sufficere.” 116 Arthur Hannequin, Études dҲhistoire des sciences et dҲhistoire de philosophie, vol. 1 (Paris: Félix Alcan, 1908), 229. Incidentally, Hannequin characterizes the operation at issue as “jugement”, which of course cannot be right in view of what was said above in ch. 1. 117 The passage is repeated with only minor changes in Reg. XI, AT X, 408–409.

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Si ..., ex. gr., per diversas operationes cognoverim primo, qualis sit habitudo inter magnitudines A & B, deinde inter B & C, tum inter C & D, ac denique inter D & E: non idcirco video qualis sit inter A & E, nec possum intelligere præcisè ex jam cognitis, nisi omnium recorder. Quamobrem illas continuo quodam [cogitationis]118 motu singula intuentis simul & ad alia transeuntis aliquoties percurram, donec à prima ad vltimam tam celeriter transire didicerim, vt ferè nullas memoriæ partes relinquendo, rem totam simul videar intueri ... (Reg. VII, AT X, 387–88).

The point is thus to eliminate, to the greatest possible extent, the part played by memory in implemented complex deductiones which are not yet reduced to intuitus. The proper objective of the exercise must be, on closer inspection, to ensure the continuity not so much of the “movement of thought” but rather of (presumably psychologically interpreted) metaphysical certainty within that “movement”: we have already seen that metaphysical certainty cannot be lost, if certain conditions are met, within immediate deductive steps in so far as they can be reduced to intuitus; and the enumeration of these immediate deductiones ensures that no evidential link within the entire series is omitted. On the other hand, Descartes sometimes uses enumeratio to refer to a type of operation the purpose of which is somehow to facilitate inventions of deductive series not yet in the possession of the given mind, and thus, in effect, to augment the extent of one‫ތ‬s scientia, with enumeratio in this sense henceforth to be called heuristic: Hîc præterea enumerationem requiri dicimus ad scientiam complementum: quoniam alia præcepta juvant quidem ad plurimas quæstiones resolvendas, sed solius enumerationis auxilio fieri potest, vt ad quamcumque animum applicemus, de illâ semper feramus judicium verum & certum, ac proinde nihil nos planè effugiat, sed de cunctis aliquid scire videamur (ibid., 388).

More specifically, it turns out that enumeration in this sense is (in Beck‫ތ‬s pertinent phrases) “the setting out ... of the conditions on which the solution of a quæstio or problem will depend, that is, a definite arrangement of antecedent data and premises ... in the initial stages of a scientific inquiry” (Beck, Method of Descartes, 119), i.e. “a preparatory 118

AT follows both A- and H-versions in reading imaginationis instead of cogitationis. I follow Crapulli in emending this place in accord with the N-version of the Regulæ. The amendment accords well with all the other places in the Regulæ in which motus ingenij is employed in connection with deduction: cf. Reg. VII, AT X, 387.12; 21; Reg. XI, AT X, 407.4; 408.25; 409.4.

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marking-out of the field of knowledge in which a proposed investigation of some particular problem is presently to take place” (ibid., 130). Heuristic enumeration thus seems to amount to the very first methodical step recommended by Descartes in the course of augmenting the range of one‫ތ‬s scientia. There is no doubt that although the two types of enumeration share the general character mentioned at the beginning of the present section, they embrace two different operations with different aims and presumably with specific requirements. Indeed, it will be observed that the two meanings of enumeratio correspond to the two régimes of the method distinguished above, and Descartes confirms this in Reg. XI, AT X, 408: Cujus rei [sc. explicandi quo pacto intuitus & enumeratio se mutuò juvent & perficiant] duplicem vtilitatem designamus: nempe ad conclusionem, circa quam versamur, certiùs cognoscendam, & ad ingenium alijs inveniendis aptius reddendum.

Yet for good or ill, Descartes resolves to run enumeratio in both senses in a single harness, which causes predictable interpretative problems. In particular, it is difficult to discriminate which of several abstract statements concerning enumeratio are meant by Descartes to apply to both types and which to just one or the other. What is uncontroversial, however, is that most of what Descartes has to say concerning enumeratio is organized around the features of sufficiency and ordering, which he deploys as the features that jointly specify any given review of steps as an enumeratio in the intended technical sense.119 My general interpretative stance is that Descartes intends these features to apply to enumerationes in both senses, simply on the grounds that there seems to be nothing in Descartes‫ ތ‬texts to preclude such a reading.120 Yet, of course, the 119

See in particular Reg. VII, AT X, 387: “Ad scientiæ complementum oportet omnia & singula, quæ ad institutum nostrum pertinent, ... sufficienti & ordinatâ enumeratione complecti.” Ibid., AT X, 389–90: “Sufficientem [enumerationem] esse debere dixi .... Addidi etiam, enumerationem debere esse ordinatam ....” Reg. X is devoted to “ordinis observatio” (verbatim at AT X, 404), and in Reg. XI Descartes explains that Reg. X is a treatment “de enumeratione solâ” (AT X, 408). 120 I differ in this from, for example, Beck, who takes sufficiency and ordering as attributes of enumeration in the heuristic sense only (see his Method of Descartes, 131). However, it seems to me that in Reg. VII, AT X, 389.26–390.2 sufficientia is also clearly associated with the approbative meaning, due to employment of the metaphor of catena which is associated with the approbative meaning throughout Reg. VII and beyond: “Sufficientem hanc operationem esse debere dixi, quia sæpe defectiva esse potest .... Interdum enim, etiamsi multa quidem enumeratione

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differences between the two types are likely to impart differences into the exact significance of sufficientia and ordo within each of them and it is these specific significances that might be of interest to us.

2.6.2 Enumeration, Sufficiency, and Ordering Let us begin with the feature of sufficiency. Descartes remarks generally that per sufficientem enumerationem sive inductionem, nos tantùm illam intelligere, ex quâ veritas certiùs concluditur, quàm per omne aliud probandi genus, præter simplicem intuitum (Reg. VII, AT X, 389).

Thus it is in any case not just any (presumably ordered) list of items that Descartes wants: the lists must make a contribution to attaining scientiæ to qualify as enumerationes in the sense Descartes envisages. On the other hand, this requirement is virtually all that the intended meaning of the attribute of sufficientia seems to convey. Descartes makes it clear enough in Reg. VII that this vagueness, indeed virtual emptiness, of the notion of sufficiency is intentional; he clearly has it that the criteria of sufficiency are, in general, relative to both the nature of the task to which enumeration is applied and the cognitive situation of the cognizing mind: [I]nterdum enumeratio hæc esse debet completa, interdum distincta, quandoque neutro est opus; ideoque dictum tantùm est, illam esse debere sufficientem. Nam si velim probare per enumerationem, quot genera entium sint corporea, sive aliquo pacto sub sensum cadant, non asseram illa tot esse, & non plura, nisi priùs certò noverim, me omnia enumeratione fuisse complexum, & singula ab invicem distinxisse. Si verò eâdem via ostendere perlustremus, quæ valde evidentia sunt, si tamen vel minimum quid omittamus, catena rupta est, & tota conclusionis labitur certitudo.” It is also suggested in Beck, Method of Descartes, 119 that “denombremens entiers and reueuës generales” of DM 2, AT VI, 19 are intended by Descartes to refer, respectively, to the heuristic and approbative senses of enumeration. Beck supports his reading with the Latin version of the fourth precept (the signs in square brackets are added by me): “Ac postremum, ut [A] tum in quærendis mediis, [B] tum in difficultatum partibus percurrendis, [a] tam perfectè singula enumerarem & [b] ad omnia circumspicerem, ut nihil à me omitti essem certus” (AT VI, 550). He clearly has it that [A] refers to what I have called heuristic enumeration and [B] to what I have called approbative enumeration, which I readily endorse. However, he suggests that [A] is to be correlated with [a] and [B] with [b], whereas I believe both [a] and [b] are to be correlated with [A] as well as with [B].

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The passage is not very helpful as regards the criteria of sufficiency in the case of approbative enumeration since the adduced examples seem to bear imprimis on the case of enumeration in its heuristic variety and there seems to be no other passage from which one could learn more on this score. One can at most conjecture, in view of what has been said of approbative enumeration up to the present, that even in this case neither the requirement of completeness—in the sense of picking out distinctively each single minimal step—nor the requirement of distinctness is absolutely essential, for different minds are able to grasp intuitively different segments of the implemented deductive series; and it seems to do no harm if the enumerated segments overlap in the course of evidential approbation. The situation seems somewhat more satisfactory (though far from crystal clear) as regards the criteria and significance of sufficientia in the case of heuristic enumeration, although the texts are to be pushed by interpretation a good deal. My tentative suggestion is based on Descartes‫ތ‬ general discussion in Reg. XIII of how quæstiones are to be treated. It reads that sufficientia as attributed to heuristic enumeration denotes the fact that a given enumeration renders the quæstio at issue “perfect” or “perfectly understood”. Nevertheless, it will be well to postpone a detailed discussion of this topic until later. To deal with the topic of ordering in the context of Descartes‫ ތ‬general account of enumeratio, we should start with a simple yet important and neglected distinction between two main contexts in which ordo is put to use in Descartes‫ ތ‬overall project. On the one hand, there is the context of discoveries proper. Here what are properly ordered are quæstiones put in terms of the data, i.e., at the bottom, in terms of previously available intuitus and (eventually immediate) deductiones; and the goal of this type of ordering is to establish a system of relations of cognitive dependence such that it extends our cognition “ad hoc, vt percipiamus rem quæsitam participare hoc vel illo modo naturam eorum quæ in propositione data sunt” (Reg. XIV, AT X, 438), the natura amounting to what in Reg. VI is

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picked out as the absolutum of a given ordered series.121 This procedure, which Descartes usually calls analysis and upon which his entire method in the heuristic régime hinges, will be analyzed and discussed in detail in the following chapters. What matters now is the contrast with another context, namely that of supplying adequate sets of initial intuitus and deductiones that are to be subjected thereafter to the ordering procedure just described, or else of supplying instances to be considered in order that one can cope with some ordered question in the course of solving a given problem. The operation designed to carry this out is, of course, heuristic enumeration (while the adequacy of the sets is a function of the sufficiency of the corresponding enumerations). And it should be clear by now that when Descartes speaks of ordo and ordering in this context, he cannot mean quite the same thing as when he speaks of it in the context of discovery proper. At the very minimum, the requirement (which occurs particularly in Reg. VII and X) that ordo be put to use in order to provide the items to be treated thereafter in the analytic operation of disclosure is logically independent of, and therefore distinct from, the requirement (which occurs particularly in Reg. V and VI) that ordo be put to use in the analytic operation itself. The difference is confirmed in Reg. VIII, AT X, 392–93:122 Tres regulæ præcedentes ordinem præcipiunt & explicant; hæc autem ostendit, quandonam sit omnino necessarius, quando vtilis tantùm. Quippe quidquid integrum gradum constituit in illâ serie, per quam à respectivis ad absolutum quid, vel contra, veniendum est, illud necessariò ante omnia quæ sequuntur est examinandum. Si verò, vt sæpe fit, multa ad eumdem gradum pertineant, est quidem semper vtile, illa omnia perlustrare ordine. Hunc 121

See Reg. VI, AT X, 381–83; cf. also the passage on the anaclastic line in Reg. VIII, AT X, 393–95 which relates absolutum of the series to Reg. V and strongly suggests it is ultimately simple natures that fulfil the rôles of the absoluta. Descartes suggests in Reg. VII that in many cases of quæsita “ex levioribus hominum artificijs”, hierarchization in terms of absoluta and respectiva is unnecessary since in such cases “tota methodus in ... ordine disponendo consistit” (AT X, 391); examples are devising anagrams (ibid.) and decoding ciphers (Reg. X, AT X, 404–405). I take this as a special, degenerative case of analysis and will ignore it from now on. 122 It has been a commonplace since Jean-Paul Weber, La Constitution du texte des Regulæ (Paris: Société d‫ތ‬édition d‫ތ‬enseignement supérieur, 1964) that the quoted passage is in fact a misplaced continuation of Reg. VII (cf. Weber, Constitution des Regulæ, 87–91) but this conjecture has no significance for us in the present context. For a brief survey of Weber’s analysis of different layers in the extant versions of Reg. VIII, see ch. 4, fn. 104.

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Chapter Two tamen ita strictè & rigide non cogimur observare, & plerumque, etiamsi non omnia, sed pauca tantùm vel vnicum quid ex illis perspicuè cognoscamus, vlteriùs tamen progredi licet.

Given that the first case refers to the steps within an analytical procedure of discovery and the second eventually to heuristic enumerations (more precisely to subsets of intuitus and deductiones supplied by heuristic enumeration distributed in the course of the given analysis), the passage is plain sailing. Besides heuristic enumeration, Descartes‫ ތ‬recommendation to observe some ordo seems also to hold for the case of (presumably complex) approbative enumeration. This is confirmed with the following passage in which Descartes refers to defects with which (as he warned earlier in Reg. VII) both approbative and heuristic cognitive procedures could be inflicted; the passage contains virtually all Descartes has to say on the rôle of ordo in enumerations, of both heuristic and approbative cast: Addidi etiam, enumerationem debere esse ordinatam: tum quia ad jam enumeratos defectus nullum præsentius remedium est, quàm si ordine omnia perscrutemur; tum etiam, quia sæpe contingit vt, si singula, quæ ad rem propositam spectant, essent separatim perlustranda, nullius hominis vita sufficeret, sive quia nimis multa sunt, sive quia sæpiùs eadem occurrerent repetenda. Sed si omnia illa optimo ordine disponamus, vt plurimùm, ad certas classes reducentur, ex quibus vel vnicam exactè videre sufficiet, vel ex singulis aliquid, vel quasdam potiùs quàm cæteras, vel saltem nihil vnquam bis frustra percurremus; quod adeò juvat, vt sæpe multa propter ordinem benè institutum brevi tempore & facili negotio peragantur, quæ prima fronte videbantur immensa (Reg. VII, AT X, 390–91).

The advice to proceed in an orderly manner when dealing enumeratively with complicated issues that encumber memory and/or with assessing the relevance of the data with respect to a given quæstio is no doubt sound. However, what one would like to learn above all is, of course, the rules that would enable one to obey that advice, i.e. to learn the rules the observation of which would enable one to find or devise123 the appropriate order for any particular case. Putting to one side the arguably less momentous case of approbative enumeration, many critics have complained ever since the Discours was published that Descartes failed to 123

Cf. Reg. X, AT X, 404: “[Methodus] in ... levioribus [quæstionibus] non alia esse solet, quàm ordinis, vel in ipsâ re existentis, vel subtiliter excogitati, constans observatio ....”

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address this crucial challenge with regard to heuristic enumeration, as well as with regard to the wider task of devising an appropriate order in the course of the entire analytical procedure of discovery (enumeration being the first stage of this procedure). This general failure professedly renders the value of Descartes‫ ތ‬proclaimed method doubtful at best. This challenge—of providing a sort of step-by-step manual to obey the general methodical advice concerning the analytical procedure of discovery—will form an important part of the context of an account of Descartes‫ ތ‬universal method in ch. 4, and it will be well to postpone the discussion of this crucial point until then. For the time being, be it enough to notice that (as I will be at pains to show in ch. 4) Descartes does confront the challenge of providing “le moyene d’observer” his general methodical precepts concerning the application of appropriate orders, however difficult it might prove to extract those means from the extant portions of his Regulæ and other related texts.

2.6.3 Inductio, Deductio, and Enumeratio “Inductio” is employed by Descartes as a synonym for “enumeratio” three times in the Regulæ.124 One might be tempted to pass over it as just a casual use of words on Descartes‫ ތ‬part. Yet there are two related reasons for stopping to consider what exactly is going on here. Firstly, the phrase “enumeratio sive inductio” always occurs where Descartes seems to suggest it is enumeratio, and not deductio, that is to count as the proper operation which together with intuitus exhausts the list of possible ways of gaining scientia.125 Secondly, when Descartes lists “omnes intellectûs nostri actiones, per quas ad rerum cognitionem absque vllo deceptionis metu possimus pervenire” (Reg. III, AT X, 368) for the first time in the Regulæ, the operation that accompanies intuitus is called inductio and not

124

The phrase “enumeratio sive inductio” occurs twice in Reg. VII, AT X, 388–89 and once in Reg. XI, AT X, 408. 125 Cf. Reg. VII, AT X, 389: “Notandum præterea, per sufficientem enumerationem sive inductionem, nos tantùm illam intelligere, ex quâ veritas certiùs concluditur, quàm per omne aliud probandi genus, præter simplicem intuitum; ad quem quoties aliqua cognitio non potest reduci, omnibus syllogismorum vinculis rejectis, superest nobis vnica hæc via, cui totam fidem debeamus adhibere.” Reg. XI, AT X, 407–408: “Si verò ad [deductionem], vt jam facta est, attendamus, ... tunc nullum motum ampliùs designat, sed terminum motus, atque ideo illam per intuitum videri supponimus, quando est simplex & perspicua, non autem quando est multiplex & involuta; cui enumerationis, sive inductionis nomen dedimus ....”

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deductio.126 The discussion of how enumeratio, inductio and deductio are related in Descartes‫ ތ‬thought is thus likely to shed further light on the nature and rôle of discursive cognitive operations in his epistemological project. Descartes acknowledges the relations between enumeratio, inductio and deductio are somewhat unclear at the beginning of the explication of Reg XI and promises to clarify the issue.127 Here is his explanation: [A]d mentis intuitum duo requirimus: nempe vt propositio clarè & distinctè, deinde etiam vt tota simul & non successivè intelligatur. Deductio verò, si de illâ faciendâ cogitemus, vt in regulâ tertiâ, non tota simul fieri videtur, sed motum quemdam ingenij nostri vnum ex alio inferentis involvit; atque idcirco ibi illam ab intuitu jure distinxerimus. Si verò ad eamdem, vt jam facta est, attendamus, sicut in dictis ad regulam septimam, tunc nullum motum ampliùs designat, sed terminum motus, atque ideo illam per intuitum videri supponimus, quando est simplex & perspicua, non autem quando est multiplex & involuta; cui enumerationis, sive inductionis nomen dedimus, quia tunc non tota simul ab intellectu potest comprehendi, sed ejus certitudo quodammodo à memoriâ dependet, in quâ judicia de singulis partibus enumeratis retineri debent, vt ex illis omnibus vnum quid colligatur (ibid., AT X, 407–408).

Descartes thus draws upon the distinction between deductio as a process and as a result or product of a process. He makes it clear that with regard to deductio in the process sense, the proper contrast is indeed between intuitus and deductio as explained in Reg. III—for the distinctive feature of the operation denoted with deductio is the motus ingenij (Reg. XI, AT X, 407), i.e. the process by which, as we saw, metaphysical certainty is established in the order of succession. Once deductio is considered as a result, however, it is—Descartes suggests—either viewed per intuitum, or (in such cases as it remains, quâ the result, multiplex & involuta) it is identified with enumeratio sive inductio. He thus seems to suggest— plausibly I think—that intuitus–deductio is the proper contrast in so far as deductio is considered as a process; and that once deductio is considered as a result, the proper contrast is intuitus–enumeratio sive inductio. For in 126

Various approaches in the scholarly literature to this prima facie somewhat puzzling fact are presented in fn. 3. I will deal with it in a moment. 127 See Reg. XI, AT X, 407: “Hîc est occasio clariùs exponendi quæ de mentis intuitu antè dicta sunt, ad regulas tertiam & septimam: quoniam illum vno in loco deductioni opposuimus, in alio verò enumerationi tantùm, quam definivimus esse illationem ex multis & disjunctis rebus collectam; simplicem verò deductionem vnius rei ex alterâ ibidem diximus fieri per intuitum.”

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this latter case, either the given deductio is reduced stepwise to intuitus in the way with which we are already familiar or it must be reconsidered (perhaps iteratively) by way of enumeratio in order that the desired certainty is secured. The related passage in Reg. VII points to the same effect once it is supposed that there deductio is considered as a result: [A]d [intuitum] quoties aliqua cognitio non potest reduci, omnibus syllogismorum vinculis rejectis, superest nobis vnica hæc via [enumerationis sive inductionis], cui totam fidem debeamus adhibere. Nam quæcumque vna ex alijs immediatè deduximus, si illatio fuerit evidens, illa ad verum intuitum jam sunt reducta. Si autem ex multis & disjunctis vnum quid inferamus, sæpe intellectûs nostri capacitas non est tanta, vt illa omnia possit vnico intuitu complecti; quo casu illi hujus operationis certitudo debet sufficere (AT X, 389).

It is important to appreciate that deductio in the result sense is indeed to be identified with enumeratio sive inductio; for what is at issue there is no more establishing certainty by way of continuous motus ingenij motivated by discovered necessities between intuitively perceived terms but rather preserving the certainty thus established by way of a complete survey of the successive steps of the terminated process with the aid of memory (in the Regulæ) or memory plus the benevolent God (in the mature Descartes). There is succession and duration implied in this operation, to be sure; but the transition from one term to another need not be continuous nor is it constitutive of the certainty which is the proper objective of the entire enterprise. Perhaps the most important corollary of the present discussion is that enumeratio sive inductio amounts to the proper opposite of intuitus exactly as long as deductio is considered in the result sense and consequently as long as the approbative régime of Descartes‫ ތ‬scientific enterprise is concerned. In view of this, we are finally in a position to understand how the mutual relations between deductio, enumeratio and inductio in Descartes‫ ތ‬thought are to be handled. In so far as the régime of discovery is concerned, the only two ways of attaining scientia are intuitus and deductio, the latter being understood in the process sense. Enumeratio is employed here in its heuristic variety as a subsidiary cognitive device and is to be identified neither with deductio nor with inductio; indeed, according to the submitted reading inductio plays no part in the régime of discovery. In so far as the régime of approbation is concerned, however, the only two ways to attain (or perhaps more precisely, to conserve) scientia are intuitus and enumeratio sive inductio, i.e. enumeration in the approbative sense, which for Descartes coincides with deductio in the

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result sense. As a result, it can readily be seen that while Descartes‫ތ‬ contrasting intuitus to inductio in his very first list in the Regulæ of the “intellectûs nostri actiones, per quas ad rerum cognitionem absque vllo deceptionis metu possimus pervenire” (Reg. III, AT X, 368) need not be amended, it does not qualify as a perfectly general rendering of the situation as viewed by Descartes himself. The adequate, complete list of possible ways to scientia should read as follows: intuitus and deductio in both the process and the result sense (being identified with enumeratio sive inductio in the latter sense); or—which comes to the same thing— intuitus, deductio and enumeratio sive inductio, the latter being identified with deductio in the result sense. The question of what might have moved Descartes to call enumeration in the approbative sense inductio remains to be discussed. It is well known that “inductio” is the term originally chosen by Cicero to translate the term “ਥʌĮȖȦȖȒ” as employed in Aristotle‫ތ‬s Organon, and it is most probable that if anything, it is the Aristotelian usage that is relevant to our present question. It has commonly been observed that Aristotle seems to associate the term with several distinct cognitive operations, “having only this in common, that in all there is an advance from one or more particular judgements to a general one”:128 (1) an advance from (relative) particulars —individuals or species—that have induced a more or less probable conjecture concerning a general principle; (2) an advance from particular sense perceptions to a direct insight of a general principle, the so-called intuitive induction; and (3) a deductive proof whose middle term is a set of individuals or species129 and which is devised to establish the major premise of a demonstrative syllogism, the so-called perfect induction.130 Descartes‫ ތ‬general notion of what enumeratio sive inductio is supposed 128

William Ross, Introduction and Commentary to Prior and Posterior Analytics by Aristotle (Oxford: Oxford University Press, 1949), 48. 129 Aristotle in fact discusses only the case of a set of species (cf. An. Pr. II, 23) but his discussion can easily and unproblematically be expanded to the case of a set of individuals—see e.g. John McCaskey, “Freeing Aristotelian EpagǀgƝ from Prior Analytics II 23,” Apeiron 40, no. 4 (2007): 347. 130 The three-part classification, as well as the terms “intuitive induction” and “perfect induction,” are taken from a classical discussion by Ross, Introduction and Commentary to Prior and Posterior Analytics, 47–51, in which the references to the loci in Aristotle‫ތ‬s corpus are listed. The present study is not, of course, the place to provide a substantive discussion of Aristotle and the Aristotelian notion of inductio, and it is not necessary for our purposes to decide whether Ross is right in holding that the three types of inductio in Aristotle are distinct operations or whether Aristotle‫ތ‬s notion of inductio can be shown to be more unified (as e.g. McCaskey has argued recently in his “Freeing Aristotelian EpagǀgƝ”).

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to carry out comes indeed close to what is identified by Ross as the only common feature of Aristotelian inductio in all three senses: for Descartes, too, the point of enumeratio sive inductio is “vt ex [omnibus judiciis de singulis partibus enumeratis] vnum quid colligatur” (Reg. XI, AT X, 408) or to infer “ex multis & disjunctis vnum quid” (Reg. VII, AT X, 389). In searching for more specific affinities, we can safely put to one side sense (1) as inductio in this sense is clearly no less incapable of yielding scientia by Aristotle‫ތ‬s131 than by Descartes‫ ތ‬standards. Aristotle‫ތ‬s notion of intuitive induction seems to be more relevant. It is deployed by Aristotle as a device to escape Meno‫ތ‬s paradox regarding the knowledge of first principles and is identified with the operation of arriving at a single intuition (ȞȠ૨Ȣ) of a first principle by way of the iterative consideration of a number of singular sense perceptions which are typically (though arguably not essentially) retained in memory.132 However, the most likely inspiration for Descartes‫ ތ‬terminological choice comes from Aristotle‫ތ‬s notion of perfect induction. Aristotle famously discusses the case in An. Pr. II, 23. Here he considers the inference (i) C1, C2, C3 etc. are A; (ii) C1, C2, C3 etc. are B; therefore (iii) all B‫ތ‬s are A, and correctly remarks that in order that the inference be valid, premise (ii) must be rendered convertible by way of complete enumeration of the singular C‫ތ‬s, so that (ii) is rendered equivalent to All B‫ތ‬s are either C1 or C2 or C3 etc. It is here if anywhere, I submit, that the source of Descartes‫ ތ‬decision to use “inductio” interchangeably with “enumeratio” in certain contexts is to be located. These affinities being acknowledged, as far as I can see there still remains an unbridgeable gulf between the Aristotelian and Cartesian notions of inductio. Putting to one side several other differences (e.g. that the vnum quid at which Descartes‫ ތ‬enumeratio sive inductio is aimed need not amount to a general principle), the decisive fact is that while Descartes, as we saw, locates his enumeratio sive inductio unambiguously and exclusively within the régime of approbation, in Aristotle both intuitive and perfect induction clearly fall within the régime of discovery: according to him intuitive induction is designed to discover certain first principles of scientia and perfect induction is designed to establish the major premises of scientific syllogisms. Embarrassing as it might sound, I therefore conclude that the connection between Aristotle‫ތ‬s notions of intuitive and perfect induction on the one hand and the way in which the term “inductio” is employed by Descartes on the other hand is so loose 131

Cf. Ross, Introduction and Commentary, 48. Cf. Aristotle, An. Post. II, 19; it is probably this passage that Aristotle refers to in EN I, 7; VI, 3. 132

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that while it might perhaps explain Descartes‫ ތ‬decision to use the term as he does, no real interpretative profit can be gained from any attempt to push any further the affinities between the two thinkers on this score.133

133 This also affects Marion‫ތ‬s ingenious attempt to integrate Descartes‫ ތ‬usage of the clause “enumeratio sive inductio” into a complex account of Descartes‫ތ‬ substantive departure from Aristotle in the fields of ontology and epistemology, in Marion, Sur l’ontologie grise de Descartes: Science Cartésienne et savoir Aristotelicien dans les Regulae (4th ed. Paris: J. Vrin, 2000), §§ 16–17. Marion notices that Aristotle‫ތ‬s account of perfect induction is designed above all to show what is to be fulfilled in order that induction qualify as a genuinely scientific cognitive operation—viz. complete enumeration that would render premise (ii) above convertible—and that this very dependence of induction on enumeration in this respect renders induction essentially an opposite of (scientific) syllogism from Aristotle‫ތ‬s own perspective (see ibid., 102, and the reference to An. Pr. II, 23, 68b33). Considering that Descartes has independent grounds for dismissing syllogisms as vehicles of scientific cognition and acknowledges the dependence of intuitus quâ scientific vehicle on enumeration, Marion suggests that Descartes arrives at enumeratio sive inductio as the very operation to which deductive operations in the field of scientiæ reduce. Marion‫ތ‬s interpretation finds some support in the above-quoted passage from Reg. VII, AT X, 389 and it might have shed further light on the rôle of sufficient enumeration in the régime of discovery in Descartes‫ ތ‬hands. Unfortunately, however, it can hardly stand up to the textual evidence I adduced to the effect that according to Descartes‫ ތ‬explicit statements enumeratio sive inductio has its place only in the régime of approbation. For the rest, it is noteworthy, and somewhat disconcerting, that Descartes does not use the term “inductio” to refer to the operation which comes closest to Aristotle‫ތ‬s intuitive induction, viz. the above discussed consideration of a general proposition within single particular instances actually intuited by the mind. I do not know how to explain all the discrepancies and omissions we have come across concerning Descartes‫ ތ‬employment of the term “inductio”.

CHAPTER THREE THE A PRIORI IN DESCARTES: THE MATHEMATICAL LINE

The time has come now to take up the central task of the present study, viz. to determine and explicate the meaning(s) Descartes associates with the term “a priori”. As indicated in the Introduction, I assumed that the term “a priori” properly characterizes, in Descartes‫ ތ‬hands, cognitive operations or processes the proper result of which is scientia in the sense thoroughly clarified and discussed in the previous chapters; and we are in a position by now not only to verify this assumption but also to make use of the connection between scientia and the a priori in determining and clarifying the content Descartes is likely to associate with this term. It was also stated in the Introduction that Descartes seems to draw upon two distinct intellectual strains when employing the a priori–a posteriori pair in his writings, namely the mathematical strains that include a Classical tradition of mathematical analysis with modern developments in algebra on the one hand, and the Aristotelian conception of scientific reasoning on the other; and that the complex and challenging question of the relations and respective rôles of both those strains in the constitution of Descartes‫ތ‬ conception of the a priori and the a posteriori is far from resolved in any consensual way. This and the following chapter are devoted to the mathematical strains, which I take as more fundamental and dominant in Descartes‫ ތ‬thinking on the issue. The Aristotelian part will be taken up in ch. 5.

3.1 Links to Mathematical Contexts The most comprehensive passage that can plausibly be interpreted as pointing towards reading the a priori–a posteriori pair along distinctively mathematical lines is the Latin version of a well-known passage from Resp. 2, in which Descartes (prompted by a proposal to set out his

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reasoning in the Meditationes more geometrico)1 discusses what he calls the modus scribendi geometricus in the context of assessing the prospects of its application to matters metaphysical. It is worth quoting at length:2 Duas res in modo scribendi geometrico distinguo, ordinem scilicet, & rationem demonstrandi. Ordo in eo tantùm consistit, quòd ea, quæ prima proponuntur, absque ullâ sequentium ope debeant cognosci, & reliqua deinde omnia ita disponi, ut ex præcedentibus solis demonstrentur. ... Demonstrandi autem ratio duplex est, alia scilicet per analysim, alia per synthesim. Analysis veram viam ostendit per quam res methodice & tanquam a priori inventa est, adeo ut, si lector illam sequi velit atque ad omnia satis attendere, rem non minus perfecte intelliget suamque reddet, quàm si ipsemet illam invenisset. Nihil autem habet, quo lectorem minus attentum aut repugnantem ad credendum impellat; nam si vel minimum quid ex iis quæ proponit non advertatur, ejus conclusionum necessitas non apparet .... Synthesis è contra per viam oppositam & tanquam a posteriori quæsitam (etsi sæpe ipsa probatio fit in hac magis a priori quàm in illâ) clare quidem id quod conclusum est demonstrat, utiturque longâ definitionum, petitionum, axiomatum, theorematum, & problematum serie, ut si quid ipsi ex consequentibus negetur, id in antecedentibus contineri statim ostendat, sicque a lectore, quantumvis repugnante ac pertinaci, assensionem extorqueat; sed non ut altera satisfacit, nec discere cupientium animos explet, quia modum quo res fuit inventa non docet. Hac solâ Geometræ veteres in scriptis suis uti solebant, non quòd aliam plane ignorarent, sed, quantum judico, quia ipsam tanti faciebant, ut sibi solis tanquam arcanum quid reservarent. Ego verò solam Analysim, quæ vera & optima via est ad docendum, in Meditationibus meis sum sequutus; sed quantum ad Synthesim, quæ procul dubio ea est quam hîc a me requiritis, etsi in rebus Geometricis aptissime post Analysim ponatur, non tamen ad has Metaphysicas tam commode potest applicari.

1

Obj. 2, AT VII, 128: “Quamobrem fuerit operæ pretium, si ad tuarum solutionum calcem, quibusdam defînitionibus, postulatis & axiomatibus præmissis, rem totam more geometrico, in quo tantopere versatus es, concludas, ut unico velut intuitu lectoris cujuscunque animum expleas, ac ipso numine divino perfundas.” 2 I deliberately ignore, for the time being, Clerselier’s French translation, authorized by Descartes, of the quoted passage. This is because in a somewhat partisan manner Clerselier seems to force the meaning of a priori and a posteriori in the present passage into the distinctively Aristotelian usage which will not be taken up until ch. 5. I postpone the discussion of this important complication until then.

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Hæc enim differentia est, quòd primæ notiones, quæ ad res Geometricas demonstrandas præsupponuntur, cum sensuum usu convenientes, facile a quibuslibet admittantur. Ideoque nulla est ibi difficultas, nisi in consequentiis rite deducendis; quod a quibuslibet etiam minus attentis fieri potest, modò tantùm præcedentium recordentur .... Contrà verò in his Metaphysicis de nullâ re magis laboratur, quàm de primis notionibus clare & distincte percipiendis (AT VII, 155–57; my emphases).

This passage is by far the most informative single direct source of how the a priori–a posteriori pair is to be understood from Descartes‫ ތ‬perspective. Descartes trades here on the distinction a priori–a posteriori in his attempt at delineation of the difference between two demonstrandi rationes which he calls analysis and synthesis, respectively, and he seems to intend the exposition to be considerably general since he apparently intends the distinction in question to apply equally at least in the fields of metaphysics and mathematics.3 To be sure, several apparently difficult questions of interpretation emerge at first glance from the very start. For instance, Descartes qualifies both aforementioned terms with tanquam, an adverb which in his hands usually signals either that he wishes to shift the established meaning of a given term, or else that he does not understand the term literally. Moreover, with regard to synthesis the situation is somewhat perplexing as Descartes mentions both a priori and a posteriori in connection with this demonstrandi ratio in a way which is anything but clear. However, putting these complications to one side for the present,4 a few tolerably determinate features, however general and rudimentary, can be discerned in Descartes‫ ތ‬statements concerning the relations between both rationes demonstrandi on the one hand and the a priori–a posteriori pair on the other hand, once we focus on the analytic demonstrative procedure. Firstly, analysis as a general ratio demonstrandi is unambiguously associated exclusively with the a priori characterization, so that it seems to hold minimally that (x is discovered by way of analysis o x is cognized a priori). Secondly, the a priori character of the cognitions gained by way of analysis seems to be somehow associated with the analytical procedure 3

The moment of ordo as treated here, i.e. as described in terms of relations of epistemic dependence, seems to have in itself no substantial bearing on our present question. This is because presumably both analysis and synthesis are to observe the mentioned requirements of order lest they fail to count as the geometrical rationes in the first place. 4 I discuss them in ch. 5.

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being suited, unlike synthesis, to discoveries or inventions in a methodical way as understood by Descartes, i.e. with its being identified with the right methodical procedure in the régime of discovery, so that it seems to hold that (x is discovered by the right method l x is discovered by way of analysis). Thirdly, while Descartes states unambiguously that synthesis is to take as its starting point what he calls primæ notiones (and what we treated in the previous chapter under the name of simple natures), he implies quite clearly that one of the moments of the analytic search performance is to establish—by way of discovery, sometimes at least—a cognition of those very same primæ notiones or simple natures, the cognition of which, as we already know, cannot but amount to scientia once at all achieved. In view of the overall situation thus surveyed, it seems advisable to focus first on the apparently more clear-cut connections between analysis and the a priori in addressing the declared central task of the present study, and to integrate the interpretation of the a posteriori and/or synthesis only thereafter, perhaps—presumably—by way of extrapolation. This is the interim basic strategy I adopt for the present chapter and indeed—for reasons that shall emerge in the course of my argument— throughout the rest of the book. In view of all this, what the present passage suggests with regard to our central task—determining the meaning(s) of the a priori–a posteriori pair—is that first of all we must clarify, according to Descartes, (i) of what the procedure of analysis properly consists, (ii) in which sense analysis, in this meaning, deserves the title of the true method of discovery, and (iii) what exactly is the rôle that the simple natures are supposed to play in the constitution of the analytical method of discovery. Now I submit (in accord with numerous commentators) that it is the distinctively mathematical contexts that provide the most promising interpretative keys with regard to all these tasks. The present passage alone could scarcely warrant such a strong interpretative suggestion, to be sure: all it indicates in this respect is that (a) the procedures of analysis and synthesis, and by implication the characterizations of a priori and a posteriori, from Descartes’ standpoint are also applicable in the domain of geometry, and (b) that both procedures in question were already in use by certain Geometræ veteres (though they are said to have kept analysis “tanquam arcanum quid”). Yet these vague hints can assist in establishing the sought-after links in a much stronger way once they are supplied with certain other passages in Descartes‫ ތ‬writings, for these supplementary passages help to establish the following points: (1) Descartes considered mathematics (of a certain cast, to be specified) as not just one among

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several domains in which analysis, as the right method of discovery, could and should be put to use but even as providing the best paradigm of how the method of analysis should be employed in any domain whatsoever, the paradigm which should allow, by extrapolation, for derivation of the methods of investigation in the other domains. (2) The attributes of methodical procedure and of analytical procedure, and a priori and a posteriori as designations of the results of those procedures, are employed by Descartes in purely mathematical contexts in the same way as they are employed by him more generally in the quoted piece of Resp. 2. (3) By the Geometræ veteres Descartes means most probably the thinkers of the “golden era” of ancient Greek mathematics whose achievements are collected and developed above all in the Mathematicæ Collectiones of Pappus of Alexandria; and Descartes acknowledges affinities and some continuity between Pappus‫ ތ‬account and Diophantus‫ ތ‬employment of a mathematical method of analysis on the one hand and his own general account of analysis on the other. Finally, (4) Descartes interprets certain developments in algebra in the sixteenth and early seventeenth centuries as motivated by attempts to revive Classical analytical practices in mathematics and was prepared to avail himself of these developments to attain his own conception of the general method of analysis. In view of these clues in combination with the exposition in Resp. 2, the claim that the methods of analysis and synthesis as understood by Descartes have mathematical grounds begins to chime more plausibly and it is likely that a detailed explication and establishment of the interrelations between the four above points, together with the exposition in Resp. 2, will in the long run provide some distinctive content to Descartes‫ ތ‬terms “a priori” and “a posteriori”. In any case, the four points are to be established first. Let us now turn to this preliminary task.

3.1.1 Mathematics as a Paradigm of a Universal Method After declaring, in the first two precepts of the Regulæ, that “[s]tudiorum finis esse debet ingenij directio ad solida & vera, de ijs omnibus quæ occurrunt, proferenda judicia” (Reg. I, AT X, 359) and that “[c]irca illa tantùm objecta oportet versari, ad quorum certam & indubitatam cognitionem nostra ingenia videntur sufficere” (Reg. II, AT X, 362), Descartes is quick to recognize arithmetic and geometry as the only “ex disciplinis ab alijs cognitis” that are “ab omni falsitatis vel incertitudinis vitio puras” (ibid., 364). He then provides a rationale for their privileged status:

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Chapter Three quia scilicet hæ solæ circa objectum ita purum & simplex versantur, vt nihil plane supponant, quod experientia reddiderit incertum, sed totæ consistunt in consequentijs rationabiliter deducendis. Sunt igitur omnium maximè faciles & perspicuæ, ... cùm in illis citra inadvertentiam falli vix humanum videatur (ibid., 365).

Furthermore, he goes on to claim: Jam verò ex his omnibus est concludendum, non quidem solas Arithmeticam & Geometriam esse addiscendas, sed tantummodo rectum veritatis iter quærentes circa nullum objectum debere occupari, de quo non possint habere certitudinem Arithmeticis & Geometricis demonstrationibus æqualem (ibid., 366).

That is to say, he establishes arithmetic and geometry as the available prima facie paradigms of the sought-after scientific cognition in general in the sense that they provide a standard of certainty which any of the soughtafter scientiæ are bound to exemplify. Significantly, the paradigmatic status of both the disciplines in this epistemic respect is presented as a function of the simplicity and purity of their objects and of a peculiar character of the way in which the demonstrations are given in them, viz. via consequentiæ rationabiliter deducendæ. In Reg. IV, Descartes supplements this exposition with an account of the specific branches of both mathematical disciplines at issue, viz. geometrical analysis and “Arithmeticæ genus quoddam, quod Algebram vocant” (AT X, 373), as available paradigmatic instances of the envisaged general method (introduced a moment before in AT X, 371–72) in operation. The core passage is worth quoting at length: Atque hæc duo [sc. analysis quædam quâ veteres Geometras vsos fuisse, & Arithmeticæ genus quoddam, quod Algebram vocant] nihil aliud sunt, quàm spontaneæ fruges ex ingenitis hujus methodi principijs natæ, quas non miror circa harum artium simplicissima objecta felicius crevisse hactenus, quam in cæteris, vbi majora illas impedimenta solent suffocare; sed vbi tamen etiam, modò summâ curâ excolantur, haud dubiè poterunt ad perfectam maturitatem pervenire. Hoc verò ego præcipuè in hoc Tractatu faciendum suscepi; neque enim magni facerem has regulas, si non sufficerent nisi ad inania problemata resolvenda, quibus Logistæ vel Geometræ otiosi ludere consueverunt; sic enim me nihil aliud præstitisse crederem, quàm quòd fortasse subtiliùs nugarer quam cæteri. Et quamvis multa de figuris & numeris hîc sim dicturus, quoniam ex nullis alijs disciplinis tam evidentia nec tam certa peti possunt exempla, quicumque tamen attente respexerit ad meum sensum,

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facile percipiet me nihil minus quàm de vulgari Mathematicâ hîc cogitare, sed quamdam aliam me exponere disciplinam, cujus integumentum sint potiùs quàm partes. Hæc enim prima rationis humanæ rudimenta continere, & ad veritates ex quovis subjecto eliciendas se extendere debet; atque, vt liberè loquar, hanc omni aliâ nobis humanitùs traditâ cognitione potiorem, vtpote aliarum omnium fontem, esse mihi persuadeo. Integumentum verò dixi, non quo hanc doctrinam tegere velim & involvere ad arcendum vulgus, sed potiùs ita vestire & ornare, vt humano ingenio accommodatior esse possit (ibid., 373–74; my emphases).

I take this passage as indicating two more senses in which Descartes is ready to invoke geometry and arithmetic as paradigms for the gaining of scientific cognition in general. To begin with, Descartes declares his ambition to bring the fruges ex ingenitis hujus methodi principijs natæ to perfect maturity in any discipline whatsoever; and geometrical analysis and the algebra of the moderns (interpreted as Arithmeticæ genus quoddam) are regarded by him as the only disciplines hitherto to have yielded such fruits sponte, presumably thanks to the simplicity of their objects. These disciplines then serve as paradigms in the sense that the ingenita hujus methodi principia are readily at hand in them, and that they exemplify, by the same token, how those innate essentials of the soughtafter method can be put to use—albeit in a more or less rudimentary manner5—to attain cognitions that satisfy the strict standards of the scientiæ. In the second section of the above passage, Descartes then proceeds to announce a disciplina quædam alia (presumably quàm vulgaris Mathematica) the explication of which is identified as the proper task of the following precepts of the Regulæ. His claim that this discipline “ad veritates ex quovis subjecto eliciendas se extendere debet” leaves no doubt that he intends it as the vehicle for performing the declared task of bringing the “fruits of the innate principles of the method” to maturity in any suitable domain whatsoever, and (especially if the wider context of Reg. IV is taken into account) that it is identical with the proper contents (yet to be specified and explicated) of the universal method announced and sketched a few pages earlier in AT X, 371–72. Now it is vital to appreciate 5

Cf. ibid., 376: “Sed mihi persuadeo, prima quædam veritatum semina humanis ingenijs à naturâ insita, quæ nos, quotidie tot errores diversos legendo & audiendo, in nobis extinguimus, tantas vires in rudi istâ & purâ antiquitate habuisse, vt eodem mentis lumine, quo virtutem voluptati, honestumque vtili præferendum esse videbant, etsi, quare hoc ita esset, ignorarent, Philosophiæ etiam & Matheseos veras ideas agnoverint, quamvis ipsas scientias perfectè consequi nondum possent.”

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that, arguably, the stated disciplina quædam alia is not identical with any set (however complete) of the ingenita methodi principia, though these principles are contained within it (this is how I read Descartes‫ ތ‬statement that “[h]æc enim [disciplina] rationis humanæ rudimenta continere ... debet”). What Descartes will need in order to attain that alia disciplina, i.e. a fully-fledged method sufficiently efficient to elicit truth “ex quovis subjecto”, is rather to shape, enhance and explicate those principia ingenita.6 We have already seen that these principia are paradigmatically in operation in the geometrical analysis of the ancients and in the algebra of the moderns—and Descartes seems to indicate now that it is once again reference to mathematics that should facilitate most readily the implementation of the envisaged disciplina quædam alia, i.e. of a universal method, if only for the reason that mathematics should provide some prima facie striking and powerful examples of this method in operation. I suggest, then, it is in view of this envisaged transition that the rôle of figuræ & numeri—i.e. objects of, respectively, arithmetic and geometry of the “vulgar” cast—with regard to the disciplina quædam alia in the last quoted passage is to be interpreted. Far from coinciding with the paradigmatic rôle of arithmetic and geometry due to their counting as the spontaneæ fruges ex ingenitis hujus methodi principijs natæ, operations with numbers and figures seem to be invoked by Descartes, in this new context, as operations within that domain of application which is likely to render most evident and certain the way in which the precepts of the envisaged universal method could and should be put to use. Descartes expresses himself clearly to this effect in a concise statement in Reg. XIV, AT X, 442: [V]sus enim regularum, quas hîc tradam, in illis [sc. in Arithmetica & Geometria] addiscendis, ad quod omnino sufficit, longé facilior est, quàm in vllo alio genere quæstionum; hujusque vtilitas est tanta ad altiorem sapientiam consequendam, vt non verear dicere hanc partem nostræ methodi non propter mathematica problemata fuisse inventam, sed potiùs hæc ferè tantùm hujus excolendæ gratia esse addiscenda (my emphasis).

6

See especially Reg. VIII, AT X, 397: “[C]ùm in his initijs nonnisi incondita quædam præcepta, & quæ videntur potiùs mentibus nostris ingenita, quàm arte parata, poterimus invenire, non statim Philosophorum lites dirimere, vel solvere Mathematicorum nodos, illorum ope esse tentandum: sed ijsdem priùs vtendum ad alia, quæcumque ad veritatis examen magis necessaria sunt, summo studio perquirenda ...” (I read “ijsdem” for “ijdsem” in AT X, 397.21). The relation between Descartes‫ ތ‬envisaged universal method and the principia ingenita of this method is discussed in detail in ch. 4.

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In other words, the application of the precepts of the universal method is longè facilior in mathematics than anywhere else. Yet it is, at the end of the day, not for the sake of solving mathematical problems that one should learn the envisaged method; rather, one should learn how the method works in mathematics in order to learn how to employ that very same method in all the remaining genera quæstionum. Even if this important point is approved, one must still be careful to steer clear through another possible misunderstanding. It is natural to describe the situation as Leslie Beck once did, namely to the effect that “the idea of the unity of all the sciences is now linked to the extension of the method used in mathematical thinking to the whole field of knowledge” (Method of Descartes, 37–38).7 This is correct enough, I submit, if one understands the “extension” to the effect that the way in which the envisaged method (which is not peculiarly mathematical any more than, say, meteorological, optical, or metaphysical) is put to use in mathematics provides the best paradigm of how that very same universal method is to be employed in any other genus quæstionum. However, the “extension” has sometimes been read to quite another effect, namely that Descartes has come up with a certain specifically mathematical method and tries to render all other domains of possible cognition susceptible to a sort of mathematical treatment by way of an extension of that peculiar mathematical method.8 It should be clear enough, in the light of the previous considerations, that this latter reading is likely to be mistaken.9 I submit one must be wary not to succumb to it lest one misses the deepest and most stimulating aspects of what Descartes was after in his project of establishing the Sapientia vniversalis (Reg. I, AT X, 360).10 7

After all, such a statement is quite close to what Descartes said to Burman at AT V, 176–77: “[U]t autem [usus ad veritatem agnoscendam] excoli possit, opus est scientiâ mathematicâ .... ... Matheseos studio opus est ad nova invenienda, tum in Mathesi, tum in Philosophiâ.” I take it that “Philosophia” is understood by Descartes here in the same way as in his Princ. Pref., AT IX-2, 2, i.e. as synonymous with “la Sagesse”, which is explicated (among other things) as “vne parfaite connoissance de toutes les choses que l‫ތ‬homme peut sçauoir” (ibid.). 8 Perhaps the most prominent proponent of this interpretative line is Étienne Gilson. Thus he writes: “[Selon] la pensée cartésienne[,] tout ce qui est susceptible de connaissance vraie est, par définition, susceptible de connaissance mathématique” (Gilson, René Descartes’ Discours, 214). 9 I return to the issue below in ch. 4 in my discussion of Descartes‫ ތ‬notion of the Mathesis vniversalis. I offer there my own suggestion as to how Descartes‫ތ‬ universalistic claims related to Mathesis are to be interpreted. 10 Cf. also Princ. Pref., AT IX-2, 2: “[P]ar la Sagesse on n‫ތ‬entend pas seulement la prudence dans les affaires, mais vne parfaite connoissance de toutes les choses que

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The lines of thought that lead to the first and the third sense in which mathematics fulfils the rôle of a paradigm of the employment of the envisaged universal method within the project of attaining the Sapientia vniversalis from Descartes’ perspective are paralleled quite closely with certain well-known passages in the Discours. Here, once again, Descartes picks out the mathematical disciplines both as prima facie instances and standards of certainty and evidence in cognition due to the “easiness” and “simplicity” of their objects and due to the demonstrative character of the reasoning involved in them. Thus he intimates early in DM 1 that during and after his studies he delighted “surtout aux Mathematiques, a cause de la certitude & de l‫ތ‬euidence de leurs raisons” (AT VI, 7), though (as he adds immediately and significantly) he did not yet notice “leur vray vsage” (ibid.). Further, immediately after having introduced his famous four precepts in DM 2, he reports: [I]e ne fus pas beaucoup en peine de chercher par lesquelles il estoit besoin de commencer: car ie sçauois desia que c‫ތ‬estoit par les plus simples & les plus aysées a connoistre; & considerant qu‫ތ‬entre tous ceux qui ont cy deuant recherché la verité dans les sciences, il n‫ތ‬y a eu que les seuls Mathematiciens qui ont pu trouuer quelques demonstrations, c‫ތ‬est a dire quelques raisons certaines & euidentes, ie ne doutois point que ce ne fust par les mesmes qu‫ތ‬ils ont examinées; bien que ie n‫ތ‬en esperasse aucune autre vtilité, sinon qu‫ތ‬elles accoustumeroient mon esprit a se repaistre de veritez, & ne se contenter point de fausses raisons (AT VI, 19).

He then makes it clear that while his method, expressed concisely by his four precepts, proved extremely powerful in the field mathematics; and that while he was encouraged to drive forward this project by the success of his method in solving virtually all mathematical problems, his proper objective was to extend its application to all the other disciplines beyond mathematics: Comme, en effect, i‫ތ‬ose dire que l‫ތ‬exacte obseruation de ce peu de preceptes que i‫ތ‬auois choisis, me donna telle facilité a demesler toutes les questions ausquelles ces deux sciences [sc. l‫ތ‬Analyse Geometrique & l‫ތ‬Algebre] s‫ތ‬estendent, qu‫ތ‬en deux ou trois mois que i‫ތ‬employay a les examiner, ayant commencé par les plus simples & plus generales, & chasque verité que ie trouuois estant vne reigle qui me seruoit après a en trouuer d‫ތ‬autres, non seulement ie vins a bout de plusieurs que i‫ތ‬auois iugées autrefois tres difficiles, mais il me sembla aussy, vers la fin, que ie pouuois determiner, en l‫ތ‬homme peut sçauoir, tant pour la conduite de sa vie, que pour la conseruation de sa santé & l‫ތ‬inuention de tous les arts ....”

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celles mesme que i‫ތ‬ignorois, par quels moyens, & iusques où, il estoit possible de les resoudre. ... Mais ce qui me contentoit le plus de cete Methode, estoit que, par elle, i‫ތ‬estois assuré d‫ތ‬vser en tout de ma raison ...; outre que ie sentois, en la prattiquant, que mon esprit s‫ތ‬accoustumoit peu a peu a conceuoir plus netement & plus distinctement ses obiets, & que, ne lҲayant point assuiettie a aucune matiere particuliere, ie me promettois de lҲappliquer aussy vtilement aux difficultez des autres sciences, que iҲauois fait a celles de lҲAlgebre (ibid., 20–21; my emphasis).

Along with the above-quoted AT X, 442, this is perhaps the most decisive evidence against interpreting Descartes‫ ތ‬methodology as promoting a sort of mathematicism in the sense specified above. Moreover, the present passages from the Discours show that the different senses in which mathematical disciplines count as paradigms for fulfilling the scientific project in Descartes are not just a matter of his private thoughts in the Regulæ but that he was ready to declare them in his first public presentation in which, I claim, the essentials of his mature thought are contained and fixed.11 In the light of the present considerations, I take it as established that the most promising way to adequately address the questions of in what the procedure of analysis properly consists, according to Descartes, and in which sense analysis in this meaning deserves the title of the true method of discovery, would be to turn to Descartes‫ ތ‬mathematical thought. Our leading interim questions then become (i*) of what does the procedure of analysis in mathematics consist, as conceived by Descartes, and (ii*) how does Descartes establish in concreto analysis as the true method of discovery in mathematics? Both these complex questions will be taken up in due course. For the moment, let us turn to the other basic points to be established in connection with the suggested interpretation of AT VII, 155–57.

3.1.2 The A Priori in Purely Mathematical Contexts There are two passages in which Descartes unambiguously employs the a priori–a posteriori terminology in purely mathematical contexts in a 11

This is, of course, a controversial claim and several influential contemporary scholars disagree—most notably Daniel Garber (see in particular his “Descartes and Method in 1637,” in idem, Descartes Embodied: Reading Cartesian Philosophy through Cartesian Science, Cambridge: Cambridge University Press, 2001: 35–51) and John Schuster (see in particular his Descartes-Agonistes: Physico-mathematics, Method and Corpuscular-Mechanism 1618-33, Sydney: Springer, 2013, ch. 6–7). I briefly deal with their respective positions below in ch. 4.

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way sufficiently similar to his treatment of the terms in Resp. 2. In one of them, Descartes writes to Mersenne while criticizing de Fermat‫ތ‬s brief essays “Methodus ad disquirendam maximam et minimam” and “Methodus de Tangentibus linearum curvarum,”12 claiming that de Fermat‫ތ‬s professed rule for tangents of curved lines at their arbitrarily chosen points hinges upon vne fausse position, fondée sur la façon de demonstrer qui réduit a l‫ތ‬impossible, & qui est la moins estimée & la moins ingénieuse de toutes celles dont on se sert en Mathematique. Au lieu que la mienne [regle] est tirée d‫ތ‬vne connoissance de la nature des Equations, qui n‫ތ‬a iamais esté, que ie sçache, assés expliquée ailleurs que dans le troisieme Liure de ma Geometrie. De sorte qu‫ތ‬elle n‫ތ‬eust sceu estre inuentée par vne personne qui aurroit ignoré le fonds de l‫ތ‬Algebre; & elle suit la plus noble façon de demonstrer qui puisse estre, a sçauoir celle qu‫ތ‬on nomme a priori (Mers., AT I, 490; Descartes’ italics).

Then, about a year later, Descartes writes to de Beaune concerning the socalled converse problem of tangents (i.e. the problem of finding the equation of a given curve given its tangents at its arbitrarily chosen points): Ie ne croy pas qu‫ތ‬il soit possible de trouuer generalement la conuerse de ma regle pour les tangentes, ny de celle dont se sert Monsieur de Fermat non plus, bien que la pratique en soit en plusieurs cas plus aisée que de la mienne. Mais on en peut déduire à posteriori des Theoremes, qui s‫ތ‬estendent à toutes les lignes courbes qui s‫ތ‬expriment par vne équation, en laquelle l‫ތ‬vne des quantitez x ou y n‫ތ‬ait point plus de deux dimensions, encore que l‫ތ‬autre en eust mille .... Il y a bien vne autre façon qui est plus generale, & à priori, à sçauoir par l‫ތ‬intersection de deux tangentes, laquelle se doit tousiours faire entre les deux points où elles touchent la courbe, tant proches qu‫ތ‬on les puisse imaginer. Car en considerant quelle doit estre cette courbe, afin que cette intersection se fasse tousiours entre ces deux points, 12 The essays were not published until 1679 in de Fermat’s Varia opera mathematica (Toulouse: Joannes Pech). However, de Fermat achieved the results presented in the essays as early as 1629 and sent the essays to Mersenne in 1636. (I rely on Michael Mahoney, “The Mathematical Realm of Nature,” in The Cambridge History of Seventeenth-Century Philosophy, ed. Daniel Garber and Michael Ayers, vol. 1 [Cambridge: Cambridge University Press, 1998], 754, n. 56.) For a clear account of the controversy between Descartes and de Fermat concerning the rule for finding tangents, see Michael Mahoney, The Mathematical Career of Pierre de Fermat, 2nd ed. (Princeton: Princeton University Press, 1994), 170–95.

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& non au deça ny au delà, on en peut trouuer la construction ... (AT II, 514; Descartes’ italics).

Let us put to one side the AT II, 514 passage; much would have to be done, including some mathematical technicalities, to have even a chance of integrating it properly into some overall interpretative edifice. The other passage, on the other hand, provides us with some promising hints with regard to Descartes‫ ތ‬employment of the term a priori in purely mathematical contexts, hints that are congruent, in several important respects, to the way Descartes invokes the pair a priori–a posteriori in the Resp. 2 passage. The chief contrast made here is that between, on the one hand, Descartes‫ ތ‬own “la plus noble façon de demonstrer,” which is characterized as a priori, and de Fermat‫ތ‬s “façon” on the other; and the preceding lines of the quoted letter show that Descartes takes the latter‫ތ‬s façon as just a way of accidental findings “sans industrie & par hazard,” i.e. (as a quick look at the Regulæ and the Discours makes clear) as a procedure which, far from perhaps exhibiting a different method, amounts to a substantially non-methodical performance,13 and professedly an erroneous one for that matter.14 In order to appreciate the affinities of the present passage to Descartes‫ ތ‬treatment of a priori in Resp. 2, consider first that a priori is employed in AT I, 490 once again as a characterization 13

See in particular Reg. X, AT X, 404–405: “[Q]uærend[æ] esse [res] cum methodo .... ... Et maximè cavendum est, ne in similibus casu & sine arte divinandis tempus teramus; nam etiamsi illa sæpe inveniri possunt sine arte, & à felicibus interdum celeriùs fortasse, quàm per methodum, hebetarent tamen ingenij lumen, & ita puerilibus & vanis assuefacerent, vt postea semper in rerum superficiebus hæreret, neque interiùs posset penetrare.” It is due to this implied charge of a substantial lack of method that the apparent possibility of interpreting Descartes‫ ތ‬description of de Fermat‫ތ‬s façon as an instance of a posteriori procedure in the sense invoked in the Resp. 2 passage is derailed. For although, according to Descartes (as we shall see), the right method in the régime of discovery coincides with analysis (and thus with a procedure characterized unambiguously as a priori by Descartes in the Resp. 2), what he is ready to characterize as a posteriori in the Resp. 2 passage is by no means any unmethodical alternative (however inappropriate) to the correct methodical performance but rather a sort of expository supplement of the successfully finished analytical procedure. 14 See Mers., AT I, 487–88 where Descartes professedly refutes de Fermat‫ތ‬s rule. As a matter of fact, Descartes’ criticisms of de Fermat on the present point are generally unjustified as is clearly shown in Mahoney, Mathematical Career of de Fermat, 178–80. However, this fact is irrelevant to our present purposes.

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of “façon de demonstrer,” which is equivalent to the Latin “ratio demonstrandi” of AT VII, 155. Moreover, Descartes is somewhat more specific here concerning the content of the a priori feature: he makes it clear that the manner of demonstration in issues geometrical counts as a priori in so far as it is drawn from “la nature des Equations” and that knowledge of the “fonds de l‫ތ‬Algebre” is needed for this. By the same token, he explicitly refers to his Geom. III, devoted almost exclusively to theorizing on algebraic equations in the context of algebraized geometry that Descartes in fact established and developed in the preceding two books of the same work.15 Given these hints, one may conclude quite safely that at least as far as mathematics is concerned, one proceeds a priori (and eventually, if skilful enough, gains a cognition a priori) in so far as one makes use of a problem-solving method which Descartes introduces and puts to use in La Geometrie, and which has to do with solving geometrical problems via algebraic operations. It should come as no surprise in view of the alleged affinities to his treatment in Resp. 2, that the mathematical method at issue is the one Descartes standardly calls analysis in the relevant contexts.16 Another substantial question in addition to those stated at the end of sec. 3.1.1 is to be addressed, then: In which sense do the essentials of Descartes‫ ތ‬algebraic problem-solving procedure in mathematics allow him to characterize the employment of this procedure as an a priori treatment? First, however, let us turn to the relevant portions of the historical and doctrinal context of Descartes‫ ތ‬thought concerning mathematics, its methodology and its proper object(s).

15

Despite the lack of familiar terminology and notation, the rudiments of analytic geometry seem to have been practiced as early as the ancient Greeks Menaechmus (ca. 380–ca. 320 BC) and Apollonius of Perga (ca. 262–ca. 190 BC)—cf. Julian Coolidge, A History of Geometrical Methods (Oxford: Oxford University Press, 1940), 117–22. Although the invention of symbolic algebra as a problem-solving tool is broadly ascribed to François Viète (we shall deal with this issue below; for the moment, cf. e.g. Carl Boyer, “Analysis: Notes on the Evolution of a Subject and a Name,” Mathematics Teacher 47, no. 7 [1954], 65; Jacob Klein, Greek Mathematical Thought and The Origin of Algebra, transl. by Eva Brann, 2nd ed., New York: Dover Publications, 1992, ch. 11), it is generally recognized that it was not until Descartes‫ ތ‬Geometrie that the decisive steps towards algebra of indeterminate quantities was made. The point is discussed below in this chapter. 16 Cf. a Mydorge, AT II, 22; Mers., AT II, 83; 250; 308; 337; 429; 438; 524; 638; a Hardy, AT II, 172; a M. de Beaune, AT II, 511; Epist. CCXLIII. bis, AT III, 709; 713–14.

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3.1.3 The Method of Analysis in Diophantus and Pappus In the context of establishing the famous four precepts of his method in DM 2, Descartes mentions “l‫ތ‬Analyse des Geometres & ... l‫ތ‬Algebre” as “sciences qui sembloient deuoir contribuër quelque chose a [son] dessein” (AT VI, 17), viz. to “bien iuger, & distinguer le vray d‫ތ‬auec le faux” (AT VI, 2). Then, soon after introducing the precepts, he reports that in his initial attempts to employ them he took over “tout le meilleur de l‫ތ‬Analyse Geometrique & de l‫ތ‬Algebre” (AT VI, 20), having already made it clear that he meant by these disciplines “l‫ތ‬Analyse des anciens & l‫ތ‬Algebre des modernes” (AT VI, 17). Let us now focus on Descartes‫ ތ‬references to analysis; the issue of algebra will be treated in sec. 3.1.4. The cited references to analysis in the Discours seem to accord well with the hints in Resp. 2 to a secret method of analysis as practised by certain Geometræ veteres. Yet the decisive passage which allows linkage of the hints in both Resp. 2 and DM 2 firmly with the analytical procedure as professedly employed in the “golden era” of ancient mathematics, and as developed in Diophantus and summed up in Pappus, are the following statements in Reg. IV: Cum igitur hujus methodi [sc. Descartes‫ ތ‬own method as generally defined in AT X, 371–72—J.P.] vtilitas sit tanta, vt sine illâ litteris operam dare nociturum esse videatur potiùs quam profuturum, facile mihi persuadeo illam jam antè à majoribus ingenijs ... fuisse aliquo modo perspectam. ... [S]atis enim advertimus veteres Geometras analysi quâdam vsos fuisse, quam ad omnium problematum resolutionem extendebant, licet eamdem posteris inviderint. ... Cùm verò postea cogitarem, vnde ergo fieret, vt primi olim Philosophiæ inventores neminem Matheseos imperitum ad studium sapientiæ vellent admittere, tanquam hæc disciplina omnium facillima & maximè necessaria videretur ad ingenia capessendis alijs majoribus scientijs erudienda & præparanda, plane suspicatus sum, quamdam eos Mathesim agnovisse valde diversam à vulgari nostræ ætatis .... ... Et quidem hujus veræ Matheseos vestigia quædam adhuc apparere mihi videntur in Pappo & Diophanto, qui, licet non prima ætate, multis tamen sæculis ante hæc tempora vixerunt. Hanc verò postea ab ipsis Scriptoribus perniciosâ quâdam astutiâ suppressam fuisse crediderim ... (AT X, 373–76; my emphases).

The secret method of certain veteres Geometræ of which Descartes speaks is most probably the very same method of analysis to which he refers with a similar phrase in Resp. 2, AT VII, 156. Moreover, that secret method is now much more clearly characterized as an achievement containing rudiments of Descartes‫ ތ‬own method sketched in Reg. IV, AT X, 371–72, a

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method which is again most probably the very same method that Descartes acknowledges to be his own in Resp. 2. Furthermore, the vestiges of that ancient secret method are, by the same token, expressly identified here with the veræ Matheseos primæ ætatis vestigia still to be found in the leading figures of the “silver era” of Greek mathematics, viz. Pappus and Diophantus. Thus putting these threads together, Descartes‫ ތ‬core message is that he has found in Pappus and Diophantus (and indirectly in those earlier mathematicians whose achievements are probably developed in the works of the two aforementioned Alexandrian authors) something substantial enough concerning the ancient method of mathematical analysis to make him build upon it, or at least see it as a rudiment of his own views, while breaking through to his own conception of analytical method most applicable in mathematics and arguably even beyond the mathematical domain—the method he refers to throughout the Regulæ and the Discours, and in the quoted passage from Resp. 2. What was it that he found? Let us look at Pappus first. In the beginning of the Book Seven of his Mathematicæ Collectiones,17 Pappus provides an extensive methodological treatment in the course of which he explains, among other things, analysis and synthesis as complementary procedures which jointly constitute an integral geometrical method.18 Here is the core of what he has to say on 17

As announced in the Introduction, I quote Pappus in the Latin translation by Commandino which was the edition available to and actually used by Descartes. The crucial Greek terms “ਕȞȐȜȣıȚȢ” and “ıȪȞșİıȚȢ” are translated by Commandino as “resolutio” and “compositio” respectively. This fact somewhat complicates our central interpretative issue in yet another respect, as we shall see in ch. 5. 18 Erkka Maula, “An End of Invention,” Annals of Science 38, no. 1 (1981), 110 and 119–20 rightly argues that Pappus seems just to recapitulate a long tradition of the “golden era” of ancient Greek mathematics (ca. the fourth to the second century BC), during which the analytic heuristics in question are likely to have been employed most extensively and fruitfully; and that while “it is clear that Pappus‫ތ‬s general description makes a good test of any reconstruction [of the ancient analysis and synthesis] based on the fragmentary material pertaining to the right period, the golden era” (ibid., 110), one should bear in mind that “Pappus‫ތ‬s own achievements in mathematics do not warrant many assumptions, based on his practices, as to the nature of the ancient heuristics. Nor does the quantity and easy availability of the Pappian material warrant the quality and correctness of a reconstruction of the ancient analysis and synthesis” (ibid.; Maula’s italics). Michael Mahoney, “Another Look at Greek Geometrical Analysis,” Archive for History of Exact Sciences 5, no. 3 (1968), 318–48 takes a more charitable stance as to Pappus‫ ތ‬competence in reconstructing the procedures in question. By the same

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this score:19 Resolutio [ਕȞȐȜȣıȚȢ—J.P.] igitur est via a quæsito tamquam concesso per ea, quæ deinceps consequuntur ad aliquod concessum in compositione [ıȪȞșİıȚȢ—J.P.]: in resolutione enim id quod quæritur tamquam factum ponentes, quid ex hoc contingat, consideramus: & rursum illius antecedens, quousque ita progredientes incidamus in aliquod iam cognitum, uel quod sit è numero principiorum. Et huiusmodi processum resolutionem appellamus, veluti ex contrario factam solutionem (Pappus, Mathematicæ Collectiones, Book Seven, 157r/v).

Interpretative complications apart, the crucial tenet of the procedure at issue undoubtedly is to assume that what is sought (quæsitum, the unknown) is already accomplished (“id quod quæritur tamquam factum ponentes”) and therefore can be treated as if it were known; and then to look step by step for what the result thus assumed depends upon.20 While Pappus‫ ތ‬meaning concerning the nature of the dependence as well as the token, he questions the authenticity of the statements contained in the extant Pappus‫ ތ‬source texts regarding synthesis; he strongly suspects them to be interpolations of (a) later scholiast(s)—see in particular ibid., 321–26. In any case, we are concerned here with an account of Descartes‫ ތ‬own method, with a reconstruction of Descartes‫ ތ‬actual resources and with assumptions concerning his own understanding of them; and it is highly probable that as for the sources in Greek mathematics, it was (setting Euclid aside) just Pappus‫ ތ‬Collectiones and Diophantus‫ ތ‬Arithmetica with which Descartes was directly familiar. The mathematicians of the “golden era” (most notably Apollonius of Perga, to whom Descartes refers frequently in La Geometrie) seem to be known to him only through the mediation of Pappus‫ ތ‬Collectiones. Given that Descartes is well aware of the derivative nature of Pappus‫ ތ‬and Diophantus‫ ތ‬accounts (as is clearly indicated in the above-quoted Reg. IV, AT X, 376), and that Descartes exhibited little or no interest in thoroughly reading or interpreting his predecessors, I feel safe in taking at face value Pappus‫ ތ‬account of the method in question in the present interpretative context. 19 A similar treatment, only more concise, can be found in an interpolated scholium to Euclid’s Elementa XIII, 1–5 (see Euclid of Alexandria, Opera omnia, vol. 4, ed. Johann L. Heiberg, Leipzig: G. B. Teubner, 1885, 364–65). Cf. Mahoney, “Another Look,” 321; Klein, Greek Mathematical Thought, 259–60, n. 217. 20 According to Pappus‫ ތ‬subsequent account, there are two kinds of quæsita to be distinguished: either a validity test and/or a proof of a theorem (the so-called zetetics), or else a solution to a problem (the so-called poristics). In the former case the analysis is classified as theoretical, in the latter case as problematical. The distinction is irrelevant at the present stage of the exposition; it will be taken up in due course.

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order to be followed in the search are somewhat obscure and in need of further discussion,21 at the very start so much is clear from Pappus‫ތ‬ account that the ad quem of the analytic procedure proper is something already cognized or amounting to a principle (“aliquid iam cognitum, uel ... è numero principiorum”). This endpoint of the analytical treatment then serves as a starting point of the reverse stepwise procedure—called ıȪȞșİıȚȢ or compositio by Pappus—the ad quem of which is an actual accomplishment of what has been sought. As Pappus explains immediately: In compositione autem per conuersionem ponentes tamquam iam factum id, quod postremum in resolutione sumpsimus: atque hic ordinantes secundum naturam ea antecedentia, quæ illic consequentia erant; & mutua illorum facta compositione ad quæsiti finem peruenimus, & hic modus vocatur compositio (ibid., 157v).

Analysis is therefore rightly characterized in the text as a “backwards solution” (ex contrario facta solutio). No comparable methodological passages occur in Diophantus‫ތ‬ Arithmetica,22 yet Diophantus‫ ތ‬approach is remarkable in other respects that are likely to have moved Descartes. For one thing, while Pappus deliberately limited the range of the methodical procedures of analysis and synthesis to the domain of geometry, Diophantus employs these procedures brilliantly in dealing with distinctively arithmetical issues: the quæsita are (only provisionally indeterminate)23 numbers and the analytic procedure consists in looking for an equation which enables one “to ignore whether the magnitudes occurring in the problem are ‫ލ‬known‫ ތ‬or ‫ލ‬unknown‫( ”ތ‬Klein, Greek Mathematical Thought, 156). Furthermore, Diophantus‫ ތ‬treatment of particular problems in his Arithmetica provides numerous impressive and instructive examples of how both the analytic and the synthetic procedures as described by Pappus actually work. In particular, Diophantus is not content with just providing a practical manual for arriving at the desired result; his ambition is rather to show step by 21

See Florka, Descartes’s Metaphysical Reasoning, 94–104 for some interpretative controversies concerning the issues of dependence and order in Pappus‫ ތ‬account of analysis/synthesis. I discuss the issues below in this chapter. 22 Diophantus of Alexandria, Arithmeticorum libri sex, in Diophanti Alexandrini opera omnia cum Græcis commentariis, ed. Paul Tannery, vol. 1 (Leipzig: G. B. Teubner, 1893), 2–449. 23 The issue of indeterminate and determinate multitudes in the context of Diophantus’ mathematical thought is discussed in Klein, Greek Mathematical Thought, 140–43.

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step—availing himself of the device of treating the unknown(s) as if they were known—how one can arrive at the proposed rule and why the rule is a good one.24 We shall see that although Descartes arguably transforms the sense in which analysis and synthesis are two constituents of a single mathematical method, his general notion of what the analytical procedure—both in mathematics and beyond—should consist is expressed by way of a very close paraphrase of Pappus’ account. Furthermore, we shall see that in Descartes‫ ތ‬hands mathematical analysis in its most general form turns out to be most fundamentally a matter of relating proportions of quantities, such proportions being properly expressed as algebraic equations. Descartes seems to have traced the rudiments of such a conception—not wholly implausibly—back to Diophantus (or to earlier authors upon whom Diophantus draws).25 Finally, Descartes seems to credit both Pappus and Diophantus with at least a part of an account of “quare [consequentibus conclusiones] ita se habeant, & quomodo invenirentur” (Reg. IV, AT X, 375), an account which he reports to have missed completely in the bulk of the mathematical production of his own time26 and which he himself 24

I draw here upon the account in Macbeth, “Viète, Descartes,” 88–89. Both the characterizations of Diophantus‫ ތ‬above approach are illustrated well by his treatment of the problem of dividing a given number into two numbers with a given difference (as announced in the Introduction, I quote from an English translation by J. Winfree Smith in Klein, Greek Mathematical Thought, 330–31, fn. 22; I omit Diophantus‫ ތ‬peculiar symbolism [cf. a survey of this symbolism in Klein, Greek Mathematical Thought, 141–47], using instead Smith’s paraphrases in square brackets): “So, let the given number be [one hundred], and let the difference be [forty units]. To find the numbers. Let the less be taken as [one unknown]. Then the greater will be [one unknown and forty units]. Then both together become [two unknowns and forty units]. But they have been given as [one hundred units]. [One hundred units], then, are equal to [two unknowns and forty units]. And, taking like things from like: I take [forty units] from the [one hundred] and likewise [forty] from the [two] numbers and [forty units]. The [two unknowns] are left equal to [sixty units]. Then, each [unknown] becomes [thirty units]. As to the actual numbers required: the less will be [thirty units] and the greater [seventy units], and the proof is clear” (Diophantus, Arithmeticorum libri sex, 16). 25 This is not to say that Diophantus (or the earlier mathematicians whose achievements were assimilated by him) should be praised as the inventor(s) of algebra in the modern sense, at least not without serious qualifications, mostly due to the lack of the concept of general magnitude conceived as a determinable indeterminate. Cf. Klein, Greek Mathematical Thought, ch. 9–10 for a detailed treatment of the issue. 26 See Reg. IV, AT X, 375: “Cùm primùm ad Mathematicas disciplinas animum applicui, perlegi protinus pleraque ex ijs, quæ ab illarum Auctoribus tradi solent,

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professes to offer. To be sure, he suspects that the vera Mathesis “postea [a Pappo & Diophanto] perniciosâ quâdam astutiâ suppressam fuisse” (ibid., 376) in that they “maluerunt nobis ... steriles quasdam veritates ex consequentibus acutulè demonstratas, tanquam artis suæ [inveniendi] effectus ... exhibere, quàm artem ipsam docere” (ibid., 376–77);27 yet while this might well turn out true—in a sense to be clarified in due course—with regard to the latter question (“quomodo invenirentur”), the charge would be entirely unjustified with regard to the former question (“quare hæc ita se habeant”). All these connections and affinities provide, once combined with the hints in Resp. 2, additional support for the suggestion of turning to mathematical contexts when clarifying Descartes‫ތ‬ general notion of analytical treatment as the likely grounds for characterizing the cognition gained by its means as a priori.

3.1.4 The “Algebre des modernes” We saw that besides the Classical tradition of mathematical analysis, Descartes mentioned “l‫ތ‬Algebre des modernes” (DM 2, AT VI, 17) as another discipline that was likely to contribute to his plan “a chercher la vraye Methode pour paruenir a la connoissance de toutes les choses dont [son] esprit seroit capable” (ibid.). As he makes clear in Reg. IV, this was due to his belief that the algebraic projects of his recent predecessors and contemporaries were basically attempts at reviving the ancient vera Mathesis, i.e. a discipline which professedly amounted to practising the art of analysis.28 However, as he reports in DM 2 and in Reg. IV, his examinations issued in disappointment: the proponents of modern algebra failed, according to Descartes, to establish the discipline in the form in which it could really fulfil the envisaged reviving rôle, i.e. “ita ... vt non Arithmeticamque & Geometriam potissimùm excolui .... Sed in neutrâ Scriptores, qui mihi abundè satisfecerint, tunc fortè incidebant in manus: nam plurima quidem in ijsdem legebam circa numeros, quæ subductis rationibus vera esse experiebar; circa figuras verò, multa ipsimet oculis quodammodo exhibeant, & ex quibusdam consequentibus cocludebant; sed quare hæc ita se habeant, & quomodo invenirentur, menti ipsi non satis videbantur ostendere ....” 27 Cf. also ibid., 373: “[V]eteres Geometras analysi quâdam vsos fuisse, quam ad omnium problematum resolutionem extendebant, licet eamdem posteris inviderint.” And Resp. 2, AT VII, 156: “Hac solâ [sc. Synthesin] Geometræ veteres in scriptis suis uti solebant, non quòd aliam plane ignorarent, sed, quantum judico, quia ipsam tanti faciebant, ut sibi solis tanquam arcanum quid reservarent.” 28 See Reg. IV, AT X, 377: “Fuerunt denique quidam ingeniossimi viri, qui [illam veram Mathesim] hoc sæculo suscitare conati sunt: nam nihil aliud esse videtur ars illa, quam barbaro nomine Algebram vocant ....”

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ampliùs ei desit perspicuitas & facilitas summa, qualem in verâ Mathesi debere esse supponimus” (Reg. IV, AT X, 377).29 Indeed, he claims it was not until his own work that algebra in this desired form had been achieved. Furthermore, he claims (as already indicated) that it is precisely his own algebra that properly embodies the method of analysis in so far as it should be employed in mathematics. Thus given our interpretative suggestion that Descartes‫ ތ‬general conception of analysis (and by the professed implication of the a priori–a posteriori pair in at least one of its meanings) is actually derived from mathematics, it is of vital importance to clarify Descartes‫ތ‬ own notion of algebra. A look—brief and selective as it is bound to be30— at what Descartes actually found and is likely to have considered as worthy of development when he turned to the conceptions of algebra in his modern predecessors and contemporaries is likely to contribute its mite to that task. In the intellectual milieu of the late sixteenth and the early seventeenth centuries, which was formative for Descartes, two significant, interconnected yet distinct strains concerning the nature and rôle of algebra are discernible among those that are likely to have influenced Descartes. One is the notion of algebra as a (or even the) most general mathematical discipline that somehow traverses or unifies or grounds or even transcends all the other mathematical disciplines including arithmetic and geometry. The other is the claim that the way algebra, as developed during the Renaissance, is put to use is either identical with, or at least amounts to an embodiment of the use of a secret analytical method by the mathematicians of the golden age of Greek mathematics. Let us now take a closer look at both these strains. 29

Cf. ibid.: “[N]ihil aliud [quàm vera Mathesis] esse videtur ars illa, quam barbaro nomine Algebram vocant, si tantùm multiplicibus numeris & inexplicabilibus figuris, quibus orbitur, ita possit exsolvi, vt non ampliùs ei desit perspicuitas & facilitas summa, qualem in verâ Mathesi debere esse supponimus” (my emphasis). Cf. also DM 2, AT VI, 18: “[O]n s‫ތ‬est tellement assuieti, en [l‫ތ‬Algebre des modernes], a certaines reigles & a certains chiffres, qu‫ތ‬on en a fait vn art confus & obscur, qui embarasse l‫ތ‬esprit, au lieu d‫ތ‬vne science qui le cultiue.” 30 This is not the place to venture a comprehensive survey of the intellectual situation concerning the algebraic thought of the sixteenth and early seventeenth centuries. In my deliberately selective account I focus only on motives I take as indispensable to grasping and adequately assessing what is so innovative, or even revolutionary, in Descartes‫ ތ‬conception of algebraic analysis, and in which sense this novel conception might work as the paradigm for the universal method of gaining scientiæ as envisaged by Descartes. For a more comprehensive account see e.g. Carl Boyer, History of Analytic Geometry (New York: Scripta Mathematica, Yeshiva University, 1956), ch. 4.

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It is a relatively well-established fact that Algebra, by the leading Jesuit mathematician of that time, Christoph Clavius,31 was for the young Descartes by far the most important source of information concerning mathematics in general and algebra in particular.32 Although the Algebra was intended by Clavius as a textbook rather than a treatise—and therefore contained little, if any, really original material—it alerted Descartes to several important Renaissance ideas concerning the status of algebra and its relation to the established mathematical disciplines. Thus it is at the very beginning of this work that Descartes probably encountered, for the first time, the idea of algebra33 (understood quite generally as the ars the subject of which is equations consisting of known and unknown quantities) as a most general mathematical discipline that is somehow common to various mathematical disciplines. For Clavius writes:

31 Algebra Christophori Clavii Bambergensis e Societate Iesv. Rome: Bartholomæus Zannettus, 1608. The text was reprinted in Christophori Clavii Bambergensis Societate Opera mathematica, vol. 2 (Mainz: Reinhardvs Eltz, 1611), the third item in the volume with separate pagination, dated 1612 [sic]. 32 The decisive textual evidence is an indirect autobiographical note by Descartes. It is contained in a report by the mathematician John Pell to Sir Charles Cavendish of the former‫ތ‬s conversation with Descartes that took place in 1646 in Amsterdam, which includes the following: “[Descartes] says he had no other instructor for Algebra than ye reading of Clavy Algebra above 30 yeares agoe” (Helen Hervey, “Hobbes and Descartes in the Light of some Unpublished Letters of the Correspondence between Sir Charles Cavendish and Dr. John Pell,” Osiris 10 (1952), 78). Chikara Sasaki, DescartesҲs Mathematical Thought (Dordrecht: Kluwer Academic Publishers, 2003), 47 conjectures plausibly on this basis that Descartes probably read Clavius‫ ތ‬Algebra at La Flèche between 1613 and 1615. 33 It is well-known that the word “Algebra” in the present sense originates in the Islamic mathematical tradition (the Arabic word being “al-jabr”). As I have learnt from Sasaki, DescartesҲs Mathematical Thought, 73–74 and 248–53, algebra as the art of treating known and unknown quantities in equations emancipated itself from the practical commercial contexts as a theoretical discipline as early as the ninth century in the Islamic world (as far as can be textually discerned), the oldest extant treatise on algebra being Al-KhwƗrizmƯ‫ތ‬s KitƗb al-mukhta‫܈‬ar fƯҲl-‫ۊ‬isƗb al-jabr waҲl-muqƗbalah [On the Calculation by Algebra and Almuqabala] ed. ‘AlƯ MustafƗ Masharrafa and Muতammad Mursi Aতmad (Cairo: The Faculty of Science, 1939) of that century. The European world had not assimilated the theoretical elements of the Islamic art of algebra until the twelfth century and it was several centuries before the significance of algebra for the methodology and philosophy of mathematics become apparent to Western thinkers. Cf. Sasaki, DescartesҲs Mathematical Thought, 248–53 for a survey of these historical issues and for further references.

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Quam igitur scopus iste [artis quam recepto vocabulo Algebram nostrates appellarunt] late vagatur, qui nec genus vllum numerorum, nec vllius magnitudinis diuersitatem, à se alienam putat? vt non numerorum modo latebras omnes detegat, sed vniuscuiusque etiam molis finitam magnitudinem, sonorum metrum, ponderum momenta, mensurarumque certos term, inos assequatur; neque vlla Arithmeticæ quæstio subijciatur, quam non veluti suam agnoscat Algebra, atque expediat. Tam multas, tam varias, tam obscuras, tam difficiles Mathematicæ partes vna algebra pertractat vniuersas (Clavius, Algebra, 2; my emphases).

As a matter of fact, this notion of algebra is a result of putting together several strains that emerge here and there in the Algebra and that Clavius adopted partly from the tradition and partly from his contemporaries. The most important traditional strain involved is no doubt the Neo-Platonist idea of a universal mathematical discipline commonly applicable to arithmetic, geometry and all the other branches of mathematics, conceived—contrary to the abstractionist view of Aristotle and his followers34—as ontologically and epistemologically self-sufficient and as 34

Aristotle is generally interpreted as hinting briefly at the idea of a universal mathematical discipline in Met. E, 1, 1026a23–26; K, 7, 1064b8–9; and An. Post. I, 5, 74a17–24. Charles Crowley, Universal Mathematics in Aristotelian-Thomistic Philosophy: The Hermeneutics of Aristotelian Texts Relative to Universal Mathematics (Washington: University Press of America, 1980) attempts to challenge the standard reading of these passages as referring to a sort of universal mathematics and tries to show, in particular, that the crucial term “țĮșȩȜȠȣ” refers to universal or first philosophy rather than to universal mathematics. It is irrelevant to our present concerns to assess the interpretative situation and we can safely retain the standard reading. For a good survey of the epistemological status of Aristotle‫ތ‬s universal mathematics in the mainstream of the Aristotelian tradition see Ian Mueller, “Aristotle‫ތ‬s Doctrine of Abstraction in the Commentators,” in Aristotle Transformed: The Ancient Commentators and Their Influence, ed. Richard Sorabji (Ithaca: Cornell University Press, 1990), 463–80; for a contrast between the Aristotelian and Neo-Platonic (particularly Proclus‫ )ތ‬views see idem, “Mathematics and Philosophy in Proclus‫ ތ‬Commentary on Book I of Euclid‫ތ‬s Elements,” in Proclus: Lecteur et interprète des ancients, ed. Jean Pépin and Henri Saffrey (Paris: Éditions du Centre nationale de la Recherche Scientifique, 1987), 305–18. Of course I cannot venture even to canvass the extremely complex and controversial topic of Aristotle‫ތ‬s and the Aristotelian notion of mathematics in general and of universal mathematics in particular. In any case, I believe that any attempt to interpret the universality of mathematics Aristotle has in mind as being de facto on a par with the universality ascribed to algebra by Clavius and his contemporaries, or even to invoke algebra as the most adequate modern analogy to what Aristotle has in mind (for a classical exposition along these lines see Thomas Heath, Mathematics in Aristotle, Oxford: Clarendon Press, 1949), are deemed to

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the most concrete, most exact and most foundational of all the mathematical disciplines. The idea was conveyed to Early Modern Western thought mostly through Proclus‫ ތ‬In primum Euclidis Elementorum Librum commentarii which became widely known in the sixteenth century thanks to the fact that Simon Grynaeus attached an original Greek text of the Commentarii to his editio princeps of Euclid’s ȈIJȠȚȤİ߿Į (Basel: Ioan. Hervagivs, 1533) and that Francesco Barozzi translated the Commentarii into Latin in 1560.35 Here Proclus introduces and advocates a conception of “una, & tota Mathematica” as a discipline which is ontologically and epistemologically prior to all the particular mathematical disciplines, its proper objects being the principia “quæ per ea omnia, quæ sunt permeant, & omnia a seipsis gignunt” (Proclus, In primvm Evclidis Elementorum, 2), viz. Finis and Infinitum, as well as communia ... [Mathematicarum] Theoremata, & simplicia, & ab vna scientia orta, quæ cunctas simul Mathematicas cognitiones in vnum continet, [quæ] omnibus [congruentes], possintque tum in Numeris, tum in Magnitudinibus, tum in Motibus inspici .... Huiuscemodi autem sunt, omnia Proportionum, & Compositionum, & Diuisionum, & Conuersionum, & alternarum Immutationum: itemque Rationum omnium ...: & prorsus quæ circa Aequale, & Inæquale vniuerse, & communiter considerantur, non quatenus in Figuris, vel Numeris, vel Motibus sunt, sed quatenus per se vnumquodque horum naturam quandam habet communem, suique simpliciorem præbet cognitionem (ibid., 4).

Thus, mathematical beings in general, and the objects of the una, & tota Mathematica in particular, are non vtique in multis, & diuisis formis primò subsistere arbitrari, neque postremò, & ex multis ortum habere: verùm, vt præcedentia ipsas, simplicitateque, & certa quadam ratione excellentia ponere. iccirco [sic] enim cognitio quoque ipsorum multas antecedit cognitiones, ipsisque principia suggerit, & eæ multæ circa ipsam subsistunt, ad ipsamque referuntur (ibid.).

failure on both historical and epistemological grounds. This view is argued in Sasaki, DescartesҲs Mathematical Thought, ch. 6, §§ 1; 5. 35 Proclus Diadochus. In primvm Evclidis Elementorum commentariorvm ad vniversam mathematicam disciplinam ... libri IIII., ed. and transl. by Francesco Barozzi (Padua: Gratiosus Perchacinus). As announced in the Introduction, I make use of Barozzi‫ތ‬s Latin version in quoting Proclus‫ ތ‬Commentaries.

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As Klein observes, Proclus‫ ތ‬present descriptions of una, & tota Mathematica “were normally understood as a reference to the Mathesis universalis” (Klein, Greek Mathematical Thought, 181) from the sixteenth century onwards.36 The most important strain Clavius seems to adopt from his contemporaries with respect to the notion of algebra, presented in the above passage, is their inclination—interpretative details apart—to identify algebra with the foundational universal mathematical discipline promoted by Proclus. The identification is documented in the Western world at least as early as Gosselin’s De Arte magna (Paris: Gilles Beys, 1577)37 but it is most probable that the figure who directly influenced Clavius in this respect was his disciple and friend Adriaan Van Roomen. For one thing, Van Roomen in his Apologia pro Archimede (1597)38 introduces and tries to justify, in the specific context of defending the employment of arithmetical means to treat geometrical problems,39 a science (called by him vniversalis Mathesis, prima Mathesis or prima 36 To avoid confusion, it should be noted that universality in the sense just established, i.e. universality as an attribute of a discipline whose subject is a common part of all other mathematical disciplines, is to be carefully distinguished and kept apart from universality in the collective sense, i.e. as an attribute of the set that embraces the doctrines of all the mathematical disciplines taken together. Giovanni Crapulli, Mathesis universalis: Genesi di una idea nel XVI secolo (Rome: Edizioni dell’Ateneo, 1969), 8 and Sasaki, DescartesҲs Mathematical Thought, 347–48 claim that this distinction was often captured with the terms “mathesis universalis” (the foundational sense) and “mathesis universa” (the collective sense). This is also how Adriaan Van Roomen uses both terms in his Universæ mathesis idea, qua mathematicæ universim sumptæ natura, præstantia, usus et distributio brevissime proponuntur (Würzburg: Georg Fleischmann, 1602) which is discussed below and then again in ch. 4 37 See ibid., 3r. I owe the reference to Klein, Greek Mathematical Thought, 181. 38 Adriaan Van Roomen, Apologia pro Archimede ad Clariss. virum Iosephum Scaligerum, in In Archimedis circuli dimensionem Expositio & Analysis. Apologia pro Archimede ad Clariss. virum Iosephum Scaligerum. Exercitationes cyclicæ contra Iosephum Scaligerum, Orontium Finæum et Raymarum Ursum, in decem dialogos distinctæ (Würzburg: De Candolle, 1597), 19–55. 39 More specifically, Van Roomen tries to refute Josephus Scaliger‫ތ‬s critique of Archimedes‫ ތ‬attempt at squaring the circle. In particular, Scaliger blames Archimedes for using arithmetical vehicles in his solution to this geometrical problem, thus offending against the principle that in every science the conclusions should be drawn from the principles of that science. See Paul Bockstaele, “Between Viète and Descartes: Adriaan van Roomen and the Mathesis Universalis,“ Archive for History of Exact Sciences 63, no. 4 (2009), 436–40 for a detailed discussion of Scaliger‫ތ‬s criticism.

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Mathematica) which is common to both arithmetic and geometry and the principles of which are valid in each of these disciplines since its proper object is general quantity and proportions or ratios of quantities in general:40 Geometriæ, & Arithmeticæ communem esse scientiam, quæ quantitatem generaliter vti mensurabilem considerat. ... Nimirum scientiam esse quandam Mathematicam communem Arithmeticæ & Geometriæ, ad quam spectarent affectiones omnibus quantitatibus: cùm autem proportio sit omnibus quantitatibus communis, non abstractis tantum vt numeris et magnitudinibus, sed concretis etiam, vti temporibus, sonis, vocibus, locis, motionibus, potentiis (Nam [sic] hæc omnia & plura alia proportionem dicuntur habere, si eorum habitudo consideretur secundum quantitatem) .... ... [P]ropositiones earumque demonstrationes quæ Mathesi Vniversali tribuendæ sunt, non esse pure Arithmeticas, cùm in plerisque vel nulla fiat numerorum mentio, in aliis verò præter numeros assumantur quoque alterius generis quantitates, nec etiam Geometricas, cum nulla magnitudinum ... fiat mentio. Inscribemus autem scientiam hanc nomine, Prima Mathematica, seu Prima MatheseȦs [sic]... (Van Roomen, Apologia pro Archimede, 22– 23; Van Roomen’s emphases).

Furthermore, Van Roomen, in this respect probably influenced by the ideas of François Viète,41 is clear by 1602 that the envisaged vniversalis Mathesis is to be identified with symbolic algebra (most likely in the form it took in Viète‫ތ‬s so-called Logistice speciosa). The most explicit statement to this effect can be found in Van Roomen‫ތ‬s draft, probably composed in 1602, of a commentary on Al-KhwƗrizmƯ‫ތ‬s Al-KitƗb (i.e. on Al-KhwƗrizmƯ‫ތ‬s treatise on algebra):42 “Nos itaque maluimus Algebram ... revocare ad Mathesin primam, quæ quantitatem universalem considerat.”43 It is most unlikely that van Roomen had this draft circulated, yet the identification is clearly hinted even in his published Universæ mathesis 40 Cf. also Van Roomen, Universæ mathesis idea, 20–21: “Prima Mathesis est quæ versatur circa quantitate absolutè sumptam. Objectum ejus est Quantitas absolute sumpta. Finis verò, affectiones quantitatibus omnibus communes exhibere. Principia habet tantum propria. Locum in Mathesi obtinet primum.” 41 Van Roomen visited Viète personally in 1601. See Sasaki, DescartesҲs Mathematical Thought, 260–61 for the relationship of both men and further references. Viète‫ތ‬s contributions relevant to our concerns will be discussed in a moment. 42 See fn. 33. 43 Quoted from Henri Bosmans, “Le fragment du Commentaire d‫ތ‬Adrien Romain sur l‫ތ‬Algèbre de Mahumed ben Musa el-Chowârezmî,” Annales de la société scientifique de Bruxelles 30, no. 2 (1906): 271, fn. ***.

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idea in which he identifies vniversalis Mathematica with Svpputatrix sive Logistice (see ibid., 17).44 Both the way he describes it—“Svpputatrix Græcis ȜȠȖȚıIJȚțȒ dicta, est quæ beneficio canonum universalium, ex datis numeris rebus accommodatis quæsitum elicit” (ibid.)—and the very name Logistice he assigns to it clearly indicate that he has in mind Viète‫ތ‬s Logistice speciosa, which is Viète‫ތ‬s name for his own symbolic algebra.45 François Viète, no doubt the most important figure in the development of algebraic thought in the sixteenth century, was involved much less than Van Roomen in theorizing on the place of vniversalis Mathesis in the entire body of mathematical disciplines, the nature of its proper objects and the ontological and epistemological status of these objects. On the other hand, he succeeded, arguably unlike Van Roomen,46 in developing a unitary algebraic system which provided means to the effective solution of both arithmetical and geometrical problems from the classical canons, i.e. de facto to set up in practice the theoretical project of the vniversalis Mathesis as understood by Van Roomen. It is these achievements of Viète‫ތ‬s that are likely to have encouraged Van Roomen to tie his idea of vniversalis Mathesis to algebra, and Viète‫ތ‬s influence in this respect is distinctly traceable—via Van Roomen—also in Clavius‫ ތ‬Algebra.47 Let us now turn to the other strain mentioned at the beginning of the present section—an identification, or at least an assimilation, of algebra as developed during the Renaissance with the ancient Greek mathematicians’ use of a secret analytical method. Whilst in the Arabic world such an assimilation is explicitly documented by the twelfth century in the writings of Ibn YahyƗ al-Maghribi al-Samaw‫ތ‬al,48 the Western world seem to have waited for hints in this direction until Petrus Ramus‫ ތ‬mathematical 44

Sasaki sees a nuance here to the effect that “‫ލ‬first mathematics‫ ތ‬is the name of [mathematica universalis] when placing it within various mathematical sciences, whereas ‫ލ‬logistic‫ ތ‬is the name from the viewpoint of its use as an instrument for the other mathematical sciences” (DescartesҲs Mathematical Thought, 353). The difference is not important to our present concerns. 45 As opposed to Logistice numerosa, which is the term by which Viète refers to ordinary non-symbolic, casual computations that make use of rudimentary algebraic techniques with determinate numbers. Cf. François Viète, In artem analyticem Isagoge: Seorsim excussa ab Opere restitutæ Mathematicæ Analyseos, Seu, Algebrâ nouâ (Tours: Jamet Mettayer, 1591), 4. Viète‫ތ‬s symbolic algebra is introduced later in this chapter. 46 I cannot argue for this claim here. I rely on the judgment of Chikara Sasaki in his DescartesҲs Mathematical Thought, 264–65). 47 On the relationship between Clavius and Viète, see Sasaki, DescartesҲs Mathematical Thought, 81–82. 48 Cf. Sasaki, DescartesҲs Mathematical Thought, 248–52.

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writings from the mid-sixteenth century,49 and the first extant explicit statement to this effect is to be found in a commentary on Ramus‫ ތ‬Algebra by Lazarus Schöner which first appeared in 1586:50 “A quâ resolutione Algebra Græcis dicta fuit analytica, quibus absoluta Arithmetica dicebatur synthetica” (ibid., 190). However, it is once again Viète who was able to turn these Ramist insights into a precisely expressed and influential form.51 For one thing, he preferred to refer to his Logistice speciosa, i.e. his groundbreaking proposal of a general symbolic algebra, with the term “Ars analytica” rather than with “algebra,” and in his manifesto concerning his symbolic algebra, called significantly In artem analyticem Isagoge, he comments thus on this terminological shift: Ecce ars quam profero noua est, aut demum ita vetusta, & à barbaris defædata & conspurcata, vt nouam omninò formam ei inducere, & ablegatis omnibus suis pseudo-categorematis, ne quid suæ spurcitiei retineret, & veternum redoleret, excogitare necesse habuerim, & emittere noua vocabula ... (Viète, Isagoge, 2v).

Furthermore, he explicitly links, in ch. 1 of the Isagoge, his project of Ars analytica with the analysis of the ancient Greeks and professes to build upon Theon‫ތ‬s description of analysis from his scholium to Book Thirteen of Euclid‫ތ‬s Elementa XIII52 and in the Apollonius Gallus he speaks about 49

Mahoney, Mathematical Career of Fermat, 32 lists Petrus Ramus, Arithmeticæ libri duo et Geometriæ septem et viginti (Frankfurt am Main: A. Wechel, 1627), 86 and idem, Scholarum mathematicarum libri unus et triginta. Dudum quidem a Lazaro Schonero recogniti & aucti, nunc verò in postrema hac editione innumeris locis emendati & locupletati (Frankfurt am Main: A. Wechel), 35. Both items were first published in 1569. 50 In Petrus Ramus, Arithmetices libri duo et algebræ totidem a Lazaro Schonero emendati & explicati. Eiusdem Schoneri libri duo; alter De Numeris figuratis; alter De Logistica sexagenaria (Frankfurt am Main: A. Wechel). As Sasaki, DescartesҲs Mathematical Thought, 256, fn. 171 notes, Mahoney mistakenly ascribes the statement to Ramus himself in his “The Royal Road: The Development of Algebraic Analysis from 1550 to 1650, with Special Reference to the Work of Pierre de Fermat,” Ph.D. diss. (Princeton University, 1967), 187. Mahoney repeats the mistake in his Mathematical Career of Fermat, 32. 51 Ramus‫ ތ‬influence on Viète is demonstrable in Viète‫ތ‬s texts (cf. the next footnote for some examples). Both men possibly met in 1571—cf. e.g. Mahoney, Mathematical Career of Fermat, 32, fn. 19; Sasaki, DescartesҲs Mathematical Thought, 257. 52 Ibid., 4r: “Est veritatis inquirendæ via quædam in Mathematicis, quam Plato primus inuenisse dicitur, à Theone nominata Analysis, & ab eodem definita,

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the “Algebra, quam tradidere Theon, Apollonius, Pappus, & alij veteres Analistæ.”53 This perfectly fits Viète‫ތ‬s self-presentation as a man whose innovations in mathematics are but a development of traditional motifs, or alternatively a renovatio or restitutio of neglected and/or unknown ancient sources.54 It remains to be seen whether, or to what extent, Viète‫ތ‬s Ramist identification of (his version of) algebra with ancient mathematical analysis does justice to the classical doctrine of analysis, and what exactly mathematical analysis can or else is bound to amount to if it is conceived through the prism of algebraic operations. It also remains to be seen whether, or to what extent, or under which conditions, algebraic analysis as envisaged by Van Roomen and as proposed by Viète could really perform the rôle of the Mathesis vniversalis in the sense explicitly stated— along Proclean lines—by Van Roomen. In any case, it is clear that the motifs introduced in the present section scented the intellectual air in the years when Descartes, in the course of his early intellectual development, started breathing that air and grew acquainted with them in one way or another.55 The time has now come to have a look at what Descartes was able to make out of these materials.

Adsumptio quæsiti tanquam concessi per consequentia ad verum concessum. Vt contra, Synthesis, Adsumptio concessi per consequentia ad quæsiti finem & comprehensionem. Et quanquam veteres duplicem tantùm proposuerunt Analysim ȗȘIJȘIJȚțȒ & ʌȠȡȚıIJȚțȒ quas definitio Theonis maximè pertinet, constitui tamen etiam tertiam speciem, quæ dicatur ૧ȘIJȚț੽ ਲ਼ İȟȘȖȘIJȚțȒ, consentaneum est, vt sit Zetetice quâ inuenitur æqualitas proportione magnitudinis de quâ quæritur, cum ijs quæ data sunt. Poristice, quâ de æqualitate vel proportione ordinati Theorematis veritas examinatur, Exegetice, quâ ex ordinata æqualitate vel proportione ipsa de qua quæritur exhibetur magnitudo. Atque adeò tota ars Analytica triplex illud sibi vendicans officium definiatur, Doctrina bene inueniendi in Mathematicis.” As a matter of fact, Viète uses verbatim Ramus‫ ތ‬paraphrase of Theon‫ތ‬s description of analysis from Ramus, Scholarum mathematicarum libri unus et triginta, 300. Also the characterization of the Ars analytica as the “doctrina bene inveniendi in mathematicis” is distinctively Ramist. 53 François Viète, Apollonius Gallus seu exsuscitata Apollonii Pergæi ʌİȡ‫݋ ޥ‬ʌĮijࠛȞ Geometrica (Paris: David Le Clerc, 1600), Appendicvla I, no pagination. 54 Cf. Klein, Greek Mathematical Thought, 152–54. 55 The best integral textual evidence to this effect is Descartes‫ ތ‬autobiographical notes in Reg. IV (see in particular AT X, 373–78). The well-known complications concerning the constitution and historical layers of Reg. IV (discussed in ch. 4) are irrelevant in this regard.

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3.2 Analysis in Descartes‫ ތ‬Mathematics Descartes presented to the public the concrete results of his treatment of the above materials in La Geometrie, his mathematical masterpiece of 1637.56 The achievements of La Geometrie are set briefly within the broader canvas of Descartes‫ ތ‬philosophical concerns, in a very illuminating way, in Part II of the Discours.57 Yet the most important source of information concerning both the larger philosophical concerns that lie behind Descartes‫ ތ‬engagement in mathematics and the development of his mathematical ideas that are brought to maturity in La Geometrie remain the Regulæ—despite intricacies due to the private character, fragmentation and overlapping textual layers of this work.58 I intend to present and discuss Descartes‫ ތ‬achievements in the field of algebraic analysis only in so far as their grasp is indispensable to answering questions (i*) and (ii*) posed at the end of sec. 3.1.1, to the extent that the resultant answers are in the long run sufficient for an adequate understanding of what Descartes seeks in his attempts at generalizing the analytic procedure of algebraic treatment, as put to work in La Geometrie, beyond the domain of mathematics. I start with a brief exposition of by far the greatest achievement in the field of algebraic analysis immediately before Descartes—Viète‫ތ‬s so-called Logistice speciosa. A comparison with this doctrine should enable us to appreciate the real strength and novelty of the version of algebraic analysis proposed by Descartes and to grasp its theoretical potential with regard to one of the proper objects of our larger concerns—generalization of an analytic method across the entire possible field of gaining scientiæ. Descartes‫ތ‬ conception itself is then presented in two stages: first I focus on his alternative way of subjecting geometry to algebraic treatment and its implications for the issue of identifying the objects of arithmetic, geometry, and general algebra; next I turn to his de facto identification of algebraic treatment with the employment of analytical method in mathematics.

56 La Geometrie is the last of the three “essais de Methode” Descartes published together with the Discours. 57 See in particular AT VI, 16–22. 58 For the sake of completeness, Isaac Beeckman‫ތ‬s record of a part of the lost early treatment on algebra by Descartes (AT X, 333–35) should be added to the list of textual sources. This text is discussed in detail in Sasaki, DescartesҲs Mathematical Thought, 159–66.

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3.2.1 A Point of Comparison: Viète‫ތ‬s Logistice Speciosa Immediately after the appearance of La Geometrie in 1637, it was clear to everyone who cared that Descartes‫ ތ‬achievements, however novel and whatever exactly they might have turned out to be, de facto amounted, in one way or another, to a continuation of Viète‫ތ‬s groundbreaking project of developing a system of symbolic algebra as a general problem-solving tool applicable both in arithmetic and geometry. Indeed, Descartes surely must have known something about Viète and his program from indirect sources and it is even likely that he at least glanced at some of Viète’s work in 1632.59 Further, while it has become a commonplace in recent scholarship that the accusation of having plagiarized Viète (an accusation which Descartes actually faced) is groundless and that the essential innovations in both the theoretical framework of algebraic treatment and the particular solutions of mathematical problems were worked out by him independently of Viète60 and amount to major progress in the field of mathematical analysis, it still holds that as a matter of fact, Descartes‫ތ‬ version of algebraic analysis in mathematics is characterized, among other things, with acute sensitivity to various problems, drawbacks and 59 As Sasaki reports (DescartesҲs Mathematical Thought, 246–47), it can be concluded with certainty from the correspondence between Mersenne and Jean de Beaugrand that Mersenne acquired a copy of de Beaugrand‫ތ‬s edition of Viète‫ތ‬s Isagoge published together with Viète‫ތ‬s Ad Logisticem speciosam notæ priores (Paris: Guillaume Baudry, 1631) for Descartes. Descartes‫ ތ‬letter to Mersenne of 3 May 1632 confirms he received the book and had a look at it—cf. AT I, 245. Moreover, Clavius mentions Viète in his Algebra, which Descartes certainly studied, in the context of solving cubic equations (Clavius, Algebra, 26). Mersenne reported on Viète‫ތ‬s Ars analytica in some detail in his Les Verités des sciences contre les septiques [sic] ou Pyrrhoniens (Paris: Tovssainct dv Bray, 1625); and Sasaki speculates (DescartesҲs Mathematical Thought, 268) Descartes might already have had conversations about Viète with Isaac Beeckman in 1618–19. 60 It was Jean de Beaugrand, a jurist, mathematician, Sécrétaire du Chancelier and editor and commentator on Viète‫ތ‬s writings, who accused Descartes of having plagiarized Viète‫ތ‬s and/or Harriot‫ތ‬s writings in 1638. Descartes was consistent in responding to the charge not that he knew nothing of Viète prior to the composition of his Geometrie but that he achieved the results presented in La Geometrie independently of Viète‫ތ‬s ideas: see in particular Mers., AT I, 479–80; AT II, 524. Moreover, Descartes‫ ތ‬correspondence with Mersenne in the spring of 1632 (see Mers., AT I, 244–45) makes it clear that Descartes had succeeded in solving the Pappus problem—presented later in La Geometrie as the hallmark of the success of his own version of algebraic analysis in mathematics—in winter 1631–32, i.e. before he saw de Beaugrand‫ތ‬s annotated edition of Viète‫ތ‬s Isagoge and Notæ.

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limitations that seem to infest Viète‫ތ‬s version of purely formal algebraic analysis. This is why it will be well to present an account, however brief, of Viète‫ތ‬s program of algebraic analysis in order to appreciate Descartes‫ތ‬ innovative contributions to mathematics. We need not venture a detailed exposition of Viète‫ތ‬s Logistice speciosa, in the sense of going through the definitions, axioms, and precepts of this discipline, accounting for the algebraic operations employed in it, or tracing its application to various types of mathematical problems. What we do need, on the other hand, is a clear account of the nature of the generality of Viète‫ތ‬s Logistice speciosa and how its proper object is related to the objects of arithmetic and geometry. From the systematic point of view, Viète adopts as his starting point a classical example of a discipline which applies generally to both numbers and geometrical figures, viz. Eudoxus‫ ތ‬general theory of proportions as reported and developed in Euclid‫ތ‬s Elementa V.61 As Danielle Macbeth puts it pertinently, Eudoxus‫ ތ‬theory ... concerns itself ... with ratios of any sorts of entities that can stand in the relevant relationships; it concerns numbers but not quâ numbers because it applies equally to figures and motions, and it concerns figures and motions but not quâ figures or motions because it applies equally to numbers (Macbeth, “Viète, Descartes,” 91; Macbeth’s italics).

It is crucial to appreciate that in this theory, the notions of ratio and proportion are perfectly univocal. This is due to the fact—established by Aristotle—that the theory at issue deals with numbers, magnitudes etc. insofar as these objects fall under a higher universal, viz. (in Van Roomen‫ތ‬s vocabulary) general quantity.62 In view of this tradition, Viète identifies as the object of his Logistice speciosa equations (æqualitates), which he takes as reciprocal to proportions (proportiones),63 and clearly takes the perfect generality of this discipline as founded upon the universality of the notions of ratio and proportion along the Eudoxian lines as interpreted by Aristotle. 61

Eudoxus‫ ތ‬theory of proportions has been a commonplace example of such general mathematical discipline at least since Aristotle‫ތ‬s An. Post. I, 5, 74a19–24. Cf. Heath, Mathematics in Aristotle, 43; Klein, Greek Mathematical Thought, 158– 59; Sasaki, DescartesҲs Mathematical Thought, 292–93. 62 See Aristotle, An. Post. I, 5, 74a19–24. Aristotle writes that the higher universal in question is not “a single named item” (Complete Works, 1:120). Cf. also Proclus, In primvm Evclidis Elementorum, 4. 63 See Viète, Isagoge, 4v: “Itaque proportio potest dici constitutio æqualitatis. Æqualitas, resolutio proportionis.”

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Viète‫ތ‬s chief achievement is that he is able to subject proportions captured in equations to algebraic operations (i.e. addition, subtraction, multiplication, division, raising to a power, and taking a root) in purely symbolic form. While he follows Diophantus in using signs for unknown quantities and their powers (uppercase vowel letters of the Latin alphabet followed by an adjective marking the power)64 to signify indeterminate (and possibly determinable) objects,65 he breaks resolutely beyond the Diophantine logistic in that he elevates the indeterminateness to an essential feature of the unknown objects in general equations up to the final solution of a given problem.66 This, together with a symbolic treatment of the known quantities—parameters and variables—enables him to propose his Logistice speciosa as a purely formal calculus dealing with the structure of algebraic expressions set equal to one another and, in Mahoney‫ތ‬s pertinent phrase, “to treat the data of a problem as parameters and hence to treat the problem itself as a general type” (Mahoney, Mathematical Career of Fermat, 35). It is this purely formal character which renders the Logistice speciosa such a powerful general tool for finding solutions to mathematical problems.67 The crucial feature of Viète‫ތ‬s Logistice speciosa as just presented, at least in view of an envisaged comparison with Descartes, is that the formal calculus of which it consists is bound to remain essentially uninterpreted.68 At root, this is due to the fact that Viète accepts the classical Greek view that combinatory operations with magnitudes are to be interpreted in terms of geometrical figures. Unlike numbers, geometrical figures have differing dimensions according to which, at least in the classical Greek interpretation of the proper objects of geometry, they are distributed into different types (non-dimensional points, one-dimensional lines and curves, 64

See e.g. Macbeth, “Viète, Descartes,” 90–91 for a brief description of Viète’s notation. 65 This is from where his Logistice speciosa probably derives its name: Viète calls the signs for the unknowns species, which he takes as the translation of the Greek word İੇįȠȢ, employed by Diophantus to refer to the (provisionally) indeterminate quantities. See Klein, Greek Mathematical Thought, 133–40 for a detailed treatment. 66 Cf. ibid., 163–66. 67 Viète assessed his art so highly that he concluded his Isagoge with a famous dictum: “[F]astuosum problema problematum ars Analytice ... iure sibi adrogat, Quod est, NULLUM NON PROBLEMATA SOLVERE” (Isagoge, 9v). 68 I am indebted to Macbeth, “Viète, Descartes” for having clarified to me how important the present feature is for adequate assessment of Descartes‫ ތ‬achievement in mathematical analysis. I also owe much to Mahoney‫ތ‬s lucid account of Viète‫ތ‬s Logistice speciosa in his Mathematical Career of Fermat, ch. 2.

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two-dimensional plane figures and three-dimensional solid figures). The ontological differences between the kinds of figures then set severe constraints on combinatory operations with magnitudes: firstly, only homogeneous magnitudes can stand as the terms of addition and subtraction, so that algebraic additions of lines to areas (in the form x + x2), of areas to solids (in the form x2 + x3), or of lines to solids (in the form x + x3) make, strictly speaking, no sense;69 secondly, the multiplication of magnitudes was understood in terms of constructing a figure of a higher dimension, so that the product of two straight lines was understood as a rectangle, the adjacent sides being the multiplied lines, and the product of three lines, or a rectangle and a line, was the corresponding parallelepiped;70 and since solids exhaust the number of available dimensions in Euclidean space, at most three lines are allowed to be multiplied, with no multiplications of areas or solids being permissible, in algebraic operations with magnitudes of no greater power than three being allowed. By way of contrast, combinatory operations with numbers are not thus constrained: numbers being dimensionless, any basic combinatory operation with them will always yield a homogeneous result. As a consequence, the notions of algebraic combinatory operations are bound to remain essentially equivocal in Viète‫ތ‬s system: they cannot mean the same when applied to magnitudes and when applied to multitudes. Thus if the purely symbolic Logistice speciosa is to retain, under these conditions, its professed general character over arithmetic, geometry (and all the other mathematical disciplines for that matter), it cannot but count, as already stated, as an essentially uninterpreted formal calculus. As Macbeth puts it, [t]he first stage of the Analytic Art ... takes one out of a particular domain of inquiry, either arithmetical or geometrical, into a purely formal system of uninterpreted signs that are to be manipulated according to rules laid out in advance, and only at the last stage ... are the signs again provided an interpretation, either arithmetical or geometrical (Macbeth, “Viète, Descartes,” 92).71

69

Viète expresses the constraint concisely thus: “Homogenea homogeneis comparari. Nam quæ sunt heterogenea, quomodo inter se adsecta sint, cognosci non potest .... Itaque, Si [sic] magnitudo magnitudini additur, hæc illi homogenea est. Si magnitudo magnitudini subducitur, hæc illi homogenea est” (Isagoge, 4v). 70 Cf. ibid.: “Si magnitudo in magnitudinem ducitur, quæ fit, huic & illi heterogenea est. Si magnitudo magnitudini adplicatur, hæc illi heterogenea est.” 71 This is where the adverbs plano and solido get to work as annotators of the signs for known parameters (upper case consonants in Viète‫ތ‬s symbolism): they indicate, throughout the entire process of analysis, which type of figure can meaningfully be

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An extremely important theoretical implication of this situation is that the Logistice speciosa can by itself generate only problems of syntax, without, however, being able to determine of itself “which problems [of this sort] will lead to successful application of specious logistic to concrete problems. ... Hence, the analytic art must look beyond itself for the problems it will investigate” (Mahoney, Mathematical Career of Fermat, 39). This is why (as Mahoney explains and shows)72 Viète in fact devoted most of his energies to the project of extending in concreto the realm of applicability of his new art by translating the classical geometrical problems into the symbolic language of his Logistice speciosa. Yet while Viète succeeded in this in so far as the problems ended up with point constructions, the essential dimensionality of geometrical problems he retained from the ancient Greeks deemed to failure any attempt at adopting to his Logistice speciosa the vast realm of problems that themselves involve dimensionality, i.e. the realm which among others includes— significantly—locus problems.73 As already indicated, Descartes‫ ތ‬program of algebraic analysis might well be characterized as de facto stemming—whether or not Descartes was actually acquainted with Viète‫ތ‬s ideas—from efforts to steer clear through the complications in which Viète‫ތ‬s project, as we saw, became jammed. Let us now turn to Descartes‫ ތ‬own views on this complex topic.

3.2.2 Descartes‫ ތ‬Algebraization of Geometry In the course of defending himself against the accusation of plagiarizing Viète‫ތ‬s work, Descartes writes to Mersenne: Et tant s‫ތ‬en faut que les choses que i‫ތ‬ay écrites puissent estre aisément tirées de Viete, qu‫ތ‬au contraire, ce qui est cause que mon traitté est difficile à entendre, c‫ތ‬est que i’ay tasché à n‫ތ‬y rien mettre que ce que i‫ތ‬ay crû n‫ތ‬auoir point esté sceu ny par luy, ny par aucun autre. ... [I]e determine [les regles de mon Algebre] generalement en toutes équations, au lieu que [Viete] n‫ތ‬en ayant donné que quelques exemples particuliers, dont il fait toutesfois si grand estat qu‫ތ‬il a voulu conclure son liure par là, il a monstré qu‫ތ‬il ne le pouuoit determiner en general. Et ainsi i‫ތ‬ay commencé où il auoit acheué ... (AT I, 479).

assigned to the parameter should the problem in question be geometrical. Cf. ibid., 90–91. 72 See Mahoney, Mathematical Career of Fermat, 39–42. 73 I owe this perspicuous diagnosis to Mahoney, ibid., 42.

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Whatever exactly it is that Descartes means by the final sentence, what is clear is that he believes his algebraic exposition undoubtedly surpasses Viète‫ތ‬s chiefly owing to its perfect generality. Indeed, from his perspective there are at least two important and closely interconnected respects in which Viète‫ތ‬s program struggles concerning generality. We accounted for one of them in sec. 3.2.1: due to Viète‫ތ‬s acceptance of the view that combinatory operations with magnitudes are burdened with the spatial dimensionality of geometrical figures, only homogeneous magnitudes can be subjected to the algebraic operations of addition and subtraction and no greater power than three is allowed in algebraic operations with magnitudes. One implication of these constraints that compromises the professed generality of Viète‫ތ‬s problem-solving tool is that they render that tool useless with regard to the entire realm of legitimate mathematical problems which themselves involve dimensionality. The other of the troublesome respects at issue has to do with an unavoidable implication of the aforementioned dimensionality constraints, viz. that the formal calculus Viète is eventually able to offer must be taken as a system of essentially uninterpreted symbols. This implication poses no problem for one such as Leibniz, whose ideal of the method of discovery is a universal computational combinatorics of symbols; but it turns out altogether unacceptable for Descartes in the light of one of his criticisms of purely formalist procedures in the régime of discovery with which we were dealing in ch. 2, namely his rejection of arriving at conclusions in this context purely ex vi formæ, while “illis confisa ratio ... quodammodo ferietur ab ipsius illationis evidenti & attentâ consideratione” (Reg. X, AT X, 405–406). The inattention Descartes warns against clearly covers both the question of the truth or falsity of the contents represented with the corresponding formulae and the question of the validity or invalidity of the rules of manipulating the symbols involved. Descartes is able to overcome both these troublesome aspects of Viète‫ތ‬s conception in one stride, by means of a single, ingenious and dazzlingly simple move: the symbols that are entered as terms in relations subjected to algebraic operations are interpreted as unambiguously standing for rectilinear line segments, while these line segments—more exactly, their lengths—are themselves interpreted as signifying neither geometrical magnitudes, i.e. continuous quantities infinitely divisible into parts, nor arithmetical multitudes, i.e. discrete quantities divisible into a finite number of parts, but instead as signifying literally general quantities.74 As such, these algebraic line segments, in Emily Grosholz‫ތ‬s 74 Descartes makes the point, quietly and briefly, as early as the end of Reg. XIV (with the only difference that he allowed by that time not only rectilinear line

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apt phrase, “become a ... thoroughly homogeneous subject-matter with no inherent texture of its own, which only serves to instantiate a relational structure ... which captures all there is to know about placeholders, line lengths whose own geometrical unity, spatiality and geometrical content ... is of no importance.”75 The algebraic line segments then share with the objects of arithmetic—numbers—the feature of being dimension-free; and they are tied to geometry in that they can be conceived as expressing relations among such lines of certain geometrical figures that are their boundaries. This brief exposition already provides some clues as to how Descartes intends to overcome the two defects concerning generality in Viète‫ތ‬s system. As for (1), the latter trouble with the purely formalist character of Viète‫ތ‬s algebraic procedure, Descartes offers an alternative conception within which the symbolic language of algebra is always pre-interpreted in a univocal manner: what is being related and operated upon are exclusively straight line segments whose lengths signify general quantities.76 As for (2), the former trouble with dimensionality, Descartes is able to show, on the one hand, with regard to the requirement of segments but also rectangular plane figures as representative of general quantities—see fn. 77): [H]îc non minùs abstrahendas esse propositiones ab ipsis figuris, de quibus Geometræ tractant, si de illis sit quæstio, quàm ab aliâ quâvis materiâ; nullasque ad hunc vsum esse retinendas præter superficies rectilineas & rectangulas, vel lineas rectas, quas figuras quoque appellamus, quia per illas non minùs imaginamur subjectum verè extensum quàm per superficies, vt suprâ dictum est; ac denique per easdem figuras, modo magnitudines continuas, modo etiam multitudinem sive numerum esse exhibendum; neque quicquam simplicius, ad omnes habitudinum differentias exponendas, inveniri posse ab humanâ industriâ (AT X, 452; my emphases). For brief and perspicuous outlines of the difference between lines quâ geometrical magnitudes and quâ representations of pluralities, see e.g. Gaukroger, Cartesian Logic, 78–79; Macbeth, “Viète, Descartes,” 99–101. 75 Emily Grosholz, Cartesian Method and the Problem of Reduction (Oxford: Clarendon Press, 1991), 83–84. 76 Cf. Macbeth, “Viète and Descartes, 101: “In Descartes‫ ތ‬Geometry a letter such as ‫ލ‬a‫ ތ‬or a combination of signs such as ‫(ލ‬a + b)2‫ ތ‬is not an uninterpreted expression that can be interpreted either geometrically or arithmetically; it is a representation of (an indeterminate) line length where a line length is to be distinguished both from a Euclidean line segment and from a number classically conceived as a collection of units.”

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homogeneity, (a) that algebraic operations whose terms are dimension-free line segments allow of a genuinely geometrical interpretation while no spatial dimensions need be introduced, i.e. to show that even upon geometrical interpretation, all the algebraic operations with line segments again yield line segments. Furthermore, with the aid of the very same insight he is able to show, with regard to the problem of power in magnitudes, (b) that even upon geometrical interpretation, the degree of an algebraic equation has essentially nothing to do with spatial dimensionality. Let us have a closer look at all these points in order. As for (1), it has already been established that more than just a univocal interpretation of the symbolic system at each moment of its employment is to be secured in order that for Descartes the employment count as capable of yielding genuinely scientific results: that is to say, the items to which the symbols refer have to count as cognitively so simple and perspicuous that they eventually can become objects of pure intuitus. Indeed, Descartes claims that much in a well-known passage from DM 2 which in fact amounts to an illuminating prolegomenon to La Geometrie. He writes that having started considering how the Sapientia vniversalis might be attained, ie ne fus pas beaucoup en peine de chercher par lesquelles il estoit besoin de commencer: car ie sçauois desia que cҲestoit par les plus simples & les plus aysées a connoistre; & considerant qu‫ތ‬entre tous ceux qui ont cy deuant recherché la verité dans les sciences, il n‫ތ‬y a eu que les seuls Mathematiciens qui ont pu trouuer quelques demonstrations, c‫ތ‬est a dire quelques raisons certaines & euidentes, ie ne doutois point que ce ne fust par les mesmes qu‫ތ‬ils ont examinées .... Mais ... voyant qu‫ތ‬encore que [les] obiets [des toutes ces sciences particulieres, qu‫ތ‬on nomme communement Mathematiques,] soient differens, elles [sciences particulieres] ne laissent pas de s‫ތ‬accorder toutes, en ce qu‫ތ‬elles n‫ތ‬y considerent autre chose que les diuers rappors ou proportions qui s‫ތ‬y trouuent, ie pensay qu‫ތ‬il valoit mieux que i‫ތ‬examinasse seulement ces proportions en general, & sans les supposer que dans les suiets qui seruiroient a m‫ތ‬en rendre la connoissance plus aysée; mesme aussy sans les y astreindre aucunement, affin de les pouuoir d‫ތ‬autant mieux appliquer aprés a tous les autres ausquels elles conuiendroient. Puis, ayant pris garde que, pour les connoistre, i‫ތ‬aurois quelque fois besoin de les considerer chascune en particulier, & quelquefois seulement de les retenir, ou de les comprendre plusieurs ensemble, ie pensay que, pour les considerer mieux en particulier, ie les deuois supposer en des lignes, a cause que ie ne trouuois rien de plus simple, ny que ie pûsse plus distinctement representer a mon imagination & a mes sens; mais que, pour les retenir, ou les comprendre plusieurs ensemble, il falloit que ie les expliquasse par quelques chiffres, les plus courts qu‫ތ‬il seroit possible ... (DM 2, AT VI, 19–20; my emphases).

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There is no doubt that “les lignes” Descartes mentions in this passage are the rectilinear line segments designed to signify general quantities in his algebraic project of La Geometrie;77 and Descartes focuses now on the relations of these lines to his own cognitive capacities and, by extrapolation from the opening sections of the Discours, to human cognitive capacities in general.78 What now matters for him is, indeed, that such lines count as “les plus simples” items and therefore as items that are “les plus aysées a connoistre”, in the sense that there is nothing (presumably within the domain of mathematical objects) that could be represented “plus distinctement” to human cognitive faculties.79 Moreover, Descartes apparently has it that the relata of the “rappors ou proportions” that hold between the objects of all of the various branches of mathematical sciences80 can effectively be reduced to those cognitive primitives; and he makes it clear that his ultimate objective is purely and 77 As a matter of fact, Descartes somewhat mispresents the development of his mathematical thought. While he refers to the algebra as he worked it out roughly in 1619–20 (and as it is present in his Regulæ) in the quoted passage, the items that entered the proportional relations to be articulated algebraically in his writings by that time were not exclusively line segments but line segments and rectangular plane figures. This indicates that the problem of dimensionality in the context of a general science of quantities was still somehow present in Descartes‫ ތ‬conception even by the time of the Regulæ. For a careful reconstruction of Descartes‫ ތ‬early algebra upon the basis of Beeckman‫ތ‬s excerpts, see Sasaki, DescartesҲs Mathematical Thought, ch. 4, § 1. For the situation in the Regulæ in this respect, see in particular Reg. XVIII, AT X, 464–67; Reg. XIV, AT X, 452. 78 Cf. DM 1, AT VI, 2: “[L]a puissance de bien iuger, & distinguer le vray d’auec le faux, qui est proprement ce qu‫ތ‬on nomme le bon sens ou la raison, est naturellement esgale en tous les hommes; et ... la diuersité de nos opinions ne vient pas de ce que les vns sont plus raisonnables que les autres, mais seulement de ce que nous conduisons nos pensées par diuerses voyes, & ne considerons pas les mesmes choses.... Pour moy, ie n‫ތ‬ay iamais presumé que mon esprit fust en rien plus parfait que ceux du commun .... [P]our la raison, ou le sens, d‫ތ‬autant qu‫ތ‬elle est la seule chose qui nous rend hommes, & nous distingue des bestes, ie veux croyre qu‫ތ‬elle est toute entiere en vn chascun ....” 79 Perspicuitas & facilitas summa are highlighted as properties of the vera Mathesis also, for example, in Reg. IV, AT X, 377. The question of why the faculties Descartes mentions on this score in the quoted AT VI, 19–20 are perhaps somewhat surprisingly those of “imagination & sens” is addressed towards the end of this chapter. 80 Descartes surely has in mind the items of the Pythagorean quadrivium (arithmetic, geometry, music, and astronomy); yet he also seems prepared to include among such mathematical sciences optics, mechanics, “aliæque complures,” which he mentions in a similar context in Reg. IV, AT X, 377.

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simply an examination of such reductively generalized “rappors ou proportions”. As for (2)(a), Descartes makes the point tersely in a few opening sentences of his Geometrie, AT VI, 369–70: Tous les Problesmes de Geometrie se peuuent facilement reduire a tels termes, qu‫ތ‬il n‫ތ‬est besoin, par aprés, que de connoistre la longeur de quelques lignes droites, pour les construire. Et comme toute l‫ތ‬Arithmetique n‫ތ‬est composée que de quatre ou cinq operations, qui sont: l‫ތ‬Addition, la Soustraction, la Multiplication, la Diuision, & l‫ތ‬Extraction des racines, qu‫ތ‬on peut prendre pour vne espece de Diuision; ainsi n‫ތ‬a-t-on autre chose a faire, en Geometrie, touchant les lignes qu‫ތ‬on cherche, pour les preparer a estre connuës, que leur en adiouster d‫ތ‬autres, ou en oster; ou bien, en ayant vne que ie nommeray l‫ތ‬vnité pour la rapporter d‫ތ‬autant mieux aux nombres, & qui peut ordinairement estre prise a discretion, puis en ayant encore deux autres, en trouuer vne quatriesme, qui soit a l‫ތ‬vne de ces deux comme l‫ތ‬autre est a l‫ތ‬vnité, ce qui est le mesme que la Multiplication; ou bien en trouuer vne quatriesme, qui soit a l‫ތ‬vne de ces deux comme l‫ތ‬vnité est a l‫ތ‬autre, ce qui est le mesme que la Diuision; ou enfin trouuer vne, ou deux, ou plusieurs moyennes proportionelles entre l‫ތ‬vnité & quelque autre ligne, ce qui est le mesme que tirer la racine quarrée, ou cubique, &c.

That is to say, the ingenious general strategy Descartes announces in offering a geometrical interpretation of algebraic operations with quantities such that the products would always remain homogeneous with the terms of the operations is to take the length of an arbitrarily chosen line segment as the unit within the given problem, then to set this unit in a certain number of known proportions with the lengths of several other line segments whose mutual proportions are given, and finally to interpret the product of a given operation as the length of a given line segment with a fixed, proportional relationship to the rest. Whilst the operations of addition and subtraction pose no problem with regard to the requirement of homogeneity in so far as executed in terms of line lengths, the geometrical interpretations Descartes offers for the remaining operations are extremely illuminating. As for multiplication and division, consider similar triangles ABC and DBE:81

81

Adapted from Geom. I, AT VI, 370.

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201

Fig. 3-1

Their similaarity ensures thhat CB : AB = EB : DB. S So if AB is tak ken as the unit length, we get EB = CB C u DB. Thu us, we derive the product of o CB and DB as follows: take an arbitrary a AB as a the unit lenngth; from B,, draw an arbitrary straight line diff fferent from AB; A put the lenngth of BC on o it from B; connect C with A; puut the length of o DB on the line AB from m B; draw the parallel w with AC throuugh D; and the intersection with BC yield ds E. The length of BE E is the sougght-after produ uct.82 The intterpretation off division can be easilyy established by b the recipro ocal proceduree. As for ttaking a roott (and the reeciprocal operration of raissing to a power), consider the folloowing construction:83 Fig. 3-2

82

Cf. ibid. Deescartes alludess to an alternatiive way, also prreserving homo ogeneity in terms of line segments, at the end of Reg. XVIII, X AT X, 4668. 83 Adapted froom Geom. I, AT T VI, 370.

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Due to the Thales‫ ތ‬theorem, the triangles FHI, FGI and GHI are similar. Their similarity ensures that FG : GI = GI : GH. So if FG is taken as the unit length, we get GI = ξ . Thus, we derive the square root of GH as follows: take an arbitrary FG as the unit length; put the length of GH on the line FG from G; divide FH into two equal parts at K; describe a semicircle with diameter FH around K as a centre; from G draw a straight line perpendicular to FH; and the intersection with the drawn semi-circle yields I. The length of GI is the sought-after root.84 Whatever problems with which this ingenious performance might be inflicted concerning overall justification,85 Descartes thus clearly succeeded in establishing what he announced: that is to say, that a professedly perfectly general problem-solving tool in matters mathematical is not compromised with homogeneity constraints even on a genuinely geometrical interpretation. Finally, let us turn to point (2)(b). The groundbreaking insight which frees the notion of degree of an algebraic equation from spatial dimensionality even upon geometrical interpretation is presented by Descartes as early as Reg. XVI, AT X, 457: Maximè ... notandum est, radicem, quadratum, cubum, &c., nihil aliud esse quàm magnitudines continuè proportionales, quibus semper præposita esse supponitur vnitas illa assumptitia ...: ad quam vnitatem prima proportionalis refertur immediatè & per vnicam relationem; secunda verò, mediante primâ, atque idcirco per duas relationes; tertia, mediante primâ & secundâ, & per tres relationes, &c. Vocabimus ergo deinceps primam proportionalem, magnitudinem illam, quæ in Algebrâ dicitur radix; secundam proportionalem, illam quæ dicitur quadratum, & sic de cæteris.

84

Cf. ibid. Grosholz, Cartesian Method and Reduction, brilliantly highlights several “very knotty problems of justification” that Descartes‫ ތ‬present performance professedly brings about “due to his conception of method” (ibid., 84). In particular, Grosholz sees as problematic in this respect that Descartes‫ ތ‬present procedure is bound to assume the prior availability of geometrical cognition that should be established first with the aid of the algebraic techniques at hand, such as, for example, “a knowledge of triangles and the kind of unity they have which makes the equivalence relation of similarity possible” (ibid., 85) and “antecedently available curves to serve as means of construction” (ibid., 86).

85

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Furthermore, he confirms it early in Geom. I, AT VI, 371:86 Il est ... a remarquer que toutes les parties d‫ތ‬vne mesme ligne se doiuent ordinairement exprimer par autant de dimensions l‫ތ‬vne que l‫ތ‬autre, lorsque l‫ތ‬vnité n‫ތ‬est point determinée en la question: comme icy a3 en contient autant qu‫ތ‬abb ou b3, dont se compose la ligne que i‫ތ‬ay nommée ටǤ ܽଷ Ȃܾଷ  ൅ ܾܾܽ....

In establishing this, Descartes makes use of the insight, as did Viète before him,87 that the successive powers of the unknown quantities form a series of continuous proportionals, i.e. x : x2 = x2 : x3 = x3 : x4 ... etc. However, Descartes goes beyond anything achieved before him in this respect in that he reinterprets the meaning of the insight in line with his univocal algebra of straight line segments. For one thing, he observes that the series of continuous proportionals could and should be constructed as beginning with a unit quantity, so that it should generally read 1 : x = x : x2 = x2 : x3 = x3 : x4 ... etc.88 Furthermore, he is in a position to apply to the series his geometrical interpretation of multiplication in terms of proportions among the lengths of line segments. Consider the following figure:89

86

“ξǤ ‫ ”ݔ‬is Descartes‫ ތ‬way of expressing the cube root of x. Cf. Viète, Isagoge, 4r: “[I]nde constituta, ut fit, solemni magnitudinum ex genere ad genus vi suâ proportionaliter adscendentium vel descendentium serie seu scalâ, quâ gradus earundem & genera in comparationibus designentur ac distinguantur.” 88 Cf. Reg. XVIII, AT X, 463: “[V]t vnitas ad a 5, ita a 5 ad a2 sive 25; & rursum, vt vnitas ad 5, ita a2 25 ad a3 125; & denique, vt vnitas ad a 5, sic a3 125 ad a4 quod est 625, &c.: neque enim aliter fit multiplicatio, si eadem magnitudo ducatur per se ipsam, quàm si per aliam planè diversam duceretur.” (The connectives in angle brackets are omitted in the A- and H-versions of the Regulæ.) 89 Adapted from Geom. II, AT VI, 391. 87

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Fig. 3-3

The similariity of the trianngles YBC, YCD, Y YDE ... etc. ensures that t YB : YC = YC : YD = YD : YE Y ... etc. Thu us if YB is taaken as the un nit length, then 1 : YC C = YC : YD = YD : YE .... etc.; therefoore YD = (YC C)2, YE = 3 (YC) ... etcc. Thus once we name YC C, say, “x,” w we get the geometrical interpretatioon of the seriees of continuo ous proportionnals—and ipsso facto a geometrical interpretatioon of powerrs of geomeetrical quantiities (i.e. magnitudes))—in homogeeneous terms of o proportionss among the lengths of line segmennts.90 As a coonsequence, Descartes D is aable to reinteerpret the notion of ddegree of algebraic equation in geomeetry as not ex xpressing spatial dimeensions but ratther the number of relationss that are indisspensable to express thhe quantity off a given pow wer by means of relating it to a unit length (eitheer determinedd by a given problem or aarbitrarily cho osen) in a series of coontinuous propportionals. Indeed, it is thiis number off relations that is propeerly signified by “n” in the superscript xn . This seems to be the only—but eextremely signnificant—esseential differennce between Descartes‫ތ‬ D symbolism aand that of Viète. As Desccartes himselff insists, all these t reinterppretations are far from making geoometry a branch of arithm metic. His pooint is just to t render algebraic treatment univvocally and generally g appllicable in thee field of geometry.91 Let us turn noow to the remaining essentiial aspect of Descartes‫ތ‬ D complex connception of allgebra, namely y his identificcation of algeb bra as the proper emboodiment of thhe method of analysis in m mathematics. This will 90

Cf. Geom m. I, AT VI, 371: 3 “[P]ar a2 ou b3 ou seemblables, ie ne n conçoy ordinairemennt que des lignees toutes simplees, encore que, pour me seruirr des noms vsités en l‫ތ‬Algebre, ie les noomme des quarrrés, ou des cubees, &c.” 91 Cf. Geom. II, AT VI, 370; 377.

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finally put us in a position to answer adequately the leading interim questions stated at the end of sec. 3.1.1.

3.2.3 Algebra as an Analytical Problem-Solving Procedure Early in his Geometrie, Descartes offers an explicit and fairly informative account of in what his algebraic problem-solving procedure is supposed to consist. It is worth quoting at length:92 Ainsi, voulant resoudre quelque problesme, on doit d‫ތ‬abord le considerer comme desia fait, & donner des noms a toutes les lignes qui semblent necessaires pour le construire, aussy bien a celles qui sont inconnues qu‫ތ‬aux autres. Puis, sans considerer aucune difference entre ces lignes connuës & inconnuës, on doit parcourir la difficulté selon l‫ތ‬ordre qui monstre, le plus naturellement de tous, en quelle sorte elles dependent mutuellement les vnes

92

The account is paralleled in a striking way towards the end of the extant fragment of the Regulæ: Proposita difficultas directè est percurrenda, abstrahendo ab eo quòd quidam ejus termini sint cogniti, alij incogniti, & mutuam singulorum ab alijs dependentiam per veros discursus intuendo. ... Jam ... exponemus, quomodo ... difficultates ita sint subigendæ, vt quotcumque erunt in vnâ propositione magnitudines ignotæ sibi invicem omnes subordinentur, & quemadmodum prima erit ad vnitatem, ita secunda sit ad primam, tertia ad secundam, quarta ad tertiam, & sic consequentur, si tam multæ sint, summam faciant æqualem magnitudini cuidam cognitæ; idque methodo tam certâ, vt hoc pacto asseramus, illas nullâ industriâ ad simpliciores terminos reduci potuisse. ... [T]otum hujus loci artificium consistet in eo quòd, ignota pro cognitis supponendo, possimus facilem & directam quærendi viam nobis proponere, etiam in difficultatibus quantumcumque intricatis; neque quicquam impedit quominùs id semper fiat, cùm supposuerimus ab initio hujus partis, nos agnoscere eorum, quæ in quæstione sunt ignota, talem esse dependentiam à cognitis, vt planè ab illis sint determinata, adeò vt si reflectamus ad illa ipsa, quæ primùm occurrunt, dum illam determinationem agnoscimus, & eadem licet ignota inter cognita numeremus, vt ex illis gradatim & per veros discursus cætera omnia etiam cognita, quasi essent ignota, deducamus, totum id quod hæc regula præcipit, exequemur ... (Reg. XVII, AT X, 459–61; Descartes’ emphasis). Cf. also Reg. XIX, AT X, 468: “Per hanc ratiocinandi methodum quærendæ sint tot magnitudines duobus modis differentibus expressæ, quod ad difficultatem directè percurrendam terminos incognitos pro cognitis supponimus: ita enim tot comparationes inter duo æqualia habebuntur.”

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Chapter Three des autres, iusques a ce qu‫ތ‬on ait trouué moyen d‫ތ‬exprimer vne mesme quantité en deux façons: ce qui se nomme vne Equation, car les termes de l‫ތ‬vne de ces deux façons sont esgaux a ceux de l‫ތ‬autre. Et on doit trouuer autant de telles Equations qu‫ތ‬on a supposé de lignes qui estoient inconnuës. Ou bien, s‫ތ‬il ne s‫ތ‬en trouue pas tant, & que, nonobstant, on n‫ތ‬omette rien de ce qui est desiré en la question, cela tesmoigne qu‫ތ‬elle n‫ތ‬est pas entierement determinée; et lors, on peut prendre a discretion des lignes connuës, pour toutes les inconnuës ausquelles ne correspond aucune Equation. Aprés cela, s‫ތ‬il en reste encore plusieurs, il se faut seruir par ordre de chascune des Equations qui restent aussy, soit en la considerant toute seule, soit en la comparant auec les autres, pour expliquer chascune de ces lignes inconnuës, & faire ainsi, en les demeslant, qu‫ތ‬il n‫ތ‬en demeure qu‫ތ‬vne seule, esgale a quelque autre qui soit connuë, ou bien dont le quarré, ou le cube, ou le quarré de quarré, ou le sursolide, ou le quarré de cube, &c., soit esgal a ce qui se produist par l‫ތ‬addition, ou soustraction, de deux ou plusieurs autres quantités, dont l‫ތ‬vne soit connuë, & les autres soient composées de quelques moyennes proportionnelles entre l‫ތ‬vnité & ce quarré, ou cube, ou quarré de quarré, &c., multipliées par d‫ތ‬autres connuës (Geom. I, AT VI, 372–73).

The first step of the proposed problem-solving procedure is to assume that the given problem is solved and then to give a name (i.e. to assign an appropriate symbol) to each line—be it known or unknown—that is necessary for the solution (i.e., in the context of La Geometrie, for the construction, since the problems dealt with are, after all, geometrical in nature). The initial assumption of having the given problem solved is indeed of pivotal importance, for it allows one to treat the unknown lines as if they were known, thus opening the road for the groundbreaking second step of the proposed analytic procedure, viz. expressing the relations of dependence between the required lines in terms of the relations between their lengths interpreted as algebraic quantities (presumably with the help of the unit length which is either given by the problem at hand or alternatively can be chosen arbitrarily, provided that all the remaining lengths are then referred to it), and linking these expressions by means of algebraic equations.93 The procedure then continues with the third step, viz. stepwise solving (demesler) the equations thus formed until all the unknowns are fully expressed in terms of the knowns (connuës) in a 93

Cf. Geom. III, AT VI, 444: “Equations [sont les] somme composées de plusieurs termes, partie connus & partie inconnus, dont les vns sont esgaux aux autres, ou, plutost, qui, considerés tous ensemble, sont esgaux a rien ....” In Reg. XIV, Descartes ventures to claim, clearly in view of the present procedure, that “quæstiones perfectè determinatas [i.e., perhaps among other, mathematical quæstiones—J.P.] vix vllam difficultatem continere, præter illam quæ consistit in proportionibus in æqualitates evolvendis ...” (AT X, 441).

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number of equations (ddepending upon whether tthe given prroblem is “determinatee” or “indeteerminate,” i.ee. whether thhere is a finiite or an infinite num mber of solutioons, respectiveely). A simplle example might m help.94 The problem m reads: giveen a line segment AB B containing a point C, find D on a ray A AB so that the rectangle AD u DB shhall be equal to the square of CD. Whatt Descartes prroposes is first to assum me that the prroblem is solv ved, i.e. that D is found, an nd then to name all thee line lengths necessary to solve the probblem, i.e. AC C, CB and BD, e.g. witth “a” and “b”” for the know wn lengths AC C and CB, and d with “x” 9 for the unknnown and sougght-after BD:95 Fig. 3-4

Then, treatiing the unknoown as if it were known,, one has to find two expressions of x in its relations r to a and/or b acccording to the t initial problem andd set up an eqquation (and there t being juust one unknow wn in the present casee, one equationn shall be eno ough) which liinks these exp pressions. The “most nnatural” equatiion in the pressent case readds (a + b + x) u x = (b + x) x 2. The last anaalytic step is thhen to “demessler” this equaation unless on ne has the unknown qquantity exprressed solely by a propoortion of thee known quantities; inn the present case, c this amo ounts to x = b2 : ((a – b). One thinng the last quuoted passagee clearly indiccates is that Descartes D indeed wishhes to draw soomehow upon n the method oof analysis professedly 94

The exampple is adduced by Frans Van n Schooten, In Geometriam Renati R Des Cartes comm mentarii, in: René R Descartees, Geometria, transl. by Frans F Van Schooten, 2nd ed. (Amsterddam: Ludovicuss & Daniel Elzzevirii, 1659), 149. 1 I owe the referencee to Gerd Bucchdahl, Metaph hysics and thee Philosophy of o Science (Oxford: Blacckwell Publishiing, 1969), 127.. 95 Adapted froom Buchdahl, ibid. i

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invented and employed by ancient Greek mathematicians, for the essential mark of the proposed procedure is that the equations are operated on as if all the quantities, both known and unknown, are equally known, and the aim is to render known the unknown quantities by way of expressing them in terms of combinatory products of the known quantities. Thus the proposed procedure seems a comfortable fit with the classical account of analysis in Pappus; indeed, the opening words of the last quoted passage, “voulant resoudre quelque problesme, on doit d‫ތ‬abord le considerer comme desia fait,” amount to a strikingly close paraphrase of the opening words of Pappus‫ ތ‬description of analysis. By the same token, however, Descartes is clearly aware that despite these apparent generic affinities of his problem-solving procedure with the professed analytic treatment of mathematical problems by certain ancient Greeks, his mathematical achievements presented in La Geometrie amount to a major advance far beyond anything attained by his predecessors in the field. He seems to be fairly justified in such a self-assessment, be it only due to his ability to deal with several problems his predecessors were unable to solve, most notably the so-called Pappus Problem.96 Furthermore, he is surely right in his claim that his successes in solving several notorious problems amount to a telling demonstration of the power of his variant of the analytic method in mathematics.97 Now, it would be a mistake to think that Descartes’ groundbreaking achievements in mathematics were simply due to an ingenious application of the essentials of the classical method of analysis on the brand new, algebraic conception of what the proper object of vera Mathesis amounts 96

We need not go into the particular mathematics involved. For an elementary exposition of the Pappus Problem and of Descartes‫ ތ‬solution see, for example, Sasaki, DescartesҲs Mathematical Thought, ch. 5, § 1. 97 Thus he writes to Mersenne: “[I]‫ތ‬ay seulement tasché par la Dioptrique & par les Meteores de persuader que ma methode est meilleure que l‫ތ‬ordinaire, mais ie pretens l‫ތ‬auoir demonstré par ma Geometrie. Car dés le commencement i‫ތ‬y resous vne question, qui par le témoignage de Pappus n‫ތ‬a pu estre trouuée par aucun des anciens; & l‫ތ‬on peut dire qu‫ތ‬elle ne l‫ތ‬a pû estre non plus par aucun des modernes, puis qu‫ތ‬aucun n‫ތ‬en a écrit, & que neantmoins les plus habiles ont tasché de trouuer les autres choses que Pappus dit au mesme endroit auoir esté cherchées par les anciens ...” (AT I, 478). Cf. also ibid., 480: “Au reste, ayant determiné comme i‫ތ‬ay fait en chaque genre de questions tout ce qui s‫ތ‬y peut faire, & monstré les moyens de le faire, ie pretens qu‫ތ‬on ne doit pas seulement croire que i‫ތ‬ay fait quelque chose de plus que ceux qui m‫ތ‬ont precedé, mais aussi qu‫ތ‬on se doit persuader que nos neueux ne trouueront iamais rien en cette matiere que ie ne pusse auoir trouué aussi bien qu‫ތ‬eux, si i‫ތ‬eusse voulu prendre la peine de le chercher.” It is not our task to assess Descartes‫ ތ‬claims of success in mathematics.

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to and of how the mathematical problems are to be articulated. Rather, Descartes‫ ތ‬new technique of symbolic algebra in fact amounts not just to a principal improvement on but—at least in certain essential respects—to a structural departure from the Classical way of doing mathematics. What he in fact envisages is not merely the establishment of symbolic algebra as a sample of mathematical analysis along classical lines; rather, he (like Viète before him)98 in fact manages to establish algebra as the normative paradigm of mathematical analysis: the algebraic form has become the sign that designates any mathematical treatment as analytical simpliciter. In short, in Descartes‫ ތ‬hands “analysis” has become virtually synonymous with “algebra” in mathematical contexts. Perhaps the most telling symptom of these groundbreaking innovations is that Descartes in fact elevates algebra all alone to set the constraints of what is to count as a legitimate object of mathematical enterprise. The criteria of the intelligibility of mathematical objects are thus exempted from the Classical bonds of spatial intuitions of physical objects or collective intuitions of counted things and delegated instead to the relational domain of algebraic operations. In Macbeth‫ތ‬s pertinent phrase, “instead of limiting the scope of his mathematical operations in light of antecedently available quantities ... [Descartes] extends the domain of quantities to include any that can be the result of such operations” (Macbeth, “Viète, Descartes,” 106). In other words, anything that counts as the root of an algebraic equation by the standards set up by algebra itself is to count as a legitimate, intelligible mathematical object. In arithmetic, this substantial shift amounts to freeing the concept of number from its tie to definite collections of counted units (conceived in effect as things by the ancient Greeks), and of allowing as legitimate numbers not only integers without qualification but also fractions, irrational and imaginary numbers. In geometry, the normative power of algebra under consideration presents a substantial revision of the traditional criteria for legitimate geometrical figures, most notably curves, and of the Classical way of classifying curves. It will be well to look more closely at the case of geometry. In place of the Classical genetic criterion of the acceptability of curves in geometry in terms of their constructability in a plane with ruler and compass only (circles) or as boundaries of intersections of a plane with a solid the construction of which requires only straight lines and circles, viz. cones (parabolas, hyperbolas, and ellipses), Descartes proposes, fundamentally, a purely algebraic criterion to the effect that any curved 98

Cf. Mahoney, Mathematical Career of Fermat, 33–34.

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line counts as geometrically legitimate (i.e. as “geometrical” in Descartes‫ތ‬ terminology, as opposed to “mechanical”, i.e. geometrically illegitimate curves)99 as long as it can be represented by an algebraic equation, such an equation amounting to an expression of the relation between all the points of the given curved line on the one hand and all the points of a straight line on the other:100 [T]ous les poins de celles [lignes courbes] qu‫ތ‬on peut nommer Geometriques, c‫ތ‬est a dire qui tombent sous quelque mesure precise & exacte, ont necessairement quelque rapport a tous les poins d‫ތ‬vne ligne droite, qui peut estre exprimé par quelque equation, en tous par vne mesme (Geom. II, AT VI, 392).

Moreover, he supplements the algebraic criterion with a genetic one: [I]l est ... tres clair que, prenant, comme on fait, pour Geometrique ce qui est precis & exact, & pour Mechanique ce qui ne l‫ތ‬est pas; & considerant la Geometrie comme vne science qui enseigne generalement a connoistre les mesures de tous les cors; on n‫ތ‬en doit pas plutost exclure les lignes les plus composées que les plus simples, pouruû qu‫ތ‬on les puisse imaginer estre descrites par vn mouuement continu, ou par plusieurs qui s‫ތ‬entresuiuent & dont les derniers soient entierement reglés par ceux qui les precedent: car, par ce moyen, on peut tousiours auoir vne connoissance exacte de leur mesure (ibid., AT VI, 389–90).

The genetic criterion enables him to directly confront the Classical approach according to which curves more complex than conic sections fail to satisfy the standards of geometrical accuracy.101 Having prepared the 99

Cf. Geom. II, AT VI, 388–90. Examples of mechanical curves adduced by Descartes include spiral and quadratrix—see ibid., AT VI, 390; 411. 100 Cf. Geom. II, AT VI, 395: “[E]n quelque autre façon qu‫ތ‬on imagine la description d‫ތ‬vne ligne courbe, pouruû qu‫ތ‬elle soit du nombre de celles que ie nomme Geometriques, on pourra tousiours trouuer vne equation pour determiner tous ses poins ....” Thus, Descartes‫“ ތ‬geometrical” curves are nowadays called algebraic; and Descartes‫“ ތ‬mechanical” curves coincide with transcendental curves in contemporary terminology. 101 As a matter of fact, Descartes unnecessarily complicates the situation as he observes that the ancient mathematicians “n‫ތ‬ont pas aussy entierement receu les sections coniques en leur Geometrie” (Geom. II, AT VI, 389). This is historically incorrect as Sasaki, DescartesҲs Mathematical Thought, 218 remarks with reference to George Molland, “Shifting the Foundations: Descartes‫ތ‬s Transformation of Ancient Geometry,” Historia Mathematica 3 (1976), 27–28.

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ground by arguing persuasively that the delineation of geometrical and mechanical curves by means of the criterion of the necessity for some special mechanism for their construction is deemed to failure, and having proposed “la iustesse du raisonnement” as the proper normative of geometrical research,102 he claims that in so far as genesis is concerned, any curved line is geometrically acceptable if its construction—as if by some ideal apparatus—can be fully traced by means of some continuous movement or by a series of ordered continuous movements fully and exactly determined by the preceding ones, provided the movements involved are conceived “nettement & ... distinctement” (ibid., AT VI, 392). More specifically, the ordered continued movements taken singularly are those that trace relatively simple curves, and their intersections determine other, more complex curves.103 This complex approach in effect enables Descartes to completely deconstruct the Classical manner of classifying geometrically legitimate curves. We have already seen how his novel criterion of algebraicity opened the door to geometry for curves representing the solution not only to so-called planar and solid but also to a variety of the ancients’ so-called linear problems. Now his genetic criterion supplements this extension of the geometrical field with a rationale for his new framing of the classes of curves: since he has it that 102

See Geom. II, AT VI, 388–89: “[I]e ne sçaurois comprendre pourquoy [les anciens] ont nommées [les lignes les plus composées que les solides] Mechaniques, plutost que Geometriques. Car, de dire que c‫ތ‬ait esté a cause qu‫ތ‬il est besoin de se seruir de quelque machine pour les descrire, il faudroit reietter, par mesme raison, les cercles & les lignes droites, vû qu‫ތ‬on ne les descrit sur le papier qu‫ތ‬auec vn compas & vne reigle, qu‫ތ‬on peut aussy nommer des machines. Ce n‫ތ‬est pas non plus a cause que les instrumens qui seruent a les tracer, estant plus composés que la reigle & le compas, ne peuuent estre si iustes: car il faudroit, pour cete raison, les reietter des Mechaniques, où la iustesse des ouurages qui sortent de la main est desirée, plutost que de la Geometrie, où cҲest seulement la iustesse du raisonnement quҲon recherche, & qui peut sans doute estre aussy parfaite, touchant ces lignes, que touchant les autres” (my emphasis). 103 See ibid., 389: “[I]l n‫ތ‬est besoin de rien supposer, pour tracer toutes les lignes courbes que ie pretens icy d‫ތ‬introduire, sinon que deux ou plusieurs lignes puissent estre meuës l‫ތ‬vne par l‫ތ‬autre, & que leurs intersections en marquent d‫ތ‬autres ....” In Geom. II, AT VI, 391–92 and again in Geom. III, AT VI, 442–44 Descartes describes a mechanical device called a mesolabe compass (circinus mesolabi; cf. Cogitationes Privatæ, AT X, 238–39), a tool for finding (an ordered series of) mean proportionals, and by implication for the orderly tracing of curves of increasing degrees (see Fig. 3-3 above). The way this instrument works shows clearly what Descartes has in mind in the quoted sentence. For a general survey of the rôles of compasses in Descartes‫ ތ‬mathematical thought, see Michel Serfati, “Les compas cartésiens,” Archives de philosophie 56, no. 2 (1993): 197–230.

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the continued movements tracing curves of degree n and n + 1 generate through intersections curves of degree n + 2 and n + 3 etc., geometrical curves are to be classified according to the degree of their equations, the genera forming an ordered series, professedly with no limit of complexity:104 Ie pourrois mettre icy plusieurs autres moyens, pour tracer & conceuoir des lignes courbes qui seroient de plus en plus composées par degrés a l‫ތ‬infini. Mais, pour comprendre ensemble toutes celles qui sont en la nature, & les distinguer par ordre en certains genres, ie ne sçache rien de meilleur que de dire que tous les poins de celles qu‫ތ‬on peut nommer Geometriques, c‫ތ‬est a dire qui tombent sous quelque mesure precise & exacte, ont necessairement quelque rapport a tous les poins d‫ތ‬vne ligne droite, qui peut estre exprimé par quelque equation, en tous par vne mesme. Et que, lorsque cete equation ne monte que iusques au rectangle de deux quantités indeterminées, ou bien au quarré d‫ތ‬vne mesme, la ligne courbe est du premier & plus simple genre, dans lequel il n‫ތ‬y a que le cercle, la parabole, l‫ތ‬hyperbole & l‫ތ‬ellipse qui soient comprises. Mais que, lorsque l‫ތ‬equation monte iusques a la trois ou quatriesme dimension des deux ou de l‫ތ‬vne des deux quantités indeterminées: car il en faut deux pour expliquer icy le rapport d‫ތ‬vn point a vn autre: elle est du second. Et que, lorsque l‫ތ‬equation monte iusques a la 5 ou sixiesme dimension, elle est du troisiesme: & ainsi des autres a l‫ތ‬infini (Geom. II, AT VI, 392–93).

The fact that while in the Classical view conic sections are toto genere different from circles (the former falling under solids and the latter under planars, according to Pappus),105 in Descartes they form a single “premier & plus simple genre” (as they are all expressible with an equation of the form ax2 + by2 + cxy + dx + ey + f = 0),106 illustrates the significance and depth of this structural change. However, it is not only the notion of the legitimate mathematical object but also—and more central to our concerns in this study—the very form of the analytical treatment that undergoes substantial structural changes in Descartes‫ ތ‬hands. It will be convenient to deal with these structural changes in the context of an issue which happens to be highly relevant to the overall concerns of the present investigation: that is to say, the 104

Descartes was mistaken in this generalization as well as in several assumptions that underlie his treatment of geometrical curves, as partly demonstrated as early as de Fermat: see Boyer, History of Analytic Geometry, 90–98. Idem, ibid., 89–90 also provides both a graphical and an algebraic general expression of how Descartes proceeds in the following quotation. 105 Cf. Pappus, Mathematicæ Collectiones, Book Three, 4v. 106 I take the formula from Macbeth, “Viète, Descartes,” 107.

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relationship of analysis and synthesis in mathematics. We shall look first at a classical account of analysis and synthesis in Book Seven of Pappus‫ތ‬ Collectiones. Against the background of this account, we shall then turn to Descartes‫ ތ‬views concerning this crucial topic.

3.2.4 Analysis and Synthesis in Pappus We saw that analysis is characterized as an ex contrario facta solutio in the classical passage from Pappus‫ ތ‬Collectiones.107 This strongly suggests that the solution proper of a given problem, or else the grounding proper of a given theorem, consists in the reverse procedure, the so-called synthesis. This squares well with another suggestion in the same passage, namely that to Pappus, synthesis seems to count as the indispensable complement of analysis, so that analysis and synthesis form two essential ordered components of a single methodical procedure in mathematics. Moreover, Pappus‫ ތ‬actual wording makes it very likely that he intends synthesis to enact, in the opposite direction, each of the steps made before in the course of the corresponding analysis; this brings to the fore the above issue of dependence relations among the steps to be completed in analysis and synthesis respectively. Let us have a closer look at these issues as conveyed by Pappus in order to appreciate the import and extent of what remains unchanged and what is transformed in Descartes‫ ތ‬hands. To begin with, according to Pappus there are two kinds of analysis, namely theoretical (or contemplatiua in Commandino‫ތ‬s Latin translation)108 and problematical analysis: Duplex autem est resolutionis genus, alterum quidem, quod veritatem perquirit, & contemplatiuum appellatur: alterum vero, quo inuestigatur id, quod dicere proposuimus, vocaturque problematicum. In contemplatiuo igitur genere quod quæritur, vt iam existens, & vt verum ponentes per ea, quæ deinceps consequuntur tamquam vera, & quæ ex positione sunt, procedimus ad aliquod concessum quod quidem si verum sit, verum erit & quæsitum; & demonstratio, quæ resolutioni ex contraria parte respondet. Si vero falso euidenti occurramus, falsum erit & quæsitum. In problematico 107

Ibid., 157v (quoted in sec. 3.1.3). Mahoney, “Another Look” holds on plausible grounds that those sentences of the passage in question in which analysis and synthesis are correlated are probably interpolations by a later scholiast. However, we can ignore this complication as Descartes—apparently following the common course—most probably took these interpolations as integral to Pappus‫ ތ‬text. This is also why, for the sake of brevity, from now on I will ascribe the sentences to Pappus. Cf. also fn. 18. 108 I retain the term “theoretical” in the subsequent discussion.

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Thus according to Pappus the difference between the two kinds of analysis eventually derives from two kinds of what is sought (quæsitum). This quæsitum can amount, in effect, either to an ordered conclusive truthvalidation of a given proposition (theoretical analysis), or else to an ordered way of solving a given problem (problematical analysis). The positive terminus of theoretical analysis is, by a safe extrapolation, some known truth which somehow “follows” (consequitur)—usually through a number of intermediate steps—from the tested theorem; in the case of problematical analysis, the positive terminus amounts to a problem which one knows how to solve and the solution of which counts as a condition— as it once again somehow “follows” from the initial problem—for the (usually stepwise) performance of the sought-after solution. Negative results of each kind of analysis, Pappus tells us, amount to an encountering of something known to be false (theoretical analysis) or something impossible to perform (problematical analysis); in such cases the tested theorem turns out false and the initial problem impossible to resolve. However, if the result of analysis is positive, then the reverse procedure, i.e. synthesis, is to be completed; and this synthetic performance amounts, as Pappus reports, to a sort of demonstration (demonstratio) in both kinds of analysis. It is worth noting, to begin with, that there is an important sense in which problematical analysis counts as more fundamental than theoretical analysis; for problematical analysis is a procedure that amounts to a heuristically prominent way of establishing the requisites for the reasonable application of theoretical analysis within a single discipline, namely theorems that are in need of proof; and this type of dependence does not obtain in the opposite direction. Certainly there are other ways in which one can obtain theorems set forth for proof; but at least sometimes such theorems are obtained as the solutions to problems worked out in a problematical analysis and certainly this is how things are normally supposed to go, especially in mathematics.109 On the other hand, it seldom if ever happens that a problem within one and the same discipline is 109

Michael Mahoney makes a similar point in his “Another Look,” 329.

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produced in the course of searching for proof by way of theoretical analysis.110 The priority of problematical analysis just highlighted contributes to an adequate understanding of Descartes‫ ތ‬stance towards the relationship of analysis and synthesis in mathematics. Another requisite for that understanding is grasping the standard reasons (which also seem to reinforce Pappus‫ ތ‬treatment of the issue) for taking synthesis as the indispensable complement of analysis. Let us have a look at theoretical analysis first. It has been observed ever since the time of Plato and Aristotle111 that there is a peculiar epistemological defect inherent in analysis in its theoretical cast: even if brought to its proposed positive terminus—a proposition known to be true—by way of the consequentiæ of the theorem under investigation, theoretical analysis fails to establish with certainty the truth of the initial theorem. To see why, consider that on the standard interpretation,112 the 110

This sort of priority of problematical analysis over theoretical analysis is pinpointed by Pappus in his Collectiones, Book Three, 1r. I take it as firmly established that along with synthesis, analysis was employed primarily and for the most part in its problematical and not in its theoretical form, i.e. as a heuristic device rather than as a method of proof. The opposite standpoint was taken most resolutely in Jaakko Hintikka and Unto Remes, The Method of Analysis: Its Geometrical Origin and Its General Significance (Dordrecht: D. Reidel, 1974). Their approach was immediately subjected to severe (and convincing) criticism in several texts by Arpád Szabó: “Working Backwards and Proving by Synthesis”, in Hintikka and Remes, Method of Analysis, Appendix I: 118–30; idem, “Analysis and Synthesis: Pappos II, p. 634ff. Hultsch,” Acta Classica Universitatis Scientiarum Debrecensis 10/11 (1975): 155–64; idem, “Zum Problem der Antiken Analysis,” in Analysis, Harmony and Synthesis in Ancient Thought ed. Jorma Mattila and Arto Siitonen (Oulu: Acta Universitatis Ouluensis, 1977), 89–99. For a similar criticism see also Maula, “End of Invention.” 111 Plato has been credited with the invention of analysis as a heuristic device since Diogenes Laërtius, Vitæ philosophorum III, 24 (ed. Miroslav Marcovich, vol. 1, Stuttgart: G. B. Teubner, 1999), probably upon the basis of some passages in Plato‫ތ‬s Meno (see in particular 75c–79d; 86e–87b; 98a–b) and Theaetetus (see in particular 196e–197a). Aristotle famously discusses analysis in An. Post. I, 12. 112 Jaakko Hintikka and Unto Remes cast doubt on the standard reading, according to which the direction of theoretical analysis is a matter of running through logical consequences, and propose a peculiar alternative to the effect that at least in its ideal form, theoretical analysis should run against the direction of logical consequence, so that one proceeds in analysis “from a hoped-for conclusion to its more and more distant premises ...” (Method of Analysis, xiv; see also ibid., 11–12; 37). I stick to the standard reading since Hintikka and Remes themselves eventually admit that, in practice, analysis usually proceeds in the standard form of

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consequentiæ from the theorem set forth for proof to a known truth run through logical consequences and are connected with one-way conditionals, thus making up sequences of the form113 [a]

T ĺ P1 ĺ P2 ĺ ... ĺ Pn ĺ E.

Sequence [a], as it stands, is sufficient to establish the negative result of the validity test at issue since it follows from [a] that (™E ĺ ™T). Thus if through a series of logically valid consequences, one ends up with a known falsehood, T is safely established as invalid. However, [a] alone does not suffice for establishing the positive result of the validity test, i.e. the truth of T, even if one reaches some E known to be true. This is because in standard propositional logic, it is possible to infer truth from falsehood, so that the converse of any one of the conditionals that constitute [a] may not hold.114 As a consequence, it still remains to be established, in order that the truth of the initial theorem under investigation be proven, that the converses of all the conditionals that constitute [a] do hold, i.e. sequences of the form [b]

E ĺ Pn ĺ ... ĺ P2 ĺ P1 ĺ T

still remain to be established; and since this is exactly what synthesis is supposed to provide, its status as the indispensable complement to theoretical analysis seems to be well-grounded.

drawing logical consequences upon the basis of one-way conditionals, and that it is only on this standard reading that synthesis comes out as necessary to attain a demonstration of the theorem under investigation. 113 T stands for the theorem under investigation, Pi for intermediate propositions, and E for a proposition known to be true. 114 The irreversibility of one-way conditionals is, of course, at bottom due to the ex falso quodlibet principle exemplified by ((p š ™p) ĺ p). The insufficiency of analysis for establishing the validity of T can be easily exemplified thus: to infer validly an evident truth from any T is as easy as inferring from a given T a disjunction in which one term is T and the other term is any evident truth F (say (p ļ p)), i.e. (T ĺ (T › F)). The validity of the converse of this conditional requires that T is true, a requirement which, however, is bound to be established on independent grounds. As Aristotle comments pertinently on this score, “[if] it were impossible to prove truth from falsehood, it would be easy to make an analysis; for they would convert from necessity” (An. Post. I, 12, 78a6–7; Complete Works, 1:127).

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However, here lurks a considerable complication. It has been observed ever since Aristotle that the relevance of the problem of reversibility, from which synthesis draws its legitimacy as the indispensable complement of theoretical analysis, is severely mitigated exactly in that field which is our present main concern, i.e. mathematics.115 Fundamentally, this is due to the fact that according to ancient Greek thinkers mathematics is concerned, for the most part, with the relations of equivalence or equality: mathematical reasoning is mainly a matter of the combining or breaking up of terms, and of the substitution of terms on the basis of their equality. Consequently, the connectors within theoretical-analytical sequences of the form [a] in mathematics are in fact, for the most part, not just one-way conditionals but biconditionals. Moreover, even if some step in the course of theoretical analysis in mathematics turns out irreversible as it stands, it can usually be rendered reversible by way of adding supplementary conditions (the so-called įȚȠȡȚıȝȠȓ or determinationes)116 which are then integrated into the statement of the initial theorem; and it is once again the task of analysis, not synthesis, to determine these įȚȠȡȚıȝȠȓ.117 The implications of these facts with regard to the status of synthesis relative to theoretical analysis are remarkable: as long as it is indeed inherent in the peculiarly mathematical variant of theoretical analysis that each step in the sequence of the form [a] is already rendered simply reversible (either as it stands or with the aid of a įȚȠȡȚıȝȩȢ) in the course the analytical treatment alone, theoretical synthesis turns out, strictly speaking, superfluous in mathematics since the work it was supposed to do—establishing the validity of the converse sequence of the form [b]— has de facto already been performed by analysis. The fact that synthesis relative to theoretical analysis in mathematics turns out superfluous even by the standards implicit in the practice of ancient Greek mathematicians118 helps to explain, among other things, the ease with 115

Cf. Aristotle, An. Post. I, 12, 78a11–13: “In mathematics things convert more because they assume nothing accidental—and in this too they differ from argumentations—but only definitions” (Complete Works, 1:127). 116 Cf., respectively, Proclus Diadochus, In primum Euclidis Elementorum Librum commentarii ed. Gottfried Friedlein (Leipzig: B. G. Teubner, 1873), 203, and idem, In primvm Evclidis Elementorum, 116. To highlight in the present context the technical meaning of įȚȠȡȚıȝȩȢ/determinatio, which is very important for our purposes, I will refer to it by the Greek term. 117 See Mahoney, “Another Look,” 327–29 for an illuminating discussion of įȚȠȡȚıȝȩȢ and for some examples. 118 Mahoney, ibid., 326–27 documents this convincingly with an illuminating example of quasi-algebraic treatment in the interpolation to Euclid’s Elementa XIII, 1. The interpolation is expressly structured in terms of analysis and

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which Pappus passes over the problem of reversibility in his discussion of theoretical analysis; far from naïveté, his easy approach turns out to reflect the peculiarities of the use of theoretical analysis in mathematics. The situation is quite different in the case of problematical analysis as here the quæsita are answers to questions rather than proofs of theorems, and what is at issue is—by the lights of ancient Greek mathematicians— not the truth of a theorem but rather either the construction of a particular figure or the calculation of a particular number. Rather than stepwise drawing consequences (logical or otherwise) from the initial theorem to be proven, the basic operation within problematical analysis is the stepwise reduction of the initial problem to be solved to another problem or a number of other problems, the solution of which is already available, the most usual pattern of reduction being to establish the initial problem under investigation as a special case of a more general problem.119 Since, therefore, no truth-functional connectors are involved in sequences of steps in problematic analysis, the reversibility problem is of no substantial import and it is not from here that synthesis could even vaguely draw its prima facie legitimacy as the indispensable complement of analysis. However, a much better case than that for theoretical analysis can be made for the indispensability of synthesis relative to problematic analysis, even in mathematics—for a given question is not yet answered, or a given problem not yet solved, only by recognizing what other questions or problems are to be answered or solved in order to address the initial question or problem: it still remains to be seen how these answers or solutions bear upon the answers or solutions to the initial question or problem set forth to be addressed. Consequently, at least as long as it is the solution of a particular problem or the answer to a particular question that problematical analysis is supposed to provide, the problems still remain to subsequent synthesis, the analysis proceeding strictly in terms of the combination, division and substitution of equal terms, and the analysis is entirely reversible into synthesis with no worries concerning reversibility. 119 A standard example is the reduction of the problem of the duplication of the cube to the problem of finding the first of two mean proportionals between two magnitudes a and b, where a is the edge of the cube to be duplicated and b = 2a. The continued proportion a : x = x : y = y : b can then be divided into a pair of equations x2 = ay and xy = 2a2, which by elimination of y yields a single equation x3 = 2a3 which expresses the initial problem. The solution of the last equation then amounts to finding the intersection of the conic sections (parabola and hyperbola respectively) determined by the former pair of equations. Cf. e.g. Judith Grabiner, “Descartes and Problem-Solving,” Mathematics Magazine 68, no. 2 (1995): 84. Mahoney, “Another Look,” 331–37 adduces other examples along with the present one and supplies them with an instructive discussion.

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be solved to which the analytical procedure eventually leads, and to resolve in terms of these solutions step by step, in the reverse order, the other problems one came across in the course of the analytical procedure, until one thus actually resolves the initial problem. It is this procedure— stepwise performing in reverse order the solutions to problems stated in the course of analysis—that amounts to synthesis relative to problematic analysis in the classical Pappian account. In the special case of mathematics, the synthetic treatment amounts (as already indicated) either to carrying out constructions of particular figures that lead step by step to the construction of the initially sought-after figure (geometry), or in computations of particular numbers that lead step by step to the initially sought-after number (arithmetic).120 Thus unlike the case of theoretical analysis, the status of synthesis as the indispensable complement of problematical analysis is indeed well-established in the context of classical Greek mathematics.

3.2.5 Descartes‫ ތ‬Approach One thing these historical preliminaries provide us with is the means of evaluating the plausibility of rather strong claims concerning the groundbreaking character of Descartes‫ ތ‬establishment of general algebra as the only paradigm of mathematical analysis. Perhaps the best illustration of this substantial shift is the fact that the procedure which is essential to Descartes‫ ތ‬notion of what constitutes analysis, to wit, resolving the algebraic equations set up to link the known and unknown quantities at the initial stage of the problem-solving procedure, has no analogue in the Classical Pappian notion of problematical analysis according to the Pappian account.121 What is not immediately clear, however, is what 120

For a classical discussion of this aspect of Greek mathematical practice see Klein, Greek Mathematical Thought, 163–64. As he puts it succinctly, “while analysis is immediately concerned with the generality of the procedure, synthesis is, in accordance with the fundamental Greek conception of the objects of mathematics, obliged to ‫ލ‬realize‫ ތ‬this general procedure in an unequivocally determinate object ...” (ibid., 164; Klein’s emphasis). 121 The shift is even more clearly visible in Viète, who completely changes the meaning of the Pappian procedures of zetetics and poristics and supplies them with the stage of exegetics which denotes the resolution of algebraic equations and has no equivalent in the Pappian account—see Viète, Isagoge, 4r. Mahoney, Mathematical Career of Fermat, 34 nicely draws the contrast between Viète and Pappus in this respect. Descartes does not use the Greek terms but his notions of the stages of analysis are similar to Viète‫ތ‬s.

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Descartes took to be the implications of these substantial shifts for the status of problematical synthesis122 in mathematics. Since he is never explicit on the issue apart from the above Resp. 2, AT VII, 155–57, his specific views concerning the status of analysis and synthesis in their mathematical use are to be gleaned from his mathematical practice within appropriate contexts. To begin with, Descartes exhibits no interest in theoretical analysis at all either in La Geometrie or in the mathematical layers of the Regulæ or anywhere else in his texts devoted to mathematical issues; he clearly conceives of mathematics as a discipline the proper and essential task of which is problem-solving. Such an approach is, of course, far from trivial; yet what matters for the present is just that (i) given this approach, it should come as no surprise that no samples of theoretical synthesis in mathematics are to be found in Descartes; and that (ii) this absence of synthetic treatment therefore implies by itself nothing regarding Descartes‫ތ‬ considered views on the status of synthesis relative to theoretical analysis. Of course, given that (as we saw) Descartes takes general quantity as the proper object of mathematics and algebra as the proper tool of mathematical reasoning, one may conjecture quite safely that were he to have been pushed on the question, he would have been bound to hold something similar to the above discerned commitment implicit in the “geometrical algebra”123 of the ancient Greeks124—viz. that at least as far as mathematics is concerned, synthesis is eventually superfluous as the 122

Here and in the following I mean the phrase “problematical synthesis” as a shorthand for “synthesis related to problematical analysis.” 123 The phrase “geometrical algebra” is due to Paul Tannery’s 1882 paper “De la solution géométrique des problèmes du second degré; avant Euclide” in idem, Mémoires scientifiques ed. Johan Heiberg and Hieronymus Zeuthen, vol. 1 (Toulouse: Édouard Privat, 1912): 254–80 and to Hieronymus Zeuthen, Die Lehre von den Kegelschnitten im Altertum (Copenhagen: A. F. Höst & Sohn, 1886), sec. 1. It was established in the historiography of mathematics in particular due to commentaries by Thomas Heath in his translation of Euclid’s Elementa (The Thirteen Books of Euclid’s Elements, 3 vols., Cambridge: Cambridge University Press, 1908). The question of the historical adequacy of employing the notion of algebra in interpretations of ancient mathematics remains controversial. It is not our task, however, to assess the credentials of such interpretative approaches. 124 Pace Gaukroger who seems to claim in his Cartesian Logic, 80–81 that the simple reversibility of ordered sequences revealed in theoretical analysis was not established, even in mathematics, until Descartes‫ ތ‬explicit treatment of geometrical matters by means of algebraic equations. Mahoney‫ތ‬s discussion in his “Another Look,”, 326–27 of the interpolation to Euclid, Elementa XIII, 1 convinced me that Gaukroger cannot be right.

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entire task it is supposed to perform is already accomplished in analysis. Yet the only real issue to be considered here with regard to Descartes‫ތ‬ actual mathematical practice is the status of problematical synthesis. Let us turn to this topic. On the one hand, there is some evidence to the effect that due to his innovative views regarding towards what the mathematical enterprise as a whole should be aimed, Descartes‫ ތ‬attitude towards problematical synthesis is strongly negative. Consider the following passage from Reg. XVI, AT X, 455–58: [A]dvertendum est, Logistas consuevisse singulas magnitudines per plures vnitates, sive per aliquem numerum designare, nos autem hoc in loco non minus abstrahere ab ipsis numeris, quàm paulò ante à figuris Geometricis, vel quâvis aliâ re. Quod agimus, tum vt longæ & superfluæ supputationis tædium vitemus, tum præcipuè, vt partes subjecti, quæ ad difficultatis naturam pertinent, maneant semper distinctæ, neque numeris inutilibus involvantur: vt si quæratur basis trianguli rectanguli, cujus latera data sint 9 & 12, dicet Logista illam esse ξʹʹͷ vel 15; nos verò pro 9 & 12 ponemus a & b, inveniemusque basim esse ξܽଶ ൅  ܾଶ , manebuntque distinctæ duæ illæ partes a2 & b2 quæ in numero sunt confusæ. ... Logistæ ... contenti sunt, si occurrat illis summa quæsita, etiamsi non animadvertant quomodo eadem dependeat ex datis, in quo tamen vno scientia propriè consistit (my emphases).

Here Descartes makes it sufficiently clear that for him mathematics, in so far as it aspires to the title of scientia, is properly to be concerned with a distinct grasp of the structural dependences between the data and the quæsita of a given problem; and he indicates in which sense his algebraic analysis is perfectly suited to this concern: the symbolic algebraic equations that result from successful analysis do indeed keep the dependence relations between the general quantities as distinctly as one may wish.125 However, what most matters regarding our present concerns is that Descartes goes beyond merely implying that the problem-solving procedure that preserves the dependence relations essential to the problem is as substantial a part of the problem-solving procedure as the gaining of the determinate solution itself; instead, he implies that the determinate solution even hides or obliterates those dependencies, thus 125

This is clearly a part of the thrust of Viète‫ތ‬s contrasting his Logistice speciosa to the Logistice numerosa (see in particular Viète, Isagoge, 4r) and of his characterization of the former as “quæ per species seu rerum formas exhibetur” (ibid., 5r).

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straightforwardly impeding the professed proper task of the mathematical enterprise. Now it will be remembered that attaining such a determinate solution is exactly the task of Classical problematic synthesis, and Descartes‫ ތ‬arithmetical example can surely be generalized over the whole field of mathematics via his identification of the proper object of mathematics with general quantities. Thus the passage, as it stands, suggests not only that problematical synthesis is superfluous but even that it is harmful enough to warrant ruling it out of mathematics altogether.126 On the other hand, there is both direct and indirect evidence that points towards a somewhat more favourable assessment of problematical synthesis by Descartes. For one thing, Descartes tells Mersenne quite generally that “quantum ad Synthesim ..., in rebus Geometricis aptissime post Analysim ponatur” (Resp. 2, AT VII, 156).127 More importantly, the above AT X, 455–58 is qualified in a significant way in the very same Reg. XVI: Denique advertendum est, etiamsi hîc à quibusdam numeris abstrahamus difficultatis terminos ad examinandam ejus naturam, sæpe tamen contingere, illam simpliciori modo resolvi posse in numeris datis, quàm si ab illis fuerit abstracta ...; ac proinde, postquam illam generalibus terminis expressam quæsivimus, oportere eamdem ad datos numeros revocare, vt videamus vtrùm forte aliquam simpliciorem solutionem nobis ibi suppeditent: verb. gr., postquam basim trianguli rectanguli ex lateribus a & b vidimus esse ξܽଶ ൅  ܾଶ , pro a2 ponendum esse 81, & pro b2, 144, quæ, addita, sunt 225, cujus radix sive media proportionalis inter vnitatem & 225 est 15; vnde cognoscemus basim 15 esse commensurabilem lateribus 9 & 126

It might be tempting to adduce in support of Descartes‫ ތ‬rejection of problematical synthesis his scornful remarks in Reg. IV to the effect that instead of showing “quare hæc ita se habeant, & quomodo invenirentur” (AT X, 375), the ancient mathematicians “maluerunt ... nobis in ejus locum steriles quasdam veritates ex consequentibus acutulè demonstratas, tanquam artis suæ effectus, vt illos miraremur, exhibere ...” (AT X, 376). However, while the demonstrations of truths ex consequentibus clearly amount to synthesis, the charge of sterility points to the superfluousness rather than the harmfulness of synthesis. Moreover, and more importantly, the synthesis thus hinted is clearly of the theoretical and not of the problematical cast, and it is contrasted in Reg. IV with the problem-solving procedure simpliciter, with no clear indication as to the status of synthesis relative to problematical analysis. 127 This direct evidence alone is of course far from conclusive as it is not altogether clear whether what Descartes has in mind is problematical or theoretical synthesis or both. Yet it can be adduced, under a certain interpretation, in support of my present point once it is established on stronger grounds to be considered immediately hereafter.

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12, non generaliter ex eo quòd sit basis rectanguli trianguli, cujus vnum latus est ad aliud, vt 3 ad 4. Quæ omna distinguimus, nos qui rerum cognitionem evidentem & distinctam quærimus, non autem Logistæ ... (AT X, 457–58; my emphases).

Thus Descartes apparently does not intend to dismiss the substitution of determinate numbers for symbols in algebraic formulae without qualification, and according to him it is not such a substitution simpliciter that marks the contrast between nos qui rerum cognitionem evidentem & distinctam quærimus and the Logistæ. Therefore, in so far as the substitution of determinate numbers in the present example counts as a part of the problematical synthesis—and if it does not then I do not know what else could—the most one can conclude from the whole passage in Reg. XVI under investigation is that according to Descartes it is not synthesis simpliciter but just some particular type of synthesis that is to be ruled out of problem-solving treatment in mathematics. Descartes‫ ތ‬actual performance in La Geometrie—an eminent exercise in mathematical problem-solving—points to a similar conclusion. Immediately after having proposed to his own satisfaction, toward the end of Geom. I, a general solution to the Pappus problem in a purely algebraic analytical manner for any number of lines,128 he goes on (as we saw above) to consider at length the nature of curved lines.129 It turns out in Geom. III that he is not interested in curves in their own right but as the loci for the pointwise construction of the determinate solutions to the Pappus problem for various numbers of initial lines.130 However, once again construction is the matter of synthesis in geometry according to the Pappian account; and I cannot see how one can avoid the conclusion, even from the Cartesian perspective of La Geometrie, that considerations concerning the constructability of curves as the determinants of algebraic solutions in a geometrical interpretation are to count as the synthetic half of the problemsolving procedure.

128

As a matter of fact, Descartes‫ ތ‬way of generalizing his solution was successfully challenged as early as Fermat; see Emily Grosholz, “Descartes‫ތ‬ Unification of Algebra and Geometry,” in Gaukroger, Descartes: Philosophy, Mathematics and Physics, 163–64. However, this defect is irrelevant to our present issue. 129 The title of the whole Geom. II reads “De la nature des lignes courbes” (AT VI, 388). 130 Cf. Grosholz, “Descartes’ Unification,” 160–62.

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I take it as established, therefore, that pace some respected commentators, Descartes does not dismiss problematical synthesis altogether.131 To be sure, in so far as the problem-solving procedure sticks to the symbolic algebraic form, the solution of a given problem precisely coincides with the end-point of the analytical part of the enterprise. However, we have just seen that Descartes is prepared to interpret the resultant algebraic equations either arithmetically or geometrically and to deploy such interpretations in a manner which should count as synthetic by the standards set up by Pappus. On the other hand, the evidence indicates that for Descartes the purpose of problematical synthesis is different from the purpose ascribed to it by Pappus—but what exactly is this difference? The quoted passages from Reg. XVI provide some assistance in addressing this question. Descartes makes it clear there that even to put down a determinate number is worthless, even harmful, to the sought-after scientia. Yet he has it, by the same token, determinate numbers replacing the data in the terminal algebraic expression and attaining the quæsitum do also help to obtain some “cognitionem evidentem & distinctam”, i.e. some scientiam concerning certain aspects of “quomodo [summa quæsita] dependeat ex datis” (Reg. XVI, AT X, 458). The point seems to be that while the symbolic algebraic expression helps to keep apart the data and the quæsitum and thus to render evident and distinct the dependence relation in question in so far as general quantities are concerned, the expression fails to reveal (perhaps among other things) whether the data (the sides of a right-angled triangle) and the quæsitum (the hypotenuse) will be commensurable or incommensurable given a particular determinate ratio of the lengths of the sides. The answer is ready at hand once the algebraic symbols are replaced with the determinate numbers—but only with the help of referring them back to the symbolic algebraic expression in which they were substituted. Thus it seems to be, at the end of the day, that the ability of one‫ތ‬s mind to move back and forth (so to speak) between the algebraic equation (the terminal stage of the analytic part of the problemsolving procedure) and the corresponding numerical interpretation (the 131

The most prominent commentator to make this strong claim is Gaukroger, Cartesian Logic, 78–86. Also Michael Mahoney, “The Beginnings of Algebraic Thought in the Seventeenth Century,” in Gaukroger, Descartes: Philosophy, Mathematics and Physics, 147 virtually commits himself to such a view. On the other hand, Sasaki, DescartesҲs Mathematical Thought, 216–22 consistently assumes Descartes does proceed synthetically in Geom. II, without, however, even mentioning the complications which induced Gaukroger, Mahoney and others to question the assumption.

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synthetic part) is required to attain certain desired scientiæ concerning the solution of a given problem. In this sense, then, according to Descartes synthesis has indeed been established as indispensable to attaining (some portions of) scientia in the mathematical problem-solving treatment. A closer look at Descartes‫ ތ‬actual performance in Geom. II concerning his professedly general solution to the Pappus problem eventually yields a similar picture. As already noted, Descartes exhibits considerable interest in curves in Geom. II, and proposes such substantial changes as a huge expansion in the scope of geometrically acceptable curves and a brandnew frame for the classification of these “geometrical” curves; and taken broadly enough, according to the Pappian account the construction of figures, including curves, no doubt belongs to the synthetic part of the problem-solving procedure in geometry. Descartes deploys his achievements concerning the theory of curves in his generalization of the solution to the Pappus problem as follows. After succeeding in completely translating the Apollonian theory of conic sections into his symbolic algebra, thus providing the solution to the Pappus problem for three or four lines, he continues: Au reste, a cause que les equations qui ne montent que iusques au quarré sont toutes comprises en ce que ie viens d‫ތ‬expliquer, non seulement le problesme des anciens en 3 & 4 lignes est icy entierement acheué, mais aussy tout ce qui appartient a ce qu‫ތ‬ils nommoient la composition des lieux solides, &, par consequent, aussy a celle des lieux plans, a cause qu‫ތ‬ils sont compris dans les solides. Car ces lieux ne sont autre chose sinon que, lorsqu‫ތ‬il est question de trouuer quelque point auquel il manque vne condition pour estre entierement determiné, ainsi qu‫ތ‬il arriue en cete exemple, tous les poins d‫ތ‬vne mesme ligne peuuent estre pris pour celuy qui est demandé. Et si cete ligne est droite ou circulaire, on la nomme vn lieu plan. Mais si c‫ތ‬est vne parabole, ou vne hyperbole, ou vne ellipse, on la nomme vn lieu solide. Et toutefois & quantes que cela est, on peut venir a vne Equation qui contient deux quantités inconnuës & est pareille a quelqu‫ތ‬vne de celles que ie viens de resoudre. Que si la ligne, qui determine ainsi le point cherché, est d‫ތ‬vn degré plus composée que les sections coniques, on la peut nommer, en mesme façon, vn lieu sursolide: & ainsi des autres. Et s‫ތ‬il manque deux conditions a la determination de ce point, le lieu où il se trouue est vne superficie, laquelle peut estre, tout de mesme, ou plate ou spherique ou plus composée (Geom. II, AT VI, 406–407).

Descartes is clearly not interested in constructing a particular locus for a given case in a given number of lines for its own sake (just as he is not interested in computing a particular number as the solution to a given problem in the case of arithmetic). Rather, I submit, his point is once again

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(as in the case of arithmetic) to secure certain pieces of scientia in geometry that are not easily (or else at all) accessible through contemplation of algebraic equations—more specifically scientiam concerning the type of figure which amounts to the sought-after locus. It will be remembered that Descartes proposes a genetic criterion of geometrical curves in terms of their constructability by a series of ordered continuous movements tracing relatively simpler curves.132 A closer look at the corresponding passages warrants the suggestion that what Descartes is after when he gets involved in this synthetic enterprise is to endow the apprehension of the locus in question with an epistemic characterization that de facto amounts to scientia as conceived by him. Thus he insists that as long as one (presumably attentively) follows the procedure of ordered, continuous movements tracing simpler curves in the production of more complex curves, “on peut tousiours auoir vne connoissance exacte de leur mesure” (ibid., 390; my emphasis), thus satisfying the requirement of “iustesse du raisonnement” (ibid., 389). Further, he intimates immediately after having explained how his mesolabe compass works in this context that he cannot see ce qui peut empescher qu‫ތ‬on ne conçoiue aussy nettement & aussy distinctement la description de cete première [courbe composée], que du cercle ou, du moins, que des sections coniques; ny ce qui peut empescher qu‫ތ‬on ne conçoiue la seconde, & la troisiesme, & toutes les autres qu‫ތ‬on peut descrire, aussy bien que la premiere; ny, par consequent, qu‫ތ‬on ne les reçoiue toutes en mesme façon, pour seruir aux speculations de Geometrie (ibid., 392; my emphasis).

Once again, however—and this is crucial—the scientia at issue is gained not by barely carrying out the construction but consists rather in the ability of one‫ތ‬s mind to move back and forth between the algebraic equation and the corresponding diagram which expresses the geometrical interpretation of the solution to a given problem. To sum up, I have tried to establish the claim that contrary to what some prominent commentators hold, Descartes does avail himself, in his mathematical practice, of the synthetic complement of the problematical analysis as understood by Pappus. Unlike the Pappian account, however, the proper aim of synthetic procedures is for him no longer to compute a determinate number or to carry out a determinate construction. Rather, the aim is to enrich the scientia obtained through an algebraic solution to a given problem in analysis, by way of considering the dependence relations between the data and the quæsita, expressed in the equation, under a 132

Cf. Geom. II, AT VI, 389–90.

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specific arithmetical or geometrical interpretation. With this account, I take as virtually accomplished the task of addressing our leading interim question (i*) of in what the procedure of analysis consists in mathematics as conceived by Descartes. A good deal has also been done by now to address the interim question (ii*) of how Descartes establishes in concreto analysis as the true method of discovery in mathematics. What remains to be done regarding (ii*) is to tackle the issue of the rôle of imagination in Descartes‫ ތ‬conception of mathematics. This will be the topic of the following section.

3.2.6 Scientia and Imagination in Descartes’ Mathematics There is abundant evidence to the effect that from Descartes’ perspective, the employment of imagination, and even of external sensation, is very useful and perhaps even indispensable for doing mathematics correctly. We have already noted that in DM 2 Descartes‫ތ‬ purported reason for having chosen (the lengths of) rectilinear line segments to signify general quantities in his algebraic project is that he could find “rien de plus simple, ny que ie pûsse plus distinctement representer a mon imagination & a mes sens” (AT VI, 20; my emphasis). Descartes‫ ތ‬motives for ascribing to imagination and sensation such a significant rôle in his mathematical project can be found in the Regulæ. He states in Reg. VIII that quidem in nobis advertimus, solum intellectum esse scientiæ capacem; sed à tribus alijs facultatibus hunc juvari posse vel impediri, nempe ab imaginatione, sensu, & memoriâ (AT X, 398; my emphases)

and he specifies this double capacity of these faculties with respect to understanding in Reg. XII, AT X, 416–17: [S]i intellectus de illis agat, in quibus nihil sit corporeum vel corporeo simile, illum non posse ab istis facultatibus [sc. ab imaginatione vel a sensibus] adjuvari sed contra, ne ab ijsdem impediatur, esse arcendos sensus, atque imaginationem, quantum fieri poterit, omni impressione distinctâ exuendam. Si verò intellectus examinandum aliquid sibi proponat, quod referri possit ad corpus, ejus idea, quàm distinctissimè poterit, in imaginatione est formanda; ad quod commodiùs præstandum, res ipsa quam hæc idea repræsentabit, sensibus externis est exhibenda (my emphasis).

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Since Descartes defines geometry as “vne science qui enseigne generalement a connoistre les mesures de tous les cors” (Geom. II, AT VI, 389), so that geometrical matters unequivocally instantiate the feature of quod referri possit ad corpus just mentioned, it should come as no surprise that he commits himself to attributing some significant function to the faculties of imagination and even to external sensation in his geometry, and more generally in his algebraized mathematical enterprise. Descartes‫ ތ‬actual performance both in the subsequent rules of the Regulæ and in La Geometrie confirms this commitment. He states in Reg. XIII that any quæstio perfectè intellecta133 “est ab omni superfluo conceptu abstrahenda” (AT X, 430); and he tells the reader shortly thereafter what he means, in general, by observing the rule at issue: [A]pparet etiam, quo modo hæc regula possit observari, ad difficultatem benè intellectam ab omni superfluo conceptu abstrahendam, eoque reducendam, vt non ampliùs cogitemus nos circa hoc vel illud subjectum versari, sed tantùm in genere circa magnitudines quasdam inter se componendas ... (ibid., 431).

However, when Descartes takes up the issue again in Reg. XIV, he insinuates that imagination is to be reinstalled in the picture: [P]ostquam juxta regulam præcedentem difficultatis termini ab omni subjecto abstracti sunt, hîc tantùm deinceps circa magnitudines in genere intelligamus nos versari. Vt verò aliquid etiam tunc imaginemur, nec intellectu puro vtamur, sed speciebus in phantasiâ depictis adjuto: notandum est denique, nihil dici de magnitudinibus in genere, quod non etiam ad quamlibet in specie possit referri. Ex quibus facilè concluditur, non parù profuturum, si transferamus illa, quæ de magnitudinibus in genere dici intelligemus, ad illam magnitudinis speciem, quæ omnium facillimè & distinctissimè in imaginatione nostrâ pingetur: hanc verò esse extensionem realem corporis abstractam ab omni alio, quàm quod sit figurata ... (AT X, 440–41; my emphases).

These prescripts are put to use in La Geometrie. The very institution of rectilinear line segments whose lengths are used as representations of general quantities was introduced, as Descartes indicates in DM 2, AT VI, 20, due to their accessibility to the imagination and senses; and as we saw, the geometrical dimension-free interpretation of combinatory algebraic 133 Descartes‫ ތ‬notions of perfectly and imperfectly understood quæstiones are discussed in ch. 4, in which references are given.

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operations in terms of these line segments is introduced by means of geometrical figures and diagrams. Furthermore, Descartes deliberately and consistently refers to acts of the imagination in his pointwise constructions of relatively more complex (or higher-degree) curves by means of ordered motions tracing relatively simpler (lower-degree) curves throughout Geom. I–II.134 Finally, Descartes feels apparently no scruples in invoking the significant participation even of external senses in his overall mathematical enterprise. Having mentioned figuras “per [quas] modò magnitudines continuas, modò etiam multitudinem sive numerum esse exhibendum” (Reg. XIV, AT X, 452), i.e. what was to be reduced later to rectilinear line segments whose lengths represent general magnitudes in his hands, he continues: Juvat etiam plerumque has figuras describere, & sensibus exhibere externis, vt hac ratione faciliùs nostra cogitatio retineatur attenta. ... [I]llæ pingendæ sint, vt distinctiùs, dum oculis ipsis onentur, illarum species in imaginatione nostrâ formentur ... (Reg. XV, AT X, 353; my emphases).

In addition, he frequently promotes the simplest possible material symbols to help the memory and to avoid distracting the mind in vain.135 The situation is somewhat complicated due to the fact that Descartes sometimes criticizes the common mathematical practice of both ancient and contemporary mathematicians precisely in terms of their undue focus on the imagination and the senses. Thus he reports in Reg. IV finding out that “circa figuras,” the writers on geometry he came across “multa oculis ... quodammodo exhibeant, & ex quibusdam consequentibus concludebant” (AT X, 375) and comments with contempt on this score that revera nihil inanius est, quàm circa ... [figuras] imaginarias ita versari ..., atque superficiarijs istis demonstrationibus, quæ casu sæpius quàm artè inveniuntur, & magis ad oculos & imaginationem pertinent quàm ad intellectum, sic incumbere, vt quodammodo ipsâ ratione vti desuescamus ... (ibid.; my emphases).

134

See in particular Geom. I, AT VI, 390; 393; Geom. II, AT VI, 408; 413; 416. Cf. e.g. Reg. XVI, AT X, 454: “Quæ verò præsentem mentis attentionem non requirunt, etiamsi ad conclusionem necessaria sint, illa melius est per brevissimas notas designare quàm per integras figuras: ita enim memoria non poterit falli, nec tamen interim cogitatio distrahetur ad hæc retinenda, dum alijs deducendis incumbit.”

135

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Further, in DM 2 he charges that “[l‫ތ‬Analyse des anciens] est tousiours si astrainte a la consideration des figures, qu‫ތ‬elle ne peut exercer l‫ތ‬entendement sans fatiguer beaucoup l‫ތ‬imagination” (AT VI, 17–18). Thus it might seem that contrary to his own programmatic commitments in the Regulæ and to his actual practice in La Geometrie, he implies in these criticisms that to linger in mathematics over drawn or imagined figures and over their manipulation in demonstrations is not only idle but even harmful as it tends to clog the imagination and consequently to hinder the due operations of the understanding. Yet the apparent contradiction in Descartes‫ ތ‬position can be resolved in terms of what we have learned in the previous sections. We saw that according to Descartes the chief burden of scientific cognition in mathematics lies in the clear and distinct grasp of the dependence relations between the data and the quæsita of a given problem, and that the terms of these relations are, at bottom, general quantities which form the proper object of Descartes‫ ތ‬symbolic algebra. Whilst for Descartes both these general quantities and their relations as expressed in algebraic equations and as treated in algebraic operations are objects of the operations of pure understanding, he is prepared, as we saw, to deploy imagination, and even sensation, as auxiliary devices which aid these operations of pure understanding in various ways. Of the various purposes, amongst the more prominent are keeping the mind away from the inattentive mechanical manipulation of uninterpreted symbols in algebraic analysis and aiding the mind to attend exclusively to what is immediately required to push the investigation of general quantities further at any given moment.136 In view of this, Descartes‫ ތ‬criticisms in question can plausibly be read as aiming at conceptions in which the proper objects of mathematics are identified not with general quantities susceptible to symbolic algebraic treatment but rather with determinate geometrical figures possessing spatial dimensions or with determinate numbers conceived as multitudes of units. For it is plausible that such objects are properly grasped with the faculty of imagination; and we are in a position by now to appreciate Descartes‫ތ‬ charge that such a conception of mathematical objects hinders the proper understanding of what the mathematical enterprise truly consists in, and leads the mind astray into idle computations, constructions and manipulations of uninterpreted symbols, so that “vt quodammodo ipsâ ratione vti desuescamus” (Reg. IV, AT X, 375). It might seem so far that while Descartes is ready to employ the imagination and the senses as auxiliary devices whenever they prove 136

Cf. in particular Reg. XVI, AT X, 454–55.

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useful in his mathematical practice, these faculties still remain altogether external to, and therefore strictly speaking inessential for, the constitution of mathematical scientia as conceived by him. Yet whereas this surely holds with regard to sensation, there are places in which Descartes seems to commit himself to a considerably stronger tie between imagination and pure understanding in gaining mathematical scientia. Thus the precept of Reg. XIV reads “[Propositio data] est ad extensionem realem corporum transferenda, & tota per nudas figuras imaginationi proponenda: ita enim longè distinctiùs ab intellectu percipietur” (AT X, 438; my emphasis). Having introduced magnitudines in genere a little later in the same rule, he states: Vt verò aliquid etiam tunc imaginemur, nec intellectu puro vtamur, sed speciebus in phantasiâ depictis adjuto: notandum est denique, nihil dici de magnitudinibus in genere, quod non etiam ad quamlibet in specie possit referri (ibid., 440–41; my emphasis).

Again, he remarks still later that “imaginationis adjumento nos vti posse & debere ...” (ibid., 445; my emphasis). To be sure, these hints are inconclusive as they stand since Descartes‫ ތ‬invocations of imagination could still be interpreted in terms of high practical desirability and not of the strict indispensability of the aid of imagination. I will argue now, however, that the stronger reading is to be adopted in view of both the context of Reg. XIV and other places in Descartes‫ ތ‬chief texts. In Reg. XIV, Descartes announces a long passage expressly devoted to the issue of the proper rôle of imagination: Quia verò nihil deinceps sine imaginationis auxilio sumus acturi, operæ pretium est cautè distinguere, per quas ideas singulæ verborum significationes intellectui nostro sint proponendæ (ibid., 443).

The core of the explanation Descartes offers on this score, after some preliminaries, reads as follows: [S]i dicatur: extensio non est corpus [Descartes’ italics], tunc ... in hac significatione nulla [extensionis vocabuli] peculiaris idea in phantasiâ correspondet, sed tota hæc enuntiatio ab intellectu puro perficitur, qui solus habet facultatem ejusmodi entia abstracta separandi. ... Ac magni est momenti distinguere enuntiationes, in quibus ejusmodi nomina: extensio ... &c., tam strictam habent significationem, vt aliquid excludant, à quo revera non sunt distinctæ [my emphasis], vt cùm dicitur: extensio ... non est corpus [Descartes’ italics] ..., &c. Quæ omnes & similes propositiones ab imaginatione omnino

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Chapter Three removendæ sunt, vt sint veræ .... Notandumque est diligenter, in omnibus alijs propositionibus, in quibus hæc nomina, quamvis eamdem significationem retineant, dicanturque eodem modo à subjectis abstracta, nihil tamen excludunt vel negant, à quo non realiter distinguantur [my emphasis], imaginationis adjumento nos vti posse & debere: quia tunc, etiamsi intellectus præcisè tantùm attendat ad illud quod verbo designatur, imaginatio tamen veram rei ideam fingere debet [my emphasis], vt ad ejus alias conditiones vocabulo non expressas, si quando vsus exigat, idem intellectus possit converti, nec illas vnquam imprudenter judicet fuisse exclusas (ibid., AT X, 444–45).

Two interconnected distinctions of the scholastic origin that Descartes employs here seem to provide the key to his position: viz., on the one hand, an epistemic distinction between omissio and exclusio as different abstractive acts of the understanding; and on the other hand, an ontic distinction between the real and less-than-real distinctions between entities.137 As for the former, epistemic distinction, its meaning should be clear from the very passage just quoted: while exclusio commits one outright to hold that the given object be in reality necessarily devoid of the feature (conditio) from which one abstracts, omissio (to which act Descartes usually refers with the scholastic technical term “præcisio”) amounts just to a neglect of the features from which one abstracts while one is still committed to hold that the feature at issue in reality pertain or at least can pertain to the given object.138 As for the latter, ontic distinction, the situation is somewhat more complex, since the most authoritative place in which Descartes defines the various kinds of distinctiones is not to be found in the Regulæ but in the much later Principia: 137

To be sure, the extent to which Descartes is faithful to the more or less established scholastic meanings of the distinctions in question is far from clear; yet we can safely put these issues to one side as one can gather a fairly clear notion of what Descartes means from his own texts. 138 Cf. an example Descartes offers to illustrate his present point: “Idem, ... si [agamus] de superficie, concipiamus idem, vt longum & latum, omissâ profunditate, non negatâ ...” (ibid., 446; my emphases). Also Descartes‫ ތ‬contention in a letter to Clerselier, which includes an explanation of præcisio, clearly confirms the present suggestion: “[I]‫ތ‬ay dit ... que, pendant que l‫ތ‬ame doute de l‫ތ‬existence de toutes les choses materielles, elle ne se connoist que précisement, præcise tantùm, comme vne substance immaterielle; &, sept ou huit lignes plus bas, pour montrer que, par ces mots præcise tantùm, ie n‫ތ‬entens point vne entiere exclusion ou negation, mais seulement vne abstraction des choses materielles, i‫ތ‬ay dit que nonobstant cela, on n‫ތ‬estoti pas assuré qu‫ތ‬il n‫ތ‬y a rien en l‫ތ‬ame qui soit corporel, bien qu‫ތ‬on n‫ތ‬y connoisse rien ...” (AT IX-1, 214–15; Descartes‫ ތ‬italics).

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[D]istinctio [in rebus] triplex est, realis, modalis, & rationis. Realis propriè tantùm est inter duas vel plures substantias: & has percipimus à se mutuò realiter esse distinctas, ex hoc solo quòd unam absque alterâ clarè & distinctè intelligere possimus. ... Distinctio modalis est ... inter modum proporiè dictum, & substantiam cujus est modus .... [Quæ] ex eo cognoscitur, quòd possimus quidem substantiam clarè percipere absque modo quem ab illâ differe dicimus, sed non possimus, viceversâ, modum illum intelligere sine ipsâ. ... Denique distinctio rationis est inter substantiam & aliquod ejus attributum, sine quo ipsa intelligi non potest .... Atque agnoscitur ex eo, quòd non possimus claram & distinctam istius substantiæ ideam formare, si ab eâ illud attributum excludamus ... (Princ. I, 60–62, AT VIII-1, 28–30).

Moreover, in the next article of the same work Descartes immediately continues in a way highly significant for our present treatment: Cogitatio & extensio spectari possunt ut constituentes naturas substantiæ intelligentis & corporeæ; tuncque non aliter concipi debent, quàm ipsa substantia cogitans & substantia extensa, hoc est, quàm mens & corpus; quo pacto clarissimè ac distinctissimè intelliguntur. Quin & faciliùs intelligimus substantiam extensam, vel substantiam cogitantem, quàm substantiam solam, omisso eo quòd cogitet vel sit extensa. Nonnulla enim est difficultas, in abstrahendâ notione substantiæ à notionibus cogitationis vel extensionis, quæ scilicet ab ipsâ ratione tantùm diversæ sunt ... (Princ. I, 63, AT VIII-1, 30–31; my emphases).

A comparison of these quotations with the following passage, which immediately precedes the quoted AT X, 444–45, should ensure that the treatment in Princ. I is indeed compatible with, and relevant to, the sections of Reg. XIV with which we are dealing: Jam pergamus ad hæc verba: corpus habet extensionem, vbi extensionem aliud quidem significare intelligimus quàm corpus; non tamen duas distinctas ideas in phantasiâ nostrâ formamus, vnam corporis, aliam extensionis, sed vnicam tantùm corporis extensi; non aliud est, à parte rei, quàm si dicerem: corpus est extensum; vel potiùs: extensum est extensum. Quod peculiare est istis entibus quæ in alio tantùm sunt, nec vnquam sine subjecto concipi possunt; aliterque contingit in illis, quæ à subjectis realiter distinguuntur: nam si dicerem, ver. gr.: Petrus habet divinitas, planè diversa est idea Petri ab illâ divitiarum; item si dicerem: Paulus est dives, omnino aliud imaginarer, quàm si dicerem, dives est dives (Reg. XIV, AT X, 444; Descartes’ italics).

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To begin with, Descartes clearly implies in Princ. I, 63 that it is a distinctio rationis that holds between the (attribute of) extension on the one hand and a body or an extended substance on the other. Despite the complication brought about by “[q]uod peculiare est istis entibus quæ in alio tantùm sunt, nec vnquam sine subjecto concipi possunt” (a phrase which points towards the distinctio modalis as defined in Princ. I, but can easily be explained away),139 the way in which Descartes draws the equivalences between corpus habet extensionem, corpus est extensum, and extensum est extensum in the last quoted passage clearly commits him to the same standpoint in the Regulæ. Given this, we can also see that the cognitive criteria for the distinctio realis and the distinctio rationis in Princ. I and Reg. XIV, respectively, virtually coincide as well. As for the Reg. XIV, the criteria are stated in terms of the number of ideas formed in the imagination: thus while in the case of the distinctio rationis between body and extension, “non tamen duas distinctas ideas in phantasiâ nostra formamus ..., sed vnicam tantùm corporis extensi” (AT X, 444), in the case of the distinctio realis between Peter and wealth, on the other hand, “planè diversa est idea Petri ab illâ divitiarum” (ibid.).140 Once we consider that “clarè & distinctè intelligere” is clearly synonymous with “claram & distinctam ideam formare” in Princ. I, 60–63, and that Descartes, as we saw, establishes imagination (together with sensation and pure intellection) as a mode of operatio intellectûs in Princ. I, 32,141 so that the intelligere responsible for the formation of a relevant idea in the context of Princ. I, 60–63 might readily be read as a matter of exercising the faculty of imagination, we can see that the criteria at issue also work in 139 Right after having defined the distinctio modalis and the distinctio rationis in Princ. I, Descartes writes at the end of Princ. I, 62, AT VIII-1, 30: “Memini quidem me alibi hoc genus distinctionis cum modali conjunxisse, nempe in fine responsionis ad primas objectiones in Meditationes de prima Philosophiâ: sed ibi non erat occasio de ipsis accuratè differendi, & sufficiebat ad meum institutum, quòd utramque à reali distinguerem.” I suggest Descartes would be prepared to apply the same remark to the AT X, 444 just quoted had he ever referred to the Regulæ in his other writings, since all the other relevant commitments of his in the AT X, 444 passage are clearly incompatible with the distinctio modalis reading. 140 Some work would be needed to render this example of Descartes‫ ތ‬in accord with his explicit denial of real accidents (see e.g. Resp. 6, AT VII, 435). I shall not venture to perform this task here. I will just suppose that it can be done, so that a sufficient affinity holds between the case of a subject and a real accident belonging to it on the one hand, and the case of substances on the other. 141 Cf. also a similar treatment in Reg. XII, AT X, 415–16. It is plausible to take the “operatio intellectûs” of Princ. I, 32 as synonymous with the “operatio ingenii” of Reg. XII.

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a similar way in the Principia: while in the case of distinctio realis, “unam [substantiam] absque alterâ clarè & distinctè intelligere possimus,” i.e. we are able to form (presumably in the imagination) two different ideas of the given substances, in the case of distinctio rationis, on the other hand, “non possimus claram & distinctam istius substantiæ ideam formare, si ab eâ illud attributum excludamus,” i.e. we are not able to form (presumably in the imagination) but one idea which involves both the given substance and its principal attribute. Having thus established the relevance of Descartes‫ ތ‬treatment of the distinctiones in the Principia to the part of Reg. XIV under examination, we are in a position to grasp the real significance of Descartes‫ ތ‬fairly strong hints concerning the rôle of imagination in mathematical cognition. His treatment in the above-quoted AT X, 444–45 hinges upon the insight that there is something wrong with abstractive acts of exclusio when dealing with items (specifically with body and extension) that are distinct merely ratione; and that—complementally—only abstractive acts of omissio are in order in such cases. Since, according to Descartes, this latter kind of abstraction (unlike the former which is, as he states explicitly, a matter of intellectus purus) essentially requires the working of imagination, the conclusion reads that imagination is indispensable whenever one deals with any item abstracted away from something from which it differs only ratione—and a fortiori in the field of geometry, a discipline which Descartes conceives, as we saw, as “vne science qui enseigne generalement a connoistre les mesures de tous les cors” (Geom. II, AT VI, 389). Yet from Descartes’ perspective, what exactly is wrong with exclusio in such cases? Three critical remarks can be extracted from the passage in Reg. XIV under examination, yet as far as I can see none of them seems conclusive. First, Descartes warns that such exclusio plerique erroris occasio est, qui non animadvertentes extensionem ita sumptam non posse ab imaginatione comprehendi, illam sibi per veram ideam repræsentant (Reg. XIV, AT X, 444);

but once that sort of error is pointed out to those who are prone to committing it, nothing objectionable seems to remain. Second, Descartes remarks that it is precisely the exclusio in question (implying the complete removal of imagination) that licenses the truth of essentially negative predications whose terms are names signifying items which are distinct

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merely ratione;142 yet there seems to be nothing wrong with this fact in itself. Third, Descartes states that the imagination ensures that the understanding never “imprudenter” takes what is ratione distinct as “excluded”, i.e. (as I understand it) as realiter distinct;143 once again, however, it might seem that such a tendency to error could easily be prevented, so that no problem of principle would remain. However, I believe Descartes has a point, indeed a point of vital importance. The missing tessera, I submit, is to be found outside the Regulæ, namely in the above Princ. I, 63. The title of this article reads “Quomodo cogitatio & extensio distinctè cognosci possint, ut constituentes naturam mentis & corporis” (AT VIII-1, 30; my emphasis). As I read it within the present context, the article suggests, among other things, that once an item (e.g. extension) is abstracted by way of exclusio from another which is but ratione distinct from it (e.g. extended substance or body), no clear and distinct cognition of the item thus abstracted is ever forthcoming. That is to say, while an abstractive act of exclusio, which “ab intellectu puro perficitur” (Reg. XIV, AT X, 444), is perhaps not outright impossible here,144 it never yields any conception that would count as clarissima ac distinctissima. It is then this insight, I suggest, that prompts Descartes to integrate the operations of imagination as a strictly essential component of the mathematical cognition in Reg. XIV: the corresponding abstractive act of omissio, which implies imagination at work, not only is far from hindering pure understanding in its business of “præcisè tantùm [attendere] ad illud quod verbo designatur” (Reg. XIV, AT X, 445), i.e. eventually of focusing exclusively on the magnitudines in genere gained by means of abstraction (of the omissio kind as we can see by now) according to the precept of Reg. XIII; rather, it is not until this abstractive act, which comprises imagination as an essential component, that one is able to retain clear and distinct ideas of the objects under investigation at all. This is strictly required in so far as the mathematical cognition gained by pure understanding is to count as scientia. Thus, Descartes really means what he says in the precept of Reg. XIV, AT X, 438: once the problem is proposed “tota per nudas figuras imaginationi,” it “longè distinctius ab 142

See ibid., 445: “Quæ omnes & similes propositiones [vt extensio, vel figura non est corpus; numerus non est res numerata; ..., &c.] ab imaginatione omnino removendæ sunt, vt sint veræ ...” (Descartes‫ ތ‬italics). 143 See ibid.: “[I]maginatio tamen veram rei ideam fingere debet, vt ad ejus alias conditiones vocabulo non expressas, si quando vsus exigat, idem intellectus possit converti, nec illa vnquam imprudenter judicet fuisse exclusas” (my emphasis). 144 Cf. Princ. I, 63, AT VIII-1, 31: “[N]onnulla enim est difficultas, in abstrahendâ notione substantiæ à notionibus cogitationis vel extensionis” (my emphasis).

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intellectu percipietur.” To conclude, imagination has been established as an essential, absolutely indispensable component in the constitution of mathematical scientia in Descartes. We saw above that clear and distinct perception or, as Descartes prefers to say in the Regulæ, facilitas & perspicuitas,145 is at least a sine quâ non of certitudo which marks off scientific cognition, and it has now been established that certain acts which essentially involve imagination, viz. abstractive acts of omissio, are indispensable for pure understanding to obtain those clear and distinct perceptions. By the same token, some of the systematic evidence promised in ch. 2 is provided in support of taking not just pure understanding alone but rather understanding in the wider sense of passio mentis as the faculty properly responsible for (pieces of) scientia in Descartes.

145

Cf. e.g. Reg. II, AT X, 365.

CHAPTER FOUR TOWARDS A UNIVERSAL METHOD OF DISCOVERY

Having accomplished the account of how Descartes describes and employs his algebraic method of analysis and synthesis in mathematics, let us take up the issue for the sake of which this account was given, viz. the all-important task of working out a detailed interpretative reconstruction of the constitution and modi operandi of Descartes‫ ތ‬envisaged universal method, i.e. of a method that would be applicable in each and every appropriate domain of cognition. For I argued in ch. 3 that several questions must be addressed for us to have even a chance of determining the meaning(s) of the a priori–a posteriori pair in Descartes, namely (i) of what the procedure of analysis properly consists, (ii) in which sense “analysis” (as understood in this context) deserves the title of the true method of discovery, and (iii) precisely what rôle simple natures are supposed to play in the constitution of the analytical method of discovery. Once the declared present task is accomplished, the answer to these questions will be given ipso facto. My approach to this challenging task is underlain throughout by a quasi-genetical assumption concerning the status of the envisaged universal method (whatever it turns out to be), which was succinctly expressed by Bret Doyle:1 [D]espite the fact that in some passages Descartes says that his method is a well-defined procedure ..., in general we find him quite unselfconsciously showing that what he does have is a set of strategies that share some strong family resemblances. The resemblances get expressed in general precepts, and the techniques are embedded in the particular, concrete problems with which 1

In idem, “How (not) to study Descartes’ Regulae,” British Journal for the History of Philosophy 17, no.1 (2009): 3–30. My adoption of Doyle‫ތ‬s description does not imply, of course, that I endorse the overall reading of the Regulæ advanced in the quoted paper.

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Descartes was concerned (ibid., 9; Doyle’s emphasis).

The basic strategy I adopt in view of this assumption is to focus in particular on Descartes‫ ތ‬treatment of problems in the field of mathematics, and to trace the ways in which his conception of universal method is structurally related to his professedly methodical problem-solving strategies in tackling mathematical issues. The results achieved in the previous chapters should provide a good basis for carrying out the present task. The interpretative decision introduced above is warranted chiefly by the above-established fact that Descartes does take mathematics, in so far as it is correctly practised, as the best paradigm of the envisaged universal method. Of particular relevance is the contention that (correctly performed) mathematics provides extraordinarily striking examples of how the method actually works and how powerful it can be. Moreover, although from the systematic point of view Descartes is bound to count mathematics—notwithstanding its professed paradigmatic rôles—as just one among various domains of application of the universal method at issue, it is virtually incontrovertible that several elements of his account of the universal method were in fact derived by him by the generalization of certain aspects of his actual mathematical practice in view of an independent account of cognitive faculties and their operations. Furthermore, since the relevant aspects of Descartes‫ ތ‬own mathematical practice turned out relatively sharp-edged and determinate, the strategy of interpreting his often dazzlingly abstract and loose statements concerning the rules or precepts of the envisaged universal method as generalizations from mathematical cases seems prima facie reasonable. Finally, a closer look at the passages in which the issue of universal method is taken up by Descartes also encourages the declared approach, for in the only text in which Descartes condescends both to state in concreto several rules or precepts of which the envisaged universal method is to consist, and to explain and/or to exemplify how the stated rules are to be put to use—to wit, in the Regulæ—the bulk of instances Descartes adduces to render his account of those rules more specific are indeed taken from the domain of mathematical application.2 I begin with some preliminaries to the above task. First I accomplish the account of the paradigmatic rôles of mathematics with respect to the 2

There are important exceptions, in particular the examples of a search for the anaclastic line in Reg. VIII; and of an investigation into the limits of human cognition in Reg. VIII; and of an investigation into the nature of the magnet in Reg. XII–XIV. I briefly deal with each of these in due course.

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project of a universal method; in particular, I examine how the project of a discipline Descartes calls Mathesis vniversalis is related to his project of a universal method. Then I turn to his famous account of the method in DM 2 and introduce the problems this account brings about. This shall prepare the ground for an attempt to reconstruct some essential elements of the universal method Descartes seems to have had in mind.

4.1 Mathematicæ, Mathesis Vniversalis and a Universal Method It will be remembered that the account of Descartes‫ ތ‬notion of mathematics as a privileged paradigm of an envisaged universal method of discovery presented in ch. 3 was said to remain incomplete in several respects. The strategy just adopted for the task of reconstructing the essentials of Descartes‫ ތ‬envisaged universal method clearly requires that the supposed gaps be brought to light and filled; and we are well-equipped by now to perform this task. One such accomplishment is not difficult to establish. In our previous discussion in ch. 3, three different senses were introduced in which mathematics counted for Descartes as a privileged paradigm of a universal method, viz. (i) providing the standard of certainty, (ii) exemplification of the sought-after method in rudimentary use, and (iii) exemplification of the precepts of that same method in a developed form in use; and (appropriate forms of) arithmetic and geometry were jointly assigned to each of these three paradigmatic functions in the course of the aforementioned discussion. It was then made plain that Descartes‫ތ‬ considered mature view is that both arithmetic and geometry are, in a peculiar sense, derivative with respect to what he considered the truly basic mathematical discipline—namely his general algebra of essentially interpreted symbols. In view of this, it sounds reasonable to maintain that sensu stricto according to Descartes it is (his peculiar variant of) general algebra that has the right to be assigned to the paradigmatic function at issue—presumably in all the three senses established above—in the most fundamental sense. Indeed, Descartes‫ ތ‬general algebra was shown to transcend the difference between traditional geometrical analysis and modern algebra conceived as an Arithmeticæ genus quoddam (Reg. IV, AT X, 373), so that it is the former discipline that has the most genuine right to claim the status of fruges ex ingenitis hujus methodi principijs natæ (ibid.); arithmetic and geometry take the stage only derivatively once general quantities are interpreted as either multitudes or magnitudes respectively. Again, Descartes‫ ތ‬own treatment of the Pappus problem in

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La Geometrie provides an excellent example of the employment of the developed methodical precepts in the field of mathematics imprimis not in the form of figuræ & numeri—objects of geometry and arithmetic respectively—but after the manner of general algebra. Finally, given the established relations of arithmetic and geometry to Descartes‫ ތ‬general algebra, it clearly follows that the certainty of arithmetical and geometrical cognition is but a function of the certainty of cognition gained through general algebraic analysis. Descartes‫ ތ‬talk of arithmetic and geometry rather than of general algebra, in the course of picking up mathematics as a privileged paradigm of the sought-after universal method in Reg. II–IV, can plausibly be explained as being due to the fact that Descartes was not, by the Regulæ, in possession of his perfectly general algebra of line segments.3 Thus while he was ultimately after a universal method of discovery, it is virtually incontrovertible that certain portions of the text of the Regulæ reflect yet another pursuit of his, namely the pursuit of a truly general mathematics—a discipline which would assume, once established, the paradigmatic function with regard to the sought-after universal method. What is highly controversial, on the other hand, is the question of exactly which portions of the Regulæ are to be read to this effect. This brings us to another, more difficult respect in which the topic of mathematics as a privileged paradigm of the envisaged universal method is yet to be accomplished: that is to say, both the nature of what Descartes refers to as Mathesis vniversalis and the relation of this discipline to the project of establishing a universal method are yet to be clarified. There are several good reasons for focusing on Descartes‫ ތ‬conception of Mathesis vniversalis in connection with the issue of his project of a universal method. Firstly, Descartes mentions it in Reg. IV, which explicitly takes up method as the proper topic—the precept of Reg. IV reads “Necessaria est Methodus ad rerum veritatem investigandam” (AT X, 371). Secondly, Descartes introduces Mathesis vniversalis by way of abstracting from the subject matter of particular mathematical disciplines including arithmetic and geometry, and characterizes it as a generalis scientia “quæ id omne explicet, quod circa ordinem & mensuram nulli speciali materiæ addictam quæri potest” (ibid., 378). This surely suggests some sort of generalisation of procedures employed in particular mathematical disciplines, although (as we shall see in a moment) it is controversial how far such a generalisation goes. Third, Descartes‫ ތ‬way of arriving at Mathesis vniversalis via some sort of departure from the 3

Cf. ch. 3, fn. 77.

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particular mathematical disciplines seems at least prima facie structurally similar to the way in which Descartes introduced an envisaged disciplina quædam alia earlier in Reg. IV, a disciplina which commentators have (almost) unanimously identified, as we saw, with the sought-after universal method. Fourth and finally, Descartes indicates that Mathesis vniversalis is a prerequisite—as long as one wishes to proceed adequately—for embarking on research into higher or more advanced (altiores) scientiæ;4 and such a status is clearly due to a rôle (the exact nature of which is yet to be determined) that, according to Descartes, Mathesis vniversalis is supposed to play in the constitution of the envisaged universal method. The problem of Mathesis vniversalis in Descartes involves two closely interconnected yet distinct principal questions, both of which have been subject to numerous controversies in recent scholarship: there is the question (1) of what exactly Descartes means by the term “Mathesis vniversalis” and how his Mathesis vniversalis is related to his project of a universal method of discovery, and (2) of how the only passage in which the term “Mathesis vniversalis” occurs in Descartes‫ ތ‬extant writings—and which occupies a few closing pages of Reg. IV—is related to the rest of Reg. IV and indeed to the Regulæ as a whole. It will be well to consider each question in isolation.

4.1.1 A Textual Problem Let us begin with (2). It has always been recognized all across the board that the only passage in which Mathesis vniversalis is explicitly discussed by Descartes,5 viz. Reg. IV, AT X, 374.16–379.13, which following Weber6 has been termed IV-B by the bulk of commentators, is a considerably autonomous piece of text which fits somewhat uneasily with both the preceding portions of the development of Reg. IV (AT X, 371.4– 375.15, likewise termed IV-A) and to what comes immediately after, from 4

See Reg. IV, AT X, 378–79. In the “Index général,” AT XII, 84 several other loci are referred to under the heading of Mathesis universalis, the most significant for us being Reg. VI, AT X, 384–85 (but see fn. 8) and DM 2, AT VI, 19–20 (but strangely omitting Reg. XIV, AT X, 450–52, and the entire Reg. XVIII). Nonetheless, the term “Mathesis vniversalis” or its derivative forms occur exclusively in the indicated portion of Reg. IV. 6 See Jean-Paul Weber, “Sur la composition de la Regula IV de Descartes,” Revue philosophique de la France et de lҲétranger 154 (1964): 1–20; idem, Constitution des Regulæ, ch. 1. 5

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Reg. V onwards. After having picked up, in IV-A, ancient geometrical analysis and modern algebra as spontaneæ fruges ex ingenitis methodi principijs natæ (Reg. IV, AT X, 373), Descartes, as we saw, announces a disciplina quædam alia designed to ensure a rich harvest of those fruges even “in cæteris [scientiis], vbi majora illas impedimenta solent suffocare” (ibid., 373–74). However, then he abruptly embarks, in IV-B, on a sort of autobiographical narrative in the course of which he criticises the usual practice of doing arithmetic and geometry of his times, praises Pappus and Diophantus as thinkers in whose writings “veræ Matheseos vestigia quædam adhunc apparere ... videntur” (ibid., 376)—though, Descartes suspects, that vera Mathesis “ab ipsis ... perniciosâ quâdam astutiâ suppressam fuisse” (ibid.)—and he finally introduces, as the terminal point of his search after the vera Mathesis, a Mathesis vniversalis as a generalis scientia. Thus while IV-B obviously involves some repetition of and structural and thematic parallels with IV-A, it differs from IV-A, at least prima facie, as to both its declared topic and terminal intention.7 Moreover, Descartes states towards the end of IV-B: “priusquam hinc [sc. a Mathesi vniversali excolendâ] migrem, quæcumque superioribus studijs notatu digniora percepi, in vnum colligere & ordine disponere conabor” (ibid., AT X, 379). Yet the rules that immediately follow seem to contain, at least prima facie, little or nothing of the collectio thus announced8 while they seem to advance, albeit loosely and generally, the project of the disciplina quædam alia which is declared towards the end of IV-A and which presumably refers to a universal method of discovery envisaged by 7

See in particular Marion, Sur l’ontologie grise, § 9 and Pamela Kraus, “From Universal Mathematics to Universal Method: Descartes‫ތ‬s ‫ލ‬Turn‫ ތ‬in Rule IV of the Regulae,” Journal of the History of Philosophy 21, no. 2 (1983): 159–74 for detailed and persuasive presentations of the indicated similarities and differences. 8 The following passage from Reg. VI might well be taken as referring to the Mathesis vniversalis as described in IV-B: “[D]eduxerim, vt facilè sit, eamdem esse proportionem inter 3 & 6, quæ est inter 6 & 12, item inter 12 & 24, &c., ac proinde numeros, 3, 6, 12, 24, 48, &c., esse continuè proportionales: inde profectò, quamvis hæc omnia tam perspicua sint, vt propemodum puerilia videantur, attentè reflectendo intelligo, quâ ratione omnes quæstiones, quæ circa proportiones sive habitudines rerum proponi possunt, involvantur, & quo ordine debeant quæri: quod vnum totius scientiæ puræ Mathematicæ summam complectitur” (AT X, 384– 85; my emphasis). Yet the proper topic of Reg. VI is not these mathematical issues but explication of a precept with a far more universal scope. Rather, the quoted mathematical observation is invoked here just as an excellent example of “veritates [quæ] alijs priùs & faciliùs potuerimus reperire” (ibid., 384) which are to be collected (colligere) in order to have but a chance of observing the general methodical precepts set forth in Reg. V–VII.

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Descartes. Finally, in the H-version (unlike, however, the A- and Nversions), IV-B is withdrawn from Reg. IV and set at the end of the entire Regulæ as an appendix. These facts jointly render it plausible to conclude that IV-A and IV-B are indeed, at least to a certain extent, heterogeneous texts which Descartes failed to integrate into a unitary whole, and that IV-B seems to be, at least prima facie, the odd man out as far as the thematic continuity from Reg. IV to Reg. V and beyond is concerned. Yet this conclusion is perfectly compatible with various alternative conjectures concerning the mutual relations of IV-A and IV-B: perhaps IV-B was written first as an elaboration of Reg. IV and later was replaced by IV-A, serving nonetheless as a model for IV-A;9 or maybe IV-B was originally written as a continuation of the opening passage of Reg. IV up to AT X, 373.2, and Descartes decided later to substitute it with the rest of IV-A; or Descartes might first have outlined a general scheme of the elaboration of Reg. IV, and then have written both IV-A and IV-B as complementary treatments based on that scheme but failed (as he abandoned the entire project of the Regulæ) either to choose one and discard the other or else to integrate both into a single smooth text;10 or finally, perhaps IV-B was supplied by Descartes as a substantial extension of the earlier IV-A but he did not bother, once again, to integrate this addendum smoothly into the original text.11 Now to be sure, it would contribute a great deal to our understanding of Descartes‫ ތ‬conception of Mathesis vniversalis if it were possible to decide this textual polylemma on independent grounds.12 Unfortunately, this hardly seems forthcoming. Rather, it is apparently in terms of an independent understanding of what Descartes means by Mathesis vniversalis that the present textual problem could be, and actually has been, handled in a tolerably controlled manner. Since, therefore, our direct concern has been with the rôle of Mathesis vniversalis in Descartes‫ތ‬ 9

This generic conjecture is endorsed—differences of detail notwithstanding—e.g. by Weber, Constitution des Regulæ, ch. 1; Kraus, “From Universal Mathematics to Universal Method,” and Doyle, “How (not) to study Descartes’ Regulae.” 10 This possibility is considered by Doyle, “How (not) to study Descartes’ Regulae” and endorsed as most reasonable by Marion, Sur l’ontologie grise, § 9 and by Frederick Van De Pitte, “Descartes‫ތ‬s Mathesis Universalis,” in Moyal, Critical Assessments 1:61–79. 11 This option is considered seriously by Van De Pitte, ibid. 12 It has been this prospect, I suppose, that has motivated controversies on this point of textual criticism of such an extension and vigour as the Descartes scholarship witnessed after the publication of Weber’s Constitution des Regulæ.

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project of establishing a universal method of discovery, textual question (2) may safely be put to one side for our purposes, question (1) being focused upon in abstraction from question (2).

4.1.2 The Meaning of Mathesis vniversalis in Descartes Three basic generic interpretations worthy of consideration of what Descartes means by Mathesis vniversalis in Reg. IV have been proposed and advocated in recent Descartes scholarship:13 the three genera can be set forth as claiming, respectively, that (the programme of establishing) Mathesis vniversalis is co-extensive (I) with (the programme of establishing) general algebra as a fundamental discipline in the field mathematics; (II) with (the programme of establishing) an essentially mathematical treatment of material reality; and (III) with (the programme of establishing) a perfectly universal method of discovery.14 Each of these generic options obviously implies certain general commitments with regard to the question of how Descartes‫ ތ‬Mathesis vniversalis is related to his envisaged universal method; but at least in the cases of (I) and (II), further ramifications seem possible concerning this latter issue. I endorse generic option (I) and reject options (II) and (III) as prima facie reasonable yet ultimately inferior alternatives to (I). Accordingly, I first introduce and support with an extensive argument the version of (I) that I endorse, then 13

The quantity of material published on the topic since the second half of the nineteenth century is overwhelming and I am far from pretending to offer here any complete survey. I focus just on a few commentators whose contributions have commonly been acknowledged substantial and whose position and/or argument can immediately help to clarify my own suggestions. For more extensive representative bibliographies of the major contributions to the topic up to 1980, see e.g. Van De Pitte, “Descartes’s Mathesis Universalis,” 74, n. 1; Schuster, “Descartes‫ ތ‬Mathesis Universalis,” 80–81, n. 1–2. As for the period since 1980, I would add to those lists in particular Kraus, “From Universal mathematics to Universal Method;” Sasaki, DescartesҲs Mathematical Thought, ch. 4, § 3 and ch. 8, § 1; and Doyle, “How (not) to read Descartes’ Regulae.” 14 Various versions of (I) are defended in particular in Louis Liard, “La méthode et la mathématique universelle de Descartes,” Revue philosophique de la France et de lҲétranger 10 (1880): 569–600; Weber, Constitution des Regulæ, ch. 1; Jürgen Mittelstrass, “The Philosopher‫ތ‬s Conception of Mathesis Universalis from Descartes to Leibniz,” Annals of Science 36, no. 6 (1979): 593–610; Kraus, “From Universal Mathematics to Universal Method”, and Sasaki, DescartesҲs Mathematical Thought, ch. 4, § 3; likewise, of (II), in particular in Schuster, “Descartes‫ ތ‬Mathesis Universalis;” of (III), in particular in Marion, Sur l’ontologie grise, §§ 10–11; and Van De Pitte, “Descartes’s Mathesis Universalis.”

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proceed in sequence to options (II) and (III), consider some arguments adduced in their support by their advocates, and then briefly demonstrate why they are to be rejected. Finally, I propose my answer to the question of the relation of Descartes‫ ތ‬Mathesis vniversalis thus interpreted to his envisaged universal method. It is important to appreciate that the present issue is far from being merely de nomine in the larger context of our investigations: the question we have been pursuing is whether what Descartes says concerning Mathesis vniversalis could help to explain the constitution of a universal method arguably envisaged by him; and to this end, it is of vital importance to clarify to what exactly the term in question refers, since this ipso facto determines exactly which of the many things Descartes says concerning mathematics are really concerned with his project of Mathesis vniversalis. 4.1.2.1 The Preferred Reading The central passage concerned with Mathesis vniversalis, which contains virtually all Descartes has to say expressly about this discipline, is worth quoting extensively (the letters in square brackets in bold are additions of mine): Quae me cogitationes [de vestigiis veræ Matheseos in Pappo & Diophanto, & de huius sæculi Algebrâ] cùm à particularibus studijs Arithmeticæ & Geometriæ ad generalem quamdam Matheseos investigationem revocâssent, quæsivi inprimis quidnam præcisè per illud nomen omnes intelligant, & [A] quare non modo jam dictæ, sed Astronomia etiam, Musica, Optica, Mechanica, aliæque complures, Mathematicæ partes dicantur. Hic enim vocis originem spectare non sufficit; nam cùm Matheseos nomen idem tantùm sonet quod disciplina, non minori jure, quam Geometria ipsa, Mathematicæ vocarentur. [B] Atqui videmus neminem fere esse, si prima tantùm scholarum limina tetigerit, qui non facile distinguat ex ijs quæ occurrunt, quidnam ad Mathesim pertineat, & quid ad alias disciplinas. Quod attentiùs consideranti tandem innotuit, illa omnia tantùm, in quibus ordo vel mensura examinatur, ad Mathesim referri, nec interesse vtrùm in numeris, vel figuris, vel astris, vel sonis, aliove quovis objecto, talis mensura quærenda sit; [C] ac proinde generalem quamdam esse debere scientiam, quæ id omne explicet, quod circa ordinem & mensuram nulli speciali materiæ addictam quæri potest, eamdemque, non ascititio vocabulo, sed jam inveterato atque vsu recepto, Mathesim vniversalem nominari, quoniam in hac continetur illud omne, propter quod aliæ scientiæ Mathematicæ partes appellantur. ... [D] [N]omen ejus omnes nôrint, &, circa quid versetur, etiam non attendentes, intelligant ... (Reg. IV, AT X, 377–78).

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Driven by the pursuit of vera Mathesis as a paradigm of certainty in scientiæ due to its perspicuitas & facilitas summa (ibid., 377), Descartes sets out in [A]15 to determine what all the particular disciplines collectively called Mathematicæ (the Pythagorean quadrivium of arithmetic, music, geometry, and astronomy, plus optics and mechanics are explicitly mentioned as representative examples) have in common, because it is due to this common something that they are rightly characterized as ad Mathesim pertinentes, or as partes of Mathesis.16 The common feature he arrives at in [B] amounts to ordinis vel mensuræ examinatio and Descartes goes on to introduce in [C] another discipline the proper subject of which is “id omne ..., quod circa ordinem & mensuram nulli speciali materiæ addictam quæri potest” (Reg. IV, AT X, 378) and ascribes to this discipline the name Mathesis vniversalis. This much seems to be a generally uncontroversial overall description of what is going on in the quoted passage. What is controversial, and what most acutely separates generic interpretations (I) and (II) from (III), is whether the ordo vel mensura, whose examination is identified as the (abstracted) common feature which renders the particular Mathematicæ pertinent to Mathesis in [B], refers to the same items as the ordo & mensura nulli speciali materiæ addicta, which is identified as the proper subject of Mathesis vniversalis in [C]. An affirmative answer is required for generic readings (I) and (II), and the negative for (III). 15

From now until the end of the present section, capital letters in square brackets refer to the corresponding statements in the last quoted passage. 16 I fully agree with Frederick Van De Pitte (cf. his “Descartes’s Mathesis Universalis,” 66–67 and 77, n. 28; I owe the following point to him) that this point is unnecessarily obscured by Adam & Tannery‫ތ‬s decision to set the comma between complures and Mathematicæ in “quare non modo jam dictæ, sed Astronomia etiam, Musica, Optica, Mechanica, aliæque complures, Mathematicæ partes dicantur” (AT X, 377.13–16; my emphasis). Their decision is understandable as otherwise the passage would count as corrupted due to the missing genitive noun before or after the partes. Yet it runs contrary to the reading which is rendered natural by the context and according to which the complures Mathematicæ are to be assigned as partes [Matheseos]. This arguably natural reading finds further support in Baillet‫ތ‬s Extraits from the Regulæ where the passage in question is paraphrased as follows: “Les pensées qui lui vinrent sur ce sujet, lui firent abandonner l‫ތ‬étude particuliére de l‫ތ‬Arithmétique & de la Géométrie, pour se donner tout entier à la recherche de cette Science générale, mais vraye & infaillible, que les Grecs ont nommée judicieusement MATHESIS, & dont toutes les Mathématiques ne sont que des parties” (Adrien Baillet, La vie de Monsieur Des Cartes, vol. 2 [Paris: Daniel Horthemels, 1691], 115; reprinted in AT X, 484; my emphasis).

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I believe a strong case can be made for the affirmative answer. To begin with, in the course of looking for the common feature of all the Mathematicæ in [A] Descartes considers the original meaning of the Greek word ȝȐșȘıȚȢ, which initially amounted to the act or process of learning and gradually drifted rather more to meaning “something learned” to which the Latin word disciplina (derived from discere, to learn) corresponds;17 and significantly, he dismisses this as by far too wide a determination of the sought-after common feature due to which the Mathematicæ rightly fall under the heading of Mathesis (since then, Descartes observes, just anything that can be learned would be rightly assigned as a member of the class of Mathematicæ).18 This performance commits Descartes to associating the term “Mathesis”, at least throughout [A] and [B], with an essential feature that pertains to the Mathematicæ, and—crucially—only to them, in contradistinction with any other branch of possible scientiæ; as a consequence, the ordo vel mensura of [B] refers to the common object of all the possible Mathematicæ, and to these only. It will be observed that (i) the considerations in [A] and [B] were introduced by Descartes immediately after his plan to embark on a “generalis Matheseos investigatio” (Reg. IV, AT X, 377.11; my emphasis) was announced; that (ii) he initially characterizes Mathesis vniversalis, the introduction of which he proceeds to as a “generalis scientia” (ibid., 378.4–5; my emphasis) at the beginning of [C]; that (iii) he refers to the proper object of this generalis scientia with ordo & mensura, i.e. the same terms that are employed in [B]; and that (iv) the passage from [B] to [C] is effected through the connective proinde (hence) and [C] is divided from [B] only by a semi-colon. Together, all of this suggests a continuity in theme and reference of the crucial terms so strong that, it seems to me, a presumption against reading (III) is established by the present considerations alone. There is even more basis to such a presumption, however. Descartes claims in the latter half of [C] that Mathesis vniversalis is a “vocabulum jam inveteratum atque vsu receptum” (ibid., 378.7–8), and adds in [D] that as for the science at issue, “nomen ejus omnes nôrint, &, circa quid versetur, etiam non attendentes, intelligant” (ibid., 378.17–18). These are abundantly clear indications that far from announcing a discovery, Descartes wishes to use the term “Mathesis vniversalis” in an established 17

I draw upon the linguistic observations in Van De Pitte, “Descartes’s Mathesis Universalis,” 63. 18 Unfortunately, the text at AT X, 377.17–19 seems corrupted as Adam & Tannery point out in AT X, 377, fn. b. In spite of this, the meaning of Descartes‫ތ‬ reasoning has commonly been agreed even by the critics of readings (I) and/or (II).

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way and to endorse (at least in so far as the general description he has given in Reg. IV is concerned) the then common conception of with what Mathesis vniversalis is properly concerned. We have already established in ch. 3 that by far the most significant of the exponents of the vsus Descartes most probably has in mind here,19 and indeed one whose ideas concerning the issue Descartes might well have been acquainted with by the time of composing Reg. IV, is Adriaan Van Roomen.20 We have already seen van Roomen introducing Mathesis vniversalis in a manner very similar to Descartes‫ ތ‬exposition in AT X, 377–78: Geometriæ, & Arithmeticæ communem esse scientiam, quæ quantitatem generaliter vti mensurabilem considerat. ... Nimirum scientiam esse quandam Mathematicam communem Arithmeticæ & Geometriæ, ad quam spectarent affectiones omnibus quantitatibus: cùm autem proportio sit omnibus quantitatibus communis, non abstractis tantum vt numeris et magnitudinibus, sed concretis etiam, vti temporibus, sonis, vocibus, locis, motionibus, potentiis ... [P]ropositiones earumque demonstrationes ... Mathesi Vniversali tribuendæ sunt .... Inscribemus autem scientiam hanc nomine, Prima Mathematica, seu Prima MatheseȦs ... (Van Roomen, Apologia pro Archimede, 22–23).

Now crucially, Van Roomen makes it absolutely clear in his Universæ mathesis idea (1602) that Mathesis vniversalis is unambiguously a 19

For an invaluable account of the history of the concept and of the term “Mathesis universalis” from the sixteenth century onwards, see Crapulli, Mathesis universalis. Crapulli‫ތ‬s list of significant figures of the period advancing the idea includes, among others, Piccolomini, Dasyopodus, Pereira, Alsted, and Van Roomen (cf. in particular ibid., 150). 20 Chikara Sasaki states, in his comprehensive attempt at assessing the possibility of the direct influence of Van Roomen upon Descartes in his DescartesҲs Mathematical Thought, 267–70, that while “there is no definitive evidence” to this effect (ibid., 269), “Descartes most probably learned the idea of ‫ލ‬mathesis universalis‫ ތ‬from ... Van Roomen ...” (ibid., 267). It is worth noting that even if Sasaki‫ތ‬s conjecture is false as a matter of fact, the following argument is affected but little or not at all as to its import. For (i) the adduced evidence from [C] and [D] suggests Descartes must have learned about the then common conception of Mathesis vniversalis from some source, and (ii) the crucial point of the following argument—that Mathesis vniversalis itself counts as a part of the Mathematicæ—is present in all the other significant proponents of the idea of Mathesis vniversalis of whom Descartes was probably aware (I cannot document this in detail here; for evidence see Crapulli, Mathesis universalis, passim, and in particular ch. 7). In other words, Van Roomen‫ތ‬s position is explained just as a representative sample of, arguably, the then common view of the aforementioned crucial point.

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division within the genus named Vniversa mathesis, a term which refers, exactly like Mathesis in Descartes‫[ ތ‬B], to the highest genus of the scientia de qvantitate (Van Roomen, Universæ mathesis idea, 3) the extension of which comprehends all the Mathematicæ:21 Mathematicarum censu comprehenduntur, tum eæ qvæ verè Mathematicæ sunt, tum qvæ qvasi Mathematicæ dici possunt. MATHEMATICA vera est tum Princeps sive primaria, tum ministrans sive Mechanica. Mathematica princeps est qvæ qvantitatum speculationi intenta est; talis duplex est pura & impura sive mixta. Pura Mathematica est qvæ speculatur qvantitatem puram sive intelligibilem. Mixta Mathematica est qvæ speculatur qvantitatem mixtam sive sensibilem. Pura iterum duplex universalis & specialis. Vniversalis est qvæ circa omnem versatur qvantitatem nempe Logistice & prima Mathesis, illa ut organum scientiæ, hæc ut scientia. Specialis qvoque duplex est Arithmetica & Geometria ... (Universæ mathesis idea, 14–15).

As a consequence, the proper subject of Mathesis vniversalis, whatever it be, is bound to count as a part of the domain of the Mathematicæ, more exactly of the pura mathematica, i.e. as not transcending the boundaries of the domain of mathematics. Furthermore, since this view of the place of Mathesis vniversalis is common to all the thinkers to whom Descartes could plausibly refer in [D],22 the conclusion rolls out, once again, that the ordo & mensura nulli speciali materiæ addicta in Descartes‫[ ތ‬C] is coextensive with the ordo vel mensura in [B]. Finally, still further evidence in support of this conclusion is provided by the following passage from Descartes’ letter to Beeckman of March 1619:23

21 The passage just referred to reads as follows: “MATHEMATICA græcis ȝȐșȘȝĮ sive ȝȐșȘıȚȢ, omnem significat disciplinam sive doctrinam. Vsus tamen scientiæ de qvantitate accommodavit ...” (ibid.). As a matter of fact, not Mathesis vniversalis but instead Vniversalis mathematica occurs verbatim in the following passage. However, Vniversalis mathematica is equated with a scientific aspect of Prima mathesis in the same passage and again a few pages later (ibid., 20–21), and “Prima mathesis” is explicitly stipulated as synonymous with “Mathesis Vniversalis” in the above-quoted Apologia pro Archimede, 23. 22 See fn. 19. 23 For a detailed account of Descartes‫ ތ‬relationship with Beeckman and of the context of Descartes‫ ތ‬proposal of the scientia penitus nova, see Sasaki, DescartesҲs Mathematical Thought, ch. 3, § 1.

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[V]t tibi nude aperiam quid moliar, non Lullij Artem brevem, sed scientiam penitus novam tradere cupio, quâ generaliter solvi possint quæstiones omnes, quæ in quolibet genere quantitatis, tam continuæ quàm discretæ, possunt proponi (AT X, 156–57).

Descartes‫ ތ‬description of the envisaged scientia penitus nova squares perfectly with Van Roomen‫ތ‬s description of Mathesis vniversalis in the above-quoted Apologia pro Archimede, 23. Furthermore, Descartes is clear here that the universality claimed by that scientia is limited to the domain of quantities, “tam continuæ quàm discretæ.” Once it is established that Descartes refers with Mathesis vniversalis to a discipline within the domain of mathematics, it is not difficult to show that Descartes‫ ތ‬project of Mathesis vniversalis in Reg. IV is most probably coincident with his project of general algebra. To begin with, it will be remembered that what was identified as the truly pivotal element of Descartes‫ ތ‬groundbreaking conception of general algebra as mathematical analysis was treating the dependencies of general quantities involved in a given problem in terms of their involvement in series of continuous proportionals, and expressing the proportional relations thus articulated by means of algebraic equations.24 It is in view of this envisaged conception (although Descartes did not possess the working version of a perfectly general algebra by then)25 that towards the end of Reg. XIV Descartes offers a classification of habitudines sive proportiones between quantities:26 24 In Sasaki‫ތ‬s pertinent phrase, “Descartes ... attempts to translate the language of the theory of proportion into that of the theory of equations” (DescartesҲs Mathematical Thought, 196). 25 Cf. ch. 3, fn. 77. 26 That the context of the following classification is indeed an envisaged general algebra should be clear in particular from these passages in the previous portions of Reg. XIV: “[P]ræcipuam partem humanæ industriæ non in allio collocari, quàm in proportionibus ... eò reducendis, vt æqualitas inter quæsitum, & aliquid quod sit cognitum, clarè videatur. Notandum est ..., nihil ad istam æqualitatem reduci posse, nisi quod recipit majus & minus, atque illud omne per magnitudinis vocabulum comprehendi: adeò vt ... hîc tantùm dienceps circa magnitudines in genere intelligamus nos versari” (AT X, 440; my emphasis). “[Q]uæstiones perfectè determinatas vix vllam difficultatem continere, præter illam quæ consistit in proportionibus in æqualitates evolvendis; atque illud omne, in quo præcisè talis difficultas invenitur, facilè posse & debere ab omni alio subjecto separari, ac deinde transferri ad extensionem & figuras, de quibus solis idcirco deinceps ... tractabimus” (ibid., 441; my emphasis). “[V]elimus duntaxat proportiones quantumcumque involutas eò reducere, vt illud, quod est ignotum, æquale cuidam cognito reperiatur ...; ac proinde sufficit ad nostrum institutum, si in ipsâ

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Sunt ... duo duntaxat genera rerum, quæ inter se conferuntur, multitudines & magnitudines .... ... jam ... sciendum est, omnes habitudines, quæ inter entia ejusdem generis esse possunt, ad duo capita esse referendas: nempe ad ordinem, vel ad mensuram (AT X, 450–51; my emphasis).

That is to say, the two capita of habitudines sive proportiones in the context of establishing the conditions of a general algebraic treatment of quantities—ordo & mensura—square perfectly with what Descartes has determined as the proper subject of Mathesis vniversalis in Reg. IV-B. Moreover, the point is confirmed with a significant passage in DM 2, AT VI, 20: [V]oyant qu‫ތ‬encore que [les] obiets [des toutes ces sciences particulieres, qu‫ތ‬on nomme communement Mathematiques,] soient differens, elles [sciences particulieres] ne laissent pas de s‫ތ‬accorder toutes, en ce qu‫ތ‬elles n‫ތ‬y considerent autre chose que les diuers rappors ou proportions qui s‫ތ‬y trouuent, ie pensay qu‫ތ‬il valoit mieux que i‫ތ‬examinasse seulement ces proportions en general, & sans les supposer que dans les suiets qui seruiroient a m‫ތ‬en rendre la connoissance plus aysée .... ... Puis, ayant pris garde que, pour les connoistre, i‫ތ‬aurois quelquefois besoin de les considerer chascune en particulier, & quelquefois seulement de les retenir, ou de les comprendre plusieurs ensemble, ie pensay que, pour les considerer mieux en particulier, ie les deuois supposer en des lignes ...; mais ... pour les retenir, ou les comprendre plusieurs ensemble, il falloit que ie les expliquasse par quelques chiffres, les plus courts qu‫ތ‬il seroit possible ....

It is beyond doubt that what Descartes speaks about in the latter half of this passage is his general algebra of line segments, and the first half is strikingly similar, both as to theme and structure, with the crucial Reg. IV, AT X, 377–78. This allows us to conclude with a fair degree of certainty that the “rappors ou proportions” in AT VI, 20 are co-extensive with the ordo & mensura in AT X, 378; and that the context in which they are treated is that of (envisaged or possessed) general algebra. 4.1.2.2 Alternative Readings Let us turn now to generic reading (II). Perhaps the best-known version of it is due to John Schuster, “Descartes‫ ތ‬Mathesis Universalis,” who himself rightly traces the fundamental insight behind the reading back to extensione illa omnia consideremus, quæ ad proportionum differentias exponendas posunt juvare ...” (ibid., 447; my emphasis).

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Jacob Klein, Greek Mathematical Thought;27 my discussion will be concerned mostly with Schuster‫ތ‬s proposal.28 Schuster endorses one principal point which draws his reading close to (I) and in direct opposition to (III): namely that the scope of Mathesis vniversalis does not extend beyond the proper field of the mathematicæ.29 On the other hand, what sharply distinguishes his proposal from (I) and renders (I) and (II) incompatible is Schuster‫ތ‬s principal contention that the proper objects of Mathesis vniversalis as envisaged by Descartes are ultimately any and all of the corporeal images that count as extensional mensuræ with respect to any dimensio defined by Descartes as modus & ratio in respect of which any extended object “consideratur esse mensurabile” (Reg. XIV, AT X, 447) by extended objects.30 As a consequence, what eventually falls within the scope of Mathesis vniversalis proper, according to Schuster, is the agenda not only of arithmetic and geometry but also of what he calls physico-mathematics, i.e. those branches of more or less specialized physical disciplines which allow a mathematical treatment, in so far as they so allow.31 Thus instead of identifying the agenda of Mathesis vniversalis in Descartes with the most thoroughly general common features involved in the particular agendas of all of the mathematicæ (which is what reading (I) recommends), Schuster proposes identifying it with the union of the particular agendas of all the mathematicæ,32 in so far as the dimensional mensuræ the particular mathematical disciplines deal with are treated analytically (presumably

27 Cf. in particular Klein, ibid., 197–203. Klein‫ތ‬s book first appeared in the original German in 1934 and 1936. Schuster presented his view for the first time in his dissertation “Descartes and the Scientific Revolution, 1618–1634: An Interpretation” (Princeton University, 1977). 28 It is fair to say that the reaction of the leading Cartesian scholars to Schuster‫ތ‬s proposal has been mostly negative (see e.g. Sasaki, DescartesҲs Mathematical Thought, 201–202; Doyle, “How (not) to read Descartes’ Regulae,” 9–11). Yet it is well to discuss it to clarify some issues and avoid certain misunderstandings. 29 Cf. Schuster, “Descartes‫ ތ‬Mathesis Universalis,” 43. 30 Cf. Schuster, ibid., 65–72. 31 Cf. ibid., 65–66; 71; 79. The examples of physico-mathematics adduced by Schuster include music theory, hydrostatics and mechanics; cf. ibid., 48; 66. 32 Sasaki, DescartesҲs Mathematical Thought, 201–202 rightly observes that Schuster‫ތ‬s reading eventually tends to amount to a (con)fusion of what Van Roomen and others kept strictly apart, namely of universalis mathesis and universa mathesis (cf. ch. 3, fn. 36). However, Schuster himself does not invoke this distinction either when describing his position or when arguing for it.

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in the sense established in ch. 3).33 While advancing his overtly developmental account of Descartes‫ތ‬ thought, Schuster presents this conception as resulting from Descartes‫ތ‬ pursuit of “legitimizing” his early project of Mathesis vniversalis by “lend[ing] ontological certification to the objects of universal mathematics and display[ing] precisely in what the certainty of its operation consists” (ibid., 59) in response to the sceptical challenges he confronted as a part of the “intellectual turmoil” during his stay in Paris in 1625–28.34 The latter branch of this pursuit—establishing the certainty of mathematical operations—has issued, according to Schuster, in a conception of basic algebraic operations as represented by “a ‘logistic‫ ތ‬of ‘extensionsymbols‫”ތ‬, i.e. by certain concrete corporeal images (straight lines and rectangles in the Regulæ, lines only in the Discours and in La Geometrie) obtained through an abstractive operation from particular material objects and manipulated, throughout in perspicuous ways, in the imagination.35 Furthermore, the former branch—lending ontological certification to the objects of Mathesis vniversalis—has issued, Schuster holds, in Descartes‫ތ‬ advice (which draws upon the mechanical physiologico-psychological theory of cognition in Reg. XII)36 to transfer illa, quæ de magnitudinibus in genere dici intelligemus, ad illam magnitudinis speciem, quæ omnium facillimè & distinctissimè in imaginatione nostrâ pingetur: hanc verò esse extensionem realem corporis abstractam ab omni alio, quàm quod sit figurata ... (Reg. XIV, AT X, 441).

I do not wish to dispute that what Schuster underpins are indeed the principal tenets of Descartes‫ ތ‬efforts (presented most explicitly in Reg. XII–XIV) to set forth the epistemological conditions of doing mathematics, 33 Here is what I take to be Schuster‫ތ‬s most straightforward claims to this effect: “[U]niversal mathematics is not to be identified with symbolic algebra, with Descartes‫ ތ‬later views on analytic geometry, with mechanistic natural philosophy, nor even with a properly mathematical physics tout court. It is a general mathematical discipline, providing machinery for the analysis of all problems occurring in properly mathematical fields… The corporeal images are the very objects of universal mathematics—geometrical dimensions and extensional measures rendered in terms of extension-symbols” (Schuster, “Descartes‫ ތ‬Mathesis Universalis,” 71–72). 34 Cf. ibid., 55–57. 35 Cf. ibid., 64–65. 36 Cf. Reg. XII, AT X, 412–17. Schuster refers to this theory as “the opticspsychology-physiology (or o-p-p) nexus”—see idem, “Descartes‫ ތ‬Mathesis Universalis,” 62.

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and both the epistemological and the ontological conditions of treating material reality mathematically; indeed, Schuster‫ތ‬s way of associating what he calls “the logistic of extension-symbols” with the project of establishing the epistemological credentials of pure mathematics exhibits some affinity with my own account of the rôle of imagination in Descartes‫ ތ‬conception of mathematics presented towards the end of ch. 3. As far as I can see, however, Schuster has failed to substantiate with sound reasoning his proposal to take those tenets as realizations of Descartes‫ތ‬ early project of Mathesis vniversalis presented in Reg. IV-B. There is a prima facie strong reason not to take those tenets as Schuster recommends: it will be remembered that in IV-B, Mathesis vniversalis is characterized, presumably in an essential manner, as a scientia, “quæ id omne explicet, quod circa ordinem & mensuram, nulli speciali materiæ addictam quæri potest” (Reg. IV, AT X, 378; my emphasis). Schuster runs straight against the aforementioned characterization as his proposal amounts to charging the discipline in question with as many materiæ speciales as there are dimensional mensuræ capable of proper mathematical treatment. In the absence of any direct textual evidence to the effect that Descartes has dropped this essential characterization in a later period of composing the Regulæ and resolved to use the term “Mathesis vniversalis” to refer instead to what Van Roomen and others would call Vniversa Mathesis, a considerably strong reason—to be derived most likely from some aspect of Descartes‫ ތ‬actual practice in the field of the mathematicæ discernible in later portions of the Regulæ—is apparently needed to beat the textual presumption just established. Schuster tries to meet this requirement by suggesting an essential connection between, on the one hand, Descartes‫ތ‬ above-highlighted treatment of certain deliverances of corporeal imagination as indispensable attendants which aid the finite pure understanding—the faculty properly responsible, among other things, for Mathesis vniversalis—to avoid mistaken ontological extrapolations due to confusion of omissio and exclusio,37 and between, on the other hand, certain deliverances of corporeal imagination—namely those referring to extensio figurata præcisè sumpta—as (indispensable?) attendants to intellective operations concerned with magnitudines in genere.38 However, there is no such essential connection as far as I can see: it is one thing to claim that in order that pure understanding not go astray while processing operations with general quantities, the corresponding ideas are to be 37

As already indicated, Schuster‫ތ‬s treatment of these difficult issues is not quite the same as mine; yet I believe Schuster could endorse the overall description of Descartes‫ ތ‬point just presented. 38 Cf. the above-quoted Reg. XIV, AT X, 441.

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accompanied by certain corporeal images—straight lines and perhaps rectangles—derived from extensio figurata præcisè sumpta; and it is quite another thing to claim that in order that one find out easily and safely “quidquid in [mathematicis] inest veritatis” (Reg. XIV, AT X, 447), one should avail oneself of certain corporeal images—extensional mensuræ of dimensiones—derived, once again, from extensio figurata præcisè sumpta. The latter claim might hold even if the former did not, and vice versâ.39 I conclude that Schuster has failed to provide sufficient support for a radical extension of the reference of the term “Mathesis vniversalis” along the suggested lines, and that his reading is thus to be rejected. Let us now turn to generic reading (III) and see whether it fares better. The core thesis of this reading is that the programme of Mathesis vniversalis in IV-B is co-extensive with the programme not just of general algebra but of a perfectly universal method of discovery. Although several respected scholars moved hither or thither in this general direction during the twentieth century,40 I will focus on by far the most influential and clear-cut statements issued by Jean-Luc Marion in his Sur l’ontologie grise and Frederick Van De Pitte in his “Descartes’ Mathesis Universalis.”41 Marion holds that while the discipline to which Descartes refers with Mathesis vniversalis deals with what is common to all the mathématiques, its subject matter is by no means limited to the mathématiques. This is because, in Marion‫ތ‬s view, Mathesis vniversalis in Descartes is concerned with “la mathématicité non-mathématique des mathématiques” (Sur l’ontologie grise, 62; 64), and as such it is not a discipline within the mathématiques (as it is, arguably, in Van Roomen, for example) but that it rather amounts to “[l]‫ތ‬unique science, productrice d‫ތ‬universelle certitude, équipollente en d‫ތ‬infinis objets indifférents” which may rightly be called 39

Klein offers a more straightforward strategy, which, while even less satisfactory than Schuster‫ތ‬s, renders the chief problem with the reading in question much more sharp-edged. He claims that “Descartes‫ ތ‬great idea … consists of identifying … the ‘general‫ ތ‬object of [his] mathesis universalis—which can be represented and conceived only symbolically—with the ‘substance‫ ތ‬of the world, with corporeality as ‘extensio‫( ”ތ‬Klein, Greek Mathematical Thought, 197; Klein’s emphases). 40 See the survey in Van De Pitte, “Descartes’s Mathesis Universalis,” 61; and 74– 75, n. 1–2. Van De Pitte mentions Pierre Boutroux, LҲimagination et les mathématiques selon Descartes (Paris: Félix Alcan, 1900); Ernst Cassirer, Descartes: Lehre, Persönlichkeit, Wirkung (Stockholm: Bermann-Fischer, 1939); Beck, Method of Descartes; and Albert Balz, Descartes and the Modern Mind (Hamden, CT: Archon Books, 1967). 41 The first edition of Marion‫ތ‬s Sur l’ontologie grise appeared in 1975. Van De Pitte‫ތ‬s article originally appeared in 1979 in Archiv für Geschichte der Philosophie 61: 154–74.

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“méthode générale” (ibid., 62; Marion’s emphasis). Marion thus suggests that in Descartes‫ ތ‬hands—unlike in Aristotle, Proclus or Iamblichus— Mathesis vniversalis is to be divorced from what Marion calls “mathématique universelle” and what arguably coincides with prima Mathesis as a division of pura Mathematica in Van Roomen‫ތ‬s scheme presented in the above-quoted Universæ mathesis idea, 14–15; as Marion puts it, “[l]a mathématique universelle n‫ތ‬était universelle qu‫ތ‬en restant mathématique; la Mathesis universalis n‫ތ‬est universelle qu‫ތ‬en ce qu‫ތ‬elle n‫ތ‬est plus seulement mathématique” (Sur l’ontologie grise, 64). Ordo & mensura, which pertained solely to the realm of the mathematicæ according to both readings (I) and (II), are thus subject to further abstraction in degrees, in the course of which, according to Marion, the originally quantitative categories of ordo & mensura42 surpass the realm of the mathematicæ and are subject to a kind of analogical treatment whose limiting case comes close to treating ordo & mensura as the most general determinants of “l’être étant en tant qu’Etre” (Marion, ibid., 66). It is fair to say that while Marion appears to submit this interpretation as a speculative possibility which is integrated into a wider framework of his deconstructive hermeneutics, Van De Pitte has done a good job both of clarifying the connection of Mathesis vniversalis, interpreted along Marion’s lines, to universal method (via treating ordo & mensura as providing “the common basis for validity in all the sciences” [Van De Pitte, “Descartes’s Mathesis Universalis,” 65]), and even more significantly, of submitting substantial arguments that come to terms with the immediate historical context of Descartes’s notion of Mathesis vniversalis. According to Van De Pitte, Descartes’ notion of Mathesis vniversalis as transcending the limits of the mathematicæ is prompted by Descartes’ having taken seriously two closely related ideas stemming from Van Roomen’s treatment of prima mathesis: firstly, the idea that Mathesis vniversalis is to the subjects and the principles of all the other scientiæ Mathematicæ as Prima Philosophia is to the subjects and the principles of all the other scientiæ, that is to say, it comprehends their subjects and demonstrates their principles;43 and secondly, the idea that because

42

Cf. Reg. XIV, AT X, 440; 451. Cf. Van Roomen, Apologia pro Archimede, 23: “Inscribemus autem [Mathesin Universalem] nomine, Prima Mathematica, seu Prima MatheseȦs, ad similitudinem Primæ Philosophiæ. Nam sicut ea dicitur, Prima quia subiecta omnium reliquarum sub se comprehendit scientiarum, quinimò & reliquarum demonstrat principia si demonstratione egeant: Ita & hæc Prima Mathematica,

43

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Mathesis vniversalis or Prima Mathesis is characterized in this way, it is self-sufficient or (as Van De Pitte puts it) “self-contained” (“Descartes’s Mathesis Universalis,” 64) in the sense that all its principles belong to it alone.44 Van De Pitte credits Descartes with the insight that these two ideas are jointly incompatible with conceiving of Prima Mathesis as a division within the mathematicæ: [I]f prima mathesis is understood to be the most basic principles concerning quantity, it is still necessary to consider those principles which alone account for how it is that questions involving quantity arise in the first place. ... [T]hen [according to Descartes] the tradition is faced with a dilemma: either prima mathesis is a division of mathematics—in which case it is not self-contained ..., since it must relate to prior principles as well; or it is self-contained ..., and thus outside mathematics, since it is logically prior to all questions of quantification (“Descartes’s Mathesis Universalis,” 64; Van De Pitte’s emphases).

As a result, Descartes is prepared, Van De Pitte submits, to draw a fundamental distinction45 between order and measure as they are related to quantity; and, on the other hand, the more basic principles of order and measure, which are logically prior and fall outside mathematics. The former is mathesis, the latter [mathesis universalis] (ibid., 68; Van De Pitte’s italics).

According to Van De Pitte, one way to resolve the alleged dilemma which Descartes was entertaining is to limit the scope of the requirement of self-containment of Prima Mathesis/Mathesis vniversalis strictly to the domain of the mathematicæ. This is certainly the view of Van Roomen and most of his contemporaries in dealing with Mathesis vniversalis.46 Now the chief problem with Van De Pitte’s reasoning is that, as far as I can see, he fails to establish on independent grounds that Descartes indeed versatur circa subiecta omnium scientiarum Mathematicarum, & purarum & mixtarum. Probat quoque principia reliquarum scientiarum.” 44 Cf. Van Roomen, Universæ mathesis idea, 20–21: “Prima Mathesis est quæ versatur circa quantitate absolutè sumptam. Objectum ejus est Quantitas absolute sumpta. Finis verò, affectiones quantitatibus omnibus communes exhibere. Principia habet tantum propria. Locum in Mathesi obtinet primum.” 45 Instead of “Mathesis vniversalis”, Van De Pitte has “MU” (i.e. his shorthand for mathesis universalis) in the following quotation. 46 Cf. Crapulli, Mathesis universalis, ch. 7; Sasaki, DescartesҲs Mathematical Thought, 356.

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abandons this standard view. To be sure, much of what is said concerning Mathesis vniversalis in IV-B can be interpreted quite smoothly along the lines of the reading under scrutiny once one opts for this reading. However, as I showed, whereas there are several episodes in IV-B which upon closer scrutiny strongly point towards reading (I) to the exclusion of (III), no single piece of direct and independent textual evidence that would unambiguously point towards reading (III) to the exclusion of (I) is ultimately to be found either in IV-B or anywhere else in Descartes. This is why reading (III) is to be rejected in favour of reading (I) once the textual evidence is taken into account together with the immediate historical context of Descartes’s treatment.47 To sum up, a good case can be made, on both textual and historical grounds in favour of reading (I) according to which the project of Mathesis vniversalis in Descartes is co-extensive with the project of general algebra as a fundamental discipline in the field mathematics. Unfortunately, whilst being (I believe) the most plausible of the available interpretations, such a reading implies it is quite unpromising to focus on Descartes‫ ތ‬project of Mathesis vniversalis if one wishes to unfold and describe the constitution of a universal method envisaged by Descartes (and, in the long run, to 47 Van De Pitte believes he has established the required evidence to bear the present burden of proof. He points out that at the end of passage [C] of the abovequoted Reg. IV, AT X, 377–78, the ampersand between scientiæ and Mathematicæ was omitted from the AT edition despite the fact that it occurs in both the A- and H-version of the Regulæ. He suggests that the ampersand be replaced in the passage and claims that the crucial phrase then says not that “‘mathesis universalis’ ... would contain everything in virtue of which the other sciences are called parts of mathematics” (“Descartes’s Mathesis Universalis,” 69; Van De Pitte’s italics) but rather that “‘mathesis universalis’ ... would contain everything in virtue of which other sciences and mathematicæ are called parts” (ibid.). If this is the real meaning of the passage, then Van De Pitte indeed succeeds in providing strong evidence in favour of reading (III): as long as the mathematicæ—all of them—count as parts of Mathesis vniversalis, the latter cannot count as a division within the mathematicæ. I agree with Van De Pitte that there is no reason to omit the ampersand. However, I believe that the amended (in fact the restored) Latin version (“... quoniam in [Mathesi vniversali] continetur illud omne, propter quod aliæ scientiæ & Mathematicæ partes appellantur”) allows of still another rendering which fits perfectly with reading (I) and uneasily with reading (III), namely “‘mathesis universalis’ ... would contain everything in virtue of which the other sciences are also called parts of the mathematicæ.” I do not mean to argue that the rendering I propose is necessarily more adequate than Van De Pitte’s. It is enough for me to hold that it is equally as good as Van De Pitte’s, since this suffices to neutralize the only textual device submitted by Van De Pitte to overcome the presumption for reading (I) established above.

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determine the meaning of the a priori–a posteriori pair in Descartes). The reason is not so much that the project has been shown to coincide with the project of general algebra—the latter was determined as a privileged paradigmatic sample of the envisaged universal method in operation, and as such can (and will) be helpful in the task of explicating that method; the reason is rather that it is passages in which Descartes treats his general algebra that shed some light on the passages in which Mathesis vniversalis is taken up, and not vice versâ. Let us therefore put the grand topic of Mathesis vniversalis to one side and have a look elsewhere in our pursuit of what Descartes‫ ތ‬universal method is like and how its constitution might clarify his use of the a priori–a posteriori pair.

4.2 A Talk of the Method: the Discours and the Essais Descartes famously claims in DM 2 that his pursuit of a method which would comprise the advantages of logic, geometrical analysis and algebra, while avoiding their defects, has resulted in no more than four precepts whose observation should suffice—so he implies—to attain all that he is after in his all-embracing task of establishing scientiæ: Le premier estoit de ne receuoir iamais aucune chose pour vraye, que ie ne la connusse euidemment estre telle: c‫ތ‬est a dire, d‫ތ‬euiter soigneusement la Precipitation, & la Preuention; & de ne comprendre rien de plus en mes iugemens, que ce qui se presenteroit si clairement & si distinctement a mon esprit, que ie n‫ތ‬eusse aucune occasion de le mettre en doute. Le second, de diuiser chascune des difficultez que i‫ތ‬examinerois, en autant de parcelles qu‫ތ‬il se pourroit, & qu‫ތ‬il seroit requis pour les mieux resoudre. Le troisiesme, de conduire par ordre mes pensées, en commençant par les obiets les plus simples & les plus aysez a connoistre, pour monter peu a peu, comme par degrez, iusques a la connoissance des plus composez; et supposant mesme de l‫ތ‬ordre entre ceux qui ne se precedent point naturellement les vns les autres. Et le dernier, de faire partout des denombremens si entiers, & des reueuës si generales, que ie fusse assuré de ne rien omettre (AT VI, 18–19).

However, this but too scant account has commonly been found embarrassing by generations of readers, both charitable and hostile. As they stand, the four precepts have often been considered banal and useless: whilst the advice contained in the precepts sounds reasonable, one can hardly resist the feeling that at least some hint at, or even better some set of rules or some manual as to how the advice is to be obeyed is needed;

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and this is—so the complaint usually goes—exactly what Descartes fails to provide.48 This criticism is certainly justified—to a degree. To begin, Descartes himself admits in a letter to Mersenne that the precepts given in DM 2 do not amount to a full exposition of his method in a certain important sense:49 [I]e ne mets pas Traité de la Methode, mais Discours de la Methode, ce qui est le mesme que Preface ou Aduis touchant la Methode, pour monstrer que ie n‫ތ‬ay pas dessein de l‫ތ‬enseigner, mais seulement d‫ތ‬en parler. Car comme on peut voir de ce que i‫ތ‬en dis, elle consiste plus en Pratique qu‫ތ‬en Theorie, & ie nomme les Traitez suiuans des Essais de cette Methode, pource que ie pretens que les choses qu‫ތ‬ils contiennent n‫ތ‬ont pu estre trouuées sans elle, & qu‫ތ‬on peut connoistre par eux ce qu‫ތ‬elle vaut ... (AT I, 349; Descartes’ italics).

That is to say, Descartes claims that he indeed did not intend, in the Discours, to teach his method in the sense of providing general step-by48

Leibniz expresses the complaint in a memorable passage: “[P]arum abest tu dicam [Cartesianæ Methodi Regulæ quatuor] similes præcepto Chemici nescio cujus: Sume quod debes et operare ut debes, et habebis quod optas. Nihil admitte nisi evidenter verum (seu nisi quod debes admittere), divide rem in partes quot requiruntur (id est quot facere debes), procede ordine (quo debes), enumera perfecte (seu quæ debes), prorsus ut quidam inter præcepta ponunt, Bonum esse appetendum, Malum fugiendum; recte profecto, sed indicia boni malique desiderantur” (Die philosophischen Schriften, ed. Carl Gerhardt, vol. 4 [Berlin: Weidmannsche Buchhandlung, 1880], 329). Cf. also Leibniz’s charge to a similar effect in a letter to Galloys: “Ceux qui nous ont donne des methodes, donnent sans doute des beaux preceptes, mais non pas le moyen de les observer. Il faut, disentils, comprendre toute chose clairement et distinctement, il faut proceder des choses simples aux composées; il faut diviser nos pensées etc. Mais cela ne sert pas beaucoup, si on ne nous dit rien davantage. Car lorsque 1a division de nos pensees n‫ތ‬est pas bien faite, elle brouille plus qu‫ތ‬elle n‫ތ‬eclaire. Il faut qu‫ތ‬un écuier tranchant sçache les jointures, sans cela il dechirera les viandes au lieu de les couper” (Die philosophischen Schriften, ed. Carl Gerhardt, vol. 7 [Berlin: Weidmannsche Buchhandlung, 1890, 21). 49 Descartes makes a similar point to Vatier at AT I, 559 (quoted below). Once again, Leibniz estimates the situation correctly: “Cartesius ... nec publicavit methodum suam, sed tantum de ea scribere ejusque specimina dare voluit, ut ipse observat. Itaque valde falluntur, qui his quæ edidit, nimis contenti sunt, methodumque ejus se habere arbitrantur” (Die Philosophischen Schriften, 4:310). While Leibniz had no access to Descartes‫ ތ‬personal correspondence with Mersenne, he was in a better position to raise his suspicion than many of his contemporaries as he owned one of the rare copies of Descartes‫ ތ‬Regulæ.

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step manuals for carrying out the requirements contained in the four precepts. Rather, those who wish to learn something more about his method proper he refers to the three Essais which follow the Discours, since (as he puts it) his method consists “plus en Pratique qu‫ތ‬en Theorie.” One would hope for a moment that what Descartes suggests is that the sought-after manuals are to be gleaned from the ways in which he proceeds in addressing the particular questions or problems in the Essais (of which the most remarkable are, of course, the search after the anaclastic line in La Dioptrique, an explanation of the rainbow in Les Meteores, and the solution to the Pappus problem in La Geometrie). Unfortunately, such an expectation is fulfilled only to a very limited degree. While we saw that there are several important hints at how exactly one is to proceed when solving a given problem in La Geometrie, what Descartes tells Vatier on this score concerning La Dioptrique and Les Meteores seems to be a tolerably adequate description of the situation:50 [M]on dessein n‫ތ‬a point esté d‫ތ‬enseigner toute ma Methode dans le discours où ie la propose, mais seulement d‫ތ‬en dire assez pour faire iuger que les nouuelles opinions, qui se verroient dans la Dioptrique & dans les Meteores, n‫ތ‬estoient point conceuës à la legere, & qu‫ތ‬elles valoient peut-estre la peine d‫ތ‬estre examinées. Ie nҲay pû aussi monstrer lҲvsage de cette methode dans les trois traittez que iҲay donnez, à cause quҲelle prescrit vn ordre pour chercher les choses qui est assez different de celuy dont iҲay crû deuoir vser pour les expliquer. I‫ތ‬en ay toutesfois monstré quelque échantillon en décriuant l‫ތ‬arc-en-ciel .... Or ce qui m‫ތ‬a fait ioindre ces trois traittez au discours qui les precede, est que ie me suis persuadé qu‫ތ‬ils pouroient suffire, pour faire que ceux qui les auront soigneusement examinez, & conferez auec ce qui a esté cy-deuant écrit des mesmes matieres, iugent que ie me sers de quelqu‫ތ‬autre methode que le commun, & qu‫ތ‬elle n‫ތ‬est peutestre pas des plus mauuaises (AT I, 559–60; my emphasis).

50

Cf. also A *** [lҲabbé de Cerizy?], AT I, 370: “[T]out le dessein de ce que ie fais imprimer à cette fois [sc. Discours], n‫ތ‬est que de ... preparer le chemin, & fonder le guay. Ie propose à cet effet vne Methode generale, laquelle veritablement ie n‫ތ‬enseigne pas, mais ie tasche d‫ތ‬en donner des preuues par les trois traitez suiuans, que ie joins au discours où i‫ތ‬en parle, ayant pour le premier vn sujet meslé de Philosophie & de Mathematique; pour le second, vn tout pur de Philosophie; & pour le 3(e), vn tout pur de Mathematique ...; en sorte qu‫ތ‬il me semble par là donner occasion de iuger que i‫ތ‬vse d‫ތ‬vne methode par laquelle ie pourois expliquer aussi bien toute autre matiere .... Outre que pour montrer que cette methode s‫ތ‬étend à tout, i‫ތ‬ay inseré briévement quelque chose de Metaphysique, de Physique & de Medecine dans le premier discours.”

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Here Descartes speaks out unambiguously that for the most part—the only exception being his explanation of the rainbow—he does not follow the order of discovery prescribed by his method but rather the order of exposition in his first two Essais. That is to say, even these Essais are not the place to see the proclaimed method actually at work; what one finds instead are the results of the (professed) application of that method. Descartes hopes the quality of these results will persuade the reader to praise the method the employment of which professedly facilitated them. Yet as a matter of fact, he keeps the method itself hidden almost completely from the sight of the reader until Geom. I. Furthermore, while La Geometrie contains (as we saw) both a sort of description, general and detached as it is, of what one is supposed to do to observe the methodical precepts of DM 2 and a telling sample of how it is to be put to use, the reader is left with no indication as to how all this squares with the treatment of the rainbow earlier in Les Meteores, a treatment which is identified by Descartes himself as the only instance in the first two Essais of his method in operation. Furthermore, Descartes seems to be no better off in the other writings he published or in his extant correspondence. One possible response vis-à-vis this unsatisfactory situation is to adopt a sceptical stance and to claim that ultimately Descartes fundamentally failed in his efforts to correlate his undeniable groundbreaking results in natural philosophy, mathematics and metaphysics with an explicitly stated general method, and that he abandoned this project at about the time his thought grew to maturity. John Schuster, the author of arguably the bestentrenched account to date along these lines,51 thus claims that Descartes himself recognized the failure of his methodological program in about 1628 and that all the later proclamations of Descartes’ concerning method, including the entire Discours and the seminal AT VII, 155–57 of Resp.

51

See Schuster, Descartes-Agonistes, ch. 6–7. Another remarkable proponent of the generic sceptical stance at issue is Garber, “Descartes and Method in 1637.” However, Garber’s version, interesting as it is, can hardly stand up to both the textual and the systematic evidence and was justly criticized in Roger Florka, “Problems with the Garber-Dear Theory of the Disappearance of Descartes‫ތ‬s Method,” Philosophical Studies 117, no. 1-2 (2004): 131–41. I would add to Florka‫ތ‬s arguments that Garber strangely ignores the contents of La Geometrie, and even more importantly the significant manner in which the methodical aspects incorporated in La Geometrie are deployed in the highly authoritative passage in Resp. 2 on the relation between methodical proceeding in metaphysics and in geometry.

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2,52 are just cynical instances of “playing for the public a method card he now knew to be flawed or counterfeit” (Descartes-Agonistes, 301), and he corroborates this provocative claim with an elaborated “mythological” account of any methodological discourse whatsoever as a merely rhetorical device which in principle cannot regulate, let alone adequately reflect, actual scientific practice in any field of inquiry. Several respected scholars have suggested an alternative approach based upon taking seriously, and putting decisive weight upon, the abovequoted indication of Descartes that his envisaged method has to do with practice rather than with theory. Far from following step-by-step precepts or rules that would enable one to obey the precepts of DM 2, the only viable way of mastering the announced method Descartes wishes to recommend—so the suggestion goes—is to cultivate one‫ތ‬s cognitive faculties by practice, i.e. presumably by observing the procedures carried out by Descartes himself or by anyone already a master of the method in question, and by appropriate exercises.53 Fortunately, we need not get involved in discussions concerning the merits of these somewhat subversive proposals in view of the leading task of the present study, i.e. determination of the meaning(s) of the a priori–a posteriori pair in Descartes. This is because even if either of the proposals just outlined was to be endorsed, still one is bound, in order to carry out the leading task just mentioned, to attempt a detailed and tolerably informative account of how exactly one is to proceed methodically in various fields of cognition according to Descartes’ official declarations. For—as regards Schuster’s sceptical approach—given the arguably established fact that the meaning of the a priori and the a posteriori in Descartes essentially draws upon Descartes’ official account of his method in mathematics and beyond, one is bound to attempt an account just mentioned anyway, regardless of whether the methodological discourse (and consequently the talk in terms of the a priori–a posteriori pair too) is denounced as mere rhetoric or not, cynical or bona fide.54 Again—as 52 I conjecture it is this passage Schuster has in mind when he mentions, in the present context, “[Descartes’] comments on the argumentative tactics in his Meditations” (Descartes-Agonistes, 302). 53 Cf. e.g. Gilson, René Descartes’ Discours, 207–208. Leslie Beck takes a similar course in the last paragraphs of his Method of Descartes—see ch. 18, §§ 2–3. 54 For similar reasons, I feel entitled, in view of the chief task of the present study, not to confront Schuster’s fundamental challenge involved in his complex position just sketched, namely his claim that it is impossible in principle to articulate an adequate, more or less general prescriptive account of how one is to proceed methodically in any given field of inquiry. Incidentally, I do not agree with

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regards the latter proposal—whilst cultivation of one‫ތ‬s mind by exercise and practical observation will indeed be recognized as an important component of Descartes‫ ތ‬view of the conditions under which his professed method can be adopted, it seems clear to me that the suggestion at issue will not work unless an independent and tolerably informative account is given of the operations, or more precisely of the determinate steps within the methodical treatment which one is actually supposed to observe in order to learn something about and practise the envisaged method. It is, of course, upon the Regulæ that one should focus if one wishes to steer clear of the approaches just outlined, both of which, each in its own way, ultimately dooms Descartes‫ ތ‬methodological project to disaster. To be sure, even in my treatment no full-blooded step-by-step manual is forthcoming. Yet I believe quite a lot can be made of the Regulæ as regards the task of specifying how exactly one is to proceed in employing Descartes‫ ތ‬universal method in various fields of cognition, especially if the Regulæ are combined—in view of the above-established paradigmatic rôles of mathematics—with La Geometrie and with certain hints and doctrinal claims from other districts of Descartes‫ ތ‬large and complex field of philosophical and scientific thought. The time has now come to engage in this task.

4.3 A Reconstruction of the Universal Method Let me begin with establishing correlations between the four precepts from DM 2 and some precepts of the Regulæ. It has been almost a commonplace in Descartes scholarship that the precepts in DM 2 incorporate essential portions of what Descartes recommends in his

Schuster’s claim that if his “mythological” diagnosis is right, the correct moral to be drawn is to abandon any further research concerning Descartes’s project of universal method and its relation to his version of general algebra (cf. in particular Schuster, Descartes-Agonistes, 289, fn. 40); for both the project and its connection to algebra are expressly there in Descartes’ writings, and they surely have some impact upon both Descartes’ metaphysical and scientific thought even if Schuster’s diagnosis is right all throughout. On the other hand, I fully agree with Schuster to be wary, in one’s interpretative attempts concerning the nature and scope of Descartes’ envisaged universal method, of drawing false analogies based upon illegitimate equivocations on terms like absolutum, respectivum, æquale etc. (cf. ibid., in particular sec. 6.7); and I take Schuster’s systematic account of professed “textual effects” of a method discourse (cf. ibid., secs. 6.6–7) as invaluable caveat in anyone’s attempt to reconstruct Descartes’ achievements in any particular domain (including mathematics) as instances of application of a universal method.

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Regulæ to lead one‫ތ‬s ingenium successfully to scientiæ, and I endorse the almost commonly held view that the doctrine of method remains essentially the same in both works.55 Indeed, it is possible to identify several precepts of the Regulæ which, once collocated with the text of DM 2 in a suitable way, virtually exhaust the content of the four precepts in the Discourse.56 The first precept of DM 2 corresponds to the precepts of Reg. II, AT X, 362 and Reg. III, AT X, 366: Circa illa tantùm objecta oportet versari, ad quorum certam & indubitatam cognitionem nostra ingenia videntur sufficere. ... Circa objecta proposita, non quid alij senserint, vel quid ipsi suspicemur, sed quid clarè & evidenter possimus intueri, vel certò deducere, quærendum est; non aliter enim scientia acquiritur.

The second precept of DM 2 is fully conveyed (with some overlap) with the precept of Reg. XIII, AT X, 430:57 55

Cf. e.g. Alexander Gibson, The Philosophy of Descartes (London: Methuen & Company, 1932), 158; Leon Roth, DescartesҲ Discourse on Method (Oxford: Clarendon Press, 1937), 64–65; Gilson, René Descartes’ Discours, 196; Beck, Method of Descartes, 149–52; Peter Schouls, The Imposition of Method: A Study of Descartes and Locke (Oxford: Clarendon Press, 1980), 57; Clarke, Descartes’ Philosophy of Science, 181; Daniel Garber, Descartes’ Metaphysical Physics (Chicago: University of Chicago Press, 1992), 49; Stephen Gaukroger, Descartes: An Intellectual Biography (Oxford: Clarendon Press, 1995), 306; Patrick Brissey, “Descartes’ Discours as a Plan for a Universal Science,” Studia UBB. Philosophia 58, no. 3 (2013): 41; Schuster, Descartes-Agonistes, 249–52. The only two respected opponents to the parallelism at issue seem to have been Léon Brunschvicg—cf. idem, “Mathématiques et métaphysique chez Descartes,” Revue de métaphysique et de morale 34, no. 3 (1927): 277–324 and Gilbert Gadoffre—cf. idem, “Introduction,” in René Descartes’ Discours de la Méthode, ed. Gilbert Gadoffre (Manchester: Editions de l’Université de Manchester, 1941), v–xlv. 56 The close textual parallels between the precepts of the Regulæ and the precepts of the Discours (in a Latin translation included in AT VI) are presented in Beck, Method of Descartes, 150. I draw upon Beck‫ތ‬s exposition but differ from him in at least one respect in establishing the parallels. Schuster, Descartes-Agonistes, 252 and fn. 66 offers an account of the parallelisms between the Regulæ and DM 2 similar to mine though differing in several minor respects. 57 The precept of Reg. XIII is omitted in Beck‫ތ‬s exposition in Method of Descartes, 150. Beck takes instead the precept of Reg. V to cover the contents of the second and the third precepts of DM 2 taken together. However, only the following passage of the precept of Reg. V corresponds to the second precept of DM 2: “[H]anc [methodum] exactè servabimus, si propositiones involutas & obscuras ad

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Si quæstionem perfectè intelligamus, illa est ab omni superfluo conceptu abstrahenda, ad simplicissimam revocanda, & in quàm minimas partes cum enumeratione dividenda.

The third precept of DM 2 corresponds to a part of the precept of Reg. V (to be sure, the precept of Reg. V does not mention the important hint at the distinction between tracing the natural order and supposing some order “mesme ... entre ceux qui ne se precedent point naturellement les vns les autres” [DM 2, AT VI, 18–19]; but the distinction is deployed by Descartes more than once in the following rules):58 [Methodum] exactè servabimus, si propositiones involutas & obscuras ad simpliciores gradatim reducamus, & deinde ex omnium simplicissimarum intuitu ad aliarum omnium cognitionem per eosdem gradus ascendere tentemus (AT X, 379).

Finally, the fourth precept of DM 2 is virtually equivalent to the precept of Reg. VII, AT X, 387:59 Ad scientiæ complementum oportet omnia & singula, quæ ad institutum nostrum pertinent, continua & nullibi interrupto cogitationis motu perlustrare, atque illa sufficienti & ordinatâ enumeratione complecti.

These close textual parallels contribute a good deal of legitimacy to engaging in the intricacies of the Regulæ and their interrelations with other simpliciores gradatim reducamus ...” (AT X, 379). The operation of divisio which is the chief topic of the second precept of DM 2, is not mentioned. On the other hand, a section or two later in Reg. XIII, one reads “[a]dditur præterea, difficultatem esse ad simplicissimam reducendam, nempe juxta regulas quintam & sextam, & dividendam juxta septimam” (AT X, 432). This seems to indicate that Descartes indeed conceived of the precept of Reg. XIII as somehow involving the precepts of Reg. V and VII. 58 Cf. in particular Descartes’ invocation of ordo rerum enumerandarum excogitandus in Reg. VII, AT X, 391; also Reg. X, AT X, 403: “... hominum artificia ... quæ ordinem explicant vel supponunt.” And ibid., 404: “... ordinis, vel in ipsâ re existentis, vel subtiliter excogitati, constans observatio.” 59 The correspondence comes out more distinctly in view of the Latin translation Descartes most probably reviewed: “Ac postremum, ut tum in quærendis mediis, tum in difficultatum partibus percurrendis, tam perfectè singula enumerarem & ad omnia circumspicerem, ut nihil à me omitti essem certus” (AT VI, 550). Some interpretative subtleties regarding this Latin passage are discussed towards the end of ch. 2.

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Descartes texts in order to supplement with some detail his all too sketchy hints at a universal method in the Discours. This is the task for the rest of the present chapter. In addressing it, I will be concerned exclusively with the methodical régime of discovery, putting to one side the régime of approbation; and I shall consider no further the issues related to the first precept of DM 2 (and to Reg. II and III given the parallels just established) as enough has been said in the preceding chapters. I submit four distinct basic issues that are jointly at work in Descartes’ complex treatment of the envisaged universal method in the Regulæ. These are (1) determination of the modus operandi of the analytical part of any particular application of the method in the régime of discovery;60 (2) identification and explication of the basic operations involved in the determined general modus operandi which are probably hinted at in the second to fourth precepts of DM 2; (3) establishing the possibility of determining the nature and constitution of the envisaged method in a specific sort of innate structure; and (4) securing certain necessary conditions of universalization of the paradigmatic case of the algebraic employment of the envisaged method. While these moments (or their aspects)—especially (1), (2)—are usually run together in the Regulæ, I will be at pains to disentangle them and keep them apart as far as possible in order to pick out the real cruces of Descartes‫ ތ‬conception. Each of the above moments will now be discussed in turn.

4.3.1 The General Modus Operandi As regards explication of the operations involved in the analytical part of any particular (non-degenerative)61 application of the method, the best place to start is Reg. XIII, in which Descartes clearly indicates how the fundamental fabric of his method proper looks, in which rules this fabric is incorporated, and how other rules are related to them: Nos hîc prærequirimus quæstionem esse perfectè intellectam. ... Additur præterea, difficultatem esse ad simplicissimam reducendam, nempe juxta regulas quintam & sextam, & dividendam juxta septimam .... Atque hæc tria tantùm occurrunt circa alicujus propositionis terminos servanda ab intellectu puro, antequam ejus vltimam solutionem aggrediamur, si sequentium vndecim regularum vsu indigeat ... (AT X, 430–32; my emphasis). 60

I will omit the specification “in the régime of discovery” from now on. I will henceforth use the term “method” exclusively in this heuristic sense unless expressly indicated otherwise. 61 See ch. 2, fn. 121.

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Putting to one side, for a moment, the requirement of understanding a given quæstio perfectly, Descartes unambiguously identifies Reg. V, VI and VII as those in which the essentials of the whole method are involved, in the sense that observing them is strictly a sine quâ non for the methodical solution of any problem whatsoever (although other precepts might turn out indispensable after the given problem is treated according to Reg. V–VII).62 Thus it seems a good start to consider the precepts of these three professedly most important rules together: Tota methodus consistit in ordine & dispositione eorum ad quæ mentis acies est convertenda, vt aliquam veritatem inveniamus. Atque hanc exactè servabimus, si propositiones involutas & obscuras ad simpliciores gradatim reducamus, & deinde ex omnium simplicissimarum intuitu ad aliarum omnium cognitionem per eosdem gradus ascendere tentemus (Reg. V, AT X, 379). Ad res simplicissimas ab involutis distinguendas & ordine persequendas, oportet in vnaquâque rerum serie, in quâ aliquot veritates vnas ex alijs directè deduximus, observare quid sit maximè simplex, & quomodo ab hoc cætera omnia magis, vel minus, vel æqualiter removeantur (Reg. VI, AT X, 381). Ad scientiæ complementum oportet omnia & singula, quæ ad institutum nostrum pertinent, continua & nullibi interrupto cogitationis motu perlustrare, atque illa sufficienti & ordinatâ enumeratione complecti (Reg. VII, AT X, 387).

I take Descartes‫ ތ‬word and suppose that these three precepts, together with their explications, indeed embody an extremely general sketch of his envisaged universal method of discovery. The following subsections up to sec. 4.4 might well be characterized as attempts at an extensive and textually supported interpretation of the three cited rules. Let me turn first to what one can learn from them with regard to the general modus operandi of the entire envisaged method. 4.3.1.1 Ordo & Dispositio I submit that the precept of Reg. V is to be read literally, so that “ordo & dispositio eorum ad quæ mentis acies est convertenda” is in fact 62

Cf. Reg. VII, AT X, 392: “[P]aucisque easdem [sc. Reg. V, VI & VII] hîc explicavimus, quia nihil aliud fere in reliquo Tractatu habemus faciendum, vbi exhibebimus in particulari quæ hîc in genere complexi sumus.”

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intended to describe, however cursorily, precisely the proper modus operandi of the entire method of discovery which we are after at present. Ordo & dispositio is thus to be read, as far as Reg. V to VII is concerned, imprimis as a general characteristic of to what each distinctive operation that counts as a constituent part of Descartes‫ ތ‬envisaged method of discovery is supposed to contribute in order that a new (piece of) scientia is facilitated.63 It is clear in view of the above-established facts that ea ad quæ mentis acies est convertenda refer fundamentally to the objects of intuitus and of immediate deductiones. Consequently, the message of the precept of Reg. V, as well as of the brief explication of the same rule, is that the most general point of the method in the régime of discovery is to “order and dispose” those (objects of) intuitus and immediate deductiones that are already available (and approved in the régime of approbation) in such a way that a new (set of) cognition(s) counting as scientia(e) can be attained. The content of the sought-after method in the régime of discovery should therefore boil down to a manual, consisting of a set of precepts as to how such an orderly disposition is to be carried out in order to arrive at the desired result, i.e. at the solution to a given problem such that it amounts to a discovery of a new (piece of) scientia. The only clue in Reg. V as to how this is supposed to be effected is that the orderly disposition in question amounts to a stepwise reductio of “propositiones involutas & obscuras ad simpliciores” (AT X, 379), and to a subsequent return “ex omnium simplicissimarum intuitu ad aliarum omnium cognitionem per eosdem gradus” (ibid.). Furthermore, Descartes himself intimates it is not until Reg. VI that one learns anything specific as to how the announced procedures are to be put to work: Sed quia sæpe ordo, qui hîc desideratur, adeò obscurus est & intricatus, vt qualis sit non omnes possint agnoscere, vix possunt satis cavere ne aberrent, nisi diligenter observent quæ in sequenti propositione exponentur (ibid., 380).

63 Descartes continues immediately after having stated the precept: “In hoc vno totius humanæ industriæ summa continetur ...” (Reg. V, AT X, 379). Taken literally, this supports the proposed reading of the precept. Moreover, ordo and dispositio indeed occur together in the context of discussing each of the submitted three main operations of the method: see Reg. VII, AT X, 391 for heuristic enumeration; Reg. XIV, AT X, 452 for reduction; and Reg. XXI, AT X, 469 for relativization to an absolute.

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The precept of Reg. VI sounds rather unhelpful as it stands;64 but Descartes does much better in the exposition of the same rule, thus doing some justice to the claim that the precept “præcipuum tamen continet artis secretum, nec vlla vtilior est in toto hoc Tractatu” (Reg. VI, AT X, 381). He starts with an important specification of towards what the envisaged dispositiones are employed: [H]æc propositio ... monet ... res omnes per quasdam series posse disponi, non quidem in quantum ad aliquod genus entis referuntur, sicut illas Philosophi in categorias suas diviserunt, sed in quantum vnæ ex alijs cognosci possunt ... (ibid., 381.7–13; my emphases).

The res involved are fundamentally, as we already know, simple natures and their compositiones subject to intuitus and immediate deductiones, and the goal is—as stated in Reg. V—to discover “aliquam veritatem” (AT X, 379), presumably having the status of a scientia. Part of what one learns from the quoted passage is thus that the general means to attain this goal is to disponere simple natures and their compositiones in series of a certain specific sort, namely such that simple natures and their compositiones vnæ ex alijs cognosci possunt, while (as already established) the only permissible way of serial organization is, fundamentally, continuous chains of immediate deductiones. The proper point of ordering tout court in the analytical régime of discovery is thus to arrange the mentioned items per deductiones as to their epistemic dependence—the notion of ordering which is retained untouched by Descartes up to the seminal passage on the modus scribendi geometricus in Resp. 2: Ordo in eo tantùm consistit, quòd ea, quæ prima proponuntur, absque ullâ sequentium ope debeant cognosci, & reliqua deinde omnia ita disponi, ut ex præcedentibus solis demonstrentur (AT VII, 155; my emphasis).

Yet the core of the message in the quoted AT X, 381.7–13 has to do with a contrast between the arrangement according to the epistemic dependence and the arrangement according to the categoriæ Philosophorum. By way of a preliminary, whatever exactly Descartes means with the latter,65 it would be a mistake to interpret the contrast just to the effect that 64 Descartes himself seems to admit this as the opening words of his explication of the precept read: “Etsi nihil valde novum hæc propositio docere videatur...” (Reg. VI, AT X, 381). 65 Marion, Sur l’ontologie grise, §§ 13–14 argues that Descartes alludes quite closely to Aristotle and his doctrine of categories throughout Reg. VI. As I see it,

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dependencies (or priorities) in essendo are distinguished from dependencies (or priorities) in cognoscendo: since only the latter sort of dependencies is clearly in question from Reg. IV onwards, such an interpretation would render the contrast in Reg. VI superfluous and hardly warranting the professedly pivotal place of Reg. VI in the whole Regulæ. Rather, the contrast is to be read as indicating, in a negative manner, how exactly the epistemic dependence, according to which the appropriate items are to be arranged, is to be understood. Descartes provides some clues for determining what he has in mind. A few lines later he specifies the sense “quo ad nostrum propositum [res omnes] vtiles esse possunt, ... vt vnæ ex alijs cognoscantur” (Reg. VI, AT X, 381), to the effect that “non illarum naturas solitarias spectamus, sed illas inter se comparamus” (ibid.); and he reiterates the point later in the same rule: “nos hîc rerum cognoscendarum series, non vniuscujusque naturam spectare” (ibid., 383; my emphasis). Now it is, of course, exactly those naturas solitarias that determine ontologically the place of the corresponding items in the categorial framework—Aristotelian or otherwise—defined by the genera entis. In view of these clues, then, Descartes‫ ތ‬point in rejecting disposing of things in quantum ad aliquod genus entis referuntur in the present context seems to be at very least that in so far as dependencies in cognoscendo are concerned, the natures of the relevant items taken singularly and piecemeal—and by implication the categorial framework in which the items are integrated due to those natures—are by no means decisive determinants of how these items are to be arranged vt vnæ ex alijs cognoscantur in the régime of scientific discovery. Descartes‫ ތ‬positive treatment of what is to take over the function of such a determinant—viz. an alternative conceptualization in terms of the complementary structure absolutum–respectivum—will be taken up later in this chapter. For the time being, it will be well to further clarify, by way of anticipation, the sense and/or extent to which Descartes commits himself to the dismissal of cognition of naturas solitarias as a possible determinant of the serial arrangement of the corresponding res in analysis. Descartes arguably never—and certainly not in the passage under scrutiny—says or implies either that things have no natures or that no single nature of any thing can ever be known by human minds. What he does imply at the beginning of Reg. VI, however, is that the rôle of cognition of naturas solitarias is to be compromised in a significant way Marion indeed succeeded in showing that Aristotle‫ތ‬s doctrine of categories forms the paradigm of the way in which “Philosophi [res] in categorias suas diviserunt” (Reg. VI, AT X, 381). Yet I argue below that Descartes‫ ތ‬point bears upon any kind of ontological classification including his own substance and/or property dualism.

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as far as the arrangement of the appropriate items in the analytical procedure of discovery is concerned. A closer scrutiny of the relevant passages (in particular of the examples Descartes adduces) unambiguously precludes a prima facie natural, strong reading of the restriction in question, according to which the consideration or cognition of naturas solitarias is to be completely dismissed as a possible relevant factor in the serial ordering at issue, in favour of some alternative conceptualizations of the ordered items. For it will be shown in due course that cognitive access to some natura(e) solitaria(e) is strictly required for each and every orderly arrangement to facilitate discoveries in the envisaged way. Fortunately, weaker alternative versions of the restriction in question are available that square well both with AT X, 381.7–13 and with the passages that preclude the above strong reading. I submit that two such alternative conceptions, logically independent of one another, are actually at work in various places in the Regulæ and elsewhere: (1) in so far as cognition of natures is utilized in the constitution of a given cognitive series, the reference of the natures at issue to different genera and/or their belonging to different categories does not preclude the possibility of arranging them within a single cognitive series; in other words, the items within a single cognitive series (objects of intuitus and immediate deductiones) may belong to different genera entis and/or categories. (2) If cognition of the nature n of any item i within a given cognitive series s turns out to be required in order that s facilitate discovery, the requirement can be substituted with another requirement, viz. to cognize i under a suitable alternative conceptualization that does not contain n. Almost needless to say, (1) stands in direct opposition to the standard Aristotelian view of the conditions of discursive reasoning that issues in a scientia, according to which stepping out of a given category in the course of relating the subject, the predicate and the middle terms of scientific syllogisms is strictly forbidden. However, (1) holds much more generally, and notably also with regard to Descartes‫ ތ‬own doctrine of two summa genera rerum, to wit, res intellectuales and res materiales.66 Indeed, Descartes himself intimates to Mersenne in the end of 1640 that67 66

See in particular Princ. I, 48, AT VIII-1, 23. There are, of course, dozens of references to Descartes‫ ތ‬so-called substance dualism and property dualism in various texts throughout his career: see e.g. Med. VI, AT VII, 78; Med. Pref., AT VII, 13; Resp. 3, AT VII, 175–76; Princ. I, 53–54, AT VIII-1, 25–26. See also fn. 121. 67 It is in view of this categorial permeability in the constitution of discursive structures that the full import of the institution of notiones communes, “quæ sunt veluti vincula quædam ad alias naturas simplices inter se conjungendas” (Reg. XII,

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As for (2), the phrase “suitable alternative conceptualization” is designed to refer in general to results of a peculiar analogical treatment— possibly crossing, once again, the borders prescribed by a categorial approach—which Descartes occasionally recommends if the direct cognition of the nature of some item is required by the order of investigation but is actually unavailable, so that the nature in question be understood at least per imitationem. As he puts it (while considering the problem of the nature of illuminatio),68 si [aliquis] statim in secundo gradu [solutionis] illuminationis naturam non possit agnoscere, enumerabit ... alias omnes potentias naturales, vt ex alicujus alterius cognitione saltem per imitationem ... hanc etiam intelligat ... (Reg. VIII, AT X, 395).

Let me sum up what we have learned so far from the submitted interpretation of Reg. V and of the opening sentences of Reg. VI. The proper task of any methodical analytical procedure in the régime of

AT X, 419), emerges distinctly. The categorial permeability just pinpointed is put in accord with Descartes’ classification of the scientiæ based upon a kind of categorial framework in the “tree comparison” passage in Princ. Pref. in the last footnote of the present chapter. 68 Cf. also a celebrated passage from La Dioptrique I, AT VI, 83: “[N]‫ތ‬ayant icy autre occasion de parler de la lumiere, que pour expliquer comment ses rayons entrent dans l‫ތ‬œil, & comment ils peuuent estre détournés par les diuers cors qu‫ތ‬ils rencontrent, il n‫ތ‬est pas besoin que i‫ތ‬entreprene de dire au vray quelle est sa nature, & ie croy qu‫ތ‬il suffira que ie me serue de deus ou trois comparaisons, qui aydent a la conceuoir en la façon qui me semble la plus commode, pour expliquer toutes celles de ses proprietés que l‫ތ‬experience nous fait connoistre, & pour deduire en suite toutes les autres qui ne peuuent pas si aysement estre remarquées ....” The meaning of comparaisons in this passage—having to do with analogies in general—is to be carefully distinguished from the meaning of comparationes— having to do with at most a very special case of analogy, viz. analogy of quantitative proportion—I will deal with in sec. 4.3.2.

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discovery was shown to amount to the “ordering and disposing” of the already available objects of intuitus and of immediate deductiones in certain series and in a certain specific way. Furthermore, we have in our possession some clues as to what one is to obey to bring to fruition such an orderly disposition. In general, the first objective is to reduce step by step “propositiones involutas & obscuras ad simpliciores” (Reg. V, AT X, 379). Reg. VI has taught us, then, that the reduction at issue is to take the form of a series of suitable intuitus and immediate deductiones ordered according to their epistemic dependence, i.e. in quantum vnæ ex alijs cognosci possunt. Finally, I have suggested two distinct ways in which Descartes can plausibly be read to mitigate the import of ontologically interpreted categorial frameworks to which the natures of the items to be ordered refer, for arranging the required cognitive series. What have not yet been encountered are any positive hints at how the following vital question, hereinafter “Q,” is to be addressed, namely, how exactly is one to proceed to arrange the available intuitus and the immediate deductiones in the cognitive series in accord with the constraints introduced so far? Before we turn to how Descartes tackles Q, however, it will be well to consider another point, and an extremely important one at that, which is what, from Descartes‫ ތ‬perspective, each and every adequately enacted discursive cognitive operation essentially involves. 4.3.1.2 Comparatio Indeed, a notion of cardinal importance has calmly strolled on stage along with the issues just discussed; that is to say, the operation called comparatio: ... res omnes, eo sensu quo ad nostrum propositum vtiles esse possunt, vbi non illarum naturas solitarias spectamus, sed illas inter se comparamus, vt vnæ ex alijs cognoscantur ... (Reg. VI, AT X, 381; my emphasis).

The pivotal place of comparationes in Descartes‫ ތ‬project of a universal method of discovery is not fully highlighted until Reg. XIV where Descartes remarks, tersely and tantalizingly, that proderit lectori, si ... concipiat omnem omnino cognitionem, quæ non habetur per simplicem & purum vnius rei solitariæ intuitum, haberi per comparationem duorum aut plurium inter se (AT X, 440; my emphases).

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Here Descartes confirms unambiguously what was just one interpretative possibility with regard to the above AT X, 381, namely that comparatio is strictly the only operation by which any adequately enacted discursive cognition whatsoever is completed; in other words, that to Descartes comparatio is the right word to describe what happens in any operation deserving the name deductio. Comparationes are ultimately all that in which the searching mind is supposed to be engaged in the course of its pursuit after new (pieces of) scientiæ. Two fundamental questions arise: (a) What are the general principles of comparationes, or in other words, what are the items in terms of which the conditions a parte rei of every comparatio can be articulated? (b) How is one to proceed to establish the conditions for the appropriate comparationes, i.e. to render the appropriate comparationes ready to be enacted? Clearly enough, (b) is virtually nothing but an alternative formulation of Q as rendered at the end of sec. 4.3.1.1; and (a) is concerned with the items in terms of which the answer to (b) and Q is to be given. The immediate continuation of the last quoted passage documents that this is how Descartes himself probably saw the situation:69 Et quidem tota fere rationis humanæ industria in hac operatione præparandâ consistit; quando enim aperta est & simplex, nullo artis adjumento, sed solius naturæ lumine est opus ad veritatem, quæ per illam habetur, intuendam. Notandumque est, comparationes dici tantùm simplices & apertas, quoties quæsitum & datum æqualiter participant quamdam naturam; cæteras autem omnes non aliam ob causam præparatione indigere, quàm quia natura illa communis non æqualiter est in vtroque, sed secundùm alias quasdam habitudines sive proportiones in quibus involvitur ... (ibid.; my emphases).

Notably, the context is expressly that of facilitating discoveries, since the terms of the comparationes are deliberately assigned as the data and the quæsita. I take comparatio aperta & simplex, i.e. an instance of operation that yields intuitus of the truth due to nothing but the lumen naturæ, as amounting to a general description of what generally happens in immediate deductiones. Furthermore, the præparatio Descartes speaks about clearly amounts precisely to that with which question (b) is concerned: one learns that a part of the purpose of this preparatory serial disposition is, in effect, to render the comparationes between the neighbouring terms of the series apertas & simplices as well. In the latter 69

Following the suggestion in Beck, Method of Descartes, 222, fn. 2, I read “vtroque” for “vtrâque” in AT X, 440.15.

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sentence, then, Descartes hints at how question (a) is to be addressed: the general principle of comparisons—the tertium comparationis—is identified with a natura that is common both to the data and the quæsita. Furthermore, the relations of the res involved (i.e., it will be remembered, eventually of objects of intuitus and of immediate deductiones) to that common nature are generally conceptualized as participationes in the given common nature. Question (a) thus boils down to the task of clarifying how exactly the relations of participatio between the natura communis on the one hand and the res to be ordered and disposed on the other hand, are to be resolved. Furthermore, question (b)/Q boils down to the question of how the relevant data and quæsita are to be treated to render them involved in such habitudines sive proportiones as would allow them to be compared with one another via participation in common nature(s).70 In view of the explications just suggested, we are finally in a position to appreciate Descartes‫ ތ‬ingenious and brave general observation concerning in what any adequately enacted analytical procedure in the régime of discovery is to consist:71 [N]otandum est, quoties vnum quid ignotum ex aliquo alio jam ante cognito deducitur, non idcirco novum aliquod genus entis inveniri, sed tantùm extendi totam hanc cognitionem ad hoc, vt percipiamus rem quæsitam participare hoc vel illo modo naturam eorum quæ in propositione data sunt (Reg. XIV, AT X, 438; my emphasis).

It is not difficult by now to determine in what, from Descartes‫ތ‬ perspective, the entire methodical procedure is to culminate. That is to say, the established, serially ordered habitudines sive proportiones are to be transformed into systems of equations which are then to be resolved (or 70

This in fact amounts, I suggest, to integration of the data and quæsita into what Descartes once called consequentiarum contextus in Reg. VI, AT X, 383: “Atque talis est vbique consequentiarum contextus, ex quo nascuntur illæ rerum quærendarum series, ad quas omnis quæstio est reducenda, vt certâ methodo possit examinari.” 71 Cf. Reg. XVII, AT X, 460–61: “[S]upposuerimus ... nos agnoscere eorum, quæ in quæstione sunt ignota, talem esse dependentiam à cognitis, vt planè ab illis sint determinata ....” Also ibid., 459: “[D]eterminatæ difficultates & perfectè intellectæ ... eò reducendæ [sint], vt nihil aliud quæratur postea, quàm magnitudines quædam cognoscendæ, ex eo quòd per hanc vel illam habitudinem referantur ad quasdam datas.” Descartes speaks here of magnitudines since in Reg. XVII, he exemplifies his general conception of the method of discovery with the paradigmatic case of solving problems in his general algebra.

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“reduced”)72 with the aim eventually of attaining an aperta & simplex comparatio—that is to say, æqualitas—between the quæsitum and something already known:73 Notandumque est ... præcipuam partem humanæ industriæ non in alio collocari, quàm in proportionibus istis eò reducendis, vt æqualitas inter quæsitum, & aliquid quod sit cognitum, clarè videatur (Reg. XIV, AT X, 440; my emphasis).

Once one arrives at this, it is ensured that one intuits the truth with the aid of the lumen naturæ alone (ibid., 440), which is the assumed goal of cognition in general, and a fortiori of mathematical cognition. Descartes offers no complete explication in his extant writings of to what the reduction in question or the sought-after æqualitas amount. All one can dredge up on this score (in particular from the Regulæ) are sketchy exemplifications taken, quite predictably, from the paradigmatic field of general algebra; and it comes as no surprise that in this field, the procedure in question boils down to the resolution of a given system of algebraic equations the purpose of which is to end up with the sought-after general quantities equal to some initially given general quantities. Descartes devotes considerable portions of Geom. III to a thorough treatment of how the resolutions of algebraic equations are to be performed: I have nothing substantial to add on this score. By way of contrast, what must be scrutinized now in detail is the præparatio comparationum introduced above. Descartes himself indicates, as we saw, that a considerable part of the extensio cognitionis mentioned in this passage coincides with this præparatio. If anything, it is an explication of how this præparatio is conceived by Descartes that promises to provide substantial portions of content for the sought-after manual for facilitating discoveries in a methodical way. I turn to this task—which in fact amounts to addressing questions (a) and (b)/Q.

72

For reducere in this sense see e.g. Reg. XIV, AT X, 447; Reg. XVII, AT X, 459. Cf. Reg. XIV, AT X, 441: “Maneat ... ratum & fixum, quæstiones perfectè determinatas vix vllam difficultatem continere, præter illam quæ consistit in proportionibus in æqualitates evolvendis ....” Reg. XVII, AT X, 460: “[T]totum hujus loci artificium consistet in eo quòd, ignota pro cognitis supponendo, possimus facilem & directam quærendi viam nobis proponere, etiam in difficultatibus quantumcumque intricatis ....” 73

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4.3.2 The Præparatio Comparationum I submit that for Descartes, three distinct basic operations are involved in the preparatory task with which (b)/Q is concerned, namely relativization to an absolute, determination of the quæsita through the data, and reduction to simplest problems.74 They will be introduced and discussed in turn in the following three subsections. It will be convenient to address question (a) concerning the nature of participatio in the course of dealing with the first of these operations. 4.3.2.1 Relativization to an Absolute A most general account of a part of how our question (b)/Q is to be addressed is given by Descartes in the first half of Reg. VI. Here is what I take to be the gros of this account: Vt autem id [sc. rerum omnium per quasdam series dispositio in quantum vnæ ex alijs cognosci possunt] rectè fieri possit, notandum est primo, res omnes, eo sensu quo ad nostrum propositum vtiles esse possunt, vbi non illarum naturas solitarias spectamus, sed illas inter se comparamus, vt vnæ ex alijs cognoscantur, dici posse vel absolutas vel respectivas. Absolutum voco, quidquid in se continet naturam puram & simplicem, de quâ est quæstio ...; atque idem primum voco simplicissimum & facillimum, vt illo vtamur in quæstionibus resolvendis. Respectivum verò est, quod eamdem quidem naturam, vel saltem aliquid ex eâ participat, secundùm quod ad absolutum potest referri, & per quamdam seriem ab eo deduci; sed insuper alia quædam in suo conceptu involvit, quæ respectus appello .... Quæ respectiva eò magis ab absolutis removentur, quò plures ejusmodi respectus sibi invicem subordinatos continent; quos omnes distinguendos esse monemur in hac regulâ, & mutuum illorum inter se nexum naturalemque ordinem ita esse observandum, vt ab vltimo ad id, quod est maximè absolutum, possimus pervenire per alios omnes transeundo. Atque in hoc totius artis secretum consistit, vt in omnibus illud maximè absolutum diligenter advertamus. ... Cæteræ autem omnes non aliter percipi possunt, quàm si ex istis deducantur, idque vel immediatè & proximè, vel non nisi per duas aut tres aut plures conclusiones diversas; quarum numerus etiam est notandus, vt agnoscamus vtrùm illæ à primâ & maximè simplici propositione pluribus vel paucioribus gradibus removeantur (AT X, 381–83; my emphases).

74

I believe it is these three operations that are hinted at (although in a different order) in the second to the fourth precepts of DM 2 introduced above. However, I need not rely on this plausible correspondence in the claims I am about to present.

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The first point of note is that the feature in view of which the items relevant to the solution of a given problem are to be classified as absoluta and respectiva is their relative cognitive simplicitas & facilitas, i.e. their relative degree of evidence and certainty which renders them ordered as to their cognitive priority.75 Two corollaries of this general point are alluded to by Descartes himself; namely that the pair of characterizations absolutum–respectivum, as employed in the present context, (i) allow of degrees, and (ii) are themselves essentially relational in a certain sense or senses.76 As for (i), Descartes observes that quædam interdum sunt verè magis absoluta quam alia, sed nondum tamen omnium maximè: vt si respiciamus individua, species est quid absolutum; si genus, est quid respectivum; inter mensurabilia, extensio est quid absolutum, sed inter extensiones longitudo, &c. (Reg. VI, AT X, 382–83; my emphasis).

It is perhaps worth emphasizing that allowing degrees in this sense does not imply that absolutum–respectivum be a relative pair of terms: the degrees hold fixed throughout any context whatsoever, as Descartes clearly indicates with the adverb verè. By way of contrast—to turn to (ii)—what to Descartes is essentially a matter of relations is that in which totius artis secretum (ibid., 381) consists, viz. the designation of something as the maximè absolutum. There are at least two different senses in which the designation in question turns out relational in Descartes‫ ތ‬hands. The more obvious of them was already hinted at: one and the same item may turn out to be the one that is cognized simplicissimè & facilissimè, and thus to qualify as the maximè absolutum, relative to set(s) of items relevant to the solution of certain problem(s), and to qualify as a more or less remote respectivum relative to set(s) of items 75

Cf. Resp. 5, AT X, 384: “Nec recte probatur unam rem aliâ esse notiorem, ex eo quod pluribus vera videatur, sed tantùm ex eo quòd illis, qui utramque, ut par est, cognoscunt, appareat esse cognitu prior, evidentior & certior.” 76 So much seems to be commonly recognized among the major commentators— cf. e.g. Beck, Method of Descartes, 164; Wolfgang Röd, Descartes’ Erste Philosophie: Versuch einer Analyse mit besonderer Berücksichtigung der Cartesianischen Methodologie (Bonn: Bouvier Herbert Grundmann, 1971), 15–16; Marion, Sur l’ontologie grise, 80–82. However, there are differences as to what exactly the relative nature of the characterizations in question amounts to, and sometimes the corollaries (i) and (ii) seem to be confused (as e.g. in Beck, ibid.). I shall not discuss the alternative views and will simply present my own positive interpretative conception.

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relevant to the solution of certain other problems. The other sense is established by Descartes as follows: Quædam enim sub vnâ quidem consideratione magis absoluta sunt quàm alia, sed aliter spectata sunt magis respectiva: vt vniversale quidem magis absolutum est quàm particulare, quia naturam habet magis simplicem, sed eodem dici potest magis respectivum, quia ab individuis dependet vt existat, &c. (ibid., 382).

That is to say (as far as I understand it) that even given one and the same set of items, one and the same item may qualify as the maximè absolutum or as a more or less remote respectivum relative to the quæstio under investigation which determines the aspect in view of which relative simplicity and ease of cognition are determined. So far, therefore, the point of the procedure Descartes recommends in the first half of Reg. VI can be put as follows. The trick is to find an item i that is designated functionally as the maximè absolutum relative to a set of items to be ordered (the set being determined with the given quæsita) and that satisfies the following two conditions: first, i is cognized, within the context defined through the given quæsita, simplicissimè & facilissimè, i.e. (perhaps among other things) independent of any other item of the given set; and second, each item of the given set can be cognized, within the context defined through the given quæsita, via a certain number of specific inferential steps which of course amount to (immediate or mediated) deductiones.77 To proceed further, Descartes also indicates in the quoted AT X, 381– 83 what is properly responsible for one item of the given set to qualify as cognized simplicissimè & facilissimè, and in terms of which the other relevant items are to be ordered as more or less remota from a given maximè absolutum. In view of the above identification of the general principle of comparationes upon the basis of Reg. XIV, AT X, 440, it comes as no surprise that all this is fundamentally a matter of various relations of the data and the quæsita to one and the same (and in this sense common)78 natura:

77

Descartes refers to deductiones also with conclusiones and consequentiæ in Reg. VI—see AT X, 383. 78 The term “natura communis” does not occur in Reg. VI; yet it seems clear to me that the naturæ in terms of which the absoluta and relativa are discriminated in Reg. VI are exactly what Descartes refers to with natura communis in Reg. XIV, AT X, 440 and 449–50.

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Chapter Four Absolutum voco, quidquid in se continet naturam puram & simplicem, de quâ est quæstio .... Respectivum verò est, quod eamdem quidem naturam, vel saltem aliquid ex eâ participat, secundùm quod ad absolutum potest referri, & per quamdam seriem ab eo deduci; sed insuper alia quædam in suo conceptu involvit, quæ respectus appello ... (Reg. VI, AT X, 381–82; my emphases).

Several things are thus sufficiently clear: one, the natura at issue is to be determined by the given quæstio; two, the crucial requirement is that this natura is to be somehow related to all the relevant items, both the quæsita and the relevant data; three, the relation in question is that of participation (participare) or containment (continere); and four, the critical differential with regard to sorting out the (maximè) absolutum on the one hand and the respectiva on the other hand has to do with how exactly the relevant items‫ތ‬ participation in or containment of the natura communis at issue is resolved. Of course, what must be finally clarified now to make something of all this is to what the relation of participation amounts and how it is supposed to fulfil the rôle of the discriminant in the present context. The first step is to clarify how the term ad quem of the relation of participation in question, i.e. the natura (communis) “de quâ est quæstio,” is related to simple natures. It surely has not passed unnoticed that common nature, in so far as contained in the resolved (maximè) absolutum, is characterized by Descartes as pura & simplex in the last quoted passage, and slightly later in the same rule Descartes writes that [n]otandum ... paucas esse duntaxat naturas puras & simplices, quas primò & per se, non dependenter ab alijs vllis ... licet intueri; atque ... sunt ... eædem, quas in vnâquâque serie maximè simplices appellamus (ibid., 383; my emphases).

There is no doubt, therefore, that the precept to search after common nature in the established sense is n by Descartes as implying a precept to search after some simple nature(s) in the strict, technical sense established in ch. 2.79 After all, this is what one would expect in view of the doctrine 79 To avoid confusion, it is perhaps worth emphasizing that natura communis in the sense in which the term is used in the present section, and in which I will use that term from now on, i.e. as referring to a sought-after nature that is to be shared by all the items relevant to the solution of a given quæstio, is to be distinguished from naturæ communes in the sense of a set of those simple natures “quæ modò rebus corporeis, modò spiritibus sine discrimine tribuuntur” (Reg. XII, AT X, 419). Even simple natures belonging to the other kinds—purè intellectuales and purè

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that omnia quæ cognoscimus amounts, in the final analysis, either to a simple nature or to a compositio or mixtura of simple natures, in combination with the present context, which is that of establishing scientific discoveries. This being granted, I submit one must be careful not to succumb to two mistaken interpretative ideas which prima facie sound appealing but would render the conception under scrutiny both unreasonably strong and at odds with certain relevant passages. One such erroneous idea is that one and only one simple nature is always to be taken as identical with any given natura communis. While such a scenario is certainly possible (and we will consider an algebraic instance of it in a moment), Descartes allows—reasonably—the possibility that the pertinent natura communis be fundamentally a compositio or mixtura of more than one simple nature, most clearly in the following passages: [S]i petatur quid sit magnetis natura ... qui cogitat, nihil in magnete posse cognosci, quod non constet ex simplicibus quibusdam naturis & per se notis, ... primo diligenter colligit illa omnia quæ de hoc lapide habere potest experimenta, ex quibus deinde deducere conatur qualis necessaria sit naturarum simplicium mixtura ad omnes illos, quos in magnete expertus est, effectus producendos; quâ semel inventâ, audacter potest asserere, se veram percepisse magnetis naturam, quantum ab homine & ex datis experimentis potuit inveniri (Reg. XII, AT X, 427; my emphasis). [S]i in magnete sit aliquod genus entis, cui nullum simile intellectus noster hactenus perceperit, non sperandum est nos illud vnquam ratiocinando cognituros; sed vel aliquo novo sensu instructos esse oporteret, vel mente divinâ; quidquid autem hac in re ab humano ingenio præstari potest, nos adeptos esse credemus, si illam jam notorum entium sive naturarum mixturam, quæ eosdem qui in magnete apparent, effectus producat, distinctissimè percipiamus (Reg. XIV, AT X, 439; my emphasis).

This commitment is compatible with the quoted AT X, 383 in which simple natures are treated as grammatical plurals; and the fact that Descartes usually treats common natures as singulars can be plausibly explained away as being due to a collective use of the term natura communis, i.e. as referring to the pertinent sets of simple natures in the relevant contexts. The other erroneous interpretation would have it that the maximè absoluta to be established are always to be strictly identified with the materiales (cf. ibid.)—can become constituents of the sought-after natura communis in the former sense.

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common nature (i.e. eventually either with a single simple nature or with a compositio of simple natures) due to which they qualify as such absoluta. Once again, such a scenario is certainly possible. Yet there are several indications that Descartes allows for cases in which common natures are just parts of a given maximè absolutum. For one thing, Descartes‫ ތ‬dictum “[a]bsolutum voco, quidquid in se continet naturam puram & simplicem, de quâ est quæstio” (Reg. VI, AT X, 381) can plausibly be read (as I have just shown) to the effect that common nature is, at least sometimes, just contained in the established absolutum. Furthermore, this textual evidence is confirmed by certain aspects of Descartes‫ ތ‬explication of how the quæsita are to be determined through the data (this topic is taken up in detail in the following subsection), namely with his requirement to abstract, in every “well-understood” problem, “ab omni superfluo conceptu” (Reg. XIII, AT X, 431) “vt ad rem non facientibus” (ibid., 437). The most telling clue as to how the relation of participation is to be understood in the present context is provided in Reg. XIV, AT X, 449–50: Vnitas est natura illa communis, quam suprà diximus debere æqualiter participari ab illis omnibus quæ inter se comparantur. Et nisi aliqua jam sit determinata, in quæstione, possumus pro illâ assumere, sive vnam ex magnitudinibus jam datis, sive aliam quamcumque, & erit communis aliarum omnium mensura; atque in illâ intelligemus tot esse dimensiones, quot in ipsis extremis, quæ inter se erunt comparanda ... (my emphasis).

The domain of discourse is unambiguously that of algebraic analysis, as is made clear both by the talk of magnitudines and of dimensiones in the quoted passage and by the immediate context in which the passage sits.80 Furthermore, vnitas is clearly designated by Descartes as the maximè absolutum in this domain. While such a designation hardly comes as surprise, it is extremely helpful in that it provides the only tolerably 80 See ibid. 446–47: “[B]revius erit exponere, quo pacto nostrum objectum concipiendum esse supponamus, vt de illo, quidquid in Arithmeticis & Geometricis inest veritatis, quàm facillime demonstremus. Hîc ... quæstiones omnes eò deductas esse supponimus, vt nihil aliud quæratur, quàm quædam extensio cognoscenda, ex eo quòd comparetur cum quâdam aliâ extensione cognitâ. Cùm enim ... velimus duntaxat proportiones quantumcumque involutas eò reducere, vt illud, quod est ignotum, æquale cuidam cognito reperiatur: certum est omnes proportionum differentias, quæcumque in alijs subjectis existunt, etiam inter duas vel plures extensiones posse inveniri; ac proinde sufficit ad nostrum institutum, si in ipsâ extensione illa omnia consideremus, quæ ad proportionum differentias exponendas possunt juvare, qualia occurrunt tantùm tria, nempe dimensio, vnitas, & figura.”

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determinate key as to how the general modus operandi of Descartes‫ތ‬ envisaged universal method under investigation is to be explicated in domains other than mathematics. For one may certainly extrapolate that maximè absoluta are to be both to the data and the quæsita in any domain as a given unit length is to both the known and the unknown general quantities in an algebraic articulation of mathematical problems; and we arrived in ch. 3 at a considerably distinct understanding of how such an articulation is supposed to work in Descartes‫ ތ‬hands. Against this general background, the meaning of the relation of participation can finally be determined, at least in the paradigmatic domain of general algebra. The last quoted passage suggests that it is precisely due to being the terminus ad quem of participation that vnitas counts as the communis aliarum omnium mensura. Moreover, Descartes writes on the next page, agnosco ..., quis sit ordo inter A & B, nullo alio considerato præter vtrumque extremum; non autem agnosco, quæ sit proportio magnitudinis inter duo & tria, nisi considerato quodam tertio, nempe vnitate quæ vtriusque est communis mensura (ibid., 451).

Together with the account of Descartes‫ ތ‬algebraic approach, the passages under scrutiny clearly yield that to participate in a given vnitas (i.e. a given unit length conceived as a general quantity) amounts to standing in a certain determinate proportion to that vnitas in an appropriate series of continuous proportionals. Furthermore, the peculiar connotation which the conceptualization of the position in that series as participatio carries with it seems to be the heuristic value: it is due to such a determinate position of each particular datum and quæsitum in an appropriate series of continuous proportionals that the comparationes can be carried out in order to solve the problem in question. By the same token, Descartes‫ ތ‬talk in Reg. VI, AT X, 381–83 of the respectiva as being more or less or equally remota from the absolutum becomes perfectly intelligible at least in its algebraic interpretation. Descartes says in that passage, it will be remembered, that “respectiva eò magis ab absolutis removentur, quò plures ... respectus sibi invicem subordinatos continent” (ibid., 382) and that the gradus at issue amounts to the number of conclusiones which are needed to pass from the given absolutum to a given datum or quæsitum by way of deductio.81 Furthermore, it should be clear by now that the number of these respectus 81

Cf. ibid., 383.

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and/or conclusiones amounts, in the algebraic realization, to the number of relationes in terms of which powers of geometrical (and eventually of general) quantities are interpreted by way of series of continuous proportionals starting with the unit length, as was shown in ch. 3.82 In view of the algebraic instance of Descartes‫ ތ‬general presentation of the præcipuum artis secretum in the opening pages of Reg. VI, we thus finally arrive at a fairly distinct notion of how the present step of the preparatory task of the methodical treatment is to be processed in the 82

The most telling in this respect is Reg. XVI, AT X, 457.1–12. The paragraph in which the passage is located begins as follows: “Advertendum est etiam, per numerum relationum intelligendas esse proportiones se continuo ordine subsequentes, quas alij in Algebrâ per plures dimensiones & figuras conatur exprimere, & quarum primam vocant radicem, secundam quadratum, tertiam cubum, quartam biquadratum, &c.” (ibid., 456). Cf. also Reg. XVIII, AT X, 462: “[S]ciendum est vnitatem, de quâ jam sumus locuti, hîc esse basim & fundamentum omnium relationum, atque in serie magnitudinum continuè proportionalium primum gradum obtinere, datas autem magnitudines in secundo gradu contineri, & in tertio, quarto, & reliquis quæsitas, si proportio sit directa; si verò indirecta, quæsitam in secundo & alijs intermedijs gradibus contineri, & datam in vltimo.” Again, in Reg. XI, Descartes illustrates how in instances of series of continued proportionals “ad mutuam simplicium propositionum dependentiam reflectendo, vsum acquiramus subitò distinguendi, quid sit magis vel minùs respectivum, & quibus gradibus ad absolutum reducatur” (AT X, 409). For the rest, the algebraic context invoked above also clarifies the somewhat obscure dictum that in vnitas “intelligemus tot esse dimensiones, quot in ipsis extremis, quæ inter se erunt comparanda” (Reg. XIV, AT X, 450). What Descartes seems to have in mind is the insight he made crystal clear later in Geom. I, AT VI, 371–72: “Il est aussy a remarquer que toutes les parties d‫ތ‬vne mesme ligne se doiuent ordinairement exprimer par autant de dimensions l‫ތ‬vne que l‫ތ‬autre, lorsque l‫ތ‬vnité n‫ތ‬est point determinée en la question: comme icy a3 en contient autant qu‫ތ‬abb ou b3, dont se compose la ligne que i‫ތ‬ay nomméeටǤ ܽଷ Ȃܾଷ  ൅ ܾܾܽ; mais que ce n‫ތ‬est pas de mesme lorsque l‫ތ‬vnité est determinée, a cause qu‫ތ‬elle peut estre sousentendue partout où il y a trop ou trop peu de dimensions; comme, s‫ތ‬il faut tirer la racine cubique de aabb – b, il faut penser que la quantité aabb est diuisée vne fois par l‫ތ‬vnité, & que l‫ތ‬autre quantité b est multipliée deux fois par la mesme.” JiĜí Fiala‫ތ‬s comment is helpful here: “That is to say, the magnitudes are to be homogeneous, for only homogeneous magnitudes, like line segments or squares, but never line segments and squares, are allowed in the operation of addition. Thus a2b2 – b can be rendered as a2b2 – b · 1 · 1 · 1, in which case both magnitudes have four dimensions; or, as Descartes suggests, as a2b2 / 1 – b · 1 · 1, in which case they have three dimensions” (JiĜí Fiala, “KomentáĜe” [Commentary], in René Descartes, Regulæ ad directionem ingenii / Pravidla pro vedení rozumu transl. VojtČch Balík [Prague: Oikoymenh, 2000], 294; my translation; Fiala’s emphasis).

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paradigmatic field of algebraic analysis. The precept might read as follows: Find or establish arbitrarily a unit length, and set in series of continuous proportionals all the data and quæsita whose treatment is required by the given problem, starting with the established unit, according as to how the data and the quæsita depend on one another once their relations are articulated in terms of (algebraically interpreted) operations with (general) quantities. The suggestion stated in these terms sound quite reasonable. Moreover, Descartes exhibited persuasively, in La Geometrie and elsewhere, his ability to put the precept to work to gain discoveries in the field of mathematical problems; and we saw him being audacious enough to declare more than once that the procedure just described could really be extended and put to effective use beyond the limits of pure mathematics.83 Let us turn now to the remaining operations devised to prepare the comparationes through which the employment of Descartes‫ ތ‬analytical method of discovery is to be accomplished. 4.3.2.2 Determination of the Quæstiones Reg. XIII starts with Descartes‫ ތ‬own version of the general prærequisita to the successful solution of any possible problem: [P]rimò, in omni quæstione necesse est aliquid esse ignotum, aliter enim frustra quæreretur; secundò, illud idem debet esse aliquo modo designatum, aliter enim non essemus determinati ad illud potiùs quàm ad aliud quidlibet investigandum; tertiò, non potest ita designari, nisi per aliud quid quod sit cognitum (AT X, 430).

The most significant of these prærequisita is the second, for it is in its terms that Descartes is able to specify further how one is to proceed in the preparatory stage of the analytical methodical procedure. This passage continues immediately as follows: Quæ omnia reperiuntur etiam in quæstionibus imperfectis: vt si quæratur qualis sit magnetis natura, id quod intelligimus significari per hæc duo vocabula, magnes & natura, est cognitum, à quo determinamur ad hoc potiùs quàm ad aliud quærendum, &c. Sed insuper vt quæstio sit perfecta, 83

Of course, it remains an open question whether Descartes really succeeded in such an extension, and if so exactly how he managed that much. I shall not confront these challenging issues in the present study for reasons specified in fn. 116.

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Chapter Four volumus illam omnino determinari, adeò vt nihil ampliùs quæratur, quàm id quod deduci potest ex datis ... (ibid., 430–31; my emphasis).

Perfectly and imperfectly understood quæstiones are introduced by Descartes in Reg. XII, AT X, 429:84 Ex quæstionibus ... aliæ intelliguntur perfectè, etiamsi illarum solutio ignoretur ...; aliæ denique non perfectè intelliguntur .... Notandum est, inter quæstiones quæ perfectè intelliguntur, nos illas tantùm ponere, in quibus tria distinctè percipimus: nempe, quibus signis id quod quæritur possit agnosci, cùm occurret; quid sit præcisè, ex quo illud deducere debeamus; & quomodo probandum sit, illa ab invicem ita pendere, vt vnum nullâ ratione possit mutari, alio immutato. Adeò vt habeamus omnes præmissas, nec aliud supersit docendum, quàm quomodo conclusio inveniatur, ... vnum quid ex multis simul implicatis dependens tam artificiosè evolvendo, vt nullibi major ingenij capacitas requiratur, quàm ad simplicissimam illationem faciendam.

Descartes indicates in the same article of Reg. XII that one is to start by dealing with perfectly understood quæstiones to get into a position to tackle methodically those quæstiones that are understood but imperfectly.85 His general reason for this ordering emerges in the course of Reg. XIII and amounts to an extremely important claim that any quæstio whatever is bound to be rendered perfect in the established sense, presumably in order that its solution qualify as scientia. Thus he writes that [a]t verò in omni quæstione, quamvis aliquid debeat esse incognitum, alioqui enim frustra quæreretur, oportet tamen hoc ipsum certis conditionibus ita esse designatum, vt omnino simus determinati ad vnum quid potiùs quam ad aliud investigandum (Reg. XIII, AT X, 434–35; my emphasis).

Furthermore, having adduced several (shortly to be introduced) examples of a certain specific way of dealing with quæstiones that are inherently 84

While Descartes speaks of (im)perfectly understood quæstiones in Reg. XII, he switches to talk of (im)perfect quæstiones in Reg. XIII. I assume these two pairs of terms are used as synonyms by him in the present context. 85 Reg. XII, AT X, 429–30: “Quam divisionem [quæstionibus quæ perfectè vel imperfectè intelliguntur] non sine consilio invenimus, tum vt nulla dicere cogamur quæ sequentium cognitionem præsupponant, tum vt illa priora doceamus, quibus etiam ad ingenia excolenda priùs incumbendum esse sentimus. ... [Q]uæstiones [quæ perfectè intelliguntur] ... parùm vtiles videbuntur imperitis; moneo tamen in hac arte addiscendâ diutiùs versari debere & exerceri illos, qui posteriorem hujus methodi partem, in quâ de alijs omnibus tradamus, perfectè cupiant possidere.”

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imperfect, he indicates that from them “facilè percipitur, quomodo omnes quæstiones imperfectæ ad perfectas reduci possint, vt fusiùs exponetur suo loco” (ibid., 431).86 In any case, then, the crucial point of difference between perfect and imperfect quæstiones is the (lack of) omnino determination or delineation of a given problem, i.e.—to follow the passages quoted so far—such a determination due to which one is strictly directed to investigate “nihil ampliùs ..., quàm id quod deduci potest ex datis” (ibid.).87 Descartes is somewhat more specific in the second half of Reg. XIII as to in what the omnino determination of quæstiones is to consist:88 [D]upliciter enim hîc falli solent humana ingenia, vel scilicet aliquid ampliùs quàm datum sit assumendo ad determinandam quæstionem, vel contrà aliquid omittendo. Cavendum est, ne plura & strictiora, quàm data sint, supponamus: præcipuè in ænigmatis alijsque petitionibus artificiosè inventis ad ingenia circumvenienda, sed interdum etiam in alijs quæstionibus, quando ad illas solvendas aliquid quasi certum supponi videtur, quod nulla nobis certa ratio, sed inveterata opinio persuasit. ... Omissione verò peccamus, quoties aliqua conditio ad quæstionis determinationem requisita, in eâdem vel expressa est, vel aliquo modo intelligenda, ad quam non reflectimus ... (ibid., 435–36).

86

The locus Descartes refers to is most probably the third part of the project of the Regulæ. However, those portions of the envisaged text were probably never written. 87 The point is repeated in the quoted AT X, 434–35. The similarity of the phrases in this passage and in AT X, 431 is striking. 88 As a matter of fact, the present approach to determinatio quæstionibus is put by Descartes as an alternative to the enumeration of the genera quæstionibus, “ad determinandum, quid circa vnamquamque præstare valeamus” (ibid., 432): “Quærimus autem vel res ex verbis, vel ex effectibus causas, vel ex causis effectus, vel ex partibus totum, sive alias partes, vel denique plura simul ex istis” (ibid., 433). Descartes sets out to comment on each of these genera in ibid., 433–34 but there is a lacuna in Reg. XIII instead of the bulk of the matter (judging from Arnauld‫ތ‬s and Nicole’s reproduction of the lost passage in their La Logique ou lҲArt de Penser: contenant, outre les regles communes, plusieurs observations nouvelles propres à former le iugement, 2nd ed., Paris: Charles Savreux, 1664; reprinted in AT X, 470–75). In any case, Descartes continues the passage as follows: “Cæterùm quia, dum aliqua quæstio nobis solvenda proponitur, sæpe non statim advertimus, cujus illa generis existat, nec vtrùm res ex verbis, vel causæ ab effectibus &c., quærantur: idcirco de his in particulari dicere plura, supervacaneum mihi videtur” (Reg. XIII, AT X, 434). He sets out instead to give the account I am about to present.

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It will be noted that Descartes‫ ތ‬initial way of putting the prerequisite of omnino determination was in fact incomplete. While the caveat “vt nihil ampliùs quæratur, quàm id quod deduci potest ex datis” (ibid., 431) refers to the case of underdetermination that is treated by Descartes as the mistake of omissio in the last quoted passage, one is bound to avoid overdetermination as well. Nonetheless, once this duality is recognized, Descartes‫ ތ‬point seems clear and reasonable. Yet if one looks closer, the issue of determinatio quæstionibus is somewhat more complicated in at least one respect. The complication concerns the procedure Descartes suggests should be carried out to render inherently imperfect quæstiones perfect. In the case of inherently perfect quæstiones, the standards of inclusion and omission of the relevant determinants are clearly given eo ipso once the quæstio is proposed to a solution, so that the mistakes of under- or overdetermination are solely due to an inadequate grasp of the content of the proposed quæstio. Consider, however, the following examples that deal with inherently imperfect quæstiones concerning the respective natures of a magnet and of sound: [S]i quæratur qualis sit magnetis natura, id quod intelligimus significari per hæc duo vocabula, magnes & natura, est cognitum, à quo determinamur ad hoc potiùs quàm ad aliud quærendum, &c. Sed insuper vt quæstio sit perfecta, volumus illam omnino determinari, adeò vt nihil ampliùs quæratur, quàm id quod deduci potest ex datis: vt si petat aliquis à me quid de naturâ magnetis sit inferendum præcisè ex illis experimentis, quæ Gilbertus se fecisse asserit, sive vera sint, sive falsa; item, si petat, quid de naturâ soni judicem præcisè tantùm ex eo quòd tres nervi A, B, C, æqualem edant sonum, inter quos ex suppositione B duplò crassior est quàm A, sed non longior, & tenditur à pondere duplò graviori; C verò non quidem crassior est quàm A, sed duplò longior tantùm, & tenditur tamen à pondere quadruplò graviori, &c. Ex quibus facilè percipitur, quomodo omnes quæstiones imperfectæ ad perfectas reduci possint ... (ibid., 430–31; my emphases).

Again, in a closely related passage from Reg. XII, Descartes writes:89 [S]i petatur quid sit magnetis natura, ... qui cogitat, nihil in magnete posse cognosci, quod non constet ex simplicibus quibusdam naturis & per se notis, non incertus quid agendum sit, primò diligenter colligit illa omnia quæ de hoc lapide habere potest experimenta, ex quibus deinde deducere conatur qualis necessaria sit naturarum simplicium mixtura ad omnes illos, quos in magnete expertus est, effectus producendos; quâ semel inventâ, audacter potest asserere, se veram percepisse magnetis naturam, quantùm ab homine 89

Cf. also Reg. XIV, AT X, 439.

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& ex datis experimentis potuit inveniri (AT X, 427; my emphases).

In these passages Descartes seems to suggest quite straightforwardly that in the case of inherently imperfect quæstiones, the point of securing their perfect understanding, in so far as the issue of under- or overdetermination is concerned, cannot consist simply in “illa omnia, quæ in propositione data sunt, ordine perlustrare, rejiciendo illa, quæ ad rem non facere apertè videbimus, necessaria retinendo” (Reg. XIII, AT X, 438); rather, the standards of inclusion and omission of the relevant data are to be adduced by the inquiring mind ad hoc, the only given regulative eventually being the requirement that the quæstio at issue be perfectly understood. The significance of this simple manoeuvre should not pass unnoticed. It enables Descartes to relativize the criteria of solving a given problem successfully to a stipulatively determined set of initial data, so that the appropriate form of any quæstio is no more “find Ȇ” but “find Ȇ upon the basis of such-and-such a stipulated set of available data” (provided of course that the given set renders the quæstio perfectly understood). Furthermore, the real point of the procedure—as is clearly implied by the last sentence of the above-quoted AT X, 429—is that the successful solution to any problem thus relativized would always count as scientia in the strictest sense. This, I submit, is of vital importance for understanding how Descartes can feel entitled to insist (as he frequently does) that conclusions made upon the basis of hypotheses concerning observable phenomena or even per imitationem can as such count as scientiæ, without equivocation of the term “scientia”. This topic, however, must wait until another occasion for a detailed treatment. For the rest, I submit that the issue of the omnino determination of quæstiones provides a promising key to understanding the point of designating heuristic enumerations as sufficientes in Descartes.90 His gloss on the magnet and sound examples in Reg. XIII, AT X, 431 suggests the aforementioned issue is indeed relevant in this respect: Si magnetem examinem ex pluribus experimentis, vnum post aliud separatim percurram; item si sonum, vt dictum est, separatim inter se comparabo nervos A & B, deinde A & C &c., vt postea omnia simul sufficienti enumeratione complectar (AT X, 432; my emphasis).

90

This also seems to be the view held by Leslie Beck. He writes while commenting on Reg. XIII in his Method of Descartes, 246: “The problem is entirely specified and limited by the data enumerated.”

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Furthermore, Descartes refers to a given quæstio, once it is omnino determined, as to quæstio sufficienter intellecta in Reg. XIII, AT X, 437 (my emphasis). It is thus not until now that the interpretation of heuristic enumeration in Descartes has been accomplished. 4.3.2.3 Division to Simplest Problems To sum up the two preparatory methodical operations just discussed— the determination of the quæstiones and the relativization to an absolutum—they amount, respectively, to the exact specification of the set of both the data and the quæsita to be ordered and disposed in appropriate series (determination), and to the actual ordering and disposing of the members of the determined set in series organized through the relations of participation in a picked-up absolutum (relativization). The last preparatory methodical task introduced by Descartes consists in splitting up the obtained series into less complex ones by way of referring them to a small number of template problems. His point is clearly to simplify matters by the reduction of complex problems, related to obtained series, to a number of relatively simpler problems whose solution can be given more readily.91 We saw that Descartes‫ ތ‬abstract account of relativization to an absolute in Reg. VI is in fact derived from the algebraic instance of constructing series of continued proportionals. Furthermore, Descartes expressly relies on the same instance in his explanation of the present operation of division. He intimates that continued proportionals, once reflected upon attentively, omnes quæstiones, quæ circa proportiones sive habitudines rerum proponi possunt, involvantur, & quo ordine debeant quæri ... (Reg. VI, AT X, 385).

It was already established that questions concerning proportiones sive habitudines are generally concerned with relations of given data and quæsita to a determined absolutum in the context of preparing comparationes as ultimately the only proper cognitive operation. The last quoted claim may therefore well be read as being of perfectly universal scope. 91

Thus referring to an algebraic instance of the reduction in question, Descartes writes in Reg. XI, AT X, 410: “Ad quæ & similia qui reflectere consuevit, quoties novam quæstionem examinat, statim agnoscit, quid in illâ pariat difficultatem, & quis sit omnium simplicissimus modus; quod maximum est ad veritatis cognitionem adjumentum” (AT X, 410; the inserted solvendi is a plausible conjecture proposed in AT).

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Descartes takes a reasonable course in explaining how continued proportionals might involve literally all the questions that can be asked in any cognitive enterprise whatsoever that aims at a scientia. That is to say, he (i) offers a professedly complete list of easily solvable template problems to which a considerable portion of possible problems concerning series of continued proportionals constructed through reference to an established absolutum can be safely reduced, and (ii) he indicates how the remaining problems are to be handled in an orderly manner. Systematically, the most general distinction Descartes draws here is that between the direct and the indirect examination of a problem (or the direct and the indirect deductio of a solution to a problem)92 concerning series of continued proportionals. The direct kind coincides with the following general problem: given two ordered quantities, find a determinate number of the succeeding quantities such that a series of continued proportionals is established; in other words, the problem is to find an xi such that a : b = b : x1 = x1 : x2 = ··· = xn-1 : xn. The indirect kind, on the other hand, coincides with the general problem of finding a determinate number of the mean proportionals when the extremities of the series are given, i.e. to find an xi such that a : x1 = x1 : x2 = ··· = xn-1 : xn = xn : b. Descartes grasps perspicuously the most remarkable peculiar feature of the direct kind of examination:93 [A]dverto, non difficiliùs inventum fuisse duplum senarij, quàm duplum ternarij; atque pariter in omnibus, inventâ proportione inter duas quascumque magnitudines, dari posse alias innumeras, quæ eamdem inter se habent proportionem; nec mutari naturam difficultatis, si quærantur 3, sive 4, sive plures ejusmodi, quia scilicet singulæ seorsim & nullâ habitâ ratione ad cæteras sunt inveniendæ (Reg. VI, AT X, 385; my emphases).

Indeed, what properly specifies the direct kind of problem is the fact that the number of the unknown members of a given series is strictly irrelevant to the way of finding any single one of them: all that is needed for this is either to know the position of the sought-after unknown in a given series regardless of how long the series is or of what the values of the preceding 92

See Reg. VI, AT X, 386.17–25 and AT X, 387.3–5, respectively. Cf. Reg. XI, AT X, 409: “[S]i datæ sint prima & secunda tantùm, facile possim invenire tertiam & quartam, & cætera: quia scilicet hoc fit per conceptus particulares & distinctos.” I take the phrase Descartes uses here to express the reason of the easiness at issue to be virtually synonymous with the phrase Descartes uses for the same purpose in the quoted AT X, 385. 93

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and succeeding members of the series amount to, since it holds that xi = bi+1 : ai (for i = 1, 2, ..., ad infinitum); or else to know the value of the immediately preceding or succeeding members of a given series, since it holds that xi-1 : xi = a : b (again for i = 1, 2, ..., ad infinitum). This, then, amounts to the first sort of template problems at which Descartes‫ ތ‬dividing procedure aims. By way of contrast, the indirect kind of problem is generally distinguished, fundamentally, by the fact that knowledge of the number of the unknown members of a given series is strictly vital to the way of finding any single one of them. Each unknown can be expressed in terms ೙శభ of the knowns with the following formula: xi = ξܽ௡ାଵି௜ ܾ௜ (for i = 1, 2, ..., n), and n is ineliminable as a determinant of the general solution. Descartes himself never puts the point in such a general way but his discussion of the problems of finding one and two mean proportionals in Reg. VI strongly suggests that this is what he has in mind:94 [V]t medium proportionale inveniatur, oportet simul attendere ad duo extrema & ad proportionem quæ est inter eadem duo, vt nova quædam ex ejus divisione habeatur; quod valde diversum est ab eo, quod datis duabus magnitudinibus requiritur ad tertiam in continua proportione inveniendam. Pergo etiam & examino, datis magnitudinibus 3 & 24, vtrùm æquè facilè vna ex duabus medijs proportionalibus, nempe 6 & 12, potuisset inveniri; hîcque adhuc aliud difficultatis genus occurrit, prioribus magis involutum: quippe hîc, non ad vnum tantùm aut ad duo, sed ad tria diversa simul est attendendum, vt quartum inveniatur (AT X, 385–86).

It would therefore seem that to find the mean proportionals amounts to an individual, irreducible problem for each n. Descartes, however, is quick to show that such a conclusion is too rash. For he continues the above as follows: Licet adhuc vlteriùs progredi, & videre vtrùm, datis tantùm 3 & 48, difficilius adhuc fuisset vnum ex tribus medijs proportionalibus, nempe 6, 12 & 24, invenire; quod quidem ita videtur primâ fronte. Sed statim postea occurrit, hanc difficultatem dividi posse & minui: si scilicet primò quæratur vnicum tantùm medium proportionale inter 3 & 48, nempe 12; & postea quæratur aliud medium proportionale inter 3 & 12, nempe 6, & aliud inter 12 & 48, nempe 24; atque ita ad secundum difficultatis genus antè expositum reduci (ibid., AT X, 386).

94

Cf. also Reg. XI, AT X, 409–10.

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Indeed, it holds generally that for any odd number of unknown mean proportionals, n = 2m + 1, the problem can be reduced to that of finding one mean proportional y between the extremities a and b, and of finding m mean proportionals between a and y, and between y and b, respectively.95 This procedure can be reiterated for each m > 2, so that any problem of finding an odd number of the mean proportionals between extremities can be reduced to an ordered series of problems of finding one or two mean proportionals. We thus arrive at two more template problems Descartes is able to provide to carry out his dividing preparatory procedure. Unfortunately, no similar reduction seems forthcoming with regard to problems regarding finding any even number n > 2 of the mean proportionals. All one can hope for here is to find a general method of ordered resolution of equations of the corresponding degrees. It is one of Descartes‫ ތ‬chief tasks in Geom. III to meet this challenge. We need not pursue this issue further. To sum up, Descartes has succeeded in showing how one might split up considerable portions of all possible problems in the field of mathematical algebra: having ordered and disposed both the knowns and the unknowns in series of continued proportionals, he has shown how certain questions embodied in these series can be referred to a small number of template problems, namely the problem of finding the next quantity given two ordered quantities, the problem of finding one mean proportional between the extremities, and the problem of finding two mean proportionals between the extremities. What has proven resistant to such a dividing procedure are problems of finding any even number of the mean proportionals greater than two. Having thus finished a detailed examination—heavily corroborated with algebraic instances—of the anatomy of the professedly universal method suggested by Descartes, the time has now come to turn to the aforementioned larger issues concerning the status of the suggested method, viz. the questions of the possibility, justification, and conditions of extension beyond the realm of mathematics. These issues will occupy us for the rest of the present chapter.

4.4 Justification and Possibility of Method Like any substantial methodological proposal, Descartes’ method as just described is bound to meet a rudimentary demand for its justification: one is certainly entitled to ask why the method Descartes proposes should 95

I owe this generalization to Fiala, “KomentáĜe,” 240.

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be preferred to other complexes of prescripts and guidelines devised to facilitate scientific discoveries. Descartes somewhat muddies the waters here with his conspicuous claim that his “dessein n’est pas d’enseigner ... la Methode que chascun doit suiure pour bien conduire sa raison, mais seulement de faire voir en quelle sorte i’ay tasché de conduire la miene” (DM 1, AT VI, 4).96 Yet such a qualification eventually makes the justificatory demand no less urgent: the demand must still be confronted at least by Descartes himself, and also, for that matter, by anyone who wishes to follow (or “imitate”) him. Another challenge, in a sense more fundamental, that Descartes is obliged to confront in this context is the question of how, given his commitments, it is possible to come up with any method whatsoever (including the one he himself proposes). For if any correctly devised method is to count as a scientia, then such a method is to be invented by means of the employment of intuitus and simple deductiones alone; but these operations are, as we saw, essentially beyond any regulation, which implies, in particular, that no method and no instructions are available to facilitate such an invention; as a consequence, one may wonder how any method could ever be established under such circumstances. The problem is particularly pressing for anyone who, like Descartes, despises the pretensions of haphazard trial-and-error procedures to count as acceptable approaches to the establishment of scientiæ.97 The two aforementioned tasks might well be deemed disparate as they stand; indeed, we saw that Descartes himself is occasionally content to justify his version of the method ex post, viz. just to claim that the method he offers has proven as faring better than any other existing method,98 a justification which is no doubt entirely independent of anything having to do with the latter challenge concerning the possibility of any method. Yet at the end of the day, such a strategy can by no means bear the burden of the justification demanded of Descartes, due to the above view of his that trial-and-error procedures are incapable of establishing anything valuable in scientiæ. This seems to be at least a part of what prompted Descartes to 96

Cf. also e.g. DM 2, AT VI, 15: “Iamais mon dessein ne s’est estendu plus auant que de tascher a reformer mes propres pensées, & de bastir dans vn fons qui est tout a moy. Que si, mon ouurage m’ayant assez pleu, ie vous en fais voir icy le modelle, ce n’est pas, pour cela, que ie veuille conseiller a personne de l’imiter.” 97 See in particular Reg. IV, AT X, 371: “[L]ongè satius est, de nullius rei veritate quærendâ vnquam cogitare, quàm id facere absque methodo: certissimum enim est, per ejusmodi studia inordinata, & meditationes obscuras, naturale lumen confundi atque ingenia excæcari; & quicumque ita in tenebris ambulare assuescunt, adeò debilitant oculorum aciem, vt postea lucem apertam ferre non possint ....” 98 Cf. in particular Mers., AT I, 478.

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tie the two tasks in question closely together in a passage where he takes them up most explicitly, viz. in the well-known blacksmith analogy in Reg. VIII. The passage is worth quoting at length:99 Hæc methodus siquidem illas ex mechanicis artibus imitatur, quæ non aliarum ope indigent, sed tradunt ipsæmet quomodo sua instrumenta facienda sint. Si quis enim vnam ex illis, ex. gr., fabrilem vellet exercere, omnibusque instrumentis esset destitutus, initio quidem vti cogeretur duro lapide, vel rudi aliquâ ferri massâ pro incude, saxum mallei loco sumere, ligna in forcipes aptare, aliaque ejusmodi pro necessitate colligere: quibus deinde paratis, non statim enses aut cassides, neque quidquam eorum quæ fiunt ex ferro, in vsus aliorum cudere conaretur; sed ante omnia malleos, incudem, forcipes, & reliqua sibi ipsi vtilia fabricaret. Quo exemplo docemur, cùm in his initijs nonnisi incondita quædam præcepta, & quæ videntur potiùs mentibus nostris ingenita, quàm arte parata, poterimus invenire, non statim Philosophorum lites dirimere, vel solvere Mathematicorum nodos, illorum ope esse tentandum: sed ijsdem priùs vtendum ad alia, quæcumque ad veritatis examen magis necessaria sunt, summo studio perquirenda; cùm præcipuè nulla ratio sit, quare difficilius videatur hæc eadem evenire, quàm vllas quæstiones ex ijs quæ in Geometriâ vel Physicâ alijsque disciplinis solent proponi (AT X, 397).

The instrumenta in the analogical case correspond, of course, to methods; and the chief point of the analogy is, of course, that just as once an adept of the blacksmith’s craft obtains his provisional instruments out of natural materials, he should first of all use them not to fabricate what a blacksmith is properly supposed to produce for the use of others, but rather to fabricate all the instruments which are necessary for the adept himself to run his craft in a fully-fledged manner, so once an adept of the scientiæ obtains his provisional methodical precepts out of something analogous to natural materials, he should first of all use them not to attain scientiæ in any discipline outside the method itself, but rather to employ them to work out the fully-fledged method which is necessary for attaining truly scientific results in any discipline whatsoever beyond the method itself.100 99

At AT X, 397.21, I read “ijsdem” for “ijdsem”. There are several hints at this thought elsewhere in the Regulæ. Cf. in particular Reg. VI, AT X, 384: “Notandum ... studiorum initia non esse facienda à rerum difficilium investigatione; sed, antequam ad determinatas aliquas quæstiones nos accingamus, priùs oportere absque vllo delectu colligere spontè obvias veritates, & sensim postea videre vtrum aliquæ aliæ ex istis deduci possint, & rursum aliæ ex his, atque ita consequenter.” Also cf. Reg. X, AT X, 403–404: “[Q]uod me vnum

100

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What are of interest regarding our present questions—i.e. how any method is possible at all and how one might justify the method Descartes recommends—are certain less obvious aspects of the analogy at issue. As for the question of possibility, the situation invoked by Descartes of a would-be blacksmith destitute of any instruments corresponds perfectly to the situation of a would-be scientist101 destitute of any method; so Descartes’ account of how it is possible to obtain any instruments at all for a would-be blacksmith (which can easily be interpolated from the analogy at issue) could help to account for how it is possible to obtain any method at all for a would-be scientist. Of the relevant elements of the blacksmith case, perhaps the most obvious is that what is to be used to fabricate the provisional tools are materials one finds ready-made in nature (stones, sticks etc.). Now, the analog of natural materials in the scientist’s case is, I suggest, the suitable ready-made intuitus and simple deductiones one finds in one’s own mind, i.e. the operations of a bona mens; it is therefore these operations that are to be employed in establishing simple problem-solving procedures or patterns. What is crucial is that the incondita præcepta one obtains as the analog of the blacksmith’s provisional tools are characterized by Descartes as “potiùs mentibus nostris ingenita, quàm arte parata” in the quoted AT X, 397. It is this, of course, which opens the door for invoking a kind of innateness as a fundamental factor which is absolutely indispensable if one is ever to come up with any real method whatsoever. Yet more than this is needed to establish even the most salient requirements of possibility in both the blacksmith and the scientist cases. To stretch the blacksmith analogy just a bit, a would-be blacksmith must be able, in the first place, to discern which of the vast range of natural items around him might serve his purpose of fabricating the required provisional tools; and any account of in what such an ability is grounded cùm juvenem adhuc ad scientias addiscendas allexisset, quoties novum inventum aliquis liber pollicebatur in titulo, antequam vlteriùs legerem, experiebar vtrùm forte aliquid simile per ingenitam quamdam sagacitatem assequerer .... Quod toties successit, vt tandem animadverterim, me non ampliùs, vt cæteri solent, per vagas & cæcas disquisitiones, fortunæ auxilio potiùs quàm artis, ad rerum veritatem pervenire; sed certas regulas, quæ ad hoc non parùm juvant, longâ experientiâ percepisse, quibus vsus sum postea ad plures excogitandas. Atque ita hanc totam methodum diligenter excolui, meque omnium maximè vtilem studendi modum ab initio sequutum fuisse mihi persuasi.” 101 I just remind the reader that I do not use the word “scientist” in the narrow sense which is common nowadays but as a short form of “possessor of scientiæ,” scientia being understood in the sense established in the course of the present study.

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will presumably draw, in some manner, upon preliminary understanding, or insight, on the part of the would-be blacksmith, into (i) the nature of what his craft is properly supposed to produce, (ii) the nature of the material(s) with which he is to work and (iii) the way in which the available natural materials might serve. By way of analogy, any account of the possibility of a method that could serve Descartes’ purpose must draw upon preliminary insights, on the part of a would-be scientist, into (i’) the nature of the scientiæ in general, (ii’) the nature of the mental operations the envisaged method is to regulate, and (iii’) the way in which the available bona mens cognitive operations—intuitus and simple deductiones—might serve. Once again, it is innateness to which one ultimately seems bound to have recourse in order to build all these insights upon firm foundations. So I submit it is all these facets, in addition to the innateness of intuitus and simple deductiones one initially finds in one’s own mind, that Descartes is committed to take into account in the context of grounding any method in general, and his own method in particular. Furthermore, by implication I submit that all these facets are also to be involved in Descartes’ talk, both in Reg. IV and several other significant places,102 of “prima quædam veritatum semina humanis ingenijs à naturâ insita” (Reg. IV, AT X, 376) and of “nescio quid divini, in quo prima cogitationum vtilium semina ... jacta sunt” (ibid., 373), these semina being identified with certain “ingenita methodi principia” (ibid.).

4.4.1 The Justification Task We shall soon see how the founding of method in innate structures is connected with the idea of the universality of the method in Descartes’ thought via his notion of bona mens. For the moment, however, let us proceed to the task of justification with regard to Descartes’ method. I have claimed that Descartes ties this task closely to the above task of establishing that a method is possible; and we are by now in a position to understand in what the alleged connection consists. We have just seen that Descartes tries to secure the possibility of any method through invoking the innate capacities of the bona mens to provide a set of incondita præcepta which are then employed in a certain way to attain a fullyfledged method.103 Now, I suggest that to render possible any justification 102

Cf. in particular DM 6, AT VI, 64: “certaines semences de Veritez qui sont naturellement en nos ames.” 103 It is, I suggest, in view of this figure that Descartes’ claim that his disciplina quædam alia (i.e. his envisaged universal method) “prima rationis humanæ rudimenta continere ... debet” (Reg. IV, AT X, 374) is to be understood.

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of his method, Descartes seems to insist that (i) a particular way in which the incondita præcepta are to be employed to attain a fully-fledged method is by no means arbitrary, and that (ii) the fully-fledged method he intends to present is simply the result of such a non-arbitrary and controlled employment of the incondita præcepta at issue. This, I suppose, is what the following passage (as well as the other two kickoffs in Reg. VIII concerning the extent and limits of human cognition),104 which is placed 104

Cf. in particular ibid., AT X, 395–96: “Si quis pro quæstione sibi proponat, examinare veritates omnes, ad quarum cognitionem humana ratio sufficiat (quod mihi videtur semel in vitâ faciendum esse ab ijs omnibus, qui seriò student ad bonam mentem pervenire), ille profectò per regulas datas inveniet nihil priùs cognosci posse quàm intellectum, cùm ab hoc cæterorum omnium cognitio dependeat, & non contra; perspectis deinde illis omnibus quæ proximè sequuntur post intellectûs puri cognitionem, inter cætera enumerabit quæcumque alia habemus instrumenta cognoscendi præter intellectum, quæ sunt tantùm duo, nempe phantasia & sensus.” Also ibid., AT X, 396–97: “Atqui ne semper incerti simus, quid possit animus, neque perperam & temerè laboret, antequam ad res in particulari cognoscendas nos accingamus: oportet semel in vitâ diligenter quæsivisse, quarumnam cognitionum humana ratio sit capax. Quod vt meliùs fiat, ex æquè facilibus, quæ vtiliora sunt, semper priora quæri debent.” It has been a widely acknowledged fact since the publication of Weber’s Constitution des Regulæ that Reg. VIII consists (like in particular Reg. IV) of several textual layers. The first paragraph of Reg. VIII in the AT edition (AT X, 392.14–393.2; Reg. VIIIA in Weber; I will use Weber’s nomenclature for the layers of Reg. VIII from now on) is in fact (as I already remarked in ch. 2) a misplaced continuation of Reg. VII (cf. Weber, Constitution des Regulæ, 87–91). The precept of Reg. VIII (AT X, 392.10–14) and the second paragraph (ibid., 393.3–21), jointly named Reg. VIII-B in Weber, are most probably integral parts of the initial draft of the opening portions of the Regulæ composed by Descartes in 1619–20. Paragraphs three to seven of Reg. VIII in the AT edition (AT X, 393.22–397.26; Reg. VIII-C in Weber) contain the famous anaclastic line example and the first two versions of the omnium nobilissimum exemplum (ibid., 395) concerning the extent and limits of human cognition; the two passages quoted in this footnote are the opening sentences of the first and the second version, respectively, and the above blacksmith analogy is a development of the second version. In the H-version (unlike, however, the A- and N-versions) of the Regulæ these portions of the text (i.e. Weber’s VIII-C) are withdrawn from the text of Reg. VIII and set at the end of the Regulæ as an appendix. Finally, the remaining four paragraphs of Reg. VIII in the AT edition (AT X, 397.27–400.11; Reg. VIII-D in Weber) contain the clearest (and most probably the final) version of the omnium nobilissimum exemplum as well as hints at the topics Descartes takes up later in Reg. XII; the quoted AT X, 397–98 is the opening passage of this final version of the exemplum at issue. I should emphasize that although I accept (with some minor reservations) this partition of the layers of Reg. VIII, I do not endorse Weber’s influential

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next to the last quoted AT X, 397 in the AT edition, is intended to insinuate: At verò nihil hîc vtilius quæri potest, quàm quid sit humana cognitio & quousque extendatur ... idque semel in vitâ ab vnoquoque ex ijs, qui tantillùm amant veritatem, esse faciendum, quoniam in illius investigatione vera instrumenta sciendi & tota methodus continentur (ibid., 397–98; my emphasis).

Although the fact that this passage opens the Reg. VIII-D layer of the text renders the case less than perfectly certain, I take the hîc in its opening clause as intended by Descartes to refer exactly to what the last sentence of Reg. VIII-C appeals, namely to alia, quæcumque ad veritatis examen magis necessaria sunt, summo studio perquirenda (ibid., 397) with the help of the incondita præcepta of the provisional method. Once the supposition is adopted, various elements relevant for the present question of justification suddenly fall into place. In particular, the vtilitas the quæstio under investigation is to exhibit is then an vtilitas for attaining a method in its fully-fledged form, and the content of the quæstio is then to be derived from what the envisaged method is to achieve, i.e. for what it is properly designed. Now we already know what these goals are: in the régime of approbation, it is “ingenij directio ad solida & vera, de ijs omnibus quæ occurrunt, proferenda judicia” (Reg. I, AT X, 359); in the régime of discovery, it is “ad veram cognitionem eorum omnium quorum [quicumque] erit capax pervenire” (Reg. IV, AT X, 372). To devise a fullyfledged method capable of achieving the former approbationary task, it is therefore necessary above all to clarify—arguably with the help of the præcepta incondita—how the de ijs omnibus quæ occurrunt judicia are constituted and how the cognitive faculties with which men are endowed are involved in them, i.e., in brief, quid sit humana cognitio; and similarly to devise a fully-fledged method with regard to the latter heuristic task, it is necessary above all to clarify what falls within and without the interpretation of the particular layers as embodying considerably separate and dramatically different stages of Descartes’ intellectual development concerning the nature of his overall philosophical project and the rôle of method in it. By implication, I reject the more recent developmental theses that build upon Weber’s interpretation, proposed most prominently by Garber and Schuster, to the effect that the development of layers VIII-B to VIII-D mirrors Descartes’ development that culminates in rejection of the method as sketched in Reg. I–VII to Reg. VIII-B. I briefly deal with Garber’s and Schuster’s readings later in the present section, and also (indirectly) in sec. 4.5.

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limitations of human cognition, i.e. quousque humana cognitio extendatur. A fully-fledged method, i.e. Descartes’s universal method in its final form, is then to be understood predominantly as a result of the controlled employment of the provisional methodical tools (the præcepta incondita) in addressing these two “most useful” questions; and the wonder as to how it is that no-one before Descartes had come up with a method of a similar nature in spite of the fact that both questions at issue had been stated and dealt with throughout the entire philosophical tradition and that the tools (as well as the know-how as to their employment) for the correct resolutions are declared innate, might then be satisfied in terms of a failure to employ the tools appropriately, presumably above all due to “communi quodam gentis humanæ vitio” (Reg. II, AT X, 362) which causes that people “cognitiones ... nimis faciles & vnicuique obvias ... reflectere neglexerunt” (ibid.).105 Although perhaps put more explicitly than usual in the literature, I take the proposed interpretation as virtually incontrovertible as far as it goes. Yet some immediately relevant explanatory gaps remain unclosed, as well as some momentous interpretative issues. As for the explanatory gaps, perhaps the most significant of them concerns the fact that Descartes still owes us a reasonably accurate account of exactly how addressing the two vtiles quæstiones of AT X, 397 issues in a fully-fledged method, whatever that is. This is a topic I shall not venture to pursue here. In any case, there is an interpretative issue which would have to be decided first if one were even to determine where in Descartes the materials relevant to addressing the aforementioned explanatory task are to be sought: namely the question of what class of precepts Descartes intends to be referred to by the præcepta incondita in his drawing the moral from the blacksmith analogy, and—correlatively—what class of precepts Descartes wishes to associate with his envisaged fully-fledged method in these contexts. This issue must be taken up now, however briefly, as it has immediate bearing upon the overall argument of the present study. My own stance on the present question should be clear from what I have said so far in the present chapter concerning the issue of the project of a universal method in Descartes and from the ways in which I have attempted to reconstruct the principal tenets of the method and certain details of the procedures of which it is to consist. That is to say, I hold that 105

Cf. also e.g. ibid., 364: “[M]ulti faciunt, quæcumque facilia sunt negligentes, & nonnisi in rebus arduis occupati ...” And Reg. V, AT X, 380: “[M]ulti ... sæpe adeò inordinatè difficillimas examinant quæstiones, vt mihi videantur idem facere, ac si ex infimâ parte ad fastigium alicujus ædificij vno saltu conarentur pervenire, vel neglectis scalæ gradibus, qui ad hunc vsum sunt destinati, vel non animadversis.”

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the gros of the envisaged fully-fledged method, which Descartes himself announces in the Regulæ and publicly in the Discours and of which he still speaks in the Meditationes and the conversation with Burman, coincides with an appropriately elaborated complex of procedures introduced and described in Reg. V–VII (which, for the sake of brevity, I will henceforth call the “default method” in the rest of the present subsection); and that— correlatively—the præcepta incondita are intended to refer to a class of very rudimentary precepts or advice one derives from those operations of intuitus, immediate deductio, and their efficient ordering which one simply finds oneself using in everyday activities, both practical and theoretical; Descartes seems to refer to these operations with the umbrella terms “perspicacitas” and “(ingenita quædam) sagacitas” in the Regulæ. 106 On my reading, then, the proper reason why the initial præcepta of the method count as incondita or provisional is that they are underdeveloped in terms of the conceptual articulation of what exactly is going on in everyday use 106

The terms are introduced in the most disciplined way in Reg. IX, AT X, 400: “[P]ergimus ... explicare, quâ industriâ possimus aptiores reddi ad [operationes intuitu & deductionis] exercendas, & simul duas præcipuas ingenij facultates excolere, perspicacitatem scilicet, res singulas distinctè intuendo, & sagacitatem, vnas alijs artificiosè deducendo.” Perspicacitas is then exemplified thus: “Artifices illi, qui in minutis operibus exercentur, & oculorum aciem ad singula puncta attentè dirigere consueverunt, vsu capacitatem acquirunt res quantumlibet exiguas & subtiles perfectè distinguendi; ita etiam illi, qui varijs simul objectis cogitationem nunquam distrahunt, sed ad simplicissima quæque & facillima consideranda totam semper occupant, fiunt perspicaces” (ibid., 401). Similarly, sagacitas is exemplified in Reg. X as follows: “Verùm, quia non omnium ingenia tam propensa sunt à naturâ rebus proprio marte indagandis, hæc propositio docet ..., levissimas quasque artes & simplicissimas priùs esse discutiendas, illasque maximè, in quibus magis ordo regnet, vt sunt artificum qui telas & tapetia texunt, aut mulierum quæ acu pingunt, vel fila intermiscent texturæ infinitis modis variatæ; item omnes lusus numerorum & quæcumque ad Arithmeticam pertinent, & similia: quæ omnia mirum quantùm ingenia exerceant, modò non ab alijs illorum inventionem mutuemur, sed à nobis ipsis. Cùm enim nihil in illis maneat occultum, & tota cognitionis humanæ capacitati aptentur, nobis distinctissimè exhibent innumeros ordines, omnes inter se diversos, & nihilominus regulares, in quibus rite observandis fere tota consistit humana sagacitas” (AT X, 404). The expression “ingenita quædam sagacitas” occurs in a relevant context in Reg. X, AT X, 403. Cf. also Reg. VI, AT X, 384; 387; Reg. XII, AT X, 428. My present interpretation is further corroborated by the fact that it is the præcepta incondita that Descartes seems to have in mind when he speaks, in Reg. XII, of “non alia præcepta ... quàm quæ vim cognoscendi præparant ad objecta quævis distinctiùs intuenda & sagaciùs perscrutanda, quoniam hæ sponte occurrere debent, nec quæri possunt” (ibid., 428).

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of the faculties of perspicacitas and sagacitas—an articulation general enough to allow for proposing the prescripts of a method under development in the form of a set of serviceable instructions; in short, an articulation instantiated by the explications under the headings of relativization to an absolute, determination of the quæstiones, and division to simplest problems (all this being in the service of establishing ordo and dispositio allowing for appropriate comparationes) as presented above.107 Likewise, those explications should at least provide clues and hints to how the desired articulation might be facilitated by proto-methodically conducted reflections upon the questions quid sit humana cognitio & quousque extendatur. At least two influential recent commentators, John Schuster and Daniel Garber, have advanced an alternative interpretation of to what the provisional and the fully-fledged stages of the method correspond in the moral of the blacksmith analogy in the context of Reg. VIII-D. According to them, the præcepta incondita in Reg. VIII are to be identified with exactly what I have just claimed coincide with the gros of the fullyfledged method, viz. with the more or less elaborated set of procedures of the default method.108 The chief argument both these authors adduce in support of such an interpretation turns upon their reading of the blacksmith 107

This account of mine—of precisely what renders the initial precepts provisional—differs from the account of Patrick Brissey, who has recently defended, in his “Rule VIII of Descartes’ Regulae ad directionem ingenii,” Journal of Early Modern Studies 3, no. 2 (2014): 9–31, a similar interpretation, in several respects, regarding with which class of precepts Descartes envisages the fullyfledged method being associated. According to Brissey, what counts as provisional in the blacksmith analogy is not at all the procedures of the method but rather the objects upon which the methodical procedures are to operate, i.e. (as Brissey puts it) “simple ideas” (ibid., 23) or “absolute natures” characterized by him as “general features of reality ... suitable to deduce answers in the sciences” (ibid., 24). I am not sure if Brissey’s proposal amounts to a real alternative to my reading as he exhibits no interest in how the method sketched chiefly in Reg. V–VII is to be explicated. I am inclined to think it does not, being rather a supplement to my proposal (just as mine might be taken as a supplement to his). 108 Cf. in particular Garber, Descartes’ Metaphysical Physics, 41: “One lesson Descartes draws from [the blacksmith] analogy is that, like the provisional tools that the would-be blacksmith has fashioned out of sticks and stones, things immediately at hand, the rules of method previously laid down must be regarded as being merely provisional.” Schuster, “Descartes and the Scientific Revolution,” 457: “The early Rules are thus demoted to ‘rough precepts’, despite the fact that Rule 7 indicated that the whole of the method consists in the heuristic aids included in the first few rules.”

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analogy and of the VIII-D version of the omnium nobilissimum exemplum as an expression, or illustration, of a substantial shift in Descartes’ assessment of the scope and epistemological legitimacy of the method sketched in the earlier rules, a shift that allegedly took place towards the end of the 1620s and that contributed considerably, in their view, to Descartes’ decision to abandon his work on the Regulæ: in short, the reason for relegating that method to the status of mere præcepta incondita is—so the present interpretation goes—Descartes’ recognition, towards the end of the 1620s, that it is severely limited in scope and epistemologically unwarranted in its foundations. The developmental assumption, which underlies this generic interpretation, effectively nullifies any textual evidence to the contrary occurring in the Regulæ or in the autobiographical narrative of DM 2: any such evidence can always be gerrymandered, of course, to pertain to the earlier, “uncritical” period of Descartes’ dealing with method.109 Yet since there seems to be no independent textual evidence that would provide direct and unambiguous positive support to the reading in question,110 and since the onus probandi lies quite clearly upon the shoulders of its proponents, its plausibility depends entirely upon the credentials of the alleged connection between Descartes’ alleged worries concerning the 109

This holds, in particular, with regard to Descartes’ statement of the four precepts in DM 2 and to an important passage in Reg. XIII, AT X, 432: “Additur ... difficultatem esse ad simplicissimam reducendam, nempe juxta regulas quintam & sextam, & dividendam juxta septimam .... Atque hæc tria tantùm occurrunt circa alicujus propositionis terminos servanda ab intellectu puro, antequam ejus vltimam solutionem aggrediamur, si sequentium vndecim regularum vsu indigeat ...” (my emphasis). Such manoeuvres seem to me instances of what Doyle, “How (not) to study Descartes’ Regulae,” 5–6 has pinpointed as a “viciously selfconfirming” character of any “patchwork theory” (of which the developmental theory now at issue is an instance). 110 Schuster, “Descartes‫ ތ‬Mathesis Universalis,” 58–59 seems to imply that the opening passage of Reg. VIII-D could be taken as such a positive support: he puts emphasis on the fact that according to that passage the seminal questions quid sit humana cognitio & quousque extendatur (AT X, 397) should be examined per regulas jam antè traditas (ibid., 398; in the translation by Elizabeth Haldane and G. R. T. Ross [Cambridge: Cambridge University Press, 1911] used by Schuster: “with the aid of the rules which we have already laid down” [ibid., 26]). This sole textual point in support of the interpretation under investigation, powerful as it is at first glance, could be explained away through insisting that numerous elements which properly constitute the rudimentary or provisional layer of the envisaged fully-fledged method are introduced in Reg. I–VII along with the sketch of the fully-fledged method in Reg. V–VII, and that it is only those rudimentary elements Descartes has in mind in the invoked AT X, 397–98.

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scope and warrant of the method described in Reg. V–VII and DM 2 on the one hand and on the other, Descartes’ alleged relegation of that version of the method to mere præcepta incondita. I find these credentials inadequate to justify abandoning the standpoint I have endorsed above. As to the alleged worry concerning the scope of the default method, Garber diagnoses it as eventually hinging upon Descartes’ alleged recognition that the default method, being made possible (among other things) by assuming the interconnectedness of all the scientiæ but being devised, by the same token, for dealing just with disparate, individual quæstiones, is eventually ill-suited to attaining the declared goal of the entire scientific enterprise, viz., as Garber puts it, “to encompass everything capable of being known” (“Descartes and Method in 1637,” 48). For, Garber maintains, “if all knowledge is interconnected, then what we should be doing is not solving individual problems, but constructing the complete system of knowledge” (ibid.). Setting to one side the fact that there seems to be no warrant for Garber’s crucial claim that the default method is suited only to the resolution of individual quæstiones and as such cannot do justice to the interconnectedness of all the scientiæ,111 Garber’s thesis fits poorly with Descartes’ declaration in the highly authoritative Resp. 2 that “[e]go ... solam Analysim, quæ vera & optima via est ad docendum, in Mediationibus meis sum sequutus” (AT VII, 156):112 for although the method of analysis Descartes mentions here is described by him, as we saw in ch. 3, in much the same terms as it was earlier in the Regulæ, Garber still presents the Meditationes as an eminent fruit of Descartes’ alleged “systematic” approach, a fruit which does not display the default method.113 There seems to be no worry concerning the scope of the default method, at least none such as would be needed to effectively motivate the alleged relegation of the default method to mere præcepta incondita. As for the other alleged worry concerning epistemological foundations, as far as I can see the situation is quite similar. Both Garber and Schuster, 111

I believe that my positive account of Descartes’ notion of the unity of the scientiæ (see sec. 4.5) implies there is no tension between the modus operandi of the default method on the one hand and the professed interconnectedness of the scientiæ on the other. For an alternative way of rebutting Garber’s present point, see Florka, “Problems with the Garber-Dear Theory,” 133–37. 112 Cf. also Med., Epistola, AT VII, 3: “[Q]uoniam nonnulli quibus notum est me quandam excoluisse Methodum ad quaslibet difficultates in scientiis resolvendas, non quidem novam, quia nihil est veritate antiquius, sed quâ me sæpe in aliis non infœliciter uti viderunt, hoc a me summopere stagitarunt: ideoque officii mei esse putavi nonnihil hac in re conari.” 113 See in particular Garber, “Descartes and Method in 1637,” 45–47.

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each in his own way, identify as the essential point that while the default method just assumes the veracity of intuitus and deductiones, Descartes later recognized that their veracity itself needs foundation, a shift which resulted in one way or another—so the respective stories go—in the abandonment of the default method by the mature Descartes.114 I maintain that put in this way, the epistemological worry Descartes did entertain after he dropped his work on the Regulæ is misstated at best. For while it is indeed undeniable, as we saw in ch. 2, that with regard to the issue of the veracity of the elementary cognitive operations capable of yielding scientiæ, the situation of the Regulæ differs considerably from that of the mature writings, I have argued extensively in the closing sections of ch. 1 that even in the mature Descartes, neither intuitus nor (simultaneously apprehended sequences of) immediate deductiones are in need of any further validation in so far as these operations are actually executed; and that an ultimate validation—the divine guarantee—is needed exclusively in so far as those metaphysically certain deliverances are represented in the corresponding veracious recollections. However, if this is so then there is every reason to consider the entire machinery of embedding the epistemological foundations of scientiæ in the divine guarantee as just a development of and not a replacement for the default method.115 Once again, therefore, the epistemological worry, once put right, fails to motivate the alleged relegation of the default method.

4.5 Universality of the Method and the Unity of the Scientiæ It should be clear by now in which respect Descartes’ tantalizing vision of a perfectly universal method is relevant to the central topic of the present study, viz. Descartes’ notion of the a priori and the a posteriori: 114

See in particular Garber, Descartes’ Metaphysical Physics, 54–57; idem, “Descartes and Method in 1637,” 49–50; Schuster, “Descartes‫ ތ‬Mathesis Universalis,” 70–79. My brief account is based upon Garber’s treatment, but a similar position can arguably be extracted from Schuster’s extremely complex and multi-layered account. 115 Patrick Brissey arrives at a similar conclusion in his “Rule VIII of Descartes’ Regulae,” 25–26. Interestingly, he concedes to Garber (and presumably to Schuster) that even intuitus is in need of validation from the perspective of the mature Descartes, and adduces independent arguments to substantiate his rejection of Garber’s thesis under that concession (see ibid., 26). As far as I can see, these arguments of his are compatible with my overall position and therefore further strengthen my present case.

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given the above-established crucial interpretative suggestion that Descartes employs the a priori as a characterization of exactly those cognitions which are arrived at by way of analysis as the only adequate method of discovery, the extent to which Descartes is able to establish that the analytic procedure, derived from the problem-solving methodology of his version of general algebra, is applicable at least in principle beyond the realm of mathematics determines the possible scope of the applicability of the term “a priori” in Descartes’ established peculiar usage. I will focus on just one of the numerous aspects of the complex and controversial topic of the envisaged universality of the method in Descartes, an aspect which seems to bear most directly upon the central topic of the present study: that is to say, the connection of Descartes’ notion of universal method with his claim of a certain substantive unity of all the scientiæ.116 This topic will help to fulfil the task of embedding Descartes’ pursuit of universal method—and by implication Descartes’ notion of the a priori and the a posteriori—in the robust fabric of his thought on the nature and conditions of scientia. A plausible case can be made to show that Descartes’ commitment to the universality of his method of discovery is essentially connected to his commitment to a peculiar unity of all the scientiæ. To put it briefly, I take 116

In particular, I set to one side the hotly disputed issue of how the algebraic procedures which serve as a privileged paradigmatic basis for the explication of the allegedly general methodical treatment in Descartes might possibly be extended beyond the purely quantitative realm of mathematics. It is one thing to ask what Descartes’ project of universal method actually was and how Descartes hoped the universal method could have been constituted; and it is quite another thing to ask whether or to what extent Descartes succeeded in executing that project; and it is only the former cluster of issues that is immediately relevant to the leading topic of determining the meaning of the terms “a priori” and “a posteriori” in Descartes. Furthermore, it seems that the only viable strategy in addressing the latter cluster of issues (provided, of course, that one believes the extension beyond mathematics amounts to be a possible feat; for the most elaborated standpoint to the contrary, see Schuster, Descartes-Agonistes, especially ch. 6) is to examine in detail Descartes’ extant physical and/or metaphysical writings and either to reconstruct Descartes’ achievements embodied in them as instances of the application of a universal method in a given domain (for the most remarkable attempts to this effect, see Charles Serrus, La méthode de Descartes et son application à la métaphysique, Paris: Félix Alcan; Beck, Method of Descartes, ch. 18; Schouls, Imposition of Method, ch. 4–5; Daniel Flage and Clarence Bonnen, Descartes and Method: A Search for a Method in Meditations, London: Routledge, 1999), or— less ambitiously but more promisingly—to trace more or less scattered and disparate elements of such an application in those writings. This is clearly a task huge enough to deserve a separate, dedicated book.

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it as firmly established in the preceding chapters that according to Descartes any scientia whatsoever is facilitated either exclusively through the operations of intuitus and immediate deductio, which are essentially beyond any regulation, or through the chains of complex deductiones which require, besides the operations of intuitus and immediate deductio, a regulation amounting to an application of the appropriate method. The crucial point of Descartes’ conception of the unity of the scientiæ is arguably (as will be shown in a moment) the claim that the cognitive operations involved in both these cases are, in the relevant respects, strictly uniform regardless of the nature of their particular objects (which henceforth I will term “the Uniformity Claim”, using “Uniformity” to refer to the uniformity which the Uniformity Claim is about). However, it seems to be beyond dispute that if even those cognitive operations that are in need of regulation count as strictly uniform, then the regulation itself should proceed in a strictly uniform manner regardless of the nature of the particular content upon which the operations to be regulated are employed; in other words, that the method supplied to facilitate any scientia should count as strictly universal. What most must be done to substantiate the argument sketched above is, of course, to establish and clarify the Uniformity Claim; and to show how the uniformity at issue is connected with the nature of the alleged peculiar unity (yet to be specified) of the scientiæ. This should help to highlight, from yet another angle, the groundbreaking nature of Descartes’ overall conception of cognition and scientia. The best place to start is the following passage from Reg. I, AT X, 360–61: [Homines scientias] pro diversitate objectorum ab invicem distinguentes, singulas seorsim & omnibus alijs omissis quærendas esse sunt arbitrati. In quo sanè decepti sunt. Nam cùm scientiæ omnes nihil aliud sint quàm humana sapientia, quæ semper vna & eadem manet, quantumvis differentibus subjectis applicata, nec majorem ab illis distinctionem mutuatur, quàm Solis lumen à rerum, quas illustrat, varietate, non opus est ingenia limitibus vllis cohibere ... Et profectò mirum mihi videtur, plerosque hominum mores, plantarum vires, siderum motus, metallorum transmutationes, similiumque disciplinarum objecta diligentissimè perscrutari, atque interim fere nullos de bonâ mente, sive de hac vniversali Sapientiâ, cogitare, cùm tamen alia omnia non tam propter se, quàm quia ad hanc aliquid conferunt, sint æstimanda... Credendumque est, ita omnes [scientiæ] inter se esse connexas, vt longè facilius sit cunctas simul addiscere, quàm vnicam ab alijs separare (my emphases).

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Although the passage will turn out somewhat vague in certain important respects, so much seems clear from it, to begin with, that the particular scientiæ are to be understood as the respective results of the application of a single item, viz. the so-called humana sapientia, to various subjecta, i.e. subject-matters; and that the fact that this sapientia “semper vna & eadem manet” on each and every occasion it is applied to different subjects should suffice to establish that all the scientiæ rightly count as somehow fundamentally united, to the effect that it is desirable to deal with all of them simul rather than separately. Yet to see the real import of the Uniformity Claim involved in the conception just outlined, the notion of sapientia and its relations to the bona mens (interpreted, it will be remembered, as the generic ability to carry out intuitus and immediate deductiones) and to the (body of) scientiæ remains to be explicated and clarified. For in the last quoted passage, sapientia is simply identified, in a somewhat perplexing manner, first with each and every scientia and then with the bona mens. Such vague identifications are in need of some rectification, something which can be done once certain other relevant passages are taken into account.117 We saw in ch. 2 in the course of introducing Descartes’ notion of bona mens (or bon sens) that the term refers to a capacity or power “de bien iuger, & distinguer le vray d’auec le faux” in so far as its employment is beyond any regulation; also that Descartes, in DM 1, AT VI, 1–2 draws a sharp contrast between the possession and the correct application of this capacity; and that this latter ability “d’appliquer [le bon sens] bien” is conceived by him as a direct function of the possession of a method. Now several of Descartes’ remarks point strongly towards associating sapientia with this ability to apply the bona mens well through a method, rather than with the bona mens simpliciter. Thus the precept of Reg. I: “Studiorum finis esse debet ingenij directio ad solida & vera, de ijs omnibus quæ occurrunt, proferenda judicia” (AT X, 359; my emphasis) signals it is directio of a power of proferenda judicia—apparently identical to what Descartes refers to with “le bon sens” etc. in DM 1—rather that the power itself that is the proper topic of Reg. I, and the rhetoric in the text of Reg. I to the effect that the sapientia could be enhanced and/or contributed to seems to corroborate the inclination not to identify sapientia with bona mens simpliciter. Furthermore, the suggested connection of sapientia with method is confirmed quite straightforwardly in Reg. XIV, AT X, 442:

117

I am indebted here, as well as elsewhere in the present subsection, to Robert McRae’s lucid discussion in ch. 1 and 3 of his The Problem of the Unity of the Sciences: Bacon to Kant (Toronto: University of Toronto Press, 1961).

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[V]tilitas [regularum, quas hic tradam] est tanta ad altiorem sapientiam consequendam, vt non verear dicere hanc partem nostræ methodi non propter mathematica problemata fuisse inventam, sed potiùs hæc ferè tantùm hujus excolendæ gratiâ esse addiscenda (my emphases).

I conclude on the strength of these considerations that despite some wavering, the term “sapientia” is best read, in Descartes’ hands, as referring to the state of a given human mind in which the mind in question is capable of regulating, to a certain considerable degree—namely in such a way that the end-product of the regulation be some (piece of) scientia— its natural abilities of judging well (i.e. its bona mens, its abilities to carry out intuitus and simple deductiones) by the means of a devised method.118 Whilst Descartes is thus clearly not entitled simply to identify sapientia (universal or otherwise) either with bona mens simpliciter or with the scientiæ simpliciter, one can see by now the basis for such loose identifications in Reg. I. For it should be clear by now that (i) sapientia has two components, viz. the bona mens which is common to all humans and whose operations are in themselves beyond regulation, and a set of methodical rules devised to regulate the complex operations of deductiones;119 and that (ii) Descartes is committed to taking every possible (piece of) scientia as an end-product of an exercise of sapientia in the submitted sense. By the same token, and more importantly with regard to our present purposes, the present analysis shows clearly it is the Uniformity Claim upon which the quoted comparison of sapientia with the Solis lumen really hinges: for given that (a) once the Uniformity Claim is established, the universality of the corresponding method follows immediately, and that (b) sapientia is constituted by the bona mens plus a set of regulative devices amounting to a method, then the uniformity of sapientia—which is the core tenet of the comparison under investigation—is a function of the Uniformity Claim. Given that the scientiæ are unexceptionally endproducts of an exercise of sapientia, the nature of the alleged unity (yet to be specified) of the scientiæ is also thus determined, in the last analysis, by the Uniformity. 118

McRae makes (as far as I can see) a similar point in terms of the different modi considerandi of the bona mens: “It is bona mens, conceived as a natural faculty but applying itself with method, which gives rise to bona mens conceived as wisdom, the highest perfection of our nature” (Unity of the Sciences, 48). 119 The present account of sapientia is compatible also with Descartes’ mature account of la Sagesse in Princ. Pref.; cf. in particular AT IX-2, 2: “[P]ar la Sagesse on n’entend pas seulement la prudence dans les affaires, mais vne parfaite connoissance de toutes les choses que l’homme peut sçauoir ...”

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Taking a closer look, it is far from clear that the Uniformity Claim (i.e. the claim that particular cognitive operations whose principle is the bona mens should count as uniform independently of to which [kind of] subjectmatter they are applied) which is so central to the overall conception under investigation, is compatible with certain of Descartes’ commitments. One prima facie possible way of grounding the Uniformity in question is to conceive of it as contingent upon the unity of subject matter; perhaps the most straightforward way of securing this is to take one entity type as basic and to reduce all the rest to it, so that the apparently specific diversity of subject-matters turns out to be in reality just a diversity of conceptualizations or descriptions of a single basic type.120 All the referring terms in sentences expressing any cognition would then eventually denote entities of one and the same kind under various conceptualizations and the Uniformity would be secured via the plausible principle that the homogeneity of the cognized referents imply the uniformity of the corresponding cognitive acts. For good or ill, however, such a strategy is, of course, definitely unavailable to Descartes due to his firmly held substance dualism.121 By the same token, however, Descartes’ distinctively dismissive attitude towards formalisms of any sort, discussed in ch. 2, also precludes an available prima facie interpretation of the Uniformity as compatible with an irreducibly specific diversity of the basic items of reality, namely to conceive Uniformity as contingent upon the unity of a system of general formulae, and a supplementary logical apparatus, applicable to all cognitions concerning any and all types of entities which are open to scientific treatment.

120

I was prompted to this line of thought by McRae’s analogical discussion of establishing the unity of sciences as contingent upon the eventual specific unity of their subject-matter, in his Unity of Sciences, 5. 121 Cf. in particular Princ. I, 53–54, AT VIII-1, 25–26: “[U]na tamen est cujusque substantiæ præcipua proprietas, quæ ipsius naturam essentiamque constituit, & ad quam aliæ omnes referuntur. Nempe extensio in longum, latum & profundum, substantiæ corporeæ naturam constituit; & cogitatio constituit naturam substantiæ cogitantis. Nam omne aliud quod corpori tribui potest, extensionem præsupponit, estque tantùm modus quidam rei extensæ; ut & omnia, quæ in mente reperimus, sunt tantùm diversi modi cogitandi. ... Atque ita facile possumus duas claras & distinctas habere notiones, sive ideas, unam substantiæ cogitantis creatæ, aliam substantiæ corporeæ, si nempe attributa omnia cogitationis ab attributis extensionis accuratè distinguamus.” And Med. VI, AT VII, 78: “[S]cio omnia quæ clare & distincte intelligo, talia a Deo fieri posse qualia illa intelligo, satis est quòd possim unam rem absque altera clare & distincte intelligere, ut certus sim unam ab altera esse diversam...” See also fn. 66.

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So how is the Solis lumen comparison in Reg. I to be interpreted to steer a clear passage between the irreducible diversity of subject-matters on the one hand and, on the other, Descartes’ rejection of the machinery of general reasoning which would abstract from the concrete content to be considered? The clue is provided, I submit, in the opening sentences of the explication of Reg. I, AT X, 359–60: [Homines] scientias, quæ totæ in animi cognitione consistunt, cum artibus, quæ aliquem corporis vsum habitumque desiderant, malè conferentes, videntesque non omnes artes simul ab eodem homine esse addiscendas, sed illum optimum artificem faciliùs evadere, qui vnicam tantùm exercet, quoniam eædem manus agris colendis & citharæ pulsandæ, vel pluribus ejusmodi diversis officijs, non tam commode quàm vnico ex illis possunt aptari: idem de scientijs etiam crediderunt, illasque pro diversitate objectorum ab invicem distinguentes, singulas seorsim & omnibus alijs omissis quærendas esse sunt arbitrati. In quo sanè decepti sunt.

Here Descartes attacks the established, presumably Aristotelian practice of isolating the particular scientiæ for specialized treatment according to their specific subject-matter; this practice is denounced by him as resting upon a false analogy between scientiæ and those artes which “aliquem corporis vsum habitumque desiderant” (I will call this kind of artes “corporeal artes” from now on).122 From the Aristotelian perspective, the analogy is substantiated by a common definition of both artes and scientiæ as the active powers (habiti) of animated beings to carry out purposeful acts, the particular habiti being individuated in both cases by the particular ends towards which the acts at issue are directed. In what corporeal artes and scientiæ mutually differ is the immediate subiectum they inhere in: whereas the subiectum of the corporeal artes is an animated body, the subiectum of the scientiæ is the intellective component of the soul. Accordingly, whereas the corporeal artes are individuated by the ends of the actions of an animated body, the scientiæ are individuated by the ends of the actions of the (theoretical)123 understanding. The end of the actions of the understanding in general is (to put it very briefly and in a heavily simplified way) to cognize what holds necessarily, which in fact amounts 122

I do not pretend here to give an adequate, let alone a comprehensive account of Aristotle’s and/or the Aristotelian conception of artes, corporeal or not, in their relations to scientiæ. All that matters to me at present is the fact that Aristotle and his followers provided a conceptual background to corroborate the prima facie plausible analogy in question. For the loci classici, see Aristotle, EN I, 1; VI, 4. 123 I will drop this proviso from now on.

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to the cognition of everything which necessarily follows from the essence of the investigated entity. It is here that the professed Aristotelian analogy takes the stage: just as different ends in the domain of the corporeal artes clearly require different types of bodily actions (cf. e.g. the farming and harp-playing mentioned by Descartes), so, by analogy, different ends in the domain of the scientiæ—i.e. different essences to be cognized— require different types of actions of the understanding. A corollary of this claim reads that just as there is no single type of bodily action which could ever provide for all the ends in the domain of the corporeal artes, so there is no single type of action of the understanding which could ever provide for the cognition of all the essences in the domain of the scientiæ.124 It is, I submit, precisely the rejection of this last claim that is the real point of Descartes’ attack upon the analogy between the corporeal artes and the scientiæ in the opening lines of Reg. I, and that also provides a clue to the sought-after interpretation of the Solis lumen analogy and of the Uniformity Claim; for substituting the Aristotelian “the cognition of all the essences” with Descartes’ phrase “solida & vera, de ijs omnibus quæ occurrunt, proferenda judicia” (Reg. I, AT X, 359), Descartes is thus bound to reject the claim that there is no single type of action of the understanding which could ever provide for solida & vera, de ijs omnibus quæ occurrunt, proferenda judicia. Consequently, Descartes commits himself to the claim that there is one single type of action of the understanding which could provide for such solida & vera judicia de omnibus. Now given Descartes’ well-documented claim that the actions, or operations, of the human understanding are strictly speaking just two in number, namely intuitus and deductio,125 and given his arguable commitment (discussed in ch. 2) that, strictly speaking, deductiones are in principle completely reducible to intuitus, it seems virtually unassailable it is intuitus, taken as a type, that stands for the Solis lumen in the analogy under investigation126 and that is therefore properly responsible for the

124

This is not to say, however, that the analogy just presented precludes any notion whatsoever of the unity of the scientiæ or that the scientiæ are generally conceived as absolutely disparate in the Aristotelian tradition. For a classical account of the unity of the scientiæ based upon the notion of one governing scientia see e.g. Thomas Aquinas, “Proœmium” to Sententia libri Metaphysicæ, http://www.corpusthomisticum.org/cmp00.html. 125 For references see the opening section of ch. 2. 126 Besides the argument from elimination just presented, the following passage supports the present interpretation:

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unity of the scientiæ as understood by Descartes. It has not been until now that we have been in a position to fully appreciate both the peculiarity of Descartes’ notion of intuitus and the sense in which intuitus counts as a condition of his groundbreaking notion of what scientia essentially is and how scientiæ are to be practised and advanced with the aid of a universal method. For we can see now that Descartes is bound to conceive intuitus as a type of operation that should navigate clear between both the Scylla of abstracting from concrete content and the Charybdis of the irreducible diversity of subject-matters in the context of the Uniformity Claim, and that he has succeeded in this. On the one hand, our above discussion in ch. 2 of Descartes’ treatment of both the simple and the composite objects of intuitus makes it clear that intuitus in Descartes’ hands is immune to the inducements of abstractive formalisms. On the other hand, the uniformity of intuitus as a concrete type of intellectual activity vis-à-vis the irreducible diversity of subjectmatters (i.e. ultimately vis-à-vis the relevant implications of Descartes’ substance dualism) is secured by the fact that perspicuitas and distinctio, i.e. the essential characterizations of intuitus quâ operations upon which any scientia whatsoever is based, are absolutely independent of the contents of the particular acts of intuitus. By the same token, it has not been until now that the above discussed possibility, and even indispensability, of categorial permeability in the discursive chains of deductiones has been traced back to Descartes’ bottom-line considerations concerning the nature of scientia simpliciter: the categorial permeability [Herbert de Cherbury] veut qu’il y ait en nous autant de facultez qu’il y a de diuersitez a connoistre, ce que ie ne puis entendre autrement que comme si, a cause que la cire peut receuoir vne infinité de figures, on disoit qu’elle a en soy vne infinité de facultez pour les receuoir. Ce qui est vray en ce sens la; mais ie ne voy point qu’on puisse tirer aucune vtilité de cete façon de parler, & il me semble plutost qu’elle peut nuire en donnant suiet aux ignorans d’imaginer autant de diuerses petites entitez en nostre ame. C’est pourquoy i’ayme mieux conceuoir que la cire, par sa seule flexibilité, reçoit toutes sortes de figures, & que l’ame acquert toutes ses connoissances par la reflexion qu’elle fait, ou sur soy mesme pour les choses intellectuelles, ou sur les diuerses dispositions du cerueau auquel elle est iointe, pour les corporelles, soit que ces dispositions dependent des sens ou d’autres causes (Mers., AT II, 598; my emphasis). In a similar vein, Descartes explains experientia as a means to the intuitive cognition of naturas compositas in Reg. XII: “Experimur ... generaliter quæcumque ad intellectum nostrum, vel aliunde perveniunt, vel ex suî ipsius contemplatione reflexâ” (AT X, 422).

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has turned out to be conditioned by the Uniformity Claim, a claim which in turn hinges upon Descartes’ peculiar notion of intuitus just explained.127 Furthermore mutatis mutandis the same holds, of course, with regard to Descartes’ commitment to a universal method, which has been the chief topic of the present chapter.

127 These insights seem to form at least a part of what Descartes might have in mind when he writes in his Cogitationes Privatæ that “[l]arvatæ nunc scientiæ sunt: quæ, larvis sublatis, pulcherrimæ apparerent. Catenam scientiarum pervidenti, non difficilius videbitur, eas animo retinere, quam seriem numerorum” (AT X, 215). In any case, it is worth noting that both the submitted interpretation of the unity of the scientiæ and the commitment to categorial permeability are perfectly compatible with Descartes’ occasional classification and ordering of the scientiæ according to their subject-matter and certain peculiar principles located within a kind of strict categorial framework, which can be found most remarkably in the “tree comparison” in Princ. Pref. (AT IX-2, 14; cf. also DM 6, AT VI, 63– 64). It is not difficult to figure out that such a “categorial” ordering might turn out useful for post facto organization of the already acquired knowledge and/or for purposes of exposition. But Descartes’ chief concern is the régime of discovery, in which categorial permeability and Uniformity are all-important and definitely not to be subject to spurious dictates due to a categorial parcelling up of reality. After all, even in the mentioned comparison in Princ. Pref., “la Philosophie”, which is used there as referring to the study of “toutes les choses que l’homme peut sçauoir” (AT IX-2, 2), is compared to a tree, i.e. to a unified whole which, in so far as it counts as a tree, is ontologically prior to its parts.

CHAPTER FIVE THE A PRIORI IN DESCARTES: INTEGRATING THE ARISTOTELIAN LINE

My hitherto positive account of the meaning of the term “a priori” in Descartes’ thought has been based upon the hypothesis (itself motivated by a certain general reading of the seminal Resp. 2, AT VII, 155–57 in the original Latin version) that Descartes employs the a priori as a characterization of exactly those cognitions which are attained by means of analysis as the only adequate method of discovery, and that what served him as the paradigm of that methodical procedure was the Classical method of mathematical analysis, perhaps further developed and/or transformed by the proponents of modern algebraic thought. I have offered upon this basis a comprehensive account of in what, for Descartes, the analysis properly consists, of the sense in which analysis as conceived by Descartes counts as the method of discovery, and of the rôle of simple natures in the constitution of Descartes’ analytical method. In view of that account, we have attained a considerably complex understanding of to what the term “a priori” is to be attributed according to Descartes’ hints in the mentioned passage of Resp. 2. However, apart from the fact that an account of a posteriori—presumably interconnected with Descartes’ notion of synthesis as a supplement to analysis—is still pending, it remains hitherto unexplained why Descartes resolved to employ the term “a priori” in this peculiar way. This latter question has to do, of course, with the intension rather than with the extension of the term a priori as used by Descartes, and it can accordingly be stated as follows: Exactly which feature(s) of the analytical procedure in question and/or of the cognitions gained by its means render(s) it correct to attribute the characterization of a priori in this particular manner alone? It is the chief goal of the present chapter to address this question.1 After this task is 1

In setting forth the task in this way, I have of course implicitly dismissed an interpretative possibility that at least in so far as the term “a priori” refers to Descartes‫ ތ‬analytic method, its meaning is strictly identical with the meaning of

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accomplished, Descartes’ notions of synthesis and a posteriori will briefly be considered. As a matter of historical fact, by the time of Descartes the only commonly established and tolerably unitary meaning of the a priori–a posteriori pair was due to the Aristotelian tradition.2 There are several prima facie reasons, both historical and textual, to adopt this established Aristotelian meaning as the point of departure in dealing with Descartes’ usage of the terms in question. For one thing, Descartes was exposed to a strong current of Aristotelian scholasticism during his studies at the Jesuit college in La Flèche and it is very likely that there he came across the standard Aristotelian theory of scientific reasoning with which the Aristotelian usage of the terms in question is associated.3 Moreover, we shall see that there are several passages in Descartes’ writings where he explains what he means with the a priori and/or the a posteriori in a fairly Aristotelian manner. Finally, although Descartes’ occasional habit of endowing traditional technical terms with novel content in order to indicate (often quite significant) doctrinal breaks with the tradition4 is the “per Analysin” in the sense thus far established. I believe (and the following account should justify this belief) that Descartes intends the a priori to refer to certain salient features of the analytic procedure as understood by him, not to the analytic procedure tout court. 2 Given the nature and goals of the present study, I cannot but employ the term “Aristotelian tradition” very widely and loosely. In fact my criterion is no stronger than that every thinker interested in methodology who has drawn upon the relevant texts forming Aristotle’s Organon, in particular upon An. Post. and Top., qualifies as falling within the Aristotelian tradition. What is of particular interest for our purposes is, of course, the situation in the Aristotelian camp thus loosely delimitated during the sixteenth and the first decades of the seventeenth century. Being no expert in the field, I rely throughout the present chapter on John Randall’s classic “The Development of Scientific Method in the School of Padua,” in idem, The School of Padua and the Emergence of Modern Science (Padua: Editrice Antenore, 1961), 13–68 for a basic background to the relevant intellectual history of those times. 3 It is most probable that Descartes’ direct source was Franciscus Toletus, Commentaria una cum quæstionibus in universam Aristotelis logica (Rome, 1572) which was an element of the Jesuit Ratio studiorum by the time Descartes studied in La Flèche: see Timothy Reiss, “Neo-Aristotle and Method: Between Zabarella and Descartes,” in Descartes’ Natural Philosophy, ed. Stephen Gaukroger, John Schuster and John Sutton (London: Routledge, 2000), 198. As Reiss remarks (ibid., 222, n. 11), Descartes mentions Toletus under relevant circumstances in Mers., AT III, 185. 4 It will be remembered we came across such a figure concerning the term “intuitus” in ch. 2.

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most likely at work concerning the term a priori (after all, we have already noticed that Descartes’ talk of a priori and a posteriori is qualified with tanquam in AT VII, 155–56), some overlap of content is needed for such a figure to work. In view of all this, it sounds reasonable to set out as an interpretative hypothesis that at least some constituents of the content of the a priori–a posteriori pair were taken over by Descartes from the Aristotelian tradition and that at the end of the day it is in terms of this shared content that Descartes’ own usage in the context of his analytical method is to be explained. Obviously, the meaning of the a priori–a posteriori pair in the Aristotelian tradition must first be briefly introduced in order to render the submitted interpretative hypothesis capable of being tested against both textual and systematic data. I turn to this task now.

5.1 The A Priori and the A Posteriori in the Aristotelian Tradition The Aristotelian notions of a priori and a posteriori are firmly rooted in Aristotle‫ތ‬s notorious distinction between proofs (syllogisms) įȚȩIJȚ (propter quid in Latin) and ੖IJȚ (quia) in An. Post. I, 13. Irrelevant complications apart,5 the point of the distinction that matters to us is, roughly, that whereas in proofs propter quid the middle term amounts to the concept of what is explanatorily (as we are prone to say)6 or causally

5 Aristotle makes it clear in An. Post. I, 13 that explanatory or causal priority (see the next footnote concerning the distinction) of the concept of the middle term is in fact a necessary yet insufficient condition for the proof to count as of the propter quid kind: it must also count as a proximate (“convertible”) and not just a remote cause/explanation (cf. ibid., 78b13–35). Again, certain proofs that qualify as propter quid relative to one science may qualify as quia relative to another, due to the hierarchy amongst sciences postulated by Aristotle (cf. ibid., 78b35–79a16). Yet as Descartes‫ ތ‬treatment in Resp. 2 clearly concerns demonstrations within a single discipline (metaphysics and geometry respectively), and as all that will be needed in our account is the implication from the propter quid character of any given Aristotelian proof to the explanatory or causal priority of the middle term, we can safely put all these complications to one side. 6 Following some leading scholars in the field, I take it that the meaning of Aristotle‫ތ‬s Į੅IJȚȠȞ is closer to that of explanans than to that of cause as these terms have generally been used in recent philosophical and scientific discourse—cf. e.g. Julius Moravcsik, “Aristotle on Adequate Explanations,” Synthese 28, no. 1 (1974): 3–17; Max Hocutt, “Aristotle’s Four Becauses,” Philosophy 49, no. 190 (1974): 385–99; Julia Annas, “Aristotle on Inefficient Causes,” Philosophical Quarterly 32, no. 129 (1982): 311–26. Yet I stick to the traditional terminology

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(in Aristotle‫ތ‬s peculiar sense associated with his term “Į੅IJȚȠȞ”)7 prior to that whose concept stands as the predicate term8—which is why the propter quid proofs properly qualify as scientific according to Aristotle and his followers—the relation of causal priority in the premises is outright reversed in proofs quia. It is precisely in this sense that the proofs propter quid and quia can be described as proceeding “from causes to effects” and “from effects to causes” respectively, as long as we do not wish to depart from Aristotle‫ތ‬s original meaning. Now it is vital for our purposes that the same holds for characterizations of proofs propter quid as a priori (literally “from what is prior”) and of proofs quia as a posteriori (“from what is posterior”) as they begin to appear in later Medieval scholasticism. The earliest documented occurrence of characterizations of proofs propter quid as a priori and of proofs quia as a posteriori, in the sense which retains the essentials of Aristotle’s doctrine of proofs in An. Post. I, 13,9 is in Ockham‫ތ‬s Summa Logicæ (ca. 1323):10 and employ the vocabulary of causation throughout the present chapter. Nothing in my treatment turns on the differences between causation and explanation. 7 I shall refrain from taking up the controversial issue of whether the domain of ĮੁIJȓĮ in the An. Post. is strictly co-extensive with that of ĮੁIJȓĮ discussed in Aristotle‫ތ‬s Phys. II, 3 and Met. V, 2, since the issue is also irrelevant to our present concerns. It boils down to the problem of whether essences in the An. Post. can count as the ĮੁIJȓĮ (in the sense of the Physics and the Metaphysics) of properties whose necessary inherence in subjects is to be proved by scientific syllogisms. Be it as it may, I will henceforth use “causal” and its kin in the sense established in the Physics and Metaphysics, with possible extension according to the present controversy, in the rest of the present section. 8 For the sake of brevity, I will henceforth use “the subject/middle/predicate term” as a short form of “that whose concept stands as the subject/middle/predicate term” etc. 9 I have learned from Lukáš Novák, ScotĤv dĤkaz Boží existence jako vrcholný výkon metafyziky jakožto aristotelské vČdy [Scire Deum esse: Scotus’ Proof of the Existence of God as a Supreme Achievement of Metaphysics as Aristotelian Science] (Prague: Kalich, 2011), 66–68 that it had become a commonplace in Aristotelian scholasticism quite some time before Ockham—apparently upon the basis of a narrow reading of Aristotle’s Į੅IJȚȠȞ as efficient cause—to apply the adjectives propter quid and quia, and the corresponding adjectives a priori and a posteriori, specifically in various kinds of proofs of God’s existence. However, Novák shows persuasively that such a specific usage does not do justice to the original doctrine of proofs by Aristotle. Descartes himself is reported to have used the a priori in this degenerate manner in Burm., AT V, 153, and—to make things even more complicated—in the context of his alleged characterization of the Principia as a synthetic treatment. Despite the fact that Martial Guéroult has made a great case of this issue which has been discussed especially by French scholars

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[O]portet scire quod quædam est demonstratio cuius præmissæ sunt simpliciter priores conclusione, et illa vocatur demonstratio a priori sive propter quid. Quædam est demonstratio cuius præmissæ non sunt simpliciter priores conslusione, sunt tamen notiores sic syllogizanti, per quas devenit sic syllogizans in notitiam conclusionis, et talis demonstratio vocatur demonstratio quia sive a posteriori. ... [Demonstratio] a priori et propter quid [est], quia præmissæ exprimunt causam propter quam sic est a parte rei sicut significatur esse per conclusionem. Si autem [aliquis] ... facit demonstrationem a posteriori; ... scit quia ita est, sed enscit propter quid ita est; quia tamen per propositiones sibi notas adquirit cognitionem sibi ignoti necessarii, ideo habet demonstrationem (ibid., III-2, c. 17; my emphases).

The identification of the two sets of characterizations (quia–propter quid and a posteriori–a priori, respectively) seem to have been fixed soon thereafter in Albert of Saxony‫ތ‬s Quæstiones super Libros Posteriorum Aristotelis (ca. 1355):11 Demonstratio quædam est procedens ex causis ad effectum et vocatur demonstratio a priori et demonstratio propter quid et potissima; ... alia est demonstratio procedens ab effectibus ad causas, et talis vocatur demonstratio a posteriori et demonstratio quia et demonstratio non potissima (ibid., I, q. 9).

As a consequence, in the eyes of the Aristotelians who introduced them, the expressions a priori and a posteriori are designed to refer precisely to the causal relation holding between the middle and the predicate terms in so far as this relation is represented in the premises of a given syllogism. This fairly clear-cut meaning of the a priori–a posteriori distinction which refers straight back to Aristotle‫ތ‬s An. Post. (henceforth the “narrow ever since—cf. in particular idem, Descartes selon l'ordre des raisons (Paris: Aubier, 1953), 1:22–28; 265–72; 357–58; a survey of the subsequent discussion can be found in Daniel Garber, “A Point of Order: Analysis, Synthesis, and Descartes‫ތ‬s Principles,” in idem, Descartes Embodied, 57, fn. 6—I set this complication to one side and will ignore it from now on, since both the distinction a priori–a posteriori and the distinction between analytic and synthetic order are clearly employed here in a derivative and/or degenerate sense, so this issue would be more likely to muddy the waters than help deal with the chief questions pursued in the present study 10 William Ockham, Opera Philosophica I: Summa Logicae, ed. Philotheus Boehner, Gedeon Gál and Stephen Brown (St. Bonaventure, NY: Editiones Instituti Franciscani Universitatis S. Bonaventuræ, 1974). 11 Albert of Saxony, Quæstiones subtilissimæ super Libros Posteriorum Aristotelis (Venice: Oslavianus Scotus, 1497). Quoted from Carl Prantl, Geschichte der Logik im Abendlande, vol. 4 (Leipzig: S. Hirzel, 1870), 78.

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Aristotelian meaning”) has been liable to various drifts, distortions and misconceptions both during the prolonged heyday of Aristotelian scholasticism up to Descartes’ times and in the subsequent periods when it sank into antiquity, becoming subject to purely historical scholarship. I do not intend to venture any sort of comprehensive survey of these departures. I will briefly consider only the two of them which happen to be directly relevant to our main topic. To begin with, Aristotle clearly (and plausibly) maintains (a) that in proofs quia the relation of the causally posterior middle term to the causally prior predicate term is quite often learned įȚૃ ਥʌĮȖȦȖોȢ ਲ਼ įȚૃ Įੁıș੾ıİȦȢ [through induction or through perception],12 i.e. empirically at any rate; and that—no less important—(b) this empirical piece of cognition is cognitively prior to any cognition of the fact that the causally prior (or prior in nature) predicate term can rightly be attributed to the subject term of the given syllogism (an item A being cognitively prior to an item B if B could not be cognized without A). Each of these insights of Aristotle‫ތ‬s underlies a latent conceptual departure from the narrow Aristotelian meaning of the a priori–a posteriori distinction. As for (b), it might be tempting to take a priori and a posteriori as referring to priority/posteriority within the order of cognition instead of that of causation. Obviously enough, such a conceptual shift is entirely at odds with the narrow Aristotelian meaning. Yet, as we shall see, several respected commentators tried to deploy this shift in their attempts to reconcile Descartes’ peculiar usage of the a priori–a posteriori pair with the Aristotelian way of employing the terms under investigation. As for (a), a possible misconception concerns the undue identification of proofs quia or a posteriori with empirical proofs. It is indeed quite natural to take syllogisms with empirical premises in the mentioned sense as paradigmatic instances of proofs quia or a posteriori. Yet quite often is, of course, not always, let alone essentially; and it is not difficult to construct proofs quia in which the relation of the causally posterior middle term to the causally prior predicate term is not discovered empirically. Moreover, it takes no more effort to construct propter quid or a priori proofs with empirical premises.13 One must therefore resist the temptation to take proofs a posteriori as co-extensive with empirical proofs and proofs a priori as coextensive with non-empirical proofs lest the narrow Aristotelian meaning of the a priori–a posteriori distinction be considerably distorted in yet

12

An. Post. I, 13, 78a34–35; Aristotle, Complete Works, 1:127. To appreciate the point more readily, remember the stipulation concerning the word “causal” in fn. 7. 13

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another respect.14 With this rudimentary account of the Aristotelian notions of the a priori and the a posteriori, let us return to Descartes and determine which if any of the above-established Aristotelian notions might plausibly be taken to be part of the content that Descartes is likely to have associated with such terms.

5.2 Some Aristotelian Strata in Descartes’ Conception It is to be borne in mind in the course of the entire discussion to follow that there is at least one absolutely unbridgeable gulf by which Descartes’ notions of the a priori and the a posteriori—whatever those notions should turn out to be—are separated once and for all from the general Aristotelian conception just sketched. For, as we saw in detail in ch. 2, Descartes resolutely rejects syllogistic as a serviceable device to handle discursive reasoning in any significant respect, i.e. as regards regulation no less than discovery or exposition; and what he offers instead is his own peculiar and groundbreaking conception of deductio as ordered series of immediate deductiones which are always reducible, at least in principle, to non-discursive intuitus.

14

A closely related mistake, only grosser and less frequent, is to take a posteriori/quia proofs as inductive arguments and a priori/propter quid proofs as deductive arguments in the contemporary sense. The source of the trouble is, once again, Aristotle‫ތ‬s plausible insight that cognition of the causal relations between the terms of syllogisms can often be established empirically, ਥʌĮȖȦȖȒ—translated as inductio into Latin from the times of Cicero—being one of the ways available for doing this (cf. An. Post. I, 13, 78a34–35). To be sure, such a view of the relations of propter quid and quia proofs on the one hand and deductive and inductive proofs on the other will not do for reasons already mentioned: since the premises can sometimes be established empirically—and more specifically by way of induction—even in the case of propter quid proofs, some such proofs would count as inductive in the proposed sense, too. Yet a more profound misconception is at work here, of course. We dealt in ch. 2 with the fact that all Aristotelian scientific proofs, for which alone the propter quid–quia and a priori–a posteriori classifications were originally designed, are deductive in the contemporary sense as a matter of principle. This is not so with inductive reasoning, even though (as we also saw in ch. 2) the peculiarities of the Aristotelian treatment render induction much closer to deduction than its modern interpretations. Furthermore, while the procedure of generalization from (comparatively) particular instances to universal conclusions is absolutely essential for induction (in any of its varieties), it plays strictly no part in deductive syllogisms as they stand.

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Yet within this radically transformed context in which the a priori–a posteriori pair is withdrawn from reference to relations of terms in syllogistic figures, it still remains an open question whether Descartes is prepared to retain, in some sense at least, another essential component of the Aristotelian meaning associated with the pair at issue, namely the reference to causal ordering.15 For good or ill, several passages in his writings strongly suggest that the question should be answered in the positive. Putting indirect evidence to one side for the moment, he intimates in Le Monde that ceux qui sçauront suffisamment examiner les consequences de ces veritez [eternelles, sur qui les Mathematiciens ont accoûtomé d’appuyer leurs plus certaines & plus évidentes demonstrations] & de nos regles, pourront connoistre les effets par leurs causes; &, pour mҲexpliquer en termes de lҲEcole, pourront avoir des demonstrations à Priori, de tout ce qui peut estre produit en ce nouveau Monde (AT XI, 47; my emphasis).

Again, he writes to Plempius:16 At principia siue præmissæ, ex quibus conclusiones istas [de naturâ luminis, de figurâ particularum salis & aquæ dulcis & similibus] deduco, sunt tantum 15

Danielle Cozzoli, “Beyond Mixed Mathematics: How a Translation changed the Story of Descartes‫ތ‬s Philosophy of Mathematics,” in Beyond Borders: Fresh Perspectives in History of Science ed. Josep Simon and Néstor Herran (Newcastle: Cambridge Scholars Publishing, 2008), 35–60 seems to dismiss this interpretative possibility on the basis that “[i]nterpreting a priori and a posteriori in causal terms entailed that one had to accept Aristotle‫ތ‬s causal notion of proof [in terms of syllogistic reasoning]” (ibid., 48). Her argument seems to me to be a plain non sequitur, however. For we shall see that Descartes was prepared to explicate his general conception of reasoning in a kind of causal terms and it is definitely false that the only possible form the description of the reasoning processes in causal terms can take is the Aristotelian one in syllogistic terms. It is one thing to hold that Descartes adopted the a priori–a posteriori terminology, along with all its causal connotations, only (or chiefly) because he wished to clarify his position in a way that would be comprehensible to his Aristotelian audience—which is the reading I shall defend—and it is another thing, and deeply mistaken I believe, to dismiss the a priori–a posteriori distinction as throughout at odds with Descartes‫ތ‬ own real position only because the distinction brings about—as it actually does— causal connotations. 16 In view of the question we are pursuing now, it is of no significance that Descartes assumes the effects to be sensible in the quoted passage. Remember what was said in sec. 5.1 concerning tempting distortion (a) of the narrow Aristotelian meaning of the a priori–a posteriori pair.

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illa axiomata quibus Geometrarum demonstrationes nituntur ... non tamen ab omni sensibili materia abstracta, vt apud Geometras, sed varijs experimentis sensu cognitis atque indubitatis applicata; vt cum ex eo quod particulæ salis sint oblongæ & inflexiles, deduxi figuram quadratam eius micarum, & alia quàm plurima, quæ sensu manifesta sunt: hæc quidem per illud volui explicare vt effectus per causam; nequaquam autem probare, quia iam erant satis nota, sed contra illud per hæc à posteriori demonstrare ... (AT I, 476; my emphasis).

In a similar vein, he explains to Mersenne that to gain cognition “à posteriori” of “toutes diuerses formes & essences des cors terrestres” means to learn about these “par leur effets” (Mers., AT I, 250–51). Finally, there is Clerselier’s translation into French, approved and praised by Descartes,17 of the crucial sentences in the seminal AT VII, 155–56. Here is Descartes’ original Latin: Analysis veram viam ostendit per quam res methodice & tanquam a priori inventa est .... ... Synthesis è contra per viam oppositam & tanquam a posteriori quæsitam (etsi sæpe ipsa probatio sit in hac magis a priori quàm in illâ) clare quidem id quod conclusum est demonstrat ... (ibid.; my emphases).

And here is Clerselier‫ތ‬s translation: L'analyse montre la vraye voye par laquelle vne chose a esté methodiquement inuentée, & fait voir comment les effets dépendent des causes .... ... La synthese, au contraire, par vne voye toute autre, & comme en examinant les causes par leurs effets (bien que la preuue qu‫ތ‬elle contient soit souuent aussi des effets par les causes), démontre à la vérité clairement ce qui est contenu en ses conclusions ... (Resp. 2, AT IX-1, 121–22; my emphases).

I conclude on the strength of these passages that Descartes indeed does retain several important strata of the meaning of the a priori–a posteriori pair as employed in the Aristotelian tradition: like the Aristotelians, he is prepared to employ these terms as properly attributed to discursive cognitive operations that are somehow closely related to gaining scientiæ, and also as referring to respective orderings of causes and effects in the context of those operations. Of course, he finds himself in a dramatic disagreement with the Aristotelians concerning the nature of discursiveness 17

Cf. “Le libraire au lecteur,” AT IX-1, 2.

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in the relevant cognitive operations, replacing Aristotelian syllogistic figures, designed to organize general concepts, with serial orderings, designed to organize particular intuitus. Yet he seems to have accepted the Aristotelian idea that in the contexts of establishing cognition, to proceed a priori amounts to proceeding from causes to effects, and to proceed a posteriori amounts to proceeding from effects to causes.

5.3 A Clash with the Aristotelians This is by no means the end of the story, however. As most dedicated Descartes scholars have been quick to recognize, Descartes’ welldocumented characterization of analysis as an a priori procedure and of synthesis as an a posteriori procedure—in so far as a priori and a posteriori are taken in the established Aristotelian sense—runs directly contrary to the established Aristotelian understanding of how the a priori and the a posteriori are to be associated with analysis and synthesis respectively. For although the terms “analysis” or “resolutio” or “dissolutio”, and “synthesis” or “compositio”,18 turn out ambiguous in a very important respect in the hands of Aristotle and his followers (as we shall see in a moment),19 it is beyond dispute that analysis uniformly 18 The history of the process of equating ਕȞȐȜȣıȚȢ with resolutio/dissolutio and ıȪȞșİıȚȢ with compositio in Medieval and Renaissance thought is extremely vexed and we cannot and need not linger on details. It seems that what played a decisive part in this process was a tradition of translating the terms “ਕȞȐȜȣıȚȢ” and “ıȪȞșİıȚȢ” as they occur in the prologue to Galen’s Ars medica with, respectively, “resolutio” or “dissolutio” and “compositio” after Galen’s texts were re-introduced into the Western world from the eleventh century onwards—cf. e.g. Randall, “Development of Scientific Method,” 27–34. Galen’s Ars medica was also known as Ars parva, Tegni or Microtegni. The situation concerning various manuscripts and editions of this compendium—as indeed of the entire extant Galenian corpus—is overwhelming; Randall, ibid., 31–32, fn. 5 refers to the following edition: Galieni principis medicorum Microtegni cum commento Hali, transl. Gerard of Cremona (no date or place of printing, but prior to 1479). 19 In fact, as usual, the situation is even more complicated as regards Aristotle’s own writings since the term “ਕȞȐȜȣıȚȢ” is apparently employed by Aristotle in several different meanings (see Marco Panza, “Classical Sources for the Concepts of Analysis and Synthesis,” in Analysis and Synthesis in Mathematics: History and Philosophy ed. Michael Otte and Marco Panza [Dordrecht: Kluwer Academic Publishers, 1997], 370–83 for a useful survey; Aristotle seems to come close to the later established core sense employed, for example, by Pappus in An. Post. I, 12, 78a6–8), and it is fair to say that the ambiguity alluded to is rather a result of a tidying of the conception by Aristotle’s followers. However, as these exegetical

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counts as an a posteriori movement and synthesis as an a priori movement in either of the basic meanings that analysis and synthesis assume in the Aristotelian tradition.20 Ferdinand Alquié‫ތ‬s gloss to this effect has become a classic:21 Le raisonnement a priori ... est celui qui va de la cause à l‫ތ‬effet, du principe à la conséquence, et le raisonnement a posteriori est celui qui remonte des effects aux causes, des conséquences aux principes. Mais il se trouve précisément que l‫ތ‬analyse remonte des conséquences aux principes, alors qu‫ތ‬elle est dite ici [sc. in Resp. 2, AT VII, 155] opérer comme a priori (Alquié’s italics).

Such a blatant discrepancy (which I will henceforth term the Discrepancy) calls for an explanation, of course. Putting to one side the highly implausible possibility that Descartes just completely misunderstood the thrust of the unanimous association of analysis with the a posteriori and synthesis with the a priori in the Aristotelian tradition, any possible explanation of the Discrepancy is bound to claim that either the complementary notions of analysis and synthesis, or the complementary notions of a priori and a posteriori, or perhaps both, have undergone some substantial changes in Descartes’ hands; and since the Discrepancy can hardly be explained solely through Descartes’ having released the relevant discursive cognitive operations from the vincula of the syllogistic, virtually any explanation worthy of consideration is likely to shed some further light on the central topics of the present study, viz. the notions of the a priori, a posteriori, analysis, and synthesis. In view of this, the rest of the present chapter is devoted to the development of my own solution to the Discrepancy problem. The leading idea of my account is that Descartes’ characterization of analysis as a priori and of synthesis as a posteriori is best understood as a deliberate deployment of the Aristotelian pair a priori–a posteriori in making use of

details are unnecessary for our purposes, I will largely ignore them in the following discussion. 20 According to Randall, “Development of Scientific Method,” 32, it was above all a remarkable Arab commentator on Galen, ‘Ali ibn Ridwan (died 1061), who identified what had commonly been rendered as via dissolutionis and via compositionis in Galen’s prologue to Ars medica with Aristotelian proofs ੖IJȚ and įȚȩIJȚ, respectively, in a way which is likely to have significantly influenced Aristotelian methodological thought ever since. 21 Œuvres philosophiques de Descartes ed. Ferdinand Alquié, vol. 2 (Paris: Classiques Garnier, 1967), 582.

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the above-established Aristotelian stratum in his concept of a priority; and that what Descartes intends with his provocative association of analysis with a priori is to indicate, this time through a causal vocabulary and in terms accessible to the Aristotelian audience of his time, how deep and radical is his break with the tradition concerning the nature of the true scientific method of discovery. In view of Descartes’ generally dismissive attitude towards synthesis as a legitimate and/or useful tool in the régime of discovery, I will focus predominantly on the case of analysis and of the a priori, and will integrate the case of synthesis and of the a posteriori only after the essentials of my positive account of analysis as a priori are provided.

5.3.1 Analysis as an Approbative Tool in the Aristotelian Tradition First of all, we must deal with the systematic ambiguity in the Aristotelian tradition between analysis as an approbative device and analysis as a heuristic device. It was virtually uncontroversial amongst mainstream Aristotelian thinkers at least as early as Boëthius through such figures as Eriugena, Albertus Magnus and Aquinas,22 and right up to Descartes’ time,23 that in so far as scientia in the strict sense is concerned, i.e. cognition marked with certainty and with the necessity of its contents, it is synthesis (compositio) and not analysis (resolutio) that counts as the real tool of discovery. Analysis, on the other hand, is designed, in the realm of strict scientiæ, to assess or (as the scholastic authors commonly put it) to iudicare the conclusions discovered through synthetic procedures, the assessment being executed by way of resolutio of complex judgments down to their first principles.24 Far from enjoying the reputation 22

For references see Timmermans, “Originality of Descartes’s Conception,” 435, fn. 9–14. Almost needless to say, the situation is by far less straightforward as the present oversimplified exposition might suggest. By way of illustration, Eileen Sweeney, “Three Notions of Resolutio and the Structure of Reasoning in Aquinas,” The Thomist 58 (1994): 197–243, has distinguished no less than three different meanings of resolutio in Aquinas’ writings. 23 See in particular Eustachius a Sancto Paulo, Svmma Philosophiæ Qvadripartita, de rebvs Dialecticis, Moralibus, Physicis, & Metaphysicis, vol. 1 (3rd ed. Paris: Carolvs Chastellain, 1614), 190. As can be learned from letters to Mersenne, Descartes praised this work as a most useful compendium of the philosophy of the Schools of his time; cf. in particular Mers., AT III, 232. 24 See Thomas Aquinas, Expositio libri Posteriorum Analyticorum, lib. I, lect. 1, n. 5–6, http://www.corpusthomisticum.org/cpa1.html, for a very clear account of this.

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of a (or even the) heuristic device, the significance of analysis in the realm of scientiæ was approbative, expository, and pedagogical. As Eustachius a Sancto Paulo concisely puts it, [e]st autem aptissima hæc methodus [diuisiua] in docendis & addiscendis doctrinis: in inueniendis verò iisdem disciplinis, alia methodo, nempe compositiua seu synthetica plerumque vtendum est (Svmma Philosophiæ Qvadripartita 1:190).

What this standard Aristotelian conception is fundamentally based upon is the insight that if any discovery whatsoever is to satisfy the strict standards of the scientific cognition as set up by Aristotle, the data from which a given research starts is bound to count as both ontologically and cognitively independent of the quæsita, since otherwise the certainty and necessity of the attained conclusions are not guaranteed even if they have been reached by valid reasoning. In order that such a requirement be met—so the argument continues—the data must amount to items which are in themselves (or naturâ) simpler than the quæsita. However, to proceed discursively from what is relatively simpler in itself to what is relatively more complex in itself is, at least from the Aristotelian perspective, a job of synthesis and not of analysis. This rationale makes it clear, by the same token, why the Aristotelians have unanimously been quick to associate synthesis with reasoning propter quid, i.e. a priori, and analysis with reasoning quia, i.e. a posteriori, in the contexts of strictly scientific treatment. This is because it counted as a virtually indisputable doctrinal point derived from Aristotle’s texts that in any strictly scientific search after something as yet unknown the data is bound to possess relative simplicity and ontological as well as cognitive independence of the quæsita, which are exactly the features with which causes (in the above-established, wide Aristotelian sense) are endowed in relation to their (actual or possible) effects.25 Out rolls the conclusion that at least in so far as strict scientiæ are concerned, heuristic cognitive movement proceeds through synthesis, and synthesis proceeds from causes to effects; and mutatis mutandis, of course, that analysis proceeds from effects to causes to enlighten and/or verify the conclusions reached through backward reference to the simpler and independent starting points.

25

Cf. in particular Aristotle, An. Post. I, 2.

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5.3.2 Heuristic Analysis in Aristotle If the matter were left at this stage, the stated intention of explaining the highlighted discrepancy between Descartes’ and the Aristotelian characterizations of analysis/synthesis in terms of the a priori/a posteriori would seem pointless, for Descartes clearly and unambiguously takes analysis—in direct opposition to the above-established, standard Aristotelian view—as a heuristic procedure, indeed the heuristic procedure par excellence. The professed discrepancy would therefore seem to issue from a gross equivocation regarding the terms “analysis” and “synthesis,” with the result that the Discrepancy either turns merely on names or is rendered destitute of any common ground. In either case, no profit could be gained from dealing with the Discrepancy in this form when seeking a better understanding of the a priori–a posteriori pair in Descartes relative to the Aristotelian doctrinal background. However, the denial that analysis enjoyed the status of a heuristic tool was far from absolute in the Aristotelian tradition. As already indicated, the denial was limited to the domain of strict scientiæ. Although this domain is no doubt of eminent importance, Aristotle (and presumably his followers) certainly recognizes other domains of cognition in which somewhat relaxed standards of rigour seem appropriate; and analysis is recognized as being perfectly suited to facilitating discoveries in these domains—with the substantial qualification that the attained cognitions never count as rigorous scientia, since these domains do not involve the necessary items independent of human powers but concern only what is contingent.26 Aristotle himself makes it clear in a frequently quoted passage that such is the case, for instance, in the remarkable domain of practical deliberation: We deliberate not about ends but about what contributes to ends. For a doctor does not deliberate whether he shall heal, nor an orator whether he shall convince, nor a statesman whether he shall produce law and order, nor does any one else deliberate about his end. Having set the end they consider how and by what means it is so attained; and if it seems to be produced by several means they consider by which it is most easily and best produced, while if it is achieved by one only they consider how it will be achieved by this and by what means this will be achieved, till they come to the first cause, which in the order of discovery is last. For the person who deliberates seems to inquire and analyse in the way described as though he were analysing a geometrical construction ..., and what is last in the order of

26

Cf. e.g. Aristotle, EN III, 3, 1112a18–b11.

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analysis seems to be first in the order of becoming (Aristotle, EN III, 3, 1112b11–24; Complete Works, 2:1756–57; my emphases).

Here one starts with an idea of the end to be achieved, which is the known datum, and the quæsita are the unknown means by which the intended end can be attained. The search procedure is clearly analytical—examination of the structure of the data reveals the relations of their possible dependence upon something else which might serve to produce them. To find some such means can indeed amount to genuine discovery; and since the discovered quæsitum counts as a potential cause of the latter, and therefore precedes the initial data in the order of being (since the end to be achieved is as yet given only in conceptu and ex hypothesi cannot be brought to existence but through operations of the sought-after means), it is appropriate to characterize this procedure as the discovery of a cause through an effect. Yet the crucial limiting point of such a heuristic procedure through analysis is, as already indicated, that the discovery remains only conjectural until it is verified through the actual production of the intended end. This is why the discovery at issue, even if the process of finding it is embodied in an appropriate syllogistic form, from Aristotle‫ތ‬s perspective fails to count as scientific. Significantly, Aristotle indicates, in the celebrated opening sections of his Physics, that due to the limited capacities of human cognitive faculties, analysis is to be employed as a heuristic vehicle even in the domain of the general science of nature: When the objects of an inquiry, in any department, have principles, causes, or elements, it is through acquaintance with these that knowledge and understanding is attained. For we do not think that we know a thing until we are acquainted with its primary causes or first principles, and have carried our analysis as far as its elements. Plainly, therefore, in the science of nature too our first task will be to try to determine what relates to its principles. The natural way of doing this is to start from the things which are more knowable and clear to us and proceed towards those which are clearer and more knowable by nature; for the same things are not knowable relatively to us and knowable without qualification. So we must follow this method and advance from what is more obscure by nature, but clearer to us, towards what is more clear and more knowable by nature (Phys. I, 1, 184a10–21; Complete Works, 1:315; my emphases).

Here the context is not practical deliberation but theoretical cognition pretending to the status of strict scientia; and what Aristotle pinpoints is a complication that has haunted the methodologists of natural science ever

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since: the principles in terms of which observable phenomena of nature are to be explained in a way that satisfies scientific cognitive standards (whatever they are), although perhaps prior in nature and more knowable or obvious in themselves, are usually not ready at hand and must be established first through investigation of the very phenomena to be explained. Aristotle is quick to recognize that heuristic analysis is, once again, a suitable vehicle to carry out this vital task. However, his own commitments concerning the status of heuristic analysis obviously engender a cluster of tensions which Aristotle left to generations of commentators to come. First, given the essential limitations of human cognitive capacities, Aristotle is committed to the claim that a nonscientific procedure is required to establish the very possibility of a general scientia of nature. Second, this dependence of the general science of nature on a preliminary reasoning from effects to causes (or from explananda to explanantia) seems to render science itself inextricably conjectural or hypothetical, since heuristic analysis as conceived by Aristotle seems unsuited to the provision of anything but conjectural cognition of explanatory principles. I submit that with this notion of heuristic analysis in Aristotle, a minimal common ground has been established that renders the declared intention to explain the discrepancy concerning Descartes’ and the Aristotelian respective characterizations of analysis with the a priori–a posteriori pair at least intelligible and possibly illuminating. By the same token, the present account has highlighted another point of divergence between Aristotle and Descartes concerning analysis: it will be remembered that according to the latter, it is (heuristic) analysis that counts as the scientific method par excellence; by way of contrast, we have just seen that according to Aristotle, heuristic analysis is at best only a non-scientific vehicle designed to establish certain conditions of certain strictly scientific enterprises.

5.3.3 Heuristic Analysis in Renaissance Aristotelianism The virtual dissent just pinpointed between the Aristotelians and Descartes concerning the question of the status of heuristic analysis27 in relation to standards of scientia is significantly diminished, if it does not disappear altogether, once certain developments in the methodology of the 27

I shall drop the characterization “heuristic” from now on. Unless explicitly indicated otherwise, in the rest of the present chapter analysis is always understood to be of the heuristic sort.

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natural sciences by Renaissance Aristotelianism—which flourished above all in northern Italy, and most prominently at the University of Padua28— are taken into account.29 For one of the most significant constant features discernible in those thinkers’ deliberations on method in the fifteenth to sixteenth centuries is their preoccupation—prompted by Averroës‫ތ‬ commentaries on Aristotle and by the Galenic medical tradition that was re-introduced into the Western world through Arabic translations and commentaries and in which sophisticated methodological thought was cultivated30—with integration of the analytical (or, as they preferred to say, resolutive) procedure, by which the explanatory principles of natural scientiæ are established quoad nos, into the very body of those natural scientiæ. As Hugo of Siena (died 1439) puts it, arguably in a fairly representative manner:31 Quia omnis demonstratio aut est per causam, aut per effectum, ideo doctrina, quæ est manifestatio demonstrabilis, non fiet nisi alio duorum modorum, scilicet resolvendo vel componendo. In una completa scientia unius generis, sicut philosophia & medicina, est impossibile unico tali ordine procedere. Sed oportet ante complementum omnibus ... modis uti; quia in cognitione causarum utimur demonstratione quia, & in cognitione scientifica effectuum utimur demonstratione propter quid. Secundum omnem opinionem, & quia uterque istorum processuum est necessarius. ... Ego autem video in inventione scientiæ effectus per causam duplicem processum, & similiter in inventione scientiæ causæ per effectum: unum inventionis medii, alium autem notificationis [i.e. setting forth of consequences—J.P.]; & processus inventionis in demonstratione per causam est resolutivus, alter compositivus. Econtra autem est in demonstratione per effectum (my emphases).

28 Other significant places where Aristotelian methodology was cultivated and developed in parallel with (and often in opposition to) the Paduan scholars include in particular Bologna and Pavia in northern Italy; and abroad Paris and Oxford. Cf. Randall, “Development of Scientific Method,” 20–26. 29 For obvious reasons, my account of Renaissance Aristotelian methodological thought in the present subsection is bound to remain considerably selective. As already announced, I draw heavily upon the classical account of Randall, “Development of Scientific Method.” 30 For brief surveys see e.g. Randall, “Development of Scientific method,” 27–34; Neal Gilbert, Renaissance Concepts of Method (New York: Columbia University Press, 1960), 81–83. 31 Hugo of Siena, Expositio Ugonis Senensis Super Libros Tegni Galeni (Venice: Octavianus Scotus, 1518), Comm. I. Quoted from Randall, “Development of Scientific Method,” 37–38, fn. 12–14.

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The idea is clearly that as regards disciplines which aspire to the title of scientia and whose explanatory principles are not readily at hand, the process of establishing those principles by way of analysis is to be conceived not just as a preliminary non-scientific prelude but as an integral component of a unitary scientific enterprise (i.e., as Hugo puts it, of a scientia completa). Such a project, of course, significantly narrows the gap between the Aristotelian understanding and Descartes’ understanding of how analysis is related to scientiæ proper. Moreover, as might be expected, the proponents of this project were quick to recognize several substantial problems such an integration brings about in the Aristotelian framework and their attempts to tackle at least two such problems contribute to an even further narrowing of that gap. One is the threat of circularity. The professed circle can be construed roughly as follows: from an Aristotelian standpoint, the proper task of any given scientia is to explain, i.e. to establish a certain specific cognition of a given effect through an appropriate cause; but in order that the cause at issue can fulfil its explanatory rôle, it must itself be cognized first, and this in turn requires, ex hypothesi, that the effect whose cognition is to be established is already cognized. To put it bluntly, then, the problem is that once the analytical quia procedure, which ex hypothesi must precede the synthetic propter quid procedure, is allowed to form an integral component of a given scientia completa, the effect whose cognition is ex hypothesi yet to be established is bound to be already cognized in order that its cognition could be established in the envisaged way. This circularity charge is addressed in a very illuminating and selfsufficient way as early as another Paduan doctor, Paul of Venice (died 1429):32 Noticia scientifica causæ dependet a noticia effectus, sicut noticia scientifica effectus dependet a notitia causæ; patet quoniam prius cognoscimus causam per effectum quam effectum per causam; et ita habetur principale investigandum, scilicet quod noticia scientifica effectus naturalis exigit priorem noticiam causarum et principiorum. In processu naturali sunt tres cognitiones: prima est effectus sine discursu, et dicitur quia. Secunda causæ per noticiam eiusdem effectus; et iterum vocatur quia. Tertia est vero effectus per causam; et dicitur propter quid. Modo noticia propter quid effectus, non est noticia quia effectus. Ideo noticia effectus non dependet a seipso, sed ab alio.

32

Paul of Venice, Summa philosophiæ naturalis magistri Pauli Veneti (Paris: Iodoci Badii Ascensii, 1521), I, ch. 9. Quoted from Randall, “Development of Scientific Method,” 40, fn. 16.

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A solution of this type to the problem of circularity was widely accepted; and the procedure of establishing a cognition of effects in another, epistemically more valuable manner, that was the preliminary cognition necessary for the entire scientific procedure to take off the ground, came to be commonly called regressus33 (i.e. literally “return”, in contrast perhaps to mere repetition). Another significant challenge that the thinkers of the period felt bound to face—and that is relevant to our purposes—concerns the charge that the elevation of analysis to an integral component of scientiæ brings about a disconcerting implication that the scientiæ at issue are doomed to remain inescapably conjectural, which in turn seems to jeopardize their very status as scientiæ. The problem is pinpointed clearly by Agostino Nifo (ca. 1473–1538 or 1545):34 [V]idetur mihi in regressu, qui fit in naturalibus demonstrationibus, primum processum quo syllogizatur causæ inventio, esse syllogismum tantum, quoniam coniecturalis tantum, cum per ipsum solum coniecturabiliter syllogizetur causæ inventio; secundum vero quo syllogizatur per inventionem causæ propter quid effectus esse demonstrationem propter quid, non quia faciat scire simpliciter, sed ex conditione, dato quod illa causa sit, vel dato quod propositiones veræ sint, quæ repræsentant causam & nulla alia causa esse possit. ... Sed occurris, quia tunc scientia de natura non esset scientia. Dicendum scientiam de natura non esse scientiam simpliciter ...; est tamen scientia propter quid, quia inventio causæ, quæ habetur per syllogismum coniecturalem est propter quid effectus. ... Nunquam enim causa potest esse ita certa quia est, sicut effectus, cuius esse est ad sensum notum. Ipsum vero quia est causæ, est coniecturale, utrum tale esse coniecturale est notius ipso effectu in genere notitiæ propter quid. Nam posita inventione causæ, semper scitur propter quid effectus ....

Furthermore, Nifo is clearly aware—apparently expressing a common opinion—that in order to overcome this unwelcome limitation, a peculiar kind of notitia, called by him intellectus negotiatio, is needed:35 Recentiores volunt esse 4 notitias: prima est effectus per sensum, aut observationem; secunda inventio causæ per effectum ...; tertia est eiusdem 33

The term in the present meaning is professedly traceable back to the Averroistic tradition; cf. Randall, “Development of Scientific Method,” 41. 34 Agostino Nifo, Expositio super octo Aristotelis Stagiritae libros de physico auditu (Venice: Hieronymvs Scotvs, 1569), 4, Recognitio. Quoted from Randall, “Development of Scientific Method,” 46, fn. 21. 35 Quoted from Randall, ibid., 43, fn. 20.

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Chapter Five causæ per intellectus negotiationem, ex quâ cum prima crescit notitia causæ intantum, ut digna sit effici medium demonstrationis simpliciter; quarta est eiusdem effectus notitia propter quid, per talem causam sic certam, ut sit medium (ibid., 4; my emphasis).

Yet to recognize the need of a certain sort of negotiatio is quite another matter from determining in what such a negotiatio should consist and although Nifo makes some stabs in this direction in the continuation of the above, his meaning remains obscure.36 Both the problems just discussed are dealt with in a much more satisfactory way by Giacomo Zabarella, the culminating figure who was able to “[sum] up the collective wisdom of the Padua school” (Randall, ibid., 49). To begin with the latter problem, here is Zabarella‫ތ‬s proposal of how Nifo‫ތ‬s negotiatio is to be explicated:37 Facto ... primo processu, qui est ab effectu ad causam, antequàm ea ad effectum retrocedamus, tertium quendam medium laborem intercedere necesse est, quo ducamur in cognitionem distinctam illius causæ, quæ confusè tantùm cognita est; hunc aliqui necessarium esse cognoscentes vocarunt negociationem [sic] intellectus, nos mentale ipsius causæ examen appellare possumus, seu mentalem consyderationem [sic] ...; qualis autem sit hæc mentalis consyderatio, & quomodo fiat, à nemine vidi esse declaratum .... Duo sunt ... quæ nos iuuant ad causam distinctè cognoscendam, vnum quidem cognitio quòd est, quæ nos preparat ad inueniendum quid sit .... Alterum verò, sine quo illud non sufficeret, est comparatio causæ inuentæ cum effectu, per quem inuenta fuit, non quidem cognoscendo hanc esse causam, & illum esse effectum sed solùm rem hanc cum illa conferendo; sic enim sit ut ducamur paulatim ad cognitionem conditionum illius rei, & vna inuenta conditione iuuemur ad alium inueniendam, donec tandem cognoscamus hanc esse illius effectus causam ... (De Regressv, ch. 5, 326–27; my emphases).

In addition, Zabarella integrates the mentalis consyderatio into a general description of the regressus procedure as a solution to the former problem of circularity: 36

Here is probably virtually all Nifo has to say on the issue: “Illa negotiatio est compositio & divisio. Nam causa ipsa inventa intellectus componit & dividit donec cognoscat causam sub ratione medii. Nam ... causa ... est ... medium, quatenus definitio. ... [N]egotiatio est ad causam ut medium ac definitio. Sed ... definitio non inventur nisi compositione & divisione ... (ibid.; quoted from Randall, “Development of Scientific Method,” 43, fn. 20). 37 Giacomo Zabarella, De Regressv, in idem, Opera logica (Venice: Paulus Meietus, 1578), 322–34.

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Ex tribus igitur partibus necessariò constat regressus, prima quidem est demonstratio quòd, quâ ex effectus cognitione confusa ducimur in confusam cognitionem causæ; secunda est consyderatio illa mentalis, quâ ex confusa notitia causæ distinctam eiusdem cognitionem acquirimus; tertia verò est demonstratio potissima, quâ ex causa distinctè cognita ad distinctam effectus cognitionem tandem perducimur ... (ibid., 328).

It was chiefly in the form given to them by Zabarella that the Renaissance Aristotelian approach to method (taken over by Galilei and some other early proponents of modern methods of investigation) was communicated throughout Western Europe at the beginning of the seventeenth century, and was thus likely to exert some influence on Descartes.

5.3.4 Some Similarities to Descartes on Analysis What has been established up to the present point should alone render it plausible, I submit, to claim that the Aristotelian conception of heuristic analysis is sufficiently close to Descartes’ own notion of for what analysis is designed and how it is to proceed, and provides some credentials to my central interpretative contention that it is this Aristotelian conception that Descartes opposes on purpose. Yet the affinity between the ways in which analysis is treated by the Renaissance Aristotelians and by Descartes grows even stronger once certain aspects of Descartes’ thought are considered in the light of the above-established Aristotelian methodological points. To begin, Descartes comments in a celebrated passage in DM 6 on certain putative methodological problems in his actual treatment in La Dioptrique and in Les Meteores—in particular on the charge of professed circularity—in distinctively causal terms:38 Que si quelques vnes de celles dont i‫ތ‬ay parlé, au commencement de la Dioptrique & des Meteores, chocquent d‫ތ‬abord, a cause que ie les nomme des suppositions, & que ie ne semble pas auoir enuie de les prouuer, qu‫ތ‬on ait la patience de lire le tout auec attention, & i‫ތ‬espere qu‫ތ‬on s‫ތ‬en trouuera satisfait. Car il me semble que les raisons s‫ތ‬y entresuiuent en telle sorte que, comme les dernieres sont demonstrées par les premieres, qui sont leurs causes, ces premieres le sont reciproquement par les dernieres, qui sont leurs effets. Et on ne doit pas imaginer que ie commette en cecy la faute que les Logiciens nomment vn cercle; car l‫ތ‬experience rendant la plus part de ces effets tres certains, les causes dont ie les deduits ne seruent pas tant a les 38

Much the same point is repeated in the above-quoted letter to Plempius, AT I, 476.

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Chapter Five prouuer qu‫ތ‬a les expliquer; mais, tout au contraire, ce sont elles qui sont prouuées par eux (AT VI, 76).

It will be recognized that this way of addressing the charge strikingly resembles the regressus theory, proposed and developed from Paul of Venice to Zabarella, as a solution to the circularity problem. Moreover, it will be remembered from ch. 4 how substantial a rôle the operation of comparatio is supposed to play in the analytic procedure envisaged by Descartes, and that the entire preparatory task of Descartes’ method is designed to facilitate comparationes in an appropriate and feasible form. It will be noted that when Zabarella set out to explicate how the mentalis consyderatio is actually supposed to work, he also adopted, presumably in a novel way, the language of comparationes; and the mentalis consyderatio at issue—and comparationes along with it—is clearly deployed by Zabarella, as we saw, precisely to overcome the merely conjectural, hypothetical status of the results of heuristic analysis and thus to render those results amenable to strictly scientific treatment. Thus although Zabarella‫ތ‬s conception is still far from coincident with Descartes’ notion of how exactly the comparationes are supposed to work, what their terms are and how they are related to cognitive operations in general, like Descartes, Zabarella also seem to ascribe to comparationes a crucial rôle in rendering analysis a procedure suited to facilitate strictly scientific discoveries. These affinities provide further support to the interpretative assumption that Descartes was far from ignoring the distinctive Aristotelian conception of heuristic analysis developed by thinkers whose intention was amazingly close to his own—namely to promote analysis as a legitimate tool of scientific discovery (although for the Renaissance Aristotelians, unlike for Descartes, this would never count as the sole, selfsufficient tool in the domain of scientiæ). Yet, of course, all this makes the Discrepancy under investigation only more disturbing and, indeed, in sore need of explanation: for unlike Descartes, all the Aristotelians involved in the campaign for imbuing a scientific status to heuristic analysis—from Arabian Averroists and commentators of Galen through their successors at northern Italian universities to Zabarella and beyond—are unanimous in taking even heuristic analysis as firmly tied to the quia, i.e. to a posteriori reasoning. So, let us now finally return to Descartes to tackle head-on the discrepancy between his own and the Aristotelian characterization of analysis in terms of the a priori–a posteriori pair.

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5.4. Analysis as A Priori in Descartes I am nowhere near the first, of course, to deal with the Discrepancy in question. However, although various attempts at resolving or at least at mitigating the Discrepancy have been proposed in recent scholarship, all those attempts appear to me eventually to fail for this or that reason. I do not pretend to provide an exhaustive survey of these flawed attempts. However, I will consider some of them, only to clear the way for my own interpretative suggestion, which is the chief task of the present section.

5.4.1 Culs-de-Sac One prima facie tempting course that must be resisted from the very start is a suggestion that the Discrepancy could be dismissed simply on the grounds that it is engendered by a single passage which, au pied de la lettre, was not written by Descartes but by his French translator, Clerselier, who perhaps misunderstood whatever it was that Descartes meant by a priori and a posteriori in the context of describing the procedures of analysis and synthesis in AT VII, 155–56. There are strong reasons to regard such a suggestion as a non-starter. First, there is every reason to believe that Clerselier‫ތ‬s paraphrase is a faithful rendering of Descartes’ meaning: for one thing, it is claimed in the publisher‫ތ‬s introduction to the reader of the French translation of the Meditationes that Descartes reviewed, corrected and approved the translation;39 for another thing, a classical biographer of Descartes, Adrien Baillet, claims that it was Descartes himself who had put his hand on translations and/or explications of certain Latin technical terms, especially those of Scholastic provenance;40 finally, Clerselier‫ތ‬s paraphrases seem to square perfectly (unless subjected to some contrived gerrymander)41 with Descartes’ own explanations, quoted in sec. 5.2, of what he means by a priori and a posteriori. This final point is decisive, for even if it is put to one side that the dismissive suggestion now under scrutiny goes against all the (admittedly indirect and not entirely conclusive) evidence to the effect that Descartes did approve of Clerselier‫ތ‬s renderings, proponents of the 39

Cf. “Le libraire av lectevr,” AT IX-1, 2: “On trouuera partout cette version assez iuste, & si religieuse, que iamais elle ne s’est escartée du sens de l‫ތ‬Auteur. ... [Les traducteurs] ont (comme il estoit iuste) reserué à l‫ތ‬Auteur le droit de reueuë & de correction. Il en a vsé ....” 40 See Baillet, La vie de monsieur Des Cartes, 2:171–73. 41 See fn. 43 for a salient example of such a contrived stretch, due to Stephen Gaukroger.

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suggestion would still commit themselves, vis-à-vis the adduced textual evidence, to the claim that precisely in the particular context of characterizing analysis and synthesis, Descartes uses the terms a priori and a posteriori—contrary to what Clerselier assumed—in a sense which is so completely detached from anything established in the Aristotelian tradition that it no longer retains even the very idea of causal ordering. Admittedly some departure from the established Aristotelian usage is warranted by the qualifying tanquam inserted by Descartes; yet assuming such a gross equivocation as is needed for the present suggestion to succeed is extremely uncharitable to Descartes. Furthermore, the prospects of explicating the envisaged alternative meaning of the a priori and a posteriori in positive terms seem remote. Thus it sounds reasonable that any viable attempt at tackling the Discrepancy might take for granted that even in the contexts of dealing with analysis and synthesis, the characterizations a priori and a posteriori do retain, for Descartes, the basic connotation—however rudimentary or modified—of an ordering from causes to effects (a priori) and from effects to causes (a posteriori), respectively. As far as I can tell, almost all the attempts at resolving the discrepancy that grant this much,42 adopt the same basic strategy (although they of course vary in matters of detail): that is to say, they draw upon an Aristotelian distinction we have already encountered, viz. the distinction between what is prior in nature or essentially or in the order of being and what is prior in cognition or epistemically or in the order of knowledge; and they suggest that it is the latter sense of priority that Descartes has in mind when he characterizes analysis as a priori and synthesis as a posteriori.43 As Benoît Timmermans puts it, 42

I say “almost” as at least one scholar, viz. Roger Florka in his Descartes’s Metaphysical Reasoning, 109–17, has come up with an alternative solution which surely escapes the fatal flaws of the genus of solutions I am about to discuss. I confess I have achieved a considerably less than perfect understanding of his positive proposal, which surely hinges upon relating analysis and synthesis to the order of dependencies between what Florka calls the metaphysics of dualism and the metaphysics of the human being (ibid., 116). In any case, Florka‫ތ‬s solution of the Discrepancy seems to me entirely disparate from mine. I shall not venture any further discussion of his proposals. 43 See in particular Gaukroger, Cartesian Logic, 99–102, and Timmermans, “Originality of Descartes’s Conception,” 435–38. Gaukroger offers an especially sophisticated version of the sort of solution to be discussed, in that he supplements his proposal with a reading of the crucial passage of the French version of Resp. 2 (AT IX-1, 121–22) to the effect that “[i]n saying that analysis shows us how effects depend upon their causes, Descartes/Clerselier is not saying that we start from causes and determine their effects, but rather that we start with effects and

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since analysis is above all heuristic, our point of view [when characterizing analysis as a priori along Descartes’ lines] is that of the order of reasons, and the starting point of analysis may thus appear to be a principle that is well known ... (“Originality of Descartes’s Conception,” 437).

In other words, analysis, which by [the Aristotelian] definition always goes from the consequences to the principles, can be considered heuristic from a certain point of view when, from this point of view, it is synthetic, that is, when it starts from something that, while a consequence, is also a simple, wellknown principle from another point of view that leads us to discover unknown things (ibid., 436; Timmermans’ emphasis).

The Discrepancy is thus resolved, indeed: analysis counts as a priori in the sense that in the order of epistemic dependence, it proceeds from principles, i.e. from what is actually better known to a given mind at the beginning of investigation, to consequences, i.e. to what is epistemically dependent upon that prior cognition; yet since from the point of view of the order of nature, the consequences reached in the analytic procedure still count as principles and the principles which the analytic procedure started from as consequences, Descartes’ overall conception can after all be taken as consistent with the Aristotelian notion of analysis as an a

show what the causes that they depend upon are. Similarly, in saying that synthesis examines causes through their effects he is not saying that we start from effects and infer causes, but that we start with causes and determine their truth, generality, and so on by looking at their effects” (Gaukroger, Cartesian Logic, 100–101). That is to say, Gaukroger attempts, by the means of a considerably stretched reading of the French passage in question, to put Clerselier‫ތ‬s gloss in accord with the standard Aristotelian interpretation of analysis and synthesis, respectively, in terms of causal ordering. Yet although Gaukroger adds that his reading “is just as natural as” the standard reading he inverts (ibid., 101), I believe that this particular suggestion of Gaukroger is fallacious. We saw above when dealing with the Paduan regressus theory that the Aristotelians whose thoughts we considered clearly distinguished between showing that effects depend on their causes and showing how effects depend on their causes; and it is the former, and no means the latter, that allows of a reading along the lines Gaukroger proposes for Clerselier‫ތ‬s gloss on analysis (see also fn. 47; mutatis mutandis the same point can be made concerning synthesis). In any case, Gaukroger‫ތ‬s above proposal must be supplied with a reading of the a priori–a posteriori pair along the lines to be discussed in the rest of the present sub-section, and it stands and falls by the success of the latter.

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posteriori procedure.44 This general approach finds some support in Descartes’ texts. In ch. 2 we have already witnessed Descartes employing a distinction between considering things nisi quantùm ab intellectu percipiuntur, i.e. in ordine ad cognitionem nostram or respectu intellectûs nostri, and considering things prout revera existunt (Reg. XII, AT X, 418); and when discussing, in the continuation of the seminal passage on analysis and synthesis in Resp. 2, the respective merits of employing analysis and synthesis in metaphysics and in geometry, he states unambiguously that what counts as principles in metaphysical synthesis are the best-known items ex naturâ suâ but not quoad nos unless the analytical procedure is carried out: [Q]uantum ad Synthesim ..., etsi in rebus Geometricis aptissime post Analysim ponatur, non tamen ad has Metaphysicas tam commode potest applicari. Hæc enim differentia est, quòd primæ notiones, quæ ad res Geometricas demonstrandas præsupponuntur, cum sensuum usu convenientes, facile a quibuslibet admittantur. Ideoque nulla est ibi difficultas, nisi in consequentiis rite deducendis .... Contra verò in his Metaphysicis de nullâ re magis laboratur, quàm de primis notionibus clare & distincte percipiendis. Etsi enim ipsæ ex naturâ suâ non minus notæ vel etiam notiores sint, quàm illa quæ a Geometris considerantur, quia tamen iis multa repugnant sensuum præjudicia quibus ab ineunte ætate assuevimus ... (AT VII, 156–57).

Moreover, it does not matter that the approach in question requires one to substitute principle for cause and consequence for effect as the terms of the ordering which determines the relations of priority and posteriority relevant to the meaning of the a priori–a posteriori distinction. For although the point might turn out somewhat intricate when scrutinized in detail (due mostly to the somewhat obscure status of Aristotelian final

44

To my knowledge, the first to employ the distinction between the order of cognition and the order of being, deliberately in the context of dealing with Descartes’ conception of analysis and synthesis, was Martial Guéroult in his Descartes selon l’ordre des raisons, 1:22–28. Guéroult, drawing upon the distinction between the ordo ad cognitionem nostram and the ordo à parte rei in Reg. XII, AT X, 418, maintains that while analysis operates in the former order, synthesis operates in the latter. Although Guéroult does not associate his claim with the Discrepancy problem (since he is interested in another issue, viz. Descartes’ reported characterization of the Principia, in opposition to the Meditationes, as a synthetic treatise in Burm., AT V, 153), his interpretation is still open to criticism I will presently raise.

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causes),45 prima facie a good case can be made for the suggestion that the epistemic state which amounts to the principle in the order of cognition counts as a cause—even as an efficient cause—of epistemic states which amount to consequences in the order of cognition. Thus although the present interpretative suggestion once again hinges upon ascribing to Descartes equivocal use of the terms “a priori” and “a posteriori”, the equivocation turns out this time based upon his texts and appears by no means gratuitous. Finally, the suggestion allows for an appealing explanation of the somewhat perplexing double use of the terms “a priori” and “a posteriori” in the Latin version of Descartes’ talk of synthesis in Resp. 2: “Synthesis [procedit] per viam ... tanquam a posteriori quæsitam (etsi sæpe ipsa probatio sit in hac magis a priori quàm in illâ) ...” (AT VII, 156). The point is—so the suggestion under scrutiny goes—that !a posteriori” is used here in the epistemic sense but “a priori” in the essential sense. Also the “tanquam” with which Descartes qualifies his attribution of a priori and a posteriori to analysis and synthesis, respectively, receives a plausible explanation: in view of the present solution, the “tanquam” indicates that the order of priority referred to is that of cognition and not (as in the standard Aristotelian usage) that of nature or being. Yet despite these pros, I believe any solution of the Discrepancy based upon the present approach is eventually doomed to failure. For as Roger Florka has acutely pointed out,46 in the proposed reading of a priori and a posteriori, in so far as these terms are designed to characterize analysis and synthesis respectively in Descartes, referring to the order of cognition is strictly incompatible with the complex fact that (i) Descartes clearly takes both analysis and synthesis as legitimate instances of modus scribendi geometricus in the seminal passage of Resp. 2, that (ii) he declares both these modi scribendi be subjected to the requirement of ordo, and that (iii) being subjected to ordo amounts to obeying the requirement to proceed from what is prior to what is posterior in the order of cognition:

45

I cannot deal with these immense complications in the present study. For a good exposition of the problem, see e.g. David Furley, “What Kind of Cause is Aristotle’s Final Cause?” in Rationality in Greek Thought ed. Michael Frede and Gisela Striker (Oxford: Clarendon Press, 1999), 59–79. 46 Cf. Florka, Descartes’s Metaphysical Reasoning, 85–86. The criticism I present is just a development of ideas already contained in Florka‫ތ‬s condensed but perspicuous and acute account.

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Chapter Five Duas res in modo scribendi geometrico distinguo, ordinem scilicet, & rationem demonstrandi. Ordo in eo tantùm consistit, quòd ea, quæ prima proponuntur, absque ullâ sequentium ope debeant cognosci, & reliqua deinde omnia ita disponi, ut ex præcedentibus solis demonstrentur. ... Demonstrandi autem ratio duplex est, alia scilicet per analysim, alia per synthesim (Resp. 2, AT VII, 155; my emphasis).

As far as I can see, the declared incompatibility is nailed down by the indisputable commitments entailed in this passage: unless still another equivocation in Descartes’ use of the terms “a priori” and “a posteriori” is invoked, the lone fact that synthesis is subjected to the requirement of ordo in the above sense should suffice to characterize even synthesis as a priori rather than as a posteriori as long as the interpretation of the a priori–a posteriori pair in terms of the order of cognition is adopted.47 Putting to one side Florka‫ތ‬s positive proposal, which remains obscure to me, I know of no attempt at solving the Discrepancy problem that eventually escapes this substantive criticism. In view of this unsatisfactory situation, an alternative suggestion, tentative and speculative as it may be, is perhaps in order.

5.4.2 A Speculative Suggestion To begin with, I endorse two assumptions that turned out reasonable in the course of the preceding discussion: one, that some connotation of an ordering from causes to effects and from effects to causes, respectively, is 47

The above discussed regressus theory as a solution to the circularity threat (engendered, it will be remembered, by elevating analysis as the proper part of the scientific cognitive enterprise) can be invoked to clarify the point of the present reductio. The professed circle is engendered as long as the synthetic part of the scientific procedure is construed as a movement from the item cognized by way of heuristic analysis back to the item from which that same heuristic analysis started and which was cognized from the very beginning; and the point of the regressus response is that what was better-known at the beginning of the analytical procedure was that there is some item (effect in the order of nature) whose causal explanation is yet to be discovered; and that once an appropriate cause is found, it counts as the better-known item with regard to another task, namely to explain how the initial effect depends upon the discovered cause. So whereas the procedure at issue is strictly circular in so far as the order of nature is concerned, it counts as a continuous non-circular procedure from relatively better-known to relatively lesser-known in so far as the order of cognition is concerned. Furthermore, all this is in perfect structural accord with that to which Descartes commits himself in the last quoted passage.

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to be retained, in the context of addressing the Discrepancy problem, as a component of the meaning of a priori and a posteriori in Descartes’ usage; and two, that Clerselier‫ތ‬s French translation of the seminal passage in Resp. 2 is an approved and accurate rendering of how that connotation is to be unfolded. To obtain a fresh perspective on the issue, let us focus now on a certain neglected blind spot in the overall picture of possible interrelations between the analysis–synthesis pair on the one hand and the a priori–a posteriori pair on the other, against the background of which the Discrepancy has been constructed. The content of the a priori–a posteriori pair is, as we saw, clearly and consistently articulated in terms of causal relations in the Aristotelian tradition and I have simply adopted an assumption that any viable conception of the meaning of that pair in Descartes is also bound to some form of retained causal connotations. However, the situation is by no means that uniform regarding the former analysis–synthesis pair. We saw that the Aristotelian thinkers whose views were considered unanimously employed the very same causal conceptual devices in explicating the meaning of analysis and synthesis as they do in the case of the a priori–a posteriori pair: the data and the quæsita in both analysis and synthesis are generally interpreted, respectively, as effects or causes in the very same sense in which the terms “effect” and “cause” are employed in explications of the meaning of a priori and a posteriori. Such a uniformity of explication renders the Aristotelian conceptual framework for the characterization of analysis and synthesis in terms of the a priori–a posteriori distinction remarkably unitary: in the ontic order of being, analysis proceeds from effects to causes and counts therefore as a posteriori, whereas synthesis proceeds from causes to effects and counts therefore as a priori. By way of contrast, ch. 3 and 4 of the present study document that Descartes is for himself able to dispense completely with the causal conceptual framework in the course of the development and structural articulation of his peculiar conception of analysis and synthesis.48 The conceptual vocabulary in terms of which the data and the quæsita are articulated by him is, as we saw in ch. 4, that of absoluta and respectiva, notions which are in turn ultimately explicated in terms of simple natures and/or their compositiones, and simple natures and their compositiones are explicated, in those contexts, in by no means causal but in purely 48 It was established in ch. 3 that although Descartes does not dispose of synthesis completely, he allows as meaningful only a special sort of synthesis and that the significance of synthetic procedure is in any case contingent upon analysis. For the sake of brevity, I limit myself to dealing with Descartes‫ ތ‬conception of analysis quâ the a priori from now on, and take up synthesis only in sec. 5.5.

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epistemic terms. It is precisely this heterogeneity of conceptual frameworks within which the pairs analysis–synthesis and a priori–a posteriori are articulated in the Aristotelian tradition and in Descartes that has effectively hindered any resolution of the Discrepancy from carrying more than a brief sheen of plausibility. So let us try to overcome that heterogeneity, and look out for the clues which such an attempt might provide. Consider first the following passage from Princ. Pref., AT IX-2, 2:49 [C]e mot Philosophie signifie l‫ތ‬estude de la Sagesse, & ... par la Sagesse on ... entend ... vne parfaite connoissance de toutes les choses que l‫ތ‬homme peut sçauoir ...; & afin que cette connoissance soit telle, il est necessaire qu‫ތ‬elle soit déduite des premieres causes, en sorte que, pour estudier à l‫ތ‬acquerir, ce qui se nomme proprement philosopher, il faut commencer par la recherche de ces premieres causes, cҲest à dire des Principes; & que ces Principes doiuent auoir deux conditions: lҲvne, quҲils soient si clairs & si éuidens que lҲesprit humain ne puisse douter de leur verité, lorsquҲil sҲapplique auec attention à les considerer; lҲautre, que ce soit dҲeux que depende la connoissance des autres choses, en sorte quҲils puissent estre connus sans elles, mais non pas reciproquement elles sans eux; & qu‫ތ‬apres cela il faut tascher de déduire tellement de ces principes la connoissance des choses qui en dependent, qu‫ތ‬il n‫ތ‬y ait rien, en toute la suite des deductions qu‫ތ‬on en fait, qui ne soit tres-manifeste (my emphases).

It will be observed, to begin with, that Descartes feels free here to refer to “les Principes,” out of which all the pieces of the sought-after “parfaite connoissance de toutes les choses” are to be “déduites”, with a presumably synonymous expression, “les premieres causes”; and that the conditions the premieres causes or the Principes are bound to satisfy according to the present passage are the same as those in terms of which, as we saw in ch. 2, Descartes elsewhere describes his naturæ simplices or notions primitiues and which are arguably also applicable to complexes of simple natures quâ objects of intuitus, at least in so far as the connections between the involved simple natures are necessary. Together with Descartes’ reductive treatment of omnia quæ cognoscimus established in ch. 2, these observations alone seem to warrant the conclusion that Descartes was at very least occasionally prepared to apply causal vocabulary to simple natures and/or to (necessary) complexes of simple natures, more specifically 49

Cf. also ibid., AT IX-2, 5: “[V]n cinquiéme degré pour paruenir à la Sagesse, incomparablement plus haut & plus assuré que les quatre autres: c‫ތ‬est de chercher les premieres causes & les vrays Principes dont on puisse déduire les raisons de tout ce qu‫ތ‬on est capable de sçauoir ....”

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to characterize them as les premieres causes, in the context of searching after “vne parfaite connoissance de toutes les choses que l‫ތ‬homme peut sçauoir.” Furthermore, this crucial conclusion is confirmed with Descartes’ list of les Principes of “la premiere partie” of “la vraye Philosophie”, i.e. the principles of metaphysics, later in Princ. Pref., AT IX-2, 14:50 [La Metaphysique] contient les Principes de la connoissance, entre lesquels est l’explication des principaux attributs de Dieu, de l’immaterialité de nos ames, & de toutes les notions claires & simples qui sont en nous.

The last class of the list clearly ultimately coincides with the complete set of simple natures; and the first two items seem ultimately to boil down to explications based exclusively upon complexes of simple natures.51 50 Cf. also the following passage from a letter to Mersenne which supplements Descartes’ account of principles/first causes in Princ. Pref. with a distinction of two kinds of principles, thus allowing integration of the notiones communes into the conception at issue: “[L]e mot de principe se peut prendre en diuers sens, & ... c’est autre chose de chercher vne notion commune, qui soit si claire & si generale qu’elle puisse seruir de principe pour prouuer l’existence de tous les Estres, les Entia, qu’on connoistra par apres; & autre chose de chercher vn Estre, l’existence duquel nous soit plus connuë que celle d’aucuns autres, en sorte qu’elle nous puisse seruir de principe pour les connoistre. Au premier sens, on peut dire que impossibile est idem simul esse & non esse est vn principe, & qu’il peut generalement seruir, non pas proprement à faire connoistre l’existence d’aucune chose, mais seulement à faire que, lors qu’on la connoist, on en confirme la verité par vn tel raisonnement: Il est impossible que ce qui est ne soit pas; or ie connois que telle chose est; donc ie connois qu’il est impossible qu’elle ne soit pas. ... En l’autre sens, le premier principe est que nostre Ame existe, à cause qu’il n’y a rien dont l’existence nous soit plus notoire” (AT IV, 444; Descartes’ emphases). 51 Cf. also DM 6, AT VI, 63–64 where Descartes sums up the path he followed in Le Monde (and the essentials of which are still present in Descartes’ mature account in Princ. Pref.): “Premierement, i’ay tasché de trouuer en general les Principes, ou Premieres Causes, de tout ce qui est, ou qui peut estre, dans le monde, sans rien considerer, pour cet effect, que Dieu seul, qui l’a creé, ny les tirer d’ailleurs que de certaines semences de Veritez qui sont naturellement en nos ames. Après cela, i’ay examiné quels estoient les premiers & plus ordinaires effets qu’on pouuoit deduire de ces causes ....” Notably, les Principes are once again identified here with Premieres Causes, and Descartes reports having drawn them (“tirer”) from “certaines semences de Veritez qui sont naturellement en nos ames”, a description which I take to refer to simple natures. Note that I ascribe to “semences de Veritez qui sont naturellement en nos ames” in DM 6 a quite different extension to that which I ascribed in ch. 4 to “veritatum semina humanis

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Only one more step remains to establish the sought-after tertium comparationis in our attempt at dealing with the Discrepancy problem. It will be observed that certain (necessary complexes of) simple natures are characterized as “les premieres causes” in AT IX-2, 2 exactly in so far as some “parfaites connoissances” (i.e., as the context makes clear, pieces of scientia) can be deduced (déduites) from them. However, this is exactly the function due to which anything qualifies as an absolutum in Descartes’ most elaborated account of method in the régime of discovery in Reg. VI: Absolutum voco, quidquid in se continet naturam puram & simplicem, de quâ est quæstio .... Respectivum verò est, quod eadem quidem naturam, vel saltem aliquid ex eâ participat, secundùm quod ad absolutum potest referri, & per quamdam seriem ab eo deduci ... (AT X, 381–82; my emphasis).

In view of this, the conclusion is ready to hand that (necessary complexes of) simple natures qualify as what Descartes is sometimes prepared to call les premieres causes exactly in so far as these (necessary complexes of) simple natures fulfil the rôle of absoluta in the course of putting Descartes’ analytic method to work. Having thus established the sought-after common ground, we get our hands, by the same token, on a powerful clue to Descartes’ notion of a priori quâ an essential characterization of analysis. For if causes coincide, in the present context, with (necessary complexes of) simple natures in so far as they fulfil the rôle of absoluta, and if the term a priori is to connote, even in Descartes’ hands, some sort of priority of causes over effects, then the rationale of Descartes’ characterization of analysis as a priori might well be derived from the peculiar rôle which those (complexes of) simple natures that count as absoluta are supposed to play in the analytical procedure. More specifically, in the light of that rôle, absoluta might well turn out prior, in some substantial sense, to whatever counts as effects in the context of analysis as understood by Descartes. So let us try to follow this lead. The rôle of absoluta in the (preparatory stage of) analytical procedure was discussed at length in ch. 4, and the envisaged priority is not hard to find in view of that discussion; for it was established there, among other things, that after a given quæstio has been determined, the process of addressing the quæstio starts, ingenijs à natura insita” in Reg. IV, AT X, 376 (cf. also my parallel reading of prima cogitationum vtilium semina in ibid., 373). Although I am not particularly happy with this result, charging Descartes with equivocation in this respect seems to me a very acceptable price to pay for the submitted, hopefully consistent overall interpretation of the fundamental layers of his methodology.

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according to Descartes, neither with the data alone, nor with the quæsita alone, nor even with both the data and the quæsita just gathered together, but with the ordering and disposing of both the data and the quæsita, regardless of what counts as known or unknown, into series of somehow determined epistemic dependencies. Furthermore, absoluta, it will be remembered, are designed precisely to operate as the principles of that ordering and disposing: they amount to the terms ad quem of a certain relation—namely the relation of participation in the above-established peculiar sense—which determines the relevant relations of epistemic dependencies in terms of which the ordering and disposing at issue is to be carried out. In other words, absoluta, i.e. naturæ communes which are eventually reducible either into a single simple nature or to a complex of simple natures, count as prior items, or causes, in the analytic procedure in the sense that they condition, without themselves being conditioned by, the very constitution of those ordered series of epistemic dependencies between both the data and the quæsita which any problem-solving procedure takes as its proper point of departure. At this point it is not difficult to extrapolate from all this, with the help of the above-quoted AT IX-2, 2 and AT X, 381–82, how effects are to be conceived in the context of analysis as understood by Descartes. Most generally, just any item arrived at by any deductio which is ultimately conditioned with a corresponding absolutum—i.e. just any respectivum in the context of a given quæstio—counts as an effect relative to cause quâ absolutum. In this most general sense, effects are ordered in chains; and the ultimate effect, i.e. the goal of the analytical procedure, which coincides (as should be clear by now) with the last piece of the causal chain thus constructed, is coincident with a piece of scientia, a piece which always takes the form of a given quæsitum being æquale to (some of) the corresponding data. Thus in view of all this the Clerselier/Descartes explication of the meaning of a priori as an essential attribute of heuristic analysis, viz. the phrase “faire voir comment les effets dépendent des causes” at AT IX-1, 121, can be translated, in the long run, into something like to show how the solution to a given quæstio can be obtained through the ordering and disposing of both the data and the quæsita by means of an appropriate absolutum, an absolutum which counts as a “premiere cause” because it ultimately conditions the ordering and disposing, without which no deductio in the régime of discovery is ever possible.

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This, indeed, is the best I can do to provide some rationale for Descartes’ characterization of analysis as a priori, and to provide an account of the content of Descartes’ concept of a priori over and above the bewilderingly loose and unclear feature of proceeding from causes to effects. We are finally in a position to deal with the Discrepancy problem in the light of this interpretation. It should be clear by now that what lies behind Descartes’ characterization of analysis (in its relevant, i.e. heuristic variety) as a priori is by no means just a reversal of the order of priority between causes and effects in the context of facilitating (scientific) discoveries as treated in the Aristotelian tradition. Rather, Descartes’ move in question is to be interpreted as an expression of a thorough reconfiguration of the basic structural elements out of which the Aristotelian framework for the articulation of the conditions of discovery is composed. Details apart, the Aristotelian view of how heuristic analysis works can be put as follows: the term a quo refers to the data that amount to the ultimate effects to be explained, and the term ad quem refers to the quæsita that amount to the causes through which the ultimate effects are to be explained. This spuriously plain and tidy conception is completely destroyed and re-constituted by Descartes: for according to him, the term a quo of the analytic procedure is neither the data nor the ultimate effects but rather the ordered series of both the data and the quæsita. Again, the term ad quem of Descartes’ analysis is neither the quæsita nor the explanatory causes but rather æquationes (or comparationes apertæ & simplices) holding between the quæsita and (some of) the data. Finally, the data and the quæsita are no longer coincident either with the ultimate effects or with the (explanatory) causes; rather, the ultimate effects turn out coincident with the æquationes that amount to the terms ad quem of the analytic enterprise, and the causes in the relevant sense turn out coincident with the absoluta that amount to the principles which the initial orderings of the terms a quo—i.e. of the series of both the data and the quæsita according to epistemic dependencies—depend upon. In the light of this substantial re-configuration, Descartes’ characterization of analysis as a priori, indeed as tanquam a priori, can well be read as a deliberate and subtle hint, addressed to his Aristotelian audience, at how profound and clinching is his break with the Aristotelian tradition as regards the method of scientific inquiry. Descartes has retained just enough of the causal connotations that the term a priori has in Aristotelian thought to be able to signal his break in terms of the reversal of causal ordering in the analytic procedure as developed by him; and the qualifying tanquam, I submit, is best read as a hint at another significant aspect of Descartes’ novel conception of analysis, namely at his destruction of a

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firm tie which has proven essential to the Aristotelian appraisal of the cognitive merits of analysis and synthesis: that is to say, the tie between the data/quæsita and effects/causes pairs.

5.5 Coda: Synthesis as A Posteriori in Descartes Having thus accomplished what has turned out to be the chief task of the present study, viz. a complex account of Descartes’ conception of analysis as a priori, what remains to be briefly considered are its implications for the vexed question of Descartes’ approach to synthesis and for his notion of a posteriori. To begin, it was established in ch. 3 that whilst Descartes’ approach to synthesis as a legitimate and possibly useful scientific tool in the régime of discovery in the domain of mathematics is basically dismissive, Descartes seems to reserve certain rôles for (problematical) synthesis in gaining some pieces of scientiæ in mathematics at least. The present interpretation accommodates such a reading quite conveniently and renders it intelligible even for domains beyond mathematics, for it should be clear by now that from Descartes’ perspective the gros of the solution to a given quæstio in any domain whatsoever consists in expressing the structural dependences between the data and the quæsita in the form of appropriately reduced æquationes, and that this is exclusively due to purely analytic procedures; yet this seems to leave the door open, in principle, for supplementary operations of a synthetic cast which might somehow contribute to strengthening or accomplishing the sought-after solution.52 By the same token, the submitted interpretation of analysis as a priori in Descartes suggests a tolerably clear-cut meaning to be associated with the term “a posteriori” as a substantial characterization of problematical 52

Descartes is unfortunately never specific, let alone explicit on this score. I extrapolate that apparently the most significant field in which such supplementary synthetic procedures might turn out useful and perhaps even indispensable in the overall context of Descartes’ thought is that which Descartes calls imperfect(ly understood) quæstiones in Reg. XII–XIII, in particular perhaps in natural philosophy or physics where—as Descartes concisely and acutely observes in DM 6, AT VI, 64–65—the actual ways in which the observable phenomena are rooted in (i.e., upon standard understanding, produced by) their causes are hopelessly underdetermined by the relations of possible dependences between the data and the quæsita ever available due to purely analytic procedures. It would exceed the limits of the present study to pursue this important motif any further; see Clarke, Descartes’ Philosophy of Science, ch. 5–6 for a clear and comprehensive treatment.

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synthesis in the context of Descartes’ thought. It has been established that what counts as prior, indeed as the first, in the analytic procedure as envisaged by Descartes is an absolutum according to which both the data and the quæsita are ordered as to their cognitive dependences and that what counts as posterior are the terminal products of the analytic procedure, viz. æquationes expressing the quæsita by (some of) the data. It should be clear in view of the corresponding discussion in ch. 3 that synthesis, in so far as it is to be in Descartes’ view of any use at all and independent of for what precisely it is designed in any given particular case, is always supposed to start exactly with these posteriora. Thus this, I submit, is the genuine gros of Descartes’ own innovative use of the characterization a posteriori. However—and this is perhaps the most important corollary of the proposed interpretation as regards synthesis—the synthetic procedure, in so far as it be legitimate and useful, is bound in principle never to terminate at the systematic point of departure of the corresponding analysis. This is because the absolutum, through participation in which the data and the quæsita were ordered and disposed at the initial stage of a given analytic procedure, is doomed to be lost from sight once and for all after it has fulfilled its pivotal function: it is supposed to play no rôle in the solution in so far as the solution is incorporated in the resulting æquationes; and vague as the proper purpose of legitimate supplementary syntheses is in view of extrapolations from the conclusions established in ch. 3, such syntheses are definitely not designed to terminate in the absoluta which conditioned the entire heuristic procedure at the outset. As a consequence, the notion of the order of synthetic procedure as a reversal of the order of analysis turns out utterly inadequate, in the light of the present interpretation, as a description of how synthesis is supposed to operate in the only form in which it has any chance of finding some place in Descartes’ overall conception of heuristic method. Such a result contributes its mite to the above destruction of the Aristotelian conception of the method of scientific inquiry starting with heuristic analysis. The Aristotelian conception, even in its most elaborated regressus form, requires synthesis quâ exact reversal of the order of the corresponding analytical procedure (even though the negotiatio crucially changes the epistemic import of the reverse procedure) as an absolutely indispensable component. On the other hand, Descartes either is able to dispense with synthesis altogether, or else synthesis (whenever it turns out necessary or useful), starting as it does exactly where the corresponding analysis terminated, is not supposed by him to work its way back through the analytical steps in reverse order but rather to instantiate the structure of

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the resultant æquatio(nes) independent of how the analysis proceeded and in a way completely determined by that resultant structure alone. These consequences bring us to the last problem to be addressed in this study. It should be clear by now that although Descartes’ conception of analysis as a priori reverses the order with which heuristic analysis has been associated in the Aristotelian tradition and breaks a firm Aristotelian tie between the data/quæsita and effects/causes pairs, it still allows for a re-description in terms of proceeding from causes to effects. By way of contrast, it should be no less clear by now that Descartes’ conception of (problematic) synthesis as a posteriori should not allow for an inverse redescription, i.e. proceeding from effects to causes, without some gross equivocation in terms “cause” and “effect.” Now it will be observed that such an asymmetry fits poorly with the fact that in the seminal AT VII, 155–56, the ratio demonstrandi employed in synthesis quâ a posteriori is treated by Descartes as a reversal of the ratio demonstrandi employed in analysis quâ a priori, and that synthetic procedure is described there in terms of proceeding from effects to causes. Such a discrepancy threatens, of course, to jeopardize the submitted overall interpretation of the general meaning of the a priori–a posteriori pair in Descartes. To appreciate the solution I am about to propose, consider once again how synthesis is described in Resp. 2, AT VII, 156:53 Synthesis è contra per viam oppositam & tanquam a posteriori quæsitam (etsi sæpe ipsa probatio sit in hac magis a priori quàm in illâ) clare quidem id quod conclusum est demonstrat, utiturque longâ definitionum, petitionum, axiomatum, theorematum, & problematum serie, ut si quid ipsi ex consequentibus negetur, id in antecedentibus contineri statim ostendat, sicque a lectore, quantumvis repugnante ac pertinaci, assensioncm extorqueat ... (my emphasis).

53

The French version due to Clerselier reads: “La synthese, au contraire, par vne voye toute autre, & comme en examinant les causes par leurs effets (bien que la preuue qu’elle contient soit souuent aussi des effets par les causes), démontre à la verité clairement ce qui est contenu en ses conclusions, & se sert d’vne longue suite de definitions, de demandes, d’axiomes, de theoremes & de problemes, afin que, si on luy nie quelques consequences, elle face voir comment elles sont contenuës dans les antecedens, & qu’elle arrache le consentement du lecteur, tant obstiné & opiniastre qu’il puisse estre ...” (Resp. 2, AT IX-1, 122; my emphasis). It will be observed that the emphasized phrase “par vne voye toute autre”, which is Clerselier’s rendering of Descartes’ “per viam oppositam”, fits smoothly (and certainly more readily than Descartes’ Latin original) into the solution to the present problem I am about to offer.

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It seems to me, on reflection, that such a description scarcely resembles anything we saw Descartes claiming in ch. 3 with regard to synthesis in so far as synthesis should have even a chance of counting as legitimate and useful as a supplement to analysis in the régime of discovery, i.e. with regard to synthesis of the problematic cast. Rather, what Descartes seems to offer here is perhaps the only explicit account of theoretical synthesis to be found in his extant writings.54 However, the procedure Descartes describes right before the quoted passage in AT VII, 155–56, under the name of analysis, is beyond any doubt analysis of the problematic cast.55 We are thus in a position by now to see that there is, therefore, a good reason to suspect that here, at the place which has been adopted as the point of departure of our overall interpretation of the a priori–a posteriori pair, Descartes fails to draw the contrast between analysis and synthesis in univocal terms: while he offers here a considerably accurate and informative account of problematical analysis, i.e. of analysis in the form which has proven to be of chief importance for his purposes, he contrasts it quite misleadingly with an altogether disparate account of theoretical synthesis which he rightly dismisses as entirely useless for the purposes of his overall methodological and scientific project, with all the easily derivable equivocations in the terms “cause,” “effect,” “demonstratio,” and even “a posteriori” under way. I therefore suggest, to conclude, that the description of synthesis in Resp. 2 be simply dismissed as irrelevant, indeed misleading with regard to the determination of the meaning of the a priori–a posteriori pair in Descartes. Although I am of course not exactly happy with this move, perhaps this is eventually not too high a price to pay for what is, one hopes, the clear and plausible solution to the Discrepancy problem which I have been able to offer, a solution which I believe improves considerably upon any hitherto submitted solution both worthy of consideration and comprehensible to me and is thus likely to make some contribution to the 54

This claim is confirmed by what one finds actually happening in the Rationes which Descartes appended to Resp. 2 (AT VII, 160–70) and which, as Descartes himself intimates, incorporate a sample of the synthetic procedure described in AT VII, 155–57. As far as I can see, the Rationes contain no traces of whatever it might be that would amount to an analogue to but fragments of the problematical synthetic treatment outlined by Descartes in particular in Reg. XVI; on the other hand, the content of the Rationes seems to be organized analogically to the synthetic treatment of the theoretical cast par excellence in mathematics, namely to Euclid’s Elementa. 55 “Analysis veram viam ostendit per quam res methodice & tanquam a priori inventa est ...” (Resp. 2, AT VII, 155; my emphasis).

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understanding of both the exact meaning of the a priori and the a posteriori in Descartes’ thought and of his characterization of the analytic method as a priori and of synthesis as a posteriori.

APPENDIX: ABBREVIATIONS

An. Post. An. Pr. AT

Burm.

DM

EN Geom. Hyp.

Med.

Med.(f)

Mers. Met. Notæ

Aristotle, Analytica Posteriora. Cited by book number, section number, and Bekker numbering. Aristotle, Analytica Priora. Cited by book number, section number, and Bekker numbering. Œuvres de Descartes ed. by Charles Adam and Paul Tannery, 11 vols. (Paris: J. Vrin, 1897–1913). Cited by volume number. René Descartes, Descartes et Burman: Responsiones Renati Des Cartes ad quasdam difficultates ex Meditationibus ejus, etc., ab ipso haustæ, in AT V, 146– 79. René Descartes, Discours de la Methode: Pour bien conduire sa raison, & chercher la verité dans les sciences, in AT VI, 1–78. Cited by part number. Aristotle, Ethica Nicomachea. Cited by book number, section number, and Bekker numbering. René Descartes, La Geometrie, in AT VI, 367–485. Cited by book number. René Descartes, Letters CCXLVI. and CCL. (the “Hyperaspistes” correspondence), in AT III, 397–412; 421–35. René Descartes, Meditationes de Prima Philosophia in qva Dei existentia et animæ immortalitas demonstratvr, in AT VII, 17–90. Cited by meditation number. René Descartes, Les meditations metaphysiqves tovchant la Premiere Philosophie, in AT IX-1, 13–72. Cited by meditation number. René Descartes, Letters to Marin Mersenne, in AT I–III, passim. Aristotle, Metaphysica. Cited by book number, section number, and Bekker numbering. René Descartes, Notæ in Programma quoddam, sub finem Anni 1647 in Belgio editum, in AT VIII-2, 335–69.

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Obj.

Phys. Princ. Princ.(f) Princ. Pref.

Resp.

RV Reg. Top.

357

René Descartes, Objectiones doctorum aliquot virorum in præcedentes Meditationes, in AT VII, 91–561. Cited by set number. Aristotle, Physica. Cited by book number, section number, and Bekker numbering. René Descartes, Principia Philosophiæ, in AT VIII-1, 1–329. Cited by part number and section number. René Descartes, Les Principes de la Philosophie, in AT IX-2, 25–325. Cited by part number and section number. René Descartes, “Lettre de l’Avtheur a celvy qvi a tradvit le livre,” in idem, Les Principes de la Philosophie, AT IX-2, 1–20. René Descartes, Responsiones authoris ad Objectiones doctorum aliquot virorum in præcedentes Meditationes, in AT VII, 91–561. Cited by set number. René Descartes, La recherche de la Verité par la lumiere naturelle, in AT X, 495–532. René Descartes, Regulæ ad directionem ingenii, in AT X, 359–469. Cited by rule number. Aristotle, Topica. Cited by book number, section number, and Bekker numbering.

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GENERAL INDEX

a posteriori and analysis, 326–29 in Aristotelianism, 1, 3, 4, 318, 319–23, 326–337, 352 and causation, 319–21, 324–27, 329, 340, 353 cognition, 2–3, 325 and demonstration quia (੖IJȚ), 319–21 in Descartes, 161–65, 323–26, 351–55 and explanation, 319–20 Kantian meaning of, 1–2, 29 and synthesis, 161–65, 325, 326, 340–41, 351–55 a priori and analysis, 161–65, 180, 238, 317–18, 325, 326–28, 339–51 in Aristotelianism, 1, 3, 4, 318– 23, 326–337, 350–51 and causation, 319–21, 324–27, 329–31, 337, 340, 344–51 cognition, 2–3, 4, 174, 180, 308, 317, 326 demonstration (proof), 2, 322, 324 and demonstration propter quid (įȚȩIJȚ), 319–21 in Descartes, 161–65, 171–74, 323–26, 339–51 and explanation, 319–20 Kantian meaning of, 1–2, 29 in mathematical contexts, 3, 165, 171–74, 181 and method, 4, 162–65, 174, 260, 264, 307–308, 317, reasoning, 327, 329 and scientia, 2–4, 29, 161, 325, 328–29, 346–47, 350

and synthesis, 162–63, 343 absolutum, 265n54, 272, 292–93, 352 degrees of, 280 as first cause, 348–49, 352 lists of, 93, 95–96 maximè, 92–93, 280–84 as relational term, 280–81 relativization to, 92–93, 279–87, 292–93 (see also natura communis, participation in) and simple natures, 92–94, 98, 153, 279–87, 345, 348–50 abstraction, 97n34, 103, 254 as exclusio, 232, 235–37, 255 as omissio, 232, 235–37, 255 as præcisio, 49, 77–78, 85, 87– 88, 99, 100n40, 149, 236, 255–56, 290 activity, 25, 27, 39 cognitive, 124 intellectual, 27, 315 vs. passivity, 10, 39 (see also mind, activity vs. passivity of) Adam, Charles, ix, 247n16, 248n18 æqualitas. See equality Alanen, Lilli, 110n55 Albert of Saxony, 321 Albertus Magnus, 328 algebra and analysis, 187–89, 205–13, 219–21, 223–24 in Arabic world, 182n33, 186–87 in Clavius, 182–83, 185 in Descartes, 195–213, 223–24, 240–41, 251–52 general, 190, 219, 240–41, 245, 251–52, 260, 265n54, 277n71, 278, 285, 308

374

General Index

“geometrical,” 220 geometrical interpretation of. See algebra, and geometry and geometry, 197–98, 200–203, 209–12, 223–27, 228–29 interpreted, 197–200, 230, 240 in Islamic world, 182n33, 186– 87 as Mathesis vniversalis, 185–87, 245, 251–52, 259–60 objects of, 197–204 (see also line segments, algebra of; line segments, as signifying general quantities; proportion, in algebra; quantity, general) operations in, 196–202 paradigm of analysis, 4, 209, 219 paradigm of method, 166–68, 181n30, 239–40, 260, 265n54, 268, 277n71, 278, 285–87, 292–95, 308 paradigm of universal method. See algebra, paradigm of method pre-interpreted, 197–200, 230, 240 problem-solving procedure, 174, 187, 191–96, 205–13, 223–24, 308 (see also analysis, as problem-solving procedure) symbolic, 174n15, 186–88, 191, 209, 221, 224–25, 230, as universal mathematics, 185– 87, 194, 196–97, 240, 245, 251–52 in Viète, 174n15, 187–89, 191– 95 See also ars, of algebra; equation, algebraic; homogeneity, as constraint in algebra algebraic equation. See equation, algebraic Alquié, Ferdinand, 327 Al-KhwƗrizmƯ, Muতammad Ibn Mnjsa, 182n32, 186

anaclastic line, 153n121, 239n2, 262, 300n104 analysis a posteriori, 326–29 a priori, 161–65, 180, 238, 317– 18, 325, 326–28, 339–51 and algebra, 187–89, 205–13, 219–21, 223–24 as approbative tool, 328–29 in Aristotle, 215–17, 326n19, 330–32 in arithmetic, 178–80 art of, 180, 195 beyond mathematics, 170–71, 190, 239, 308n116, 351 complement of synthesis, 176– 78, 215–27, 351–52 in Descartes, 161–65, 205–13, 268–95, 323–26, 337–51 in Diophantus, 178–79 as dissolutio, 326, 327n20 (see also analysis, as resolutio) in Galenic tradition, 326n18, 327n20, and geometry, 162, 164–65, 167– 68, 170, 175–78, 205–208, 222–23, 243, 260, 342 as heuristic tool. See analysis, as method of discovery independent of synthesis, 217– 18, 220–22 as method of discovery, 4, 154, 155, 164–65, 171, 173n13, 227, 238, 268–95, 308, 317, 330–38 in Pappus, 176–78, 213–19 paradigm of method. See algebra, paradigm of method paradigm of universal method. See algebra, paradigm of method problematical vs. theoretical, 213–15 as problem-solving procedure, 205–209, 219–20, 224, 308 (see also algebra, problem-

The a priori in the Thought of Descartes solving procedure) in Renaissance Aristotelianism, 332–37 as resolutio, 176n17, 177–78, 213–14, 218, 326, 328 (see also analysis, as dissolutio) and reversibility problem, 215–18 theoretical. See analysis, problematical vs. theoretical in Viète, 188–89, 191–95 See also ars, of analysis Annas, Julia, ix, 319n6 Anscombe, Elizabeth, 25n71, 26n73 Apollonius of Perga, 174n15, 177n18, 189 apprehension, 36–37, 44, 113–14, 120, 226 distinct, 105–106 explicit, 42, 47, 82, 105 implicit, 145 implicit vs. explicit, 42–43, 105, 144 innate, 42 of innate ideas, 47, 145 modes of, 32 operation of understanding, 11– 13, 32–33 of simple natures 88, 92–93, 101, 105–106 See also cognition, as apprehension; content, apprehended approbation and analysis, 328–29 and enumeration, 148–50, régime of, 137–38, 148–49, 150, 153, 157, 159, 160n133, 270, 301, 328–29 Aquinas, Thomas. See Thomas Aquinas Aristotle, ix, 8n9, 128, 320n9, 324n15, 329, 333 See also a posteriori, in Aristotelianism; a priori, in Aristotelianism; analysis, in Aristotle; ars, Aristotelian

375

notion of; categories, Aristotle’s doctrine of; cause, Aristotelian meaning of; explanation, in Aristotle; inductio, in Aristotle; scientia, Aristotelian conception of; syllogism, definition of in Aristotle; syllogism, Aristotle’s notion of; universal mathematics, in Aristotle arithmetic, 224–27, 241, 243, 247, 253 in Diophantus, 178–79 objects of, 168, 190, 192, 196– 97, 209, 241 paradigm of certainty, 165–66, 240 paradigm of methodical inquiry, 168, 219, 240 paradigm of scientific cognition, 166–67 subordinate to general algebra, 181, 183, 185–86, 187, 191, 194, 240–41 Arnauld, Antoine, 36n92, 43n100, 45n103, 68, 289n88 ars of algebra, 180n28, 181n29, 182 of analysis, 180, 188, 189n52, 191n59, 193n67, 195 Aristotelian notion of, 313–14 inveniendi, 138 medica, 326n18, 327n20 vs. scientia, 313–15 art. See ars Ashwort, Earline, 128n85 assent, 78, 140 compelled. See compelled assent constituent of judgment, 17, 42n100 and will, 14, 42n100, 53 Averroës, 333

Baillet, Adrien, 247n16, 339 Balz, Albert, 256n40

376

General Index

Barnes, Jonathan, ix, 131n89 Barozzi, Francesco, x, 184 Beaugrand, Jean de, 191n59, 191n60 Beaune, Florimond de, 172 Beck, Leslie, 76n3, 81n11, 105n50, 149, 150n120, 169, 256n40, 264n53, 266nn55–57, 276n69, 280n76, 291n90, 308n116 Beeckman, Isaac, 190n58, 191n59, 199n77, 250 Bennett, Jonathan, 40n98, 48n111, 55n125, 58, 59n133, 60, 70nn153–54, 72, 97n35, 110n55 Benzi, Ugo. See Hugo of Siena blacksmith analogy, 297–99, 300n104, 302, 304–305 Bockstaele, Paul, 185n39 Boëthius, Anicius Manlius Severinus, 328 bon sens. See bona mens bona mens, 133–36 and lumen naturale, 135n95 operations of, 136, 298, 311 (see also deductio, and bona mens; intuitus, and bona mens) and possibility of method, 298– 99 and cognitive operations, 312 and sapientia, 310–11 and universal method, 310–12 Bonnen, Clarence, 308n116 Bosmans, Henri, 186n43 Bourdin, Pierre, 51, 59n134 Boutroux, Pierre, 256n40 Bouveresse, Jacques, 109n55 Boyer, Carl, 174n15, 181n30, 212n104 Bréhier, Émile, 70n153 Brissey, Patrick, 266n55, 304n107, 307n115 Broughton, Janet, 60n136 Brown, Gregory, 20n58

Brunschvicg, Léon, 266n55 Buchdahl, Gerd, 207nn94–95

Burman, Frans, 14n32, 22, 61n140, 71, 132, 142, 144, 169n7, 303

Cartesian Circle, 68–74 Cassirer, Ernst, 256n40 categories, 271–75, 315–16 Aristotle’s doctrine of, 271n65 causation. See cause cause, 20, 30–32, 36, 44–46, 319– 20, 325–27, 329, 340–41, 344–45, 353–54 Aristotelian meaning of, 319n6, 329 efficient, 320n9, 343 and explanation, 319n6, 333–34, 337–38, 350 final, 342–43 first, 330, 346–48 (see also absolutum, as first cause) as quæsitum, 331–32, 345, 350– 51, 353 See also idea, causal history of Cavendish, Charles, 182n32 certainty, 215, 241, 254, 337 and clear and distinct perception, 47–49, 53, 55, 56, 58–63 of Cogito, 28n80 of cognition, 48–49, 52–53, 55, 62–63, 70, 73, 108, 117–18, 120, 124, 165, 170, 223–24, 266, 328 of deductio, 121–22, 148, 307 descriptive. See certainty, normative vs. factual as disposition, 51n116 and evidence, 5, 48, 51–52, 120– 22, 168, 170, 280, 324 factual. See certainty, normative vs. factual and immutability, 49, 63–67, 73 and irrevisability, 51–52 and memory, 62n141, 120–23, 156–57 metaphysical, 48, 49, 50–51, 52– 53, 70–71, 91, 108, 113–14, 307

The a priori in the Thought of Descartes metaphysical vs. moral, 51 moral. See certainty, metaphysical vs. moral and motus ingenij, 117–18, 121– 23, 148–49, 156–57 normative vs. factual, 52–53, 56– 60, 69–70 paradigm of, 247 perfect. See certainty, metaphysical practical. See certainty, metaphysical vs. moral psychological. See certainty, normative vs. factual as relation, 50–51 of mathematical reasons, 170 and scientia, 2, 5, 138, 148, 328– 29 standard of, 166, 170, 240 See also compelled assent, and certainty; indubitability Chappell, Vere, 10n18, 20n57 Cicero, Marcus Tullius, 158, 323n14 Clarke, Desmond, 31n84, 40–42, 76n3, 82n14, 117nn67–69, 266n55, 351n52 Clavius, Christoph, 182–85, 187, 191n59 clarity and distinctness, 48, 57, 121, 156, 260 applied to ideas, 48n109 (see also idea, clear and distinct) and certainty, 47–49, 53, 55, 56, 58–63 criteria of, 116 definition of, 49 Gewirth’s interpretation of, 49– 50, 78, 88, 92 and God’s guarantee, 68–74 and omissio, 235–37 and scientia, 47–49, 53–74, 230, 237, 247 See also clear and distinct perception; cognition, clear and distinct; idea, clear and

377

distinct clear and distinct perception, 47–50, 70–73, 88, 103–104, 108, 113–14, 116, 163, 230–31, 237, 283, 288, 342 (see also intellectio, clear and distinct) and certainty, 47–49, 53, 55, 56, 58–63 and compelled assent, 53, 62, 113 and false propositions, 54–55 and scientia, 47–49, 53–74, 230, 237, 247 and sensation, 50n115, 77–80, 88–89 Clemenson, David, 18n48 Clerselier, Claude, 162n2, 232n138, 325, 339–41, 345, 349, 353n53 cogitatio (thinking, thought), 12, 24, 25, 31, 38n94, 43, 65, 71, 72, 78, 91, 95–96, 105n50, 120, 139, 141, 143, 149, 229 attribute vs. mode, 8–9, 10n17, 18, 20n55, 233, 236 (see also mind, essence of) and conscientia, definition of, 25–29, 36n92 faculty, 8 semina of, 299, 348n51 cognition, 4, 5–7, 29, 51, 67, 70, 82, 83n16, 84, 95–96, 125, 126, 137, 152, 155, 158, 167, 169, 184, 238, 254, 264–65, 267, 269–70, 272–75, 277–78, 281, 288n85, 309, 312, 321–22, 330, 336–37 a posteriori, 2–23, 325 a priori, 2–3, 4, 174, 180, 308, 317, 326 as apprehension, 13, 42n100 certain. See certainty, of cognition classification of, 67, 86–87, 120 clear and distinct, 62, 64–65, 75, 77n5, 91, 102, 108, 223–24,

378

General Index

236, 336–37 (see also clear and distinct perception) of compositiones, 99–100 conjectural, 332, 335, 338 discursive, 73, 156–57, 275–76, 325–27 divine, 111–16 empirical, 80, 322, 323n14 evident, 7, 49, 223–24 explicit, 144–46 extension of, 278, 301–302, 304 foundations of, 5–6, 98, 299 of God’s existence, 50n116, 62, 64, 68–75 immutable, 49, 63–67, 73 innate, 39 and judgment, 12–13, 42n100 interna, 42n100, 43 intuitive, 66, 100, 108, 133, 143, 315n126 limits of, 239n2, 300–302 mathematical, 184, 235–36, 241, 278 non-discursive, 91, 138 objects of, 14, 112 (see also judgment, objects of; reduction, of omnia quæ cognoscimus) as opinio, 61–63, 141–42 (see also cognition, as persuasio) order of. See ordo, of cognition of particulars, 143–45 per se nota, 74, 92 (see also truth, per se nota) as persuasio, 63–66, 121 (see also cognition, as opinio) probable, 62, 120, 141–42 scientific, 17, 29, 47, 51, 57, 63, 75, 78–80, 85, 98, 119, 120, 124–25, 130, 139, 160n133, 161, 164–67, 230, 237, 270, 313–14, 328–31, 333–34 true, 92, 96, 129, 137 See also activity, cognitive; certainty, of cognition; cognitive capacity; cognitive

circularity; cognitive operation; content, cognitive; faculty, cogitandi; simple natures, cognition of cognitive capacity, 123–24, 199 and memory, 123, 154 cognitive circularity, 140–42, 334– 35 cognitive faculty. See faculty, cogitandi cognitive operation, 72n158, 84, 91, 97, 116, 128, 131, 133, 148, 156, 160n133, 239, 275, 292, 299, 307, 325–27, 338 uniformity of, 309–16 Commandino, Federico, x, 176n17, 213 common nature. See natura communis common notions. See notiones communes comparatio, 304, 336, 338 analogical treatment, 274 aperta & simplex, 276, 278, 350 and deductio, 276–77 and discursive cognition, 275–76 præparatio of, 278–299 principle(s) of, 276–77, 281 two meanings of, 274n68 comparison. See comparatio compelled assent and certainty, 53, 56, 113 and clear and distinct perception, 53, 62, 113 and factuality. See compelled assent, normativity vs. factuality and false propositions, 54–56 and freedom, 53–54 and normativity vs. factuality, 52–53, 56–60, 69–70 compositio, 78, 103 cognition of, 99–100 contingent. See compositio, necessary vs. contingent necessary, 101, 104, 122 (see

The a priori in the Thought of Descartes also connection, necessary) necessary vs. contingent, 79, 86– 87, 99–100, 118 (see also connection, contingent) object of deductio, 119 object of intuitus, 99–100, 101, 315 of simple natures, 33n88, 86–87, 93, 99–100, 119, 122, 271, 283–84, 345–46 and synthesis, 176n17, 177–78, 326–29, 333 compositum. See compositio connection analytic vs. non-analytic, 101– 106 contingent, 88, 107 deductive, 115–16, 120 necessary. See necessary connections non-analytic. See connection, analytic vs. non-analytic object of intuitus, 87–88, 115, 119 (see also compositio, object of intuitus) of premises and conclusions, 117, 133n94 of simple natures, 87–88, 103– 112, 115–16, 119, 346 (see also simple natures, indivisible) conscientia definiens of cogitatio, 9, 25–27, 45n103 notion of, 27, 42–43n100 content, 83, 86, 92, 98, 103, 134, 139, 196, 309, 313, 315, 328 apprehended, 11, 13–14, 17, 20– 21, 28n81, 42n100, 82, 88, 92–93, 101, 132 (see also content, perceived) cognitive, 20, 124 direct vs. interpretative, 50, 79, 88, 92 interpretative. See content, direct vs. interpretative

379

mental, 35, 91 preceived, 48–49, 53–57, 60–63, 69–70, 87–88 (see also content, apprehended) propositional, 14, 18, 28n81, 58, 115–16 sub-propositional vs. propositional, 16, 92n28 continued proportionals. See continuous proportionals continuous proportionals, 202–204, 218n119, 243n8, 251, 285–87, 292–93, 295 geometrical interpretation of, 203–204 and powers of quantities, 285–86 series of, 203–204, 251, 285–87, 292–93, 295 See also proportion, in algebra; proportion, object of Mathesis vniversalis Coolidge, Julian, 174n15 Costa, Michael, 23n64 Cottingham, John, 26n71, 27–28, 82n12, 83n16 Cozzoli, Danielle, 324n15 Crapulli, Giovanni, xi, 76n3, 149n118, 185n36, 249nn19– 20, 258n46 Crowley, Charles, 183n34 Curley, Edwin, 6n4, 17n45, 60n136, 109n55 curve, 172, 193, 223 algebraic vs. transcendental, 210n100 geometrical vs. mechanical, 210– 11, 225–26 higher-degree, 211n103, 212, 229 legitimate, 209–11, 225 mechanical, 210–11, 225–26 transcendental, 210n100

data, 124, 152–54, 187, 193, 221, 224, 226–27, 230–31, 279,

380

General Index

281–82, 285, 287, 291–92, 329–31, 345, 348–53 as participating in natura communis, 276–77, 281–85 See also equality, of data and quæsita Davenport, Anne, 11n20 deceiver. See Demon hypothesis deductio, 87, 100, 346, 349 beyond regulation, 128–31, 135– 38, 296, 309–11 and bona mens, 134–36, 298, 310–11 certainty of, 121–22, 148, 307 and comparatio, 276–77 complex. See deductio, immediate vs. mediate and demonstration, 65 direct vs. indirect, 293 and discursivity, 65, 285, 315 (see also reasoning, discursive) and enumeration, 117n67, 147– 60 immediate vs. mediate, 123–24, 127–139, 148–49, 152–54, 270–77, 281, 296, 298–99, 303, 307, 309–11, 323 indirect, 293 and inductio, 76n3, 117n67, 147, 155–60 infallible, 119n72, 128–29 as inference, 117–19, 124, 156, 281 mediate. See deductio, immediate vs. mediate and memory, 121, 149 and motus ingenij, 120–21, 124, 148–49, 156–57 and necessary connections, 119– 20 objects of, 270–75 principle of scientia, 75–76, 117– 18, 121–22, 147, 309 reducible to intuitus, 74n160, 121–25, 134, 149, 157, 314,

323 and regulation, 128–43, 296, 309–11 and scientia, 75–76, 80, 82, 117– 18, 121–22, 147, 309 as series of intuitus, 115, 122, 275, 323 (see also series, deductive) simple vs. complex. See deductio, immediate vs. mediate and syllogistic, 125–46 and understanding, 75–76, 128– 29, 147, 314 veracity of. See deductio, certainty of deduction. See deductio Demon hypothesis, 69, 121 demonstration a posteriori. See demonstration, a priori vs. a posteriori a priori vs. a posteriori, 2, 322– 24 (see also reasoning, a priori vs. a posteriori) and deductio, 65 propter quid vs. quia, 319–22, 323n14, 333, 335 quia. See demonstration, propter quid vs. quia dependence cognitive, 134n94, 152, 163n3, 271–72, 275, 329, 341–44, 349–50, 352 in cognoscendo. See dependence, cognitive epistemic. See dependence, cognitive in essendo. See dependence, natural essential, 23–24 natural, 272, 329, 341 ontological. See dependence, natural dialectic and regulation, 128–31, 134, 323 vincula of, 119, 125–46

The a priori in the Thought of Descartes dialecticians. See dialectic dimensio, 203, 212, 253–56, 284 and geometry, 193–99, 228, 286n82 spatial, 196, 198, 202, 230 dimensionality, problem of, 193–99, 228, 286n82 Diogenes Laërtius, 215n111 Diophantus of Alexandria, ix–x, 165, 175–80, 193, 243 disciplina equivalent to mathesis, 246, 248 “quædam alia,” 167–70, 242–43, 299n103 discovery, 123, 139, 329–31, 338, 341, 350 method of. See method, of discovery order of, 263, 330 régime of, 127, 131, 137–38, 152–55, 157–60, 164, 173n13, 196, 268, 270–79, 301, 323, 328, 348–49, 351, 354 tools of, 131, 137, 143n107, 146, 215n100, 328–30, 338, 351 dispositio. See ordo, and dispositio dissolutio, 326, 327n20 (see also analysis, as resolutio) distinctio modal, 23, 233, 234 real, 30–31, 232–36 of reason, 20, 233, 234–36 rationis, 20, 233, 234–36 Doney, Willis, 69n153 doubt, 48–51 cause of, 49, 59 certainty, opposite of, 48 inability to, 53, 56–58, 91 and will, 11 See also indubitability Doyle, Bret, 238, 244nn9–10, 245n13, 253n28, 305n109 dualism, 97n35, 273, 340n42 property. See dualism, substantial vs. property substantial vs. property, 40,

381

272n65, 273n66, 312, 315

ens rationis, 90n26 enumeration approbative, 148–50, 152, 154, 157–59 complete, 152 and deductio, 147–60 definition of, 147–48 and determination of quæstio, 291–92 heuristic, 147–51, 152–54, 157– 59, 291–92 and inductio, 76n3, 117n67, 147, 151, 155–60 and order, 152–55, 267, 269 sufficient, 151–52, 153, 267, 269, 291–92 equality beyond algebra, 93, 98, 277–78, 353 of data and quæsita, 124, 178, 206, 219, 251n26, 278, 284n80, 349–51, 352 logical, 50, 78 quantitative, 217, 278 (see also equation) equation, algebraic, 172, 174, 179, 182, 193, 195, 205–206, 209– 10, 221, 224–26, 230, 251–52 definition of, 206n93 degree of, 198, 202–204, 212 object of Logistice speciosa, 189n52, 192–93 reciprocal to proportion, 192 reduction of. See equation, algebraic, resolution of resolution of, 206–207, 219, 251n26, 277–78, 295, 351 system of, 206, 277–78 Eriugena, Johannes Scottus, 328 error, 12n22, 13n28, 92, 100, 129, 137, 235–36 and falsity, 13, 17 esse obiective, 15, 19

382

General Index

Etchemendy, John, 70n153 eternal truth, 95–96, 109–17, 324 Euclid of Alexandria, x, 177nn18– 19, 183n34, 184, 188, 192, 217n118, 220nn123–24, 354n54 Eudoxus of Cnidus, 192 Eustachius a Sancto Paulo, 328n23, 329 evidence borrowed, 120–22 and certainty, 5, 48, 51–52, 120– 22, 168, 170, 280, 324 per se vs. borrowed, 120–22 exclusio, 232, 235–37, 255 experience. See experientia experimenta. See experientia experientia and imagination, 84–85 and innate ideas, 38–47 and intuitus, 82–86, 87–88, 96, 99–100, 315n126 of particulars, 141–42, 144–45 and understanding, 84, 87–88, 99–100 and scientia, 82–86, 166, 283, 290–91, 324–25, 337–38 See also will, experimenta of acts of explanation in Aristotle, 319n6, 329 and causation, 319n6, 333–34, 337–38, 350 exposition, 323 order of, 263 régime of, 127, 131–34 extension as absolutum, 280 essence of body, 44, 90, 96, 228, 231–36, 254–56 essence of matter. See extension, essence of body as simple nature, 90, 95–96, 102– 106, extension-symbol, 254–55

facultas. See faculty faculty cogitandi, 4, 21n59, 23, 30–32, 36–41, 45–47, 75, 128, 132– 35, 199, 227, 239, 264, 301, 331 cognitive. See faculty, cogitandi corporeal, 84–85n20, 228 list of, 10–11 See also cogitatio, faculty; imagination, faculty; reason, faculty; understanding, faculty; will, faculty Fermat, Pierre de, 172–73, 212n104, 223n128 Fiala, JiĜí, ix, 286n82, 295n95 Flage, Daniel, 308n116 Florka, Roger, 1n1, 117n70, 126n82, 127nn83–84, 133n93, 178n21, 263n51, 306n111, 340n42, 343, 344 formalism Descartes’ critique of, 127–34, 139, 196–97, 312, 315 and scientia, 134, 136–37 (see also algebra, interpreted) Frankfurt, Harry, 6n3, 60n136, 70nn153–54, 71n157, 109n55 Furley, David, 343n45

Gadoffre, Gilbert, 266n55 Galen of Pergamon, 326n18, 327n20, 338 Garber, Daniel, 6n4, 122n75, 171n11, 263n51, 266n55, 301n104, 304–307, 321n9 Gassendi, Pierre, 36n91, 57n129, 68, 141 Gaukroger, Stephen, 1n1, 56n126, 60n138, 110n57, 114–16, 119n74, 126n82, 127n83, 128n85, 130, 133n94, 140n102, 143n107, 197n74, 220n124, 224n131, 266n55, 339n41, 340n43

The a priori in the Thought of Descartes general notions and abstraction, 145–46 and order of cognition, 145 vs. particular instances, 141–46, 160n133, 326 general propositions. See general notions generalization of algebraic techniques, 211–12, 225, 239 inductive, 140–42 geometry and algebra, 197–98, 200–203, 209–12, 223–27, 228–29 (see also geometry, subordinate to general algebra) algebraization of, 174, 190, 195– 205 analysis in. See analysis, and geometry arithmetical means in, 185–86 and dimensionality, 193–99, 228, 286n82 objects of, 168, 190, 192, 196– 97, 209, 241 in Pappus. See analysis, in Pappus; synthesis, in Pappus paradigm of certainty, 165–66, 240 paradigm of methodical inquiry, 168, 219, 240 paradigm of scientific cognition, 166–67 subordinate to general algebra, 181, 183, 185–86, 187, 191, 194, 240–41 (see also geometry, and algebra) in Viète, 193–95 Gewirth, Alan, 49–50, 57n129, 70n155, 78–79 See also clarity and distinctness, Gewirth’s interpretation of Gibson, Alexander, 266n55 Gilson, Étienne, 6n4, 169n8, 264n53, 266n55 Gilbert, Neal, 333n30

383

Gilbert, William, 290 given, the. See data God and Cartesian Circle, 68–74 cognition of existence of, 50n116, 62, 64, 68–75 cognitively omnipotent, 115–16 and contingency. See God, and necessity vs. contingency and eternal truths, 109–17 existence of, 111 guarantee of certainty (of cognition, of truth), 60–67, 68, 70–72, 108, 113–16, 121, 124n79, 157, 307 guarantee of cognition. See God, guarantee of certainty guarantee of truth. See God, guarantee of certainty and necessity vs. contingency, 109–17 power of, 110 proof of existence of, 59n133, 71, 74, 320n9 will of, 109, 112 See also cognition, divine Gosselin, Guillaume, 185 Grabiner, Judith, 218n119 Grosholz, Emily, 196–97, 202n85, 223n128, 223n130 Grynaeus, Simon, 184 Guéroult, Martial, 8n9, 320n9, 342n44

Haldane, Elizabeth, 76n3, 305n110 Haly. See Ibn Ridwan, ‘Ali Hannequin, Arthur, 148 Harriot, Thomas, 191n60 Heath, Thomas, 183n34, 192n61, 220n123 Heffernan, George, xi Hervey, Helen, 182n32 Hill, James, 6n4, 8n9, 26n73, 27– 29, 35n91, 42n100, 47n107, 50n115, 82n12

384

General Index

Hintikka, Jaakko, 215n110, 215n112 Hobbes, Thomas, 12n22 Hocutt, Max, 319n6 homogeneity as constraint in algebra, 194, 196–200, 202 Hugo of Siena, 333–34 Humber, James, 57n129 Hyperaspistes, ix, 43

Iamblichus of Apamea, 257 Ibn Mnjsa Al-KhwƗrizmƯ, Muতammad, 182n32, 186 Ibn Ridwan, ‘Ali, 327n20 Ibn Rushd. See Averroës Ibn YahyƗ al-Maghribi alSamaw‫ތ‬al, 187 idea, 12, 19, 95–96, 231–36 abstract, 106 adventitious, 29–32, 34, 38–42, 45, 47 causal history of, 29–33, 35, 37, 44 classification of, 29–31, 34 clear and distinct, 48n109, 49– 50, 113, 233–36, 312n121 corporeal image, 22n64, 23 essentially representative, 18, 19n50 (see also representation, by ideas) factitious, 29–34, 38, 42 of imagination. See idea, factitious innate. See innate ideas material falsity of, 15, 92n28 material truth of, 15, 92n28 object of judgment, 14 realitas formalis vs. objectiva of, 18–20, 30–32, 33, 41–42, 55 realitas objectiva vs. formalis of, 18–20, 30–32, 33, 41–42, 55 reflexive awareness of, 45, 47, 82 sensory. See idea, adventitious and sub-propositional items, 14–

16, 92n38 imagination and clear and distinct perception, 234–37 (see also clear and distinct perception, and sensation) corporeal. See phantasia dependent on body, 22–24 and experientia, 84–85 and factitious ideas, 33 (see also idea, factitious) faculty, 227, 231–35, 237 fallible, 80–81 and ingenium, 135n95 and intuitus, 82–86, 87–88, 96, 99–100, 315n126 in mathematics, 227–37, 254–55 mode of cogitatio, 10–11, 22–23, 25–29, 77–78, 234 mode of understanding. See imagination, mode of cogitatio and pure understanding, 227–37 and scientia, 227–37 and simple natures, 88–89 immutability and certainty, 49, 63–67, 73 of cognition, 63 of natures, 15n35, 20n58 of truth, 70n153 indubitability and certainty, 48n111 per se, 81 psychological notion of, 81 See also certainty inductio, 152 in Aristotle, 158–60, 322, 323n14 and deductio, 76n3, 117n67, 147, 155–60 and enumeration, 76n3, 117n67, 147, 151, 155–60 intuitive, 158–60 and memory, 159 perfect, 158–60 See also generalization, inductive; syllogism, inductive

The a priori in the Thought of Descartes induction. See inductio inference, 31, 59n133, 94, 97, 116, 140, 159, 290 cognitively circular, 140–42 and deductio, 117–19, 124, 156, 281 deductive, 125 in reasoning, 25 rules of, 97, 119n74 and scientia, 134, 137 ingenium, 128, 134, 135n95, 150, 165, 167, 175, 234n141, 266, 283, 289, 299, 301, 309–10 capacity of, 124, 288 motus of, 118, 121, 124, 149n118, 156 –57 (see also deductio, and motus ingenij) innate ideas, 29–47, 105 as dispositions, 35–37, 43n100, 44–45 and implicit apprehension, 38– 47, 145 and mental reflexion, 31–33, 37, 42, 45–47, 82, 144 and sensation, 38–47 intellectio clear and distinct, 233–34, 312n121 pure, 11, 21–22, 77–79, 81–83, 234 intellectus. See understanding intuitus beyond regulation, 128–31, 135– 38, 296, 309–11 and bona mens, 134–36, 298, 310–11 certain per se, 81 and deductio. See deductio, reducible to intuitus; ordo, and dispositio evident, 81, 120, 266 and experientia, 82–86, 87–88, 96, 99–100, 315n126 of Ʀod’s existence, 74–75 and imagination, 82–86, 87–88, 96, 99–100, 315n126

385

indubitable per se, 81 infallible, 128–29 objects of, 85–107, 115, 119, 123n79, 198, 270, 273–75, 277, 309, 315, 346 (see also line segments, objects of intuitus) principle of scientia, 80, 82, 117– 18, 121–22, 147, 309 and scientia. See intuitus, principle of scientia; intuitus, and unity of scientiæ and sensation. See intuitus, and experientia and sense perception. See intuitus, and experientia series of, 124, 149, 275, 323 and understanding, 75–80, 91, 129, 147, 314–15 uniformity of, 315–16 and unity of scientiæ, 308–309, 314–16

Jesuit thought, 126n82, 128, 182– 83, 318 Jolley, Nicholas, 67n150 judgment, 48–50, 69, 73, 328 constitution of, 12–14, 17, 28n81, 42n100 evident, 2 locus of cognition, 13 locus of truth, 13, 17 objects of, 14–17, 48n109, 50 (see also cognition, objects of) pre-reflexive, 42n100 and sensation, 27–28, 78, 105– 106, suspension of, 13, 114, 116

Kant, Immanuel, 1–2, 29 Kaufman, Dan, 110n55 Kenny, Anthony, 20n58, 25n71, 63n143, 70n155 Klein, Jacob, x, 174n15, 177n19,

386

General Index

178, 179nn24–25, 185, 189n54, 192n61, 193n65, 219n120, 252–53, 256n39 known, the. See data Kraus, Pamela, 243n7, 244n9, 245nn13–14

La Flèche, 182n32, 318 LeBlond, Jean, 94n32 Leibniz, Gottfried, xi, 196, 261nn48–49 Liard, Louis, 245n14 line segments algebra of, 196–98, 200, 203, 228–29, 241, 252 as dimension-free, 197–98 and general quantities, 196–97, 199, 227–29, 255–56 as geometrical magnitudes, 196 lengths of, 196–97 (see also unit length) as multitudes, 196 (see also representation, in mathematics) as signifying general quantities, 196–97, 199, 227–29, 255–56 Loeb, Louis, 58–59, 60n135 Logistice numerosa, 187n45, 221n125 Logistice speciosa, 186–88, 190 generality of, 191, 192–93 vs. Logistice numerosa, 187n45, 221n125 objects of, 192–95 uninterpreted calculus, 193–95, 196 lumen naturæ. See lumen naturale lumen naturale, 21n59, 80–81, 83n115, 95, 132n92, 134n93, 276, 278 as bona mens, 135n95 lumen rations. See lumen naturale lux rationis. See lumen naturale

Macbeth, Danielle, 140n102, 142n106, 179n24, 192, 193n64, 193n68, 194, 197n74, 197n76, 209, 212n106 McCaskey, John, 158nn129–30 McRae, Robert, 310n117, 311n118, 312n120 magnet, nature of, 239n2, 283, 287– 88, 290–91 magnitude, 96, 178, 183, 184, 186, 192, 198, 202, 218n119, 221, 277n71, 284, 285, 293–94 algebraic operations with, 193– 94, 196 general, 179n25, 228–29, 231, 236, 254–55 geometrical, 196, 197n74, 204 homogeneous, 194, 196 vs. multitude, 194, 196, 197n74, 229, 240, 252 Mahoney, Michael, 172n12, 173n14, 176n18, 177n19, 188nn49–51, 193, 195, 209n98, 213n107, 214n109, 217–18nn117–19, 219n121, 220n123, 224n131 Marion, Jean-Luc, xi, 76n3, 110n57, 160n133, 243n7, 244n10, 245n14, 256–57, 271n65, 280n76 See also Mathesis vniversalis, Marion’s interpretation of Markie, Peter, 63n143 mathematicism, 171 mathematics and analysis. See analysis, and algebra; analysis, in arithmetic; analysis, and geometry ancient, 165, 176, 181, 187, 193– 95, 207–10, 217–19, 220 Greek. See mathematics, ancient and imagination, 227–37, 254–55 and mathesis, 247–48, 250, 257 and Mathesis vniversalis, 245–60 paradigm of method. See algebra,

The a priori in the Thought of Descartes paradigm of method; arithmetic, paradigm of methodical inquiry; geometry, paradigm of methodical inquiry paradigm of universal method. See algebra, paradigm of method; arithmetic, paradigm of methodical inquiry; geometry, paradigm of methodical inquiry as scientia, 184–86, 198–200, 221–37, 249–60, 351 universal. See Mathesis vniversalis; universal mathematics See also algebra; arithmetic; geometry mathesis and algebra, 180, 208–209 paradigm of certainty, 247 prima, 185, 250, 257–58 true, 180, 199n79, 208, 243 universal, 185n36, 249–50, 253n32, 255 vera. See mathesis, true vniversa. See mathesis, universal Mathesis vniversalis and algebra, 185–87, 245, 251– 52, 259–60 as general scientia, 241, 243, 246, 247n16, 248, 255 Marion’s intepretation of, 245n14, 256–59 and mathematics, 245–60 objects of, 186, 198–200, 243n8, 251–52, 246–58 (see also ordo & mensura, object of Mathesis vniversalis) principles of, 186, 258 Schuster’s interpretation of, 245n14, 252–56 self-sufficiency of, 257–58 and universal method, 242, 256– 60 in Van Roomen, 185–87, 249–

387

50, 251, 256–58 vs. vniversa mathesis, 185n36, 249–50, 253n32, 255 See also universal mathematics Maula, Erkka, 176n18, 215n110 mean proportional, 206, 218n119, 222, 293–95 series of, 211n103 memory and certainty, 62n141, 120–23, 156–57 and cognitive capacity, 123, 154, 229 and deductio, 121, 149 and inductio, 159 and understanding, 11, 21, 78 Menaechmus, 174n15 Meno’s Paradox, 159 mensura, 284–85 object of Mathesis vniversalis, 241, 246–48, 250, 252, 255, 257–58 Mersenne, Marin, ix, 64n144, 130, 172, 191n59, 195, 208n97, 222, 261, 273, 325, 328n23, 347n50 method of analysis. See analysis, as method of discovery of approbation. See approbation, régime of beyond mathematics, 170–71, 190, 239, 308n116, 351 and bona mens, 134–36, 310–12 definition of, 129, 131n88, 137 of discovery, 149–50, 153, 164– 65, 171, 173n13, 196, 238–45, 256, 268–96, 308, 317, 328 (see also analysis, as method of discovery) fully-fledged vs. præcepta incondita, 167–68, 297–307 generalization of. See method, beyond mathematics in geometry. See analysis, and geometry; geometry, paradigm

388

General Index

of methodical inquiry heuristic. See method, of discovery innate principles of, 166–68, 240, 243, 268, 298–99, 303 justification of, 295–97, 299–307 in Pappus. See analysis, in Pappus; synthesis, in Pappus possibility of, 296–99, 306 as præcepta incondita, 167–68, 297–307 precepts of, 5n2, 150–51n120, 155, 168–69, 170, 175, 238– 41, 243n7, 260–72, 279n74, 282, 287, 297, 302–304, 305n109 reconstruction of, 265–95 régimes of. See approbation, régime of; discovery, régime of as regulation, 137, 296, 309 and sapientia, 310–11 and scientia. See scientia, and method; scientia, and universal method talk about, 260–65 teaching, 260–65 as tool of discovery, 124–25, 131, 137–38, 142–46, 328 and unity of scientiæ, 308–16 universal. See universal method See also algebra, paradigm of method; arithmetic, paradigm of methodical inquiry; geometry, paradigm of methodical inquiry Miles, Murray, 31n84, 47n107, 145n111 Miller, Leonard, 101n44, 102n45, 108n53, 111n59, 112–13, 114n64 mind, 27–28, 35, 41–42, 50, 57–58, 61–62, 73, 91, 93, 96, 98, 102, 106, 117, 120–25, 128, 132– 35, 142, 147–49, 151–52, 160n133, 224, 226, 229–30,

265, 272, 276, 291, 298–99, 311, 341 activity vs. passivity of, 10, 39, 44–45, 99–101, 107–108, 110 divine, 111–17 embodied, 24–25, 46, 47 essence of, 8–9, 40, 44 (see also cogitatio, attribute vs. mode) finite vs. divine, 111–17 nature of, 31–33, 36, 44 passivity of. See mind, activity vs. passivity of receptivity of, 39–40, 44 reflexive, 31–33, 37, 42, 45–47, 82, 144 self-reflexive. See mind, reflexive Mittelstrass, Jürgen, 245n14 mode of apprehension, 32 vs. attribute. See cogitatio, attribute vs. mode of cogitatio. See imagination, mode of cogitatio; sensation, mode of cogitatio; understanding, mode of cogitatio; volition, mode of cogitatio of understanding, 10–11, 21–29, 32, 40–41, 80, 227, 234 Molland, George, 210n101 Moravcsik, Julius, 319n6 Morris, John, 135n96 motus cogitationis, 124, 267, 269, 329 See also certainty, and motus ingenij; ingenium, motus of Mueller, Ian, 183n34

natura. See nature natura communis, 281n78 principle of comparatio, 276–77, 282–85 participation in, 277, 279, 281– 82, 284–85, 292, 348–49, 352

The a priori in the Thought of Descartes (see also absolutum, relativization to) and simple natures, 282–84 naturæ simplices. See simple natures natural light. See lumen naturale nature common. See natura communis of mind, 31–33, 36, 44 order of, 341, 343, 344n47 science of, 331–32 simple. See simple natures necessary connections, 87, 103–112, 115–16, 117, 119–20, 346 foundations of, 107–111, 116 objects of deductio. See deductio, and necessary connections; deductio, reducible to intuitus objects of intuitus. See connection, object of intuitus; deductio, reducible to intuitus realistic conception of, 99–101, 107, 110 Neo-Platonism, 183 Newman, Lex, 10n18 Nifo, Agostino, 335–36, Nolan, Lawrence, 21n58, 64n145 notiones communes, 84 axioms, 96, 145–46, 347n50 and discovery, 146 general notions, 146, 347n50 rules of inference, 95, 273n67 as simple natures, 95 notiones simplices. See simple natures Novák, Lukáš, 320n9 number, 74, 104, 106, 179n24, 192, 223–26 dimension-free, 193–94, 197 imaginary, 209 indeterminate, 178 irrational, 209 object of arithmetic, 168, 190, 192, 196–97, 209, 241

389

objective being. See esse objective Ockham, William, 320–21 omissio, 232, 235–37, 255 O’Neil, Brian, 92n28, 97, 98nn36– 37, 101n44 opinio. See cognition, as opinio order. See ordo ordo, 95, 151, 285 of analysis vs. of becoming, 330– 31 of analysis vs. of synthesis, 344– 45, 350–53 and approbation, 154 of becoming, 330–31 of cognition vs. à parte rei, 90, 94, 97, 102, 103, 145, 162–63, 271, 322, 331, 340–45 and dispositio, 154, 269–75, 304, 349 (see also series, ordo & dispositio in) of discovery, 263, 330 and enumeration, 152–55, 267, 269 invented, 154, 267 and mensura. See ordo & mensura natural vs. invented, 154, 267 object of Mathesis vniversalis, 241, 246–48, 250, 252, 255, 257–58 à parte rei. See ordo, of cognition vs. à parte rei of reason, 274, 337, 341 respectu intellectûs. See ordo, of cognition vs. à parte rei of series of intuitus. See ordo, and dispositio of synthesis vs. of analysis, 344– 45, 350–53 as tool of discovery, 125, 153, 155 two meanings of, in Descartes, 152–54 ordo & mensura object of Mathesis vniversalis, 241, 246–48, 250, 252, 255, 257–58

390

General Index

Panza, Marco, 326n19 Pappus of Alexandria, ix–x, 165, 175–80, 189, 208, 212, 213– 19, 224, 226, 243, 326n19 Pappus problem, the, 191n60, 208, 223, 225–26, 240–41, 262 passivity. See activity, vs. passivity Patterson, Sarah, 57n129 Patzig, Günter, 133n94 Paul of Venice, 334, 338 Pell, John, 182n32 perception clear and distinct. See certainty, and clear and distinct perception; clear and distinct perception; scientia, and clear and distinct perception) constituent of judgment, 12–14, 17, 28n81, 42n100 evident, 56, 61–62, 68 operation of understanding, 9, 11, 32, 36 and understanding, 9, 11, 32, 36 persuasio. See cognition, as persuasio phantasia, 21, 22n61, 85, 87, 88, 99, 228, 231, 233–34 plagiarism affair, 191, 195 Plantinga, Alvin, 109n55 Plato, 83n16, 215 Popkin, Richard, 6n4, 141n105 præcisio. See abstraction, as omissio Prantl, Carl, 321n11 principle, 177–78, 184, 312, 316n127, 324, 327, 332, 341– 43, 349–50 acquaintance with, 65 common, 145–46 of comparatio, 276–77, 281 conditions of, 346 evident. See principle, per se notum explanatory, 332–34 first, 72, 122, 147, 159, 328, 331, 346–47 general, 158–59

innate. See method, innate principles of of indubitability, 81 logical, 97 of Mathesis vniversalis, 186, 258 per se notum, 65, 72, 147 of scientia, 75–76, 185n39, 257 true, 120, 346n49 two meanings of, 347n50 problem. See quæstio Proclus Diadochus, ix–x, 183n34, 184–85, 192n62, 217n116, 257 proof. See demonstration proportion, 141, 184, 276–78 in algebra, 124, 171, 192–93, 203–204, 251–52, 285, 292– 93 Eudoxus’ theory of, 192 object of Mathesis vniversalis, 186, 198–200, 243n8, 251–52 reciprocal to equation, 192 See also continuous proportionals Pythagorean quadrivium, 199n80, 247

quadrivium, Pythagorean, 199n80, 247 quæsitum (the sought-after, the unknown), 124, 152–54, 177– 78, 187, 213–14, 218, 221, 224, 226, 230, 278–79, 287, 292, 329–31, 345, 349, 351– 53 as participating in natura communis, 276–77, 281–82, 284–85 See also equality, of data and quæsita quæstio, 93, 147, 149, 154, 168–69, 251, 279, 281–82, 284, 292, 297, 301–302, 306, 349, 351 determination of, 287–92, 304, 348 direct vs. indirect examination of, 293

The a priori in the Thought of Descartes imperfectè intellecta. See quæstio, perfectè vs. imperfectè intellecta indirect examination of, 293 inherently imperfect, 152, 290– 91 inherently perfect vs. inherently imperfect, 152, 290–91 overdetermination of, 289–90 perfectè vs. imperfectè intellecta, 152, 228, 267, 268–69, 287– 89, 291, 351n52 reduction of, 288–92 series of, 162, 295, 353 template, 292–95 underdetermination of, 289–90 quantity continuous vs. discrete, 196 discrete vs. continuous, 196 general, 186, 192, 220–22, 224, 230, 240, 251, 278, 285–87 (see also line segments, and general quantities) signified by line segments, 196– 97, 199, 227–29, 255–56

rainbow, 262–63 Ramus, Petrus, 126n82, 187–89 Randall, John, 326n18, 327n20, 333–35nn28–35, 336 ratio. See reason reason being of, 90n26 as bona mens, 135–36 distinctio of, 20, 233, 234–36 faculty, 6n5, 8n9, 95, 112–13, 131n88, 139, 167–68, 171, 196, 229–30, 276, 296, 300n104 and justification, 58–60 light of. See lumen naturale in mathematics, 170, 198 order of, 274, 337, 341 reasoning, 128, 283 a posteriori. See reasoning, a

391

priori vs. a posteriori a priori vs. a posteriori, 2, 327, 329, 338 (see also demonstration, a priori vs. a posteriori; demonstration, propter quid vs. quia) chains of, 72 demonstrative, 95, 142n106, 146, 170 discursive, 119, 125, 273, 323 (see also deductio, and discursivity) from effects to causes, 332 general, 313 inductive, 323n14 inferential, 127 mathematical, 170, 211, 217, 220, 226 orderly, 274 (see also ordo, and dispositio) propter quid vs. quia, 329 (see also demonstration, propter quid vs. quia) quia, 329 scientific, 3, 161, 318 syllogistic, 125–27, 131, 137–41, 143–46 reduction in analysis. See analysis, as resolutio to cognitive primitives, 199–200, 229 of deductio to intuitus, 74n160, 121–25, 134, 149, 157, 314, 323 and dispositio, 154, 275 of equations, 206–207, 219, 251n26, 277–78, 295, 351 as generalization, 218, 228 of imperfect quæstiones to perfect, 288–92 of omnia quæ cognoscimus, 86– 89, 346, 349 ontological, 312 of quæstiones, 288–92 to simplest problems, 93, 268–

392

General Index

70, 275, 279, 292–95 Regius, Henricus, 10, 30, 37, 41 regressus, 335–38, 341n43, 344n47, 352 regulation, 127–136, 309, 311, 323 application of method, 137, 296, 309 approbative, 137, 138 definition of, 128 and dialectic, 128–31, 134, 323 heuristic, 131, 136–39 Reiss, Timothy, 318n3 Remes, Unto, 215n110, 215n112 representation, 321 distinct, 198–99, 227 by ideas, 19, 21 in mathematics, 196, 197, 210– 11, 228–29, 254, 256n39 in memory, 121, 307 mental, 20, 94 of things, 15 respectivum, 94, 98, 153, 265n54, 272, 279, 282, 345, 349 degrees of, 280–81, 285–86 lists of, 93, 348 resolutio. See analysis, as resolutio Risse, Wilhelm, 128n85 Röd, Wolfgang, 280n76 Romanus, Adrianus. See Van Roomen, Adriaan Rosenthal, David, 13n27, 18n46, 54n122, 55 Ross, G. R. T., 76n3, 305n110 Ross, William, 158n128, 158n130, 159 Roth, Leon, 266n55 Rozemond, Marleen, 8n9, 23n65

sapientia, 168, 169n7, 169n10, 175 and bona mens, 310–11 and method, 310–11 and philosophy, 346–47 and scientia, 310–11, 346 and unity of scientiæ, 309–11 universal, 169–70, 198, 309

Sasaki, Chikara, x, 182nn32–33, 184n34, 185n36, 186n41, 187n44, 187nn46–48, 188nn50–51, 190n58, 191n59, 192n61, 199n77, 208n96, 210n101, 224n131, 245nn13– 14, 249n20, 250n23, 251n24, 253n28, 253n32, 258n46 Schöner, Lazarus, 188 Schouls, Peter, 266n55, 308n116 Schuster, John, 84n20, 122n75, 171n11, 245nn13–14, 252–56, 263–65, 266nn55–56, 301n104, 304–307, 308n116 science. See scientia scientia a priori 2–4, 29, 161, 325, 328– 29, 346–47, 350 Aristotelian conception of, 159– 60, 161, 273, 313–15, 318, 319–23, 330–37, 338, 350 vs. ars, 313–15 and certainty, 2, 5, 47–74, 120– 22, 138, 148, 166–68, 247, 256, 328–29, 331–32, 335, 338 and clear and distinct perception, 47–49, 53–74, 230, 237, 247 and discovery, 123–25, 139–46, 149–50, 152–55, 164, 268–95, 324–28, 330–38, 348–55 and experientia, 82–86, 166, 283, 290–91, 324–25, 337–38 and formalism, 134, 136–37, 312 (see also algebra, interpreted) foundations of, 73, 307 (see also necessary connections, foundations of) general, 241, 243, 246, 247n16, 248, 255, 332 and God’s guarantee, 60–74, 108, 113–16, 121, 124n79, 157, 307 and imagination, 227–37 and inference, 134, 137 interconnection of, 67n150, 306, 316n127

The a priori in the Thought of Descartes and method, 1, 93, 129–30, 137– 39, 147–55, 167, 171, 296–99, 301, 332–37 (see also scientia, and universal method) and necessity, 108–109, 116 (see also connection, object of intuitus; deductio, and necessary connections; deductio, reducible to intuitus; necessary connections, realistic conception of) vs. opinio, 141–42 vs. persuasio, 63–67, 121 possibility of, 68–74 principles of, 75–76, 134, 159, 185n39, 257 (see also deductio, principle of scientia; intuitus, principle of scientia) and sapientia, 310–11, 346 and simple natures, 91–93, 98, 164 standards of, 130, 138, 166–67, 329, 332 and understanding, 75–80, 85– 86, 227, 236–37 unity of, 136, 169, 306n111, 307–16 and universal method, 181n30, 190, 242, 260–84, 287–92, 308, 344–55 See also cognition, scientific; mathematics, as scientia; nature, science of sensatio. See sensation sensation and adventitious ideas, 31–32, 34, 38–40 and clear and distinct perception, 50n115, 77–80, 88–89 dependent on body, 22–24 and experientia, 84–85, 325 external. See sensation, as sense perception Hill’s interpretation of, 27–28, 50n115, 82n12 and innate ideas, 38–47

393

and intuitus, 82–86, 87–88, 96, 99–100, 315n126 and mathematics, 227–31 mode of cogitatio, 10–11, 22–23, 25–29, 78, 234 (see also sense, faculty) mode of understanding. See sensation, mode of cogitatio as sense perception, 22n63, 23, 27n75, 31–34, 40, 50n115, 78–79, 84–85, 99, 158–59 and simple natures, 88–89 and phantasia, 88 scope of, 22n63 See also sense, as sensation sense communis, 21–22 faculty, 26, 29, 37, 40, 45–47, 77–80, 85–86, 88, 151, 163, 227–30, 325, 335, 342 (see also sensation, mode of cogitatio) as organ, 23, 38, 118n72, 283 as sensation, 22, 24, 80–82, 96 See also sensation, as sense perception sensus. See sense sentio. See sensation Sepper, Dennis, 21n60, 88n23, 135n95 Serfati, Michel, 211n103 series, 211, 212, 216, 226 arrangement in. See series, ordo & dispositio in of continuous proportionals, 203–204, 251, 285–87, 292– 93, 295 deductive, 146, 149, 152–53 division of, 292–95 extremities of, 293–95 of intuitus, 124, 149, 275, 323 of mean proportionals, 211n103 of numbers, 316n127 ordo & dispositio in, 93, 94n31, 98, 124, 153, 269–87, 292, 348–50

394

General Index

of quæstiones, 162, 295, 353 of simple natures, 271 in synthesis, 162, 353 Serrus, Charles, 308n116 Sextus Empiricus, ix, 140–41, 142n106 simple natures and absolutum, 92–94, 98, 153, 279–87, 345, 348–50 apprehension of, 88, 92–93, 101, 105–106 as causes, 346–48 and certainty, 91 classifications of, 94–96, 146n112 cognition of, 91–92, 96, 105– 106, 108, 164, 347 cognitive functions of, 92–93 compositions of. See compositio, of simple natures; connection, of simple natures definition of, 90 heterogeneity of, 89, 97–98 idealist reading of, 93–94, 97 and imagination, 88–89 indivisible, 91–92, 102–104, 108 and natura communis, 282–84 necessary connections of. See connection, necessary non-necessary connections of. See connection, contingent objects of intuitus, 85, 87–88, 91–92, 106–107, 109, 114–15, 123n79, 271 ontological status of, 93–97 per se notæ, 83, 91–92, 283, 290 and scientia, 91–93, 98, 109, 116, 164 and sensation, 88–89 series of, 271 unity of doctrine of, 89, 97–99 simplices notiones. See simple natures sought-after, the. See quæsitum Stout, Alan, 69n153 syllogism

Aristotle’s notion of, 125–26, 131 and cognitive circularity, 140–42 and deductive inference, 125–27 definition of in Aristotle, 125n81 demonstrative, 131, 143n107, 158 (see also reasoning, demonstrative) heuristic value of, 131, 139–46 inductive, 126 inherently discursive, 139 and particulars, 142–46 perfect, 133n94, 143n107 propter quid vs. quia, 319, 335 (see also demonstration, propter quid vs. quia) quia. See syllogism, propter quid vs. quia scientific, 126, 320 See also reasoning, syllogistic syllogistic. See syllogism synthesis a posteriori, 161–65, 325, 326, 340–41, 351–55 a priori, 162–63, 326–27, 343 complement of analysis, 176–78, 215–27, 351–52 as compositio, 176n17, 177–78, 326–29, 333 in Descartes, 219–27, 351–55 in Pappus, 178, 215–19 problematical vs. theoretical, 220–22, 353–55 and reversibility problem, 215–18 superfluous, 217–18, 220–21 theoretical. See synthesis, problematical vs. theoretical Szabó, Arpád, 215n110

Tannery, Paul, ix, 220n123, 247n16, 248n18 Theon of Alexandria, 188–89 thinking. See cogitatio Thomas Aquinas, 97n34, 314n124, 328

The a priori in the Thought of Descartes thought. See cogitatio movement of, 124, 267, 269, 329 Timmermans, Benoît, 1n2, 328n22, 340–41 Tlumak, Jeffrey, 50n116, 52nn119– 20 Toletus, Franciscus, 318n3 tree analogy, 274n67, 316n127 truth, 6–7, 13n28, 92, 129, 137, 139, 167, 170, 180, 198, 213, 241, 256, 269–71, 274, 297, 301, 325 and clear and distinct perception, 54–55, 58, 60n139, 346 and certainty, 59, 72, 100, 118, 147, 151, 166, 346 contingent. See truth, necessary vs. contingent eternal, 95–96, 109–17, 324 evident, 100–101, 216n 114 (see also truth, per se nota) formal, 17 and God’ guarantee, 64–65 immutability of, 70n153 as innate idea, 31 and judgment, 2, 13, 17, 70n153 locus of, 13, 17 material, 15n35 necessary vs. contingent, 109, 111–16 (see also necessary connections) per se nota, 41, 46n104, 60n139, 276 (see also cognition, per se nota; truth, evident) semina of, 167n5, 299, 347n51 and syllogistic, 139–41 undefinable, 130n87

understanding (intellectus) capacity of, 156–57 and deductio, 75–76, 128–29, 147, 314 divine vs. human, 111–17 and experientia, 84, 87–88, 99– 100

395

faculty, 12–13, 17, 21, 23, 55, 66, 75–77, 80, 82, 85, 91, 94–95, 97, 99, 227, 229–30, 255, 283, 300n104 human vs. divine, 111–17 and imagination, 227–37 and intuitus, 75–80, 91, 147, 314–15 and judgment, 12–13, 17, 21, 28n81 mode of cogitatio, 9–10 modes of, 10–11, 21–29, 32, 40– 41, 80, 227, 234 negotiatio of, 335–36 operations of, 9, 11–13, 18, 21– 22, 25–26, 31–33, 36, 65–66, 75–81, 85, 86, 147, 155, 158, 255, 313–14 (see also intellectio) passio mentis, 10, 76–77, 79–80, 81–83, 101, 237 pure, 10, 11, 21–22, 32–33, 36, 77–86, 228, 230–31, 234–37, 255, 268, 300n104, 305n109 and scientia, 75–80, 85–86, 155, 158, 227, 236–37, 313 See also ordo, of cognition uniformity of cognitive operations, 309–16 of intuitus, 315–16 unit length, 200–206, 285–87 (see also line segments, lengths of) unity as natura communis, 284–87 of scientiæ, 136, 169, 306n111, 307–16 as simple nature, 95 universal mathematics, 254 and algebra, 185–87, 194, 196– 97, 240, 245, 251–52 (see also Mathesis vniversalis, and algebra) in Aristotle, 183 in Proclus, 183–85 See also Mathesis vniversalis universal method

396

General Index

and Mathesis vniversalis, 242, 256–60 modus operandi of, 268–78 paradigms of, 165–71, 238–42, 245n14, 256–59, 284–87, 292–95 (see also algebra, paradigm of universal method) possibility of, 308n116 (see also method, possibility of) precepts of. See method, precepts of reconstruction of, 265–95 sceptical attitude to, 263–65, 308n116 and scientia. See scientia, and method; scientia, and universal method and unity of scientiæ, 308–16 See also method, beyond mathematics unknown, the. See quæsitum

Van Cleve, James, 70n155, 110n55 Van De Pitte, Frederick, 244n10– 11, 245nn13–14, 247– 48nn16–17, 256, 257–59 Van Roomen, Adriaan, 185–87, 189, 192, 249–51, 253n32, 255, 256–59 Van Schooten, Frans, 207n94 Viète, François, x, 174n15, 186–89, 190–97, 203, 209, 219n121, 221n125 volition actio mentis, 10, 77 constituent of judgment, 12–13,

17, 42n100 mode of cogitatio, 9–10, 26 object of experimenta, 84 operation of will, 9–10, 95–96 (see also will, operations of) voluntas. See will

Weber, Jean-Paul, ix, 153n122, 242, 244n9, 244n12, 245n14, 300n104 will divine, 109–13, 116 determination of, 12n23, 31–32, 54 and factitious ideas, 31–34 faculty, 12, 28n81, 113 freedom of, 13, 53–54, 84n18 operations of, 9–10, 11, 13–14, 17, 25–26, 28n81, 31–35, 95– 96 required for judgment, 12–13, 17, 21, 42n100 Williams, Bernard, 14n29, 15– 16n37, 25, 54n123, 58n130, 59n133, 71n157 Wilson, Margaret, 6n4, 16n40, 17n44, 20n58, 23n65, 54–55, 69n152 Winfree Smith, J., x, 179n24 wisdom. See sapientia Wohlers, Christian, xi

Zabarella, Giacomo, 336–37, 338 Zeuthen, Hieronymus, 220n123