Kant's Metaphysical Foundations of Natural Science: A Critical Guide 1108476899, 9781108476898
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KANT’S METAPHYSICAL FOUNDATIONS OF NATURAL SCIENCE
In his Metaphysical Foundations of Natural Science (), Kant accounts for the possibility of an acting-at-a-distance gravitational force, demonstrates the infinite divisibility of matter, and derives analogues to Newtonian laws of motion. The work is his major statement in philosophy of science, and was especially influential in German-speaking countries in the nineteenth century. However, this complex text has not received the scholarly attention it deserves. The chapters of this Critical Guide clarify the accounts of matter, motion, the mathematization of nature, space, and natural laws exhibited in the Metaphysical Foundations; elucidate the relationship between its metaphysics of nature and Kant’s Critical philosophy; and describe the historical context for Kant’s account of natural science. The volume will be an invaluable resource for understanding one of Kant’s most difficult works, and will set the agenda for future scholarship on Kant’s philosophy of science. is Assistant Professor in the Department of Philosophy, University of Minnesota. He has published widely on various aspects of Kant’s philosophy of science and philosophy of nature.
Titles published in this series: Nietzsche’s Thus Spoke Zarathustra - . Aristotle’s On the Soul . Schopenhauer’s World as Will and Representation Kant’s Prolegomena Hegel’s Encyclopedia of the Philosophical Sciences Maimonides’ Guide of the Perplexed Fichte’s System of Ethics Hume’s An Enquiry Concerning the Principles of Morals Hobbes’s On the Citizen Hegel’s Philosophy of Spirit . Kant’s Lectures on Metaphysics . Spinoza’s Political Treatise . Aquinas’s Summa Theologiae Aristotle’s Generation of Animals Hegel’s Elements of the Philosophy of Right Kant’s Critique of Pure Reason . ’ Spinoza’s Ethics . Plato’s Symposium e´ Fichte’s Foundations of Natural Right Aquinas’s Disputed Questions on Evil . . Aristotle’s Politics Aristotle’s Physics
(Continued after the Index)
KANT’S METAPHYSICAL FOUNDATIONS OF NATURAL SCIENCE A Critical Guide
MICHAEL BENNETT MCNULTY University of Minnesota, Twin Cities
University Printing House, Cambridge , United Kingdom One Liberty Plaza, th Floor, New York, , USA Williamstown Road, Port Melbourne, , Australia –, rd Floor, Plot , Splendor Forum, Jasola District Centre, New Delhi – , India Penang Road, #–/, Visioncrest Commercial, Singapore Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/ : ./ © Cambridge University Press This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published A catalogue record for this publication is available from the British Library. ---- Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
Contents
List of Figures List of Contributors Note on Abbreviations and Translations Introduction
page vii viii ix
Michael Bennett McNulty
Kant’s Conception of the Metaphysical Foundations of Natural Science: Subject Matter, Method, and Aim
Thomas Sturm
Kant’s Normative Conception of Natural Science
Angela Breitenbach
The Applicability of Mathematics as a Metaphysical Problem: Kant’s Principles for the Construction of Concepts
Katherine Dunlop
Phoronomy: Space, Construction, and Mathematizing Motion
Marius Stan
Space, Pure Intuition, and Laws in the Metaphysical Foundations
James Messina
Finitism in the Metaphysical Foundations
Lydia Patton
The Construction of the Concept of Space-Filling: Kant’s Approach and Intentions in the Dynamics Chapter of the Metaphysical Foundations Daniel Warren v
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Contents Beyond the Metaphysical Foundations of Natural Science: Kant’s Empirical Physics and the General Remark to the Dynamics
Michael Bennett McNulty
How Do We Transform Appearance into Experience? Kant’s Metaphysical Foundations of Phenomenology
Silvia De Bianchi
Absolute Space as a Necessary Idea: Reading Kant’s Phenomenology through Perspectival Lenses
Michela Massimi
Proper Natural Science and Its Role in the Critical System
Michael Friedman
References Index
Figures
. . . .
Newton’s diagram for his proof of Kepler’s Area Law Pairs of velocities respectively equal in size Pairs of equal speeds ’s Gravesande’s and d’Alembert’s demonstrations of the Parallelogram Rule
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Contributors
is University Lecturer in Philosophy and Fellow of King’s College at the University of Cambridge. is Research Fellow in Logic and Philosophy of Science at the University of Milan. is Associate Professor of Philosophy at the University of Texas at Austin. is Patrick Suppes Professor of the Philosophy of Science at Stanford University. is Professor of Philosophy of Science at the University of Edinburgh. is Assistant Professor of Philosophy at the University of Minnesota, Twin Cities. is Associate Professor of Philosophy at the University of Wisconsin, Madison. is Professor of Philosophy at Virginia Polytechnic Institute and State University. is Associate Professor of Philosophy at Boston College. is ICREA Research Professor in Philosophy and History of Science at the Autonomous University of Barcelona. is Associate Professor of Philosophy at the University of California, Berkeley.
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Note on Abbreviations and Translations
Citations of Kant’s own writings and student notes from his lectures refer to their production in the Akademie edition of his Gesammelte Schriften (Kant –), using a siglum to denote the work followed by the volume number and the page number. References to Kritik der reinen Vernunft utilize the standard, A/B system to refer, respectively, to the page numbers of the first and second editions of the book. Where available, English translations from The Cambridge Edition of the Works of Immanuel Kant have been used. The following lists the sigla used to refer to Kant’s works, followed by their translations in the Cambridge series. Anth BDG Br DfS DI FM
GMS GSK GUGR
Anthropologie in pragmatischer Hinsicht (Kant , –) Der einzig mögliche Beweisgrund zu einer Demonstration des Daseins Gottes (Kant b, –) Briefe (selections in Kant ) Die falsche Spitzfindigkeit der vier syllogistischen Figuren erwiesen (Kant b, –) Meditationum quarundam de igne succincta delineatio (Kant , –) Welches sind die wirklichen Fortschritte, die die Metaphysik seit Leibnitzens und Wolf’s Zeiten in Deutschland gemacht hat? (Kant , –) Grundlegung zur Metaphysik der Sitten (Kant a, –) Gedanken von der wahren Schätzung der lebendigen Kräfte (Kant , –) Von dem ersten Grunde des Unterschiedes der Gegenden im Raume (Kant b, –) ix
x KpV KrV KU Log MAN MonPh MS MSI NG NLBR
NTH OP PND Prol Refl SF UD ÜGTP V-Lo/Blomberg V-Lo/Dohna V-Lo/Wiener
Note on Abbreviations and Translations Kritik der praktischen Vernunft (Kant a, –) Kritik der reinen Vernunft (Kant ) Kritik der Urteilskraft (Kant ) Logic (Kant a, –) Metaphysische Anfangsgru¨nde der Naturwissenschaft (Kant , –) Metaphysicae cum geometria iunctae usus in philosophia naturali, cuius specimen I. continet monadologiam physicam (Kant b, –) Die Metaphysik der Sitten (Kant a, –) De mundi sensibilis atque intelligibilis forma et principiis (Kant b, –) Versuch, den Begriff der negativen Größen in die Weltweisheit einzufu¨hren (Kant b, –) Neuer Lehrbegriff der Bewegung und Ruhe und der damit verknu¨pften Folgerungen in den ersten Gru¨nden der Naturwissenschaft (Kant , –) Allgemeine Naturgeschichte und Theorie des Himmels (Kant , –) Opus postumum (selections in Kant ) Principiorum primorum cognitionis metaphysicae nova dilucidatio (Kant b, –) Prolegomena zu einer jeden ku¨nftigen Metaphysik (Kant , –) Reflexionen (selections in Kant ) Der Streit der Fakultäten (Kant b, –) Untersuchung u¨ber die Deutlichkeit der Grundsätze der natu¨rlichen Theologie und der Moral (Kant b, –) Über den Gebrauch teleologischer Principien in der Philosophie (Kant , –) Logik Blomberg (Kant a, –) Logik Dohna-Wundlacken (Kant a, –) Wiener Logik (Kant a, –)
Note on Abbreviations and Translations V-Met/Dohna V-Met/Mron V-Met/Volckmann V-Met-L/Pölitz V-Met-L/Pölitz V-Phys/Berliner V-Phys/Mron VvRM
xi
Metaphysik Dohna-Wundlacken (selections in Kant a, –) Metaphysik Mrongovius (Kant a, –) Metaphysik Volckmann (selections in Kant a, –) Metaphysik L (selections in Kant a, –) Metaphysik L (selections in Kant a, –) Berliner Physik Physik Mrongovius (Danziger Physik) Von den verschiedenen Racen der Menschen (Kant , –)
Citations of Descartes’ writings use the abbreviation “AT” and refer to their production in Oeuvres de Descartes (Descartes ), by volume and page number. Citations of Leibniz’s works refer via abbreviations to their production in the following volumes. Refences to Die Philosophischen Schriften von Gottfried Wilhelm Leibniz refer to the volume and page number. AG G L
Philosophical Essays (Leibniz a) Die Philosophischen Schriften von Gottfried Wilhelm Leibniz (Leibniz –) Philosophical Papers and Letters (Leibniz b)
Introduction Michael Bennett McNulty
I. The Metaphysical Foundations of Natural Science and Transcendental Philosophy Immanuel Kant long sought to write a metaphysics of nature. In a letter to Johann Heinrich Lambert, Kant reported that he was postponing the general project he had been working on, the “Proper Method of Metaphysics.” He would instead produce the paired “Metaphysical Foundations of Natural Philosophy” (metaphysische Anfangsgru¨nde der natu¨rlichen Weltweisheit) and “Metaphysical Foundations of Practical Philosophy” (der praktischen Weltweisheit) (Br, :; see Förster , –) as particular examples in concreto of the proper philosophical methodology. Despite Kant’s claim that the “content” of these projects was “already worked out,” they were, in turn, deferred and subsequently shelved during his writing of the Inaugural Dissertation (MSI) and the subsequent tumult to his metaphysical outlook left in its wake. Nonetheless, Kant still harbored ambition to write a metaphysics of nature, expressing his hope to return to this project in the preface of the first edition of the Critique of Pure Reason while acknowledging that a critique of reason must antecede his metaphysical ambitions: “[The] Metaphysics of Nature . . . will be not half so extensive but will be incomparably richer in content than this critique, which had first to display the sources and conditions of its possibility, and needed to clear and level a ground that was completely overgrown” (KrV, Axxi). With the foundation for the metaphysics of nature thus laid bare with the completion of the Critique of Pure Reason, Kant embarked on writing the Metaphysical Foundations of Natural Science (Metaphysische Anfangsgru¨nde der Naturwissenschaft), finally published in , over twenty years after he mentioned the project to Lambert. According to Kant, the particular metaphysics of nature discussed in the Metaphysical Foundations of Natural Science – that of corporeal nature – relates closely to the general metaphysics developed in the Critique of Pure
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Reason. In the latter Kant situates the particular metaphysical doctrines of nature in his architectonic (KrV, A–/B–). Among the subdivisions of metaphysics, in general, are the “immanent physiology of nature” alongside a “transcendent physiology” (of the world-whole and of God). The immanent physiology of nature, or what Kant elsewhere calls “the general metaphysics of nature,” includes the a priori metaphysics of nature detailed especially in the body of the Transcendental Analytic. This general metaphysics of nature describes the conditions of the possibility of nature (see also B–, A/B). But there are additionally two special metaphysical doctrines of nature, the metaphysics of “corporeal nature” (rational physics) and that of “thinking nature” (rational psychology), which contain a priori knowledge concerning these more circumscribed domains. In the preface to the Metaphysical Foundations, Kant indicates that the particular special metaphysics of nature considered in the book is this “metaphysics of corporeal nature” (:). Thus the general metaphysics of the first Critique lays the groundwork within which the project of the Metaphysical Foundations is situated – the special metaphysics of corporeal nature is one of the two chief members of the metaphysics of nature. What is more, there is are more substantial connections between this special metaphysical doctrine and the metaphysics of the first Critique. As Kant states, in a special metaphysics of nature, the “transcendental principles [sc. those that make possible the concept of nature in general] are applied to the two species of objects of our senses” (MAN, :). Thus, the special metaphysics of body proceeds on the tracks laid by transcendental philosophy, to wit, the framework of the categories developed in the first Critique. As Kant claims, the concept of matter – the empirical concept at the basis of the special metaphysics of corporeal nature (MAN, :–, ) – is “carried through all four of the indicated functions of the concepts of the understanding (in four chapters)” (MAN, :). That is, the concept is consecutively determined by the categories – quantity, quality, relation, and modality – which determination grounds the a priori principles of the special metaphysical doctrine. Along these lines, in the chapters of the Metaphysical Foundations, Kant, respectively, discusses matter’s quantity
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Kant also rehearses much of this architectonic at the outset of the Metaphysical Foundations (MAN, :). Psychology, for Kant, is impossible as a “proper natural science” – it does not adequately allow for the application of mathematics (MAN, :) – meaning that its foundations are not at issue in the Metaphysical Foundations. This claim has been much discussed in the secondary literature, such as by Mischel (), Gouaux (), Nayak and Sotnak (), Sturm (; ; , ch. ), and Kraus ().
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(Chapter : Metaphysical Foundations of Phoronomy), quality (Chapter : Metaphysical Foundations of Dynamics), relation (Chapter : Metaphysical Foundations of Mechanics), and modality (Chapter : Metaphysical Foundations of Phenomenology). Indeed, it is this dependence of the special metaphysics of corporeal nature on transcendental philosophy – the determination of the concept of matter by the categories – that guarantees its completeness (MAN, :). Furthermore, throughout the Metaphysical Foundations Kant utilizes principles and conceptual machinery from general metaphysics; most prominently, the principles of each of the Analogies of Experience are cited in the proofs of the corresponding mechanical laws (MAN, :, , –). In addition, Kant’s theories of space and time, mathematical construction, the divisibility of objects, causality, force, and continuity all play prominent roles in the arguments of the Metaphysical Foundations. Altogether, such aspects demonstrate that, for Kant, the metaphysical foundations of natural science are intimately tied to and, indeed, constitute an extension of the transcendental account of nature developed in the Critique of Pure Reason. Yet, reciprocally, the special metaphysics of nature “does excellent and indispensable service for general metaphysics,” insofar as it provides examples “in concreto” of the concepts and judgments of transcendental philosophy; that is, it “give[s] a mere form of thought sense and meaning” (MAN, :). Although such claims are not immediately transparent, in all, these references both demonstrate the intimate connection between the project of the Critique of Pure Reason and that of the Metaphysical Foundations of Natural Science and depict the special metaphysics of corporeal nature as a genuine part of the critical program.
I. Scholarship and the Critical Guide There is a long history of scholarship both on Kant’s philosophy of science, generally, and the Metaphysical Foundations, in particular. The keystone and starting point for modern commentaries on Kant’s philosophy of science is Erich Adickes’ two-volume Kant als Naturforscher (–), a
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A note on versions of MAN. The Metaphysische Anfangsgru¨nde der Naturwissenschaft is available in volume of Kant’s Gesammelte Schriften (–), originally edited by Alois Höfler and published in . A new edition for the series edited by Thomas Sturm and Bernhard Thöle is forthcoming, as part of the ongoing project of publishing revised versions of the volumes constituting division I, Werke, of the Gesammelte Schriften. Konstantin Pollok also edited a free-standing German rendition that includes his substantial and valuable introduction (Kant b). English translations are available from James Ellington (Kant ; included in Kant ) and Michael Friedman,
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far-ranging account of Kant’s philosophy of nature. Mid- to late twentiethcentury accounts of Kant’s philosophy of science appeared in both German – including Schäfer (), Hoppe (), Tuschling (), Gloy () – and English scholarship – such as Buchdahl (b) and Brittan (). Furthermore, a variety of texts specifically on the Metaphysical Foundations were penned, including Plaass’ () commentary on its Preface, Butts’ (b) edited volume celebrating the bicentennial of the book’s publication, and Pollok’s () comprehensive commentary. Additionally, the chapters of Watkins (b) provide a sweeping introduction to Kant’s general philosophy of science and views on a variety of special sciences. Anglo-American scholarship has of late refocused on Kant’s philosophy of science, especially spurred by the pioneering work of Michael Friedman (b, ). The recent surge of research includes special issues of journals, such as those edited by Massimi (), Gaukroger and Nassar (), Heidemann (), and De Bianchi and Kraus (), as well as assorted related interventions from individual authors, such as Edwards (), Watkins (), and Glezer (). Additionally, there have been many recent studies of Kant’s views on particular sciences, including (but not limited to) physics, chemistry, psychology, and biology. Since the Metaphysical Foundations is the first and primary extension of the critical philosophy into natural science, its consideration is essential to such studies. In this volume, chapters from Sturm, Breitenbach, and Friedman are particularly relevant to Kant’s general philosophy of science, especially insofar as they clarify his conception of the metaphysical foundations of natural science and situate this project with respect to the overarching critical system. Altogether the chapters of this book both clarify the general
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whose version was originally in Theoretical Writings after in the Cambridge Edition of the Works of Immanuel Kant, edited by Allison and Heath (Kant ) and later appeared as a stand-alone book in the Cambridge Texts in the History of Philosophy series (Kant ). Onnasch (), Warren (), Smith (), Stan (, a, b, , ), Sutherland (), and Kahn (). Earlier relevant work includes Palter (, ), Brittan (), Falkenburg (, ), Carrier (), Watkins (, b), Warren (), Pollok (, ), Emundts (), and Engelhard (). McNulty (, , , , ), Blomme (), Gaukroger (). Earlier work concerning Kant’s conception of chemistry includes Carrier (, ), Friedman (b, pt. ), and Lequan (). Cohen (), Sturm (), Sturm and Wunderlich (), Dyck (b), Frierson (), and Kraus (, ). Earlier texts on psychology include Mischel (), Gouaux (), Hatfield (, , ), Pa. Kitcher (), Makkreel (), and Sturm (, ). Breitenbach (a, ), Cohen (), van den Berg (), Watkins and Goy (), and Goy (). Earlier work on biology includes McLaughlin (), Ginsborg (, ), Zammito (), Quarfood (, ), and Kreines ().
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Introduction
structure of Kant’s conception of natural science and open up a variety of new issues and conceptual spaces in his theory of science. Additionally, Kant’s conception of laws – particularly, of laws of nature – has been a popular, if contentious, issue in recent scholarship. A major impetus for this literature was Michela Massimi’s Leverhulme Trust research project, “Kant and the Laws of Nature,” which produced special issues of Kant-Studien (Massimi a) and The Monist (Breitenbach and Massimi ), as well as the outstanding volume Kant and Laws of Nature (Massimi and Breitenbach ). A variety of scholars have also individually weighed in on the topic. Most prominently, Eric Watkins’ recent Kant on Laws () develops a comprehensive account of the unity and diversity of laws in Kant’s philosophy, including an account of his views on the laws of nature, particularly those of mechanics. Many others, including Kreines (, ), McNulty (), Stang (), Breitenbach (), Engelhard (), Massimi (b, , a, b), Messina (, a, b), and Patton (), have delved into Kant’s conception of laws from various angles. This is a pivotal issue in contemporary Kant scholarship, one on which the present volume has much to say. Messina and Patton’s contributions to this volume both bear explicitly on the issue of laws of nature, but, insofar as the grounding of the laws of matter is at the heart of the Metaphysical Foundations, each of the chapters bears on this central issue. Finally, the Metaphysical Foundations of Natural Science is a vexing book that inspires fascinating interpretative questions in its own right. The chapters of the present volume not only attempt to resolve some of the thorniest of these issues – such as the point and structure of Kant’s Dynamics chapter (Warren) and its General Remark (McNulty), the place of the idea of absolute space in Kant’s account of natural science (De Bianchi and Massimi), and his views on the mathematization of motion and its historical context (Dunlop and Stan) – but also prepare the way for future scholarship on them.
I. Overview of the Chapters Chapter , Thomas Sturm’s “Kant’s Conception of the Metaphysical Foundations of Natural Science: Subject Matter, Method, and Aim” orients
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Kant’s conception of laws is something of an evergreen topic, given its centrality to his metaphysics, theory of science, and account of causality. Earlier relevant literature includes Buchdahl (), Parsons (), Ph. Kitcher (, ), Guyer (), Thöle (), Friedman (a), Allison (), and Rush ().
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the reader to Kant’s project in the Metaphysical Foundations of Natural Science by relating it to transcendental philosophy. After carefully distinguishing various dimensions of the entanglement of the metaphysics of body and transcendental philosophy, Sturm considers accounts according to which the metaphysical foundations of natural science complete transcendental philosophy as well as those that posit no dependence of transcendental philosophy on natural science. The latter sort of interpretation especially aims to insulate the synthetic a priori transcendental philosophy from the apparent empirical disconfirmation of Kant’s theory of science provided by non-Euclidean geometries, general relativistic physics, and quantum mechanics. To evaluate the two sorts of interpretation, Sturm goes back to basics and meticulously characterizes the metaphysical foundations of natural science as a doctrine in Kant’s architectonic. He explains that, for Kant, sciences are essentially defined by ideas, which codify the science’s ontological domain, epistemological characteristics, and axiological features. Based on a consideration of these aspects of the metaphysics of body, Sturm ultimately concludes that such metaphysics depends on transcendental philosophy, but not vice versa. Chapter , “Kant’s Normative Conception of Natural Science,” by Angela Breitenbach, provides a fresh perspective on Kant’s conception of “properly so-called natural science.” Notoriously, Kant espoused a particularly stringent conception of sciencehood in the preface of the Metaphysical Foundations, holding that only those natural sciences that are apodictically certain, contain a priori laws, and adequately allow for the application of mathematics are properly so-called natural science (MAN, :–). Other investigations of nature, like chemistry or empirical psychology, fail to satisfy collectively these standards and are thus deemed “improper” natural sciences. In the past, Breitenbach contends, scholars have interpreted Kant’s comments on proper science either as providing a classificatory system for the sciences – in which chemistry is a “rational” but “improper” natural science, physics is a “proper natural science,” and empirical psychology is a form of “natural description” – or as presenting Kant’s demarcation standard – proper science is science, everything else is non- or pseudoscience. Breitenbach argues against these received interpretations, contending instead that Kant countenanced a broad conception of science that encompassed physics along with the “improperly so-called natural sciences.” But within this broad conception of natural science,
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See, e.g., Plaass (), Watkins (a), Van den Berg (), McNulty () and Zammito ().
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the notion of proper natural science stands as a norm to which all sciences aspire insofar as we aim for the complete systematic unity of our knowledge. Breitenbach’s “Normative Reading” of Kant’s conception of natural science hence reorients our understanding of one of the most studied and notorious topics in the Metaphysical Foundations. Katherine Dunlop’s Chapter , “The Applicability of Mathematics as a Metaphysical Problem: Kant’s Principles for the Construction of Concepts,” is a welcome intervention into the debate surrounding Kant’s account of the application of mathematics to natural science. Dunlop maintains that metaphysics’ role with respect to the mathematization of nature is largely negative: it clears away obstacles to mathematization by ruling out deficient or incoherent metaphysical pictures that would make the application of mathematics impossible or unwarranted. Thus, Kant dismisses monadology, for making impossible the application of mathematics; Newtonian absolute space, for incoherence; physica generalis (a Wolffian approach to cosmology popular in eighteenth-century Germany), for simply assuming the application of mathematics to nature, instead of explaining it; and Lambert’s empiricist account of matter, for not grounding a priori knowledge of outer objects. Ultimately, according to Dunlop, it is only the analysis of matter as an object of possible experience that provides the desired a priori foundations making possible mathematical construction. However, the determination of the specific constructions utilized in natural science is a task for which metaphysics is not liable; metaphysics’ responsibilities end with its validation of the mere possibility of mathematical construction. Chapter , “Phoronomy: Space, Construction, and Mathematizing Motion,” by Marius Stan, innovates on the slim literature on the Phoronomy by concentrating on basic questions and Kant’s historical context. Stan first examines what phoronomy, as a doctrine, really is, for Kant: he concludes that it is a “kinematics for particle collision in a forcefree vacuum,” instead of a theory of dynamics or a general doctrine of bodies. Stan proceeds to explicate Kant’s concept of speed and situate it with respect to notions utilized in mechanics of Kant’s day, arguing that Kant espoused a “pre-classical concept” of motion, which allowed for the sort of geometric representation of motions that he aspired toward. Finally, Stan examines Kant’s proof of the parallelogram law, which codifies the method for combining motions as directed quantities. According to Stan,
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The parallelogram law of motions states, roughly, that the composition of two motions is represented by the diagonal of the parallelogram produced by the lines representing the two composed motions.
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there remains an open problem for Kant’s phoronomy: namely that, despite the claimed synthetic apriority of the parallelogram law, it involves empirical information. Specifically, the concepts of relative space and motion as well as the principle of Galilean relativity are all empirical. Thus, Stan concludes the chapter with a challenge to exegesis of the Phoronomy chapter: How can its proposition be synthetic a priori, when its sources are empirical? Chapters and , by James Messina and Lydia Patton, respectively, utilize the Metaphysical Foundations of Natural Science in order to push the recently burgeoning debates on Kant’s conception of natural laws in new and fruitful directions. In his “Space, Pure Intuition, and Laws in the Metaphysical Foundations,” Messina examines how the pure intuition of space relates to the laws of nature. This issue arises because of some conflicting and enigmatic comments that Kant makes about the dependence of physical laws on space throughout the critical decade. On the one hand, in the Dynamics of the Metaphysical Foundations, Kant suggests that the laws of diffusion of the fundamental attractive and repulsive forces – those essential to matter’s filling of space – admit of a purely geometrical derivation. On the other hand, these laws must involve some empirical content, and, in the Prolegomena, Kant appears to deny that the pure intuition of space can ground laws of outer nature. Messina contends that global features of space and time play a critical, ineliminable role in grounding the modal force of Kant’s laws of physics, and his chapter thus constitutes a careful, discerning corrective to accounts of laws that ground their necessity wholly on the categories. Furthermore, by elaborating this essential role for the pure form of intuition vis-à-vis Kant’s laws of nature, Messina’s chapter also contributes to our understanding of the relation between natural science and transcendental philosophy. In her “Finitism in the Metaphysical Foundations,” Lydia Patton aims at a novel articulation and extension of the recently popular necessitarian interpretation of Kant’s account of laws and, more particularly, of her own version of this approach (Patton ). Patton especially examines the relation between natures, which serve as the basis for the necessity of empirical laws of nature according to a necessitarian account, and the purported completeness of Kant’s system of nature. Examining this relation gives rise to her “Finitist Account of Laws,” according to which the system of nature outlined in the Metaphysical Foundations is finitist in the sense that Kant eschews appeal to actual infinities, both at the level of content (no actually infinite concepts) and at the level of demonstrations (proof in Kant’s system depends on concrete, intuitive mathematical
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Introduction
constructions). According to Patton, Kant’s finitism and necessitarianism interact reciprocally with one another: on the one hand, Kant’s finitist proofs need refer only to natures of material bodies, whereas, on the other hand, such natures must not involve any actual infinities. Patton’s chapter points in a new direction for literature on Kant’s conception of laws and brings this topic into an important dialogue with that of the completeness of his system of nature. Chapters and concern foundational questions regarding the second chapter of the Metaphysical Foundations – the Dynamics – and the associated conception of matter. Warren’s “The Construction of the Concept of Space-Filling: Kant’s Approach and Intentions in the Dynamics Chapter of the Metaphysical Foundations,” answers a devilishly straightforward question: What is the aim of the Dynamics chapter? Warren’s detailed and resourceful answer provides a new and fruitful framework for understanding the Dynamics chapter. Central to Warren’s concerns is the notion of mathematical construction in relation to Kant’s dynamical theory of matter. First, Warren provides a helpful account of the possibility of such mathematical construction, despite certain apparent conceptual obstacles. Subsequently, Warren particularly digs into the mathematical laws of the dynamical theory of matter, particularly, the force laws governing the diffusion of the fundamental attractive and repulsive forces. Warren traces these laws to the “universal law of dynamics,” according to which the intensity of a force stands in inverse ratio to the space upon which it acts, building on his prior work on the Dynamics (Warren ). Warren explains that a bulk of the Dynamics is oriented toward demonstrating that the fundamental forces of matter – those of attraction and repulsion – satisfy the conditions of the universal law of dynamics. By dint of the applicability of this law to the fundamental forces, they are thereby in-principle mathematically constructible. Warren, like Dunlop in her chapter, emphasizes that metaphysics plays a preparatory role vis-à-vis mathematization, accounting for its possibility, but that it is not responsible for the specifics of the mathematization. McNulty’s Chapter , “Beyond the Metaphysical Foundations of Natural Science: Empirical Physics and the General Remark to the Dynamics,” examines the enigmatic appendix to the Dynamics chapter, the General Remark to the Dynamics, in which Kant discusses both his preferred, force-based approach to natural explanation and a slate of empirically variable material phenomena – such as density, cohesion, and elasticity – that constitute the “specific variety of matter.” McNulty seeks to understand the precise relation between the phenomena canvassed in
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the Remark and the synthetic a priori theory of matter developed in the body of the Metaphysical Foundations. He argues that, for Kant, the specific variety of matter is to be explained by appeal to the fundamental forces of matter; density, cohesion, and elasticity are understood as emerging from the complex interplay of the fundamental attractive and repulsive forces. However, such explanations require the postulation of different sorts of matter expressing the fundamental forces to distinct degrees. In particular, Kant’s account of cohesion rests upon the assumption of an everywhere present ether, whose repulsive force totally outstrips its attraction. McNulty’s chapter thus clarifies both the relation between the Remark and the body of the Metaphysical Foundations and how Kant uses the fundamental forces of matter to explain empirical phenomena. Chapters and by Silvia De Bianchi and Michela Massimi, respectively, concern the final chapter of Kant’s Metaphysical Foundations of Natural Science, the Phenomenology, which concerns possible, actual, and necessary motions. Both authors find the faculty of reason and its characteristic ideas – postulated concepts corresponding to objects beyond the possibility of experience – lurking behind the scenes of the Phenomenology, and both chapters also situate the Phenomenology historically: De Bianchi’s, especially with respect to Euler, and Massimi’s, especially with respect to Kant’s pre-critical corpus. In her “How Do We Transform Appearance into Experience? Kant’s Metaphysical Foundations of Phenomenology,” Chapter , De Bianchi centers on the “reduction” of all motion to absolute space, an idea of reason. As she understands it, such reduction achieves the greatest end of reason in rational physics, namely, the systematic unification of all relative motions, which transforms them into genuine experiences. Although De Bianchi thus highlights the essential role of absolute space with respect to the Phenomenology, she notes a discrepancy between the functions of space and time. Absolute time, as an idea of reason, does not play a role in the Phenomenology; the process of unification, rather, makes possible the actual measurement of time via the relations among co-moving reference frames. De Bianchi finds support for this reading in the account of measurement from the Critique of the Power of Judgment, bringing together passages rarely read in conjunction. In Massimi’s Chapter , “Absolute Space as a Necessary Idea: Reading Kant’s Phenomenology through Perspectival Lenses,” examines Kant’s prima facie peculiar claim that the idea of absolute space is “necessary, not as a concept of an actual object, but rather as an idea, which is to serve as a rule for considering all motion therein merely as relative”
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Introduction
(MAN, :). Massimi understands the necessity of this idea as analogous to the necessity of reason’s ideas for the empirical use of the understanding, as famously mentioned in the appendix to the Transcendental Dialectic of the Critique of Pure Reason. By playing these two difficult claims off of each other, Massimi both illuminates and well motivates her “perspectival” account of the ideas of reason. Massimi understands ideas of reason in their regulative use not as stand-ins for noumenal objects, but rather as foci imaginarii, or imaginary standpoints, whose function is to make possible intersubjective agreement among epistemic agents. It is through this framework that it is possible for particular judgments of motion to gain the unanimity and universality typically associated with scientific knowledge claims, thus explaining the “necessity” of the idea of absolute space even in the absence of absolute space as a metaphysical entity (as Newton had it). Finally, Michael Friedman’s Chapter , “Proper Natural Science and Its Role in the Critical System,” effectively bookends the volume by returning to the topic of proper natural science and looking outward from the Metaphysical Foundations of Natural Science. In his chapter, Friedman situates Kant’s conception of proper natural science in the broader critical corpus and traces its development throughout the s and ’s. The chapter focuses on the relations among proper natural science, chemistry, the teleological explanation of organized bodies, and Kant’s moral teleology. From the highest vantage point, Friedman variously locates these projects and doctrines on the bridge between nature and freedom. Among the chief contentions of the chapter are that the explanations belonging to proper natural science are those “mechanical” explanations that fall short of explaining organized bodies in the Critique of the Power of Judgment; that Kant reevaluates the status of chemistry in the later s, becoming more optimistic about it admitting of explanation in proper natural science; that explanation of organized bodies delimits the boundaries of proper natural scientific explanation; and that the explanation of organized bodies via natural teleology is part of Kant’s bridge from nature to freedom, especially as described in the Methodology of Teleological Judgment. Friedman thus distinctly displays the exterior boundaries of proper natural science within the context of Kant’s uniting of nature and freedom: proper natural science ends precisely where the transition to freedom begins. This chapter is extremely rich, brings far-flung passages into dialogue, and rewards the reader with an understanding of the broad context for Kant’s claims about proper natural science.
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On the whole, the chapters of Kant’s Metaphysical Foundations of Natural Science: A Critical well acquaint the reader with the forefront of scholarship on Kant’s philosophy of nature, resolve difficult issues in the Metaphysical Foundations of Natural Science, and open up new horizons for research on Kant’s theory of natural science.
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Kant’s Conception of the Metaphysical Foundations of Natural Science Subject Matter, Method, and Aim Thomas Sturm . Introduction In the Architectonic of Pure Reason, near the end of the Critique of Pure Reason, Kant declares: Nobody attempts to establish a science without grounding it on an idea. But in its elaboration the schema, indeed even the definition of the science which is given right at the outset, seldom corresponds to the idea; for this lies in reason like a seed, all of whose parts still lie very involuted and are hardly recognizable even under microscopic observation. (KrV, A/B)
Here Kant suggests that any science (Wissenschaft) needs to be grounded on an “idea” – a concept of reason which helps to distinguish that science from others, and which makes possible the integration of a plurality of cognitions into a systematic whole. This means that natural science, too, requires such an idea. Since Kant takes the Metaphysical Foundations of Natural Science (Metaphysische Anfangsgru¨nde der Naturwissenschaft, MAN) to be a constitutive part of natural science, it is incumbent upon him to clarify the idea upon which it is grounded. This important task is carried out in the Preface (Vorrede) to MAN (:–). While stating the task is easy, it took Kant great intellectual effort to complete it in his own characteristic way. The Preface can be divided into five main parts. It begins with general considerations concerning the concepts of () nature (paragraph , :) and () science in order to explicate the specific concept of “natural science” (Naturwissenschaft). Using a series of successively more restrictive requirements, Kant arrives at the conclusion that in order to qualify as a
Many interpreters also take the Preface to contain Kant’s official and final statement of the general concept of science, Wissenschaft (e.g., Walsh ; Van den Berg , ch. ). Following Plaass (), I have expressed my doubts about such a reading: it does not pay attention to the fact that Kant distinguishes between Wissenschaft and Naturwissenschaft (cf. Sturm , ch. ; Sturm ; Sturm and De Bianchi ). The explication given in the Metaphysical Foundations was developed
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“proper natural science” (eigentliche Naturwissenschaft; :), a discipline requires metaphysical principles or presuppositions (paragraphs –, :–; see Breitenbach, this volume). He then () casts doubts on the pretensions of empirical psychology and chemistry to claim such a status (paragraphs –, :–). Only the doctrine of matter or body, that is, physics, fully qualifies as a natural science “properly so called.” However, the argument for this claim relies not on the necessity of metaphysical but of mathematical presuppositions of natural science. Only physics satisfies (at least to a sufficient degree) the condition that a proper natural science requires the application of mathematics. () At the same time, the application of mathematics to physics itself requires an explanation, which Kant promises to provide by way of metaphysical presuppositions. For this, he introduces the table of categories of the first Critique “as the schema of the system of the metaphysics of corporeal nature” – that is, as providing the basis for a complete system of such metaphysical principles, and thereby the structure of MAN (paragraphs –, :–). That he calls the table of categories the “schema” for the system provided in MAN refers back to his claim in the first Critique that each “idea” of a science requires a “schema” for its “execution” (KrV, A/B). Stated differently, in the Preface, Kant has first given us the elements by which we can distinguish the science contained in MAN from others – its idea – and is now telling us how MAN is systematically structured. () Kant concludes the Preface by comparing his project with Newton’s Philosophiæ naturalis principia mathematica, thereby elucidating the book’s title and emphasizing the considerable importance he ascribes to his own work (paragraphs –, :–). My aim in this chapter is not to provide a complete analysis of the Preface. Instead, I shall focus on a question underlying several of the text’s
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for this book’s project, and the concepts of both science and natural science kept on changing afterwards too. In September , Kant wrote to Christian Gottfried Schu¨tz, the early promotor of his critical philosophy. Kant promised that MAN would contain an appendix on metaphysical foundations of empirical psychology (Br, :). However, no such appendix exists. Instead, Kant presents criticisms concerning psychology and chemistry – arguments not found in his previous publications (nor in other source materials) and, in the case of chemistry, later retracted (Carrier ; McNulty ). We will later see an even more important example of Kant’s changing views (Section .). The most extensive interpretations of the Preface are Plaass (), a still highly valuable book devoted only to the Preface, and Pollok’s historical-critical commentary on MAN, which devotes more than pages to the Preface alone (Pollok , –). Friedman () contains extensive remarks on the Preface in the book’s introduction (focusing on the broader historical background of the place of MAN within Kant’s critical system; pp. –) and its conclusion (discussing the “complementary” perspectives of MAN and KrV; pp. –).
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considerations concerning the “idea” and “schema” of MAN: How to understand the conception of a metaphysical foundation of natural science and the systematic structure of the doctrines developed in MAN? And how are Kant’s conceptions of a metaphysical foundation of natural science and of transcendental philosophy (as presented in the first, constructive half of KrV) related to one another? Some scholars have argued that the relation must be a very close one: that in Kant’s view, the only real point of his transcendental philosophy is to provide a foundation for science, or – what is more – that transcendental philosophy requires to be completed by the metaphysical foundations of natural science (e.g., Friedman a, b, b, ; Westphal a; Lyre ). Others insist that Kant’s transcendental philosophy in no way depends on the metaphysical foundations of natural science: there is a “looseness of fit” between these two projects (Buchdahl b; Allison ). This dispute possesses a broader significance, since it relates to the important debate concerning how we should assess the positive contributions of Kant’s philosophy, especially in the light of post-Kantian developments in philosophy and science. I will proceed in three steps. In Section ., I will consider the entanglement between metaphysics and the sciences in Kant’s thought more generally. This gives preliminary support to the claim that MAN is important for understanding the Critique of Pure Reason. In Section ., I will turn to the debate concerning Kant’s views on the relation between “metaphysical” and “transcendental” principles of cognition. By analyzing, in Section ., the defining features of the aims, methods, and object given in the Preface to MAN, I defend a moderate interpretation: the MAN presents important concretizations of Kant’s doctrines of synthetic a priori conditions of empirical cognition but is not necessary to complete transcendental philosophy.
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The Entanglement of Science and Metaphysics
Since his earliest writings, Kant was deeply interested in empirical science. He worked on cosmology, geology, and physics, and increasingly on medicine, anthropology, geography, and human history as well. From what we know, he did not undertake wide-ranging and systematic field studies, experiments, or measurements. Nonetheless, he produced novel and even lasting results, such as the famous nebular hypothesis concerning the formation of the universe, a correct explanation of the Earth’s axial rotation, and a field theory of matter (Adickes –; Falkenburg ; Schönfeld ). As Ian Hacking (, ) has remarked, Kant often
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picked winners, and he would have made a formidable member of decision panels for research projects. In addition, Kant’s lifelong interest in metaphysics and its reform originated, in part, in questions concerning science. His first publication, On the True Estimation of Living Forces (), his Physical Monadology (), and the inaugural dissertation, De mundi sensibilis atque intelligibilis forma et principiis (), among others, addressed metaphysical issues related to the concepts of matter, force, and motion, and considered the difference between the methods of mathematics and metaphysics. These views were partly revised, sometimes radically, in his critical works. We should not be surprised to find these entanglements between philosophy and the sciences. In Kant’s day, the two areas were often conjoined; one needs only to think of names like Descartes, Leibniz, or Newton to see as much. To think that Kant pursued the critical project independently of his interests in the sciences is interpretively inadequate. Increasingly, scholars have emphasized and discussed these links, and rightly so (cf., e.g., Plaass ; Buchdahl b; Gloy ; Brittan ; Ph. Kitcher ; Butts a; Röd ; Friedman a, b, b, ; Watkins , a, b; Lefèvre ; Pollok ; Lyre ; Massimi ; Watkins and Stan ; Breitenbach and Massimi ). Upon closer inspection, however, the entanglement between science and metaphysics is a complex one, especially in Kant’s critical philosophy. One can distinguish at least three basic dimensions of this entanglement (see Sturm ): () some special sciences provide a model for his project of critically reforming metaphysics; () some special sciences are supported by metaphysical presuppositions; and () all special sciences should serve the goals of philosophy according to its “world concept” (Weltbegriff) (KrV, A–/B–). Regarding (), Kant emphasizes that logic, mathematics, and parts of natural science are all “sciences of reason” (Vernunftwissenschaften). Reason here not only means the faculty that provides nonempirical, a priori knowledge and methods of justification, but also concerns the very subject matter of these sciences. So in a sense, reason “knows itself” in these sciences by knowing its a priori representations (concepts and
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At KrV A/B, Kant adds pure moral philosophy to the group of Vernunftwissenschaften. Also note that, in MAN, Kant speaks of “rational science” (rationale Wissenschaft), a term that is not identical in meaning with that of Vernunftwissenschaften. Rational science only requires that its cognitions be unified inferentially, as “grounds” and “consequences” (:). There are no restrictions concerning the subject matter or the epistemic status of cognitions contained in such a science; only the form of their connection matters.
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principles) and their legitimate uses. It does so in different ways: (general formal) logic has no specific object whatsoever; it concerns “nothing but the pure form of thinking” (KrV, A/B) or, more specifically, the formal aspects and rules of concepts, judgments, and inferences (Log, :ff.). Mathematics, again, applies reason by constructing objects a priori in pure intuition; and the rational parts of natural science make possible and guide experimentation and observation by means of various principles of reason, which are partly domain-specific. But despite these different uses of reason, it is the apriority of these sciences that has allowed them to succeed (KrV, Bvii–xiv). Kant claims that Euclidean geometry and Newtonian physics provide undeniable examples of synthetic a priori cognition and that these sciences (or their rational parts) are here to stay: such cognition is necessary and therefore secure from empirical refutation (KrV, A/B; B–; B; Prol, :, ). In addition, only through a unified and complete system of a priori cognition can the search for new knowledge in each science be framed and guided and its results be structured systematically – systematicity being the most fundamental requirement for a body of cognition to count as a science (KrV, A/ B; MAN, :–). Given that metaphysics is likewise concerned with a priori concepts and principles of reason, it must develop its own systematic a priori framework of legitimate representations and principles if it is ever to become a successful science. The other sciences of reason can provide clues for how to get metaphysics on the “secure path” of a science. I will turn to claim () below, since MAN is Kant’s outstanding and, indeed, only developed example for an a priori framework for a special natural science. Concerning (), he argues that the sciences can pursue legitimate aims of their own – but, especially when we consider the fact that sciences can be used for practical aims, this ought to be done under the guidance of metaphysics: Mathematics, natural science, even our empirical knowledge of man, have a high value as means, for the most part, to contingent ends, but yet ultimately to necessary and essential ends of humanity, but only through the mediation of a rational cognition from mere concepts, which, call it what one will, is really nothing but metaphysics. (KrV, A/B)
As is well known, Kant uses the term “metaphysics” in several ways. He rejects a “dogmatic” form of metaphysics, and he defends a critically sanitized version which is supposed to contribute to theory and practice in general. The meaning of “metaphysics” in the above quotation is clearly not the dogmatic one; only a critically sanitized metaphysics can guide
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science’s contribution to the “necessary and essential ends of humanity,” and thus to the “world concept” of philosophy. Such general metaphysics, again, he distinguishes from the domain-specific “special” metaphysics, which he subdivides into “the metaphysics of nature” and “the metaphysics of freedom”; these are then further divided into more special domains (KrV, A–/B–). These complex entanglements already indicate that the MAN possesses a close relation to the first Critique. This makes it incumbent upon us to clarify the nature of that relation. As I will show next, a crucial worry concerning the relation between the metaphysical foundations of natural science and transcendental philosophy arises from claims () and (). Claim () can be set aside in what follows (for more on this claim, see Sturm , ).
. The Entanglement of Transcendental and Metaphysical Principles .. A Classical Objection A familiar objection concerning Kant’s transcendental philosophy starts from a reading of claim (), that is, his reliance upon Newtonian physics and Euclidean geometry. This claim becomes entangled with claim (): not only does Kant take these exact sciences to be models for the development of a critically cleansed, scientific metaphysics; according to the objection, his transcendental system of forms of intuition and of concepts and principles of the pure understanding is also meant to provide a kind of permanent foundation of these sciences. Since Kant viewed Newtonian physics and Euclidean geometry as “ultimate and permanent achievements of the human mind,” the argument goes, “he naturally regarded their presuppositions as absolute” (Körner , ). However, with the advent of non-Euclidean geometries during the nineteenth century and the
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Kant sometimes uses the term “philosophy” in a broader sense, such that metaphysics is only a branch of it. He also uses the term “general metaphysics” as inclusive of critical or transcendental philosophy. I will ignore these complications here; see Baum (a, b). Another question that I will leave aside here concerns whether MAN is, in Kant’s view, to be identified with the “metaphysics of nature” or to be seen as providing only a part of it. See Br (:), Pollok (, –), Baum (c). Debates about parts of Euclid’s geometry, such as the parallel line postulate, had already begun in the eighteenth century, with works by Johann Heinrich Lambert, Abraham Gotthelf Kästner, and Johann Schultz, among others. For how far Kant took notice of these developments, see Heis ().
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Einsteinian revolution of the early twentieth century, the assumption of the unrevisability of Newton’s physics and Euclid’s geometry no longer held water. Accordingly, the synthetic a priori principles of Kant’s first Critique – the transcendental principles of the understanding – came to be seen as incapable of supporting the new scientific theories: “one would hardly expect to find all of them necessary to all thinking about matters of fact” (Körner , ). Alternative presuppositions must be discovered to support the possibility of the new theories, and the status as unrevisable synthetic a priori principles, indeed even their truth, must be reconsidered (e.g., Reichenbach ; Cassirer ; Mittelstaedt ; Friedman a; , esp. –). A standard reply to this objection is that explicating and justifying the synthetic a priori presuppositions of science is not really the aim of Kant’s transcendental philosophy in the first place; therefore, a historical change in scientific theories cannot affect it. In Peter Strawson’s words, we must not give in to a merely “historical view” of Kant’s transcendental conditions of the possibility of any experience – that is, empirical cognition, whether scientific or not (Strawson , –). Especially, though not exclusively, this concerns the relation between the first Critique’s Analogies of Experience (KrV, A–/B–) – the synthetic a priori principles of the permanence of substance, causality, and interaction – and empirical laws of science. In line with this defense, the Analogies are not interpreted as grounding primarily or exclusively scientific laws. Their function is to explain the possibility of cognition in general, not specifically scientific cognition. They can perhaps perform the latter task as well, but only, as it were, as a bonus. In any case, the validity of the Analogies is not threatened by eventual changes in scientific theories. Such is the first defense against the classical objection. .. The Analogies of Experience and Natural Science: Four Questions However, the issue cannot be handled so easily. It is more complex. For reasons of simplicity, let us continue to restrict ourselves to the Analogies of Experience and their relation to science, leaving aside other principles of the pure understanding of KrV and their counterparts in MAN. The Analogies have been most often at the center of the debate just outlined, so it should help us here too. We need to distinguish at least four questions here. First, the interpretation of the Analogies is disputed. For instance, does Kant’s Second Analogy, the principle of causality, assert that each cause of
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a certain type brings about an effect of a certain type, or that it must be covered by a general law (in Lewis White Beck’s useful terminology, the “same-cause-same-effect” rule)? Or is it the claim that all events or, in Kant’s language, all “alterations” (Veränderungen) must have a cause (the “every-event-some-cause” rule [Beck ]) – so that Kant’s principle of causality “merely” secures that there is an irreversible time order of our experience of events? The former principle is an ambitious statement of strict causal determinism, depending on unchanging laws, and highly controversial, especially in the light of post-Newtonian developments in science (e.g., Körner , –; Mittelstaedt ). The latter is a much more moderate principle of a certain causal determination of events. Second, as already indicated, Kant asserts in the Preface (MAN, :–) that he develops the systematic structure of his metaphysical principles of natural science against the backdrop of the Critique’s system of categories. This forms the backbone of his division of MAN into four chapters, each allegedly corresponding to one set of transcendental principles. But how to understand the precise relation between the Critique’s Analogies and the relevant counterparts in MAN’s Mechanics chapter, especially the three laws of motion (MAN, :–)? Are the latter meant to be derived from the former? If so, how? If not, what support does Kant provide for them, and what does that say about their epistemic status? Third, what is the relation between the laws of motion as Kant formulates them in MAN and Newton’s own laws? Did Kant copy Newton’s versions, or was he also influenced by other authors, such as Leibniz? Fourth and finally, how does the revision or replacement of Newtonian mechanics in subsequent history of science affect the critical evaluation of Kant’s views? A full interpretation and critical assessment of his claims and arguments, even if one only considers the example of causality, is accordingly complex and far-reaching. It involves issues that cannot be discussed here (see Pollok , ch. .; Friedman b; , ch. ; Watkins , ). For instance, I will leave aside the third question completely, and only touch the fourth one. Even for the first two questions, it must suffice to make a few points in order to gain a better understanding of the relation between Kant’s transcendental and metaphysical doctrines.
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Besides these options, there are other interpretations (see Watkins ; Kannisto ). Watkins (a) and Lyre (, –) note mismatches between Newton’s and Kant’s laws of motions. Watkins (, b) and Stan () argue that Leibniz and Wolff played a role in Kant’s understanding of the laws of mechanics. Friedman, who originally emphasized Newton’s influence, has since accepted this point (Friedman , xiv; cf. n).
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To begin, there are reasonable interpretations of the Second Analogy according to which Kant did not aim to argue for the strong “same-causesame-effect” rule, but only the weaker “every-event-some-cause” rule (Buchdahl b, –; Beck ; Allison ; Thöle ). We will see below that, in MAN, Kant explicitly formulates and uses the Second Analogy as the “every-event-some-cause” rule. This provides extra evidence for this reading. Such an interpretation may not provide what one wishes for, but it is not nothing either. To explain the meaning of the Second Analogy, it helps to provide a brief characterization of Kant’s transcendental inquiry. This inquiry aims to state and justify what synthetic a priori conditions we must assume for our judgments to be determinately true or false of objects of experience. Kant does not mean that the concepts and principles of the pure understanding would determine which of our cognitive judgments are “objective” in the sense of being true but, instead, which ones are determinate enough to possess a truth-value at all. His main target is cognition (Erkenntnis) rather than knowledge (Wissen): Erkenntnis, while it aims at truth, can be false, namely, when it does not correspond to its object (KrV, A/B; an approximate German equivalent today might be Erkenntnisanspruch). Wissen, however, requires more, namely, actual truth and sufficient justification (KrV, A–/B–). In this vein, Kant characterizes the Transcendental Analytic as a “logic of truth” (Logik der Wahrheit; KrV, A/B, B). This is also what he means when he speaks of the (real, not merely logical) “possibility” of objective cognition, or when he states that his inquiry aims to identify necessary conditions for a certain determinacy of the empirical world – so that by using appropriate criteria, we can figure out whether our cognitions are true or false. Of course, for this inquiry to succeed, the synthetic a priori principles of the understanding must themselves be taken to be true. To show that they are, Kant provides proofs. In line with this, the Second Analogy states, as a minimal condition for causal judgments to possess a determinate truth-value, that an alteration is brought about by the cause that precedes it in time. This order cannot be reversed, or else we are no longer talking of a causal relation. Our knowledge of causal laws presupposes that there is a determinate temporal
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Kant has a moderate correspondence concept of truth, combined with a pluralism about criteria of truth. This account of truth shapes and guides his transcendental inquiry (Sturm ). I thus follow Guyer and Wood’s translation. For a fine textual analysis of the terminology – even though I do not follow it in all its details – see Willaschek and Watkins ().
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order of relevant causal sequences. Importantly, the reverse does not hold. Now, doubtlessly, in the formulation of the Second Analogy, Kant claims that every causal relation must be covered by some law (KrV, A/B). But this does not mean that he would prove, in the chapter on this Analogy, the “same-cause-same-effect” rule. If one looks at his proof, it is the claim of a determinate temporal order of causal relations that he focuses on. What, then, is the point of the thesis of lawlikeness? Clearly, it cannot be assumed to be a mere conceptual or analytic truth. Perhaps we might read it as an epistemological claim: to be able to know the relation between a change and its cause, we must assume that some law exists, even if we do not know it (yet), let alone know it to be necessarily true. Alternatively, we can view it as a methodological claim: the principle of causality invites us to search for laws in order for us to be able to defend the assumption that the relation between a change and its cause possesses necessity – that the time order of the relation cannot be reversed. I have sympathy for such interpretations, though they are not without their problems and certainly stand in need of a number of qualifications. However, I shall not discuss this topic here. Let us accept that the Second Analogy does not guarantee any knowledge of specific causal laws, let alone its necessity, but that it directs our search for causal relation by assuming that there are such laws, and that these, when discovered, help to explain why the causal relation has the determinate time order that Kant claims it has. Beyond that, discovering causal laws is a matter of empirical inquiry – albeit an inquiry that must, in Kant’s view, be guided by rational constraints of systematicity. He discusses the question of how we can discover causal laws and their necessity in other places, and his answer is based upon a “regulative” use of ideas and maxims of reason (KrV, A–/B–; KU, :–). Granted such a moderate reading of the Second Analogy, its validity is no longer necessarily threatened by fundamental changes of scientific causal laws. Insofar as there are causal relations, Kant claims, they must exhibit a determinate temporal order, and it is necessary that there is some law behind them (though, as current theoreticians of causality know, the
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For these and other options, see Allison (, –) and Watkins (, –). As Watkins makes clear in his reconstruction of Kant’s argument for the Second Analogy (, –), the epistemic nature of the Analogy does not reduce to a merely psychological or phenomenological claim. Since Kant ultimately aims to show how cognition (Erkenntnis) is possible, and since cognition implies a claim to truth, cognizing a causal relation implies that there is something objective that makes such a claim true if it is true.
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latter is a matter of dispute; see, e.g., Kru¨ger ; Hoefer ). The historical changes in science that are typically cited in the classical objection – for instance, the replacement of Newton’s physics by Einstein’s (e.g., Friedman ) – do not imply that we must generally abandon the theses of the temporal order and the lawlikeness of causal relations. There are indeed attempts to defend suitable forms of determinism for both special and general relativity theory and even for quantum theory. This, of course, does not constitute a positive argument for the general validity of the Kant’s Second Analogy. That depends on the quality of the proof Kant provides for it in KrV; the same holds for all other transcendental principles. What do these points mean for the four questions distinguished above? As noted, I will leave aside the third question here, and can only superficially touch the fourth question. By favoring the moderate version of the general principle of causality (answer to question ), we have made room for separating its validity from that of special scientific laws. Hence, we are also able to protect this principle from far-reaching revisions of our scientific knowledge of causal laws (answer to question ). But what about the second question, concerning the fact that Kant develops the systematic structure of MAN against the background of the system of categories and principles of KrV? If we want to maintain that transcendental principles such as the Analogies of Experience are not vulnerable to scientific change, we must clarify the relation between them and their counterparts in MAN’s chapter on mechanics – since here Kant formulates (his own version) of the three Newtonian laws of motion which might be vulnerable to such change.
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Einstein claimed that we must reject quantum mechanics because of its (alleged) indeterminism; therefore, relativity theory does not look like in violation of the moderate claims about causation that I ascribed to Kant in the first place. Quantum mechanics, in contrast, has indeed often been viewed as raising problems for the universal validity of Kant’s principle of causality: among other things, because it is believed that the theory does not allow for an arbitrarily precise description of initial states of affairs and that it does not predict what will happen but only offers (objective) probabilities. But these issues are highly disputed. For balanced discussions, see Brittan (, –), Brittan (), Mittelstaedt (), and Hoefer (). A complete discussion of the interpretations that defend a close link between the Analogies of Experience and the laws of motion as Kant presents them in MAN would have to distinguish between the following three claims (at least): () The Analogies of Experience require for their meaning instances of concrete application to spatial, i.e., material objects (what might be called the “semantic thesis”). () The sole function of the Analogies of Experience is to provide a foundation for Newton’s laws of motion (the “grounding thesis”). This thesis might be understood as either claiming (a) that Kant justifies the laws of motion on the basis of the Analogies alone, and that this justification is their sole function, or, more plausibly, (b) that the sole function of the Analogies is to provide necessary conditions, or presuppositions for the validity of these laws (Friedman b,
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I will return to this issue in a moment (and again in Section .). Before that, however, it will be useful to examine a textual development in Kant’s works. Consider the Prolegomena to Any Future Metaphysics, published between the first edition of KrV and MAN. Here, Kant speaks of the transcendental principles of the understanding as “laws of nature,” and even describes them as content of “pure natural science”: Now we are . . . in possession of a pure natural science, which . . . propounds laws to which nature is subject. Here I need call to witness only that propaedeutic to the theory of nature which, under the title of universal natural science, precedes all of physics (which is founded on empirical principles) . . . among the principles of this universal physics a few are found that actually have the universality we require, such as the proposition: that substance remains and persists, that everything that happens always previously is determined by a cause according to constant laws, and so on. These are truly universal laws of nature, that exist fully a priori. There is then in fact a pure natural science. (Prol, :–; cf. , –, , )
This passage refers to the first two of the three Analogies – the principles of permanence of substance and of causality – and, due to the openness of the formulation, potentially to all transcendental principles (KrV, Aff./ Bff.). Kant describes the examples as “really general laws of nature that exist completely a priori.” The problem arises due to his conclusion: “There is then in fact a pure natural science” (Prol, :). Here, the tasks of transcendental philosophy and pure natural science are not separated. In addition, in Prolegomena, Kant describes the table of principles as “Universal Principles of Natural Science” (allgemeine[r] Grundsätze der Naturwissenschaft; Prol, :; cf. –, ), whereas in the first Critique – in both editions! – the same principles are listed under explicit exclusion of “general natural science” (der allgemeinen Naturwissenschaft; KrV, A/B). One might interpret talk of “universal principles” of natural science or of “general natural science” charitably and say that there is no conflict here. However, talk of “pure natural science” cannot be so interpreted. While in Prolegomena it is identified with transcendental
; Lyre ). () The Analogies of Experience (and, especially, their proofs) are incomplete without metaphysical laws of motion (the Newtonian laws of motion, but, again, as Kant formulates them in MAN) (the “completion thesis”). On this reading, the principles of MAN are not merely required as presuppositions of empirical science; they also help to fill a gap in Kant’s own transcendental philosophy (Westphal a). It would go beyond the scope of this chapter to discuss these interpretations.
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philosophy, in MAN it is reserved for the metaphysical principles of physics. There is much to recommend the view that, in Prolegomena, Kant had not yet fixed his set of relevant concepts and claims. Only by the time of MAN, and the second edition of KrV that appeared one year after MAN, did things settle down. This can be corroborated by pointing to a related shift in Kant’s views concerning which concepts and principles can be considered “pure.” In Prolegomena, he claims that a “propaedeutics of natural science,” while containing the synthetic a priori principles mentioned above, is actually a mixed bag: there is also much in it that is not completely pure and independent of sources in experience, such as the concept of motion, of impenetrability (on which the empirical concept of matter is based), of inertia, among others, so that it cannot be called completely pure natural science. (Prol, :)
However, in MAN, and in the B-edition of the first Critique, Kant affirms that these concepts and principles do belong to pure natural science. Thus, mathematical physicists could in no way avoid metaphysical principles, and, among them, also not those that make the concept of their proper object, namely, matter, a priori suitable for application to outer experience, such as the concept of motion, the filling of space, inertia, and so on. (MAN, :; cf. KrV, B–)
Concepts such as those of motion or impenetrability, and principles such as that of “persistence of the same quantity of matter, inertia, and of equality of action and counter-effect,” thus become an essential part of pure natural science or “pura physica (or rationalis)” (KrV, Bn). In this way, Kant distinguishes the contents of transcendental philosophy from those of metaphysical foundations of natural science. One might think that this is just a terminological improvement; no less, but also no more. But it matters in concrete ways. As indicated at the outset, in the Preface to MAN Kant declares that the table of categories in the first Critique provides “the schema of the system of the metaphysics of corporeal nature” (MAN, :–). Only on that basis can he claim that
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When this problem is not noticed, scholars (e.g., Röd , ; Lyre , ) fail to distinguish between Kant’s considered concepts of transcendental philosophy and of pure natural science. For more details, see Sturm and De Bianchi (). There remain two passages in the second edition of KrV where Kant still seems confused about a related issue: at B, he calls the transcendental principle of causality a priori but “not entirely pure,” whereas just one page later, he cites it as an example of a “pure a priori” judgment (B–). On this, see Cramer ().
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he is using the categories to formulate metaphysical principles of natural science, and the transcendental principles as premises in the proofs of those principles. But for this to be possible, Kant must be conceptually clear about the difference between transcendental philosophy and metaphysical foundations of natural science. He must not conflate the two projects. Consider again the Second Analogy. In KrV, it is stated as that all changes must have a cause. In MAN, this principle becomes projected onto the concept of matter – the movable in space (:) – thus delivering the Second Law of Mechanics as follows: “Every change in matter has an external cause” (:). It is here that the Second Analogy is formulated in the every-event-some-cause version, leading to this specific formulation of the Second Law of Mechanics. Similarly, Kant projects the First Analogy of Experience, which asserts the permanence of substance throughout all changes (KrV, B), onto the concept of matter, resulting in the “First Law of Mechanics. In all changes of corporeal nature, the total quantity of matter remains the same, neither increased nor diminished” (MAN, :). The same holds true for the Third Analogy and the Third Law of Mechanics (MAN, :–). In each case, Kant explicitly starts the proof of the respective mechanical law by citing the relevant transcendental Analogy. The three laws of mechanics are precisely those which Kant, in the B-edition of the Critique, refers to as the “persistence of the same quantity of matter, inertia, and of equality of action and countereffect” and which form an essential part of pure natural science (KrV, Bn). This all speaks for both a clear difference and a connection between transcendental and metaphysical principles. That connection must not be overstated, however. There are several difficulties with the view that Kant simply projects each of the transcendental categories and principles of the first Critique upon the concept of matter. For instance, the Mechanics chapter does not merely consist of the three laws of mechanics, also called Propositions , , and . Before them,
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I use the version of the First Analogy in the B-edition of KrV; in the A-edition (KrV, A), the Analogy is formulated differently, but the point discussed here is not affected by this. In his own copy of the A-edition, Kant added comments on the proof of this Analogy, but did not change the proof as such. Watkins (a, –, , ) rightly notes problems here. For instance, a projection of the Second Analogy must involve more than a mere “substitution”: if Kant only wanted to replace the notion of an object of cognition in general, one would only move from “all changes must have a cause” to “all changes of matter must have a cause.” That the cause must be an external one has to do with a non-trivial assumption concerning matter, namely, its inertia (cf. the addendum to the Second Law of Mechanics; MAN, :). In the case of the Third Analogy, Kant weakens the connection between the transcendental and the metaphysical levels too.
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we find a Proposition , concerning the measurement of the quantity of matter by means of the quantity of motion at a given speed (MAN, :). There is no transcendental principle that Kant would invoke here to formulate or prove that metaphysical principle. Also, the first part of MAN, the Phoronomy – approximately, a kinematics – does not connect well to the first transcendental rule, the principle of the Axioms of Intuition (KrV, A/B). This principle appeals exclusively to the possibility of making judgments about “extensive” magnitudes, while in the Phoronomy chapter Kant writes of the constructability of velocities, which he calls “intensive” magnitudes (MAN, :). In the Critique, intensive magnitudes are the topic of the second principle, the Anticipations of Perception (KrV, A/B; cf. Watkins a; Pollok , –). Thus, there are sections of MAN that do not have clear counterparts in KrV, and vice versa. For the system of MAN, Kant lets himself be guided or inspired by the system of transcendental categories, but also breaks away from it. So much for question () introduced earlier on. For the record, let us note what relevance the difference and the (problematic) connection between transcendental and metaphysical inquiries has for question (): How does the replacement of Newtonian mechanics in subsequent history of science affect the critical evaluation of Kant’s transcendental philosophy? In his view, the explication and justification of the synthetic a priori presuppositions of Newtonian mechanics is the task of MAN rather than the first Critique. Accordingly, if in the course of history, Newton’s theory had been replaced by a different one, then one might need new metaphysical foundations for the new theory. For instance, post-Kantian science gave up the thesis of the permanence of material substance, replacing it with a principle of energy conservation. In this, interpreters such as Friedman (b, b) are certainly right. However, a refutation of transcendental principles of cognition u¨berhaupt can hardly be inferred from this. The transcendental principles are premises of some of MAN’s proofs for the metaphysical principles, for example, the laws of mechanics. However, if we find the conclusions of these proofs objectionable, mistakes might be found in our modes of reasoning or in the premises – of which transcendental principles form only a part. Also, the problematic details of the connection between transcendental and metaphysical principles should give us reason for doubt too. In sum: yes, there are important connections between Kant’s transcendental philosophy and his project of metaphysical foundations of natural science; but the differences matter just as much.
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. The ‘Idea’ of Metaphysical Foundations of Natural Science ..
Systems of Sciences in General
We have considered but a small part of MAN’s content and its relation to transcendental philosophy. Even if we went through all four chapters of the book in the same way, we would not gain a clear and completely general understanding of Kant’s conception of metaphysical foundations of natural science. We could not infer from all the differences and relations a complete understanding of the boundaries of that science. Remember that Kant claims that all sciences aim at systematicity. This means, minimally and ignoring for the moment other requirements, that each doctrine must consist of () a unified set of cognitions, which () is clearly distinguishable from other sets. Kant repeatedly stresses that we “do not enhance but distort sciences, if we allow them to trespass upon one another’s territory” (KrV, Bviii). Otherwise, “none of [the sciences] can be thoroughly dealt with in a manner that suits its nature” (Prol, :; cf. MAN, :–; SF, :). One might call () the “internal” and () the “external” systematicity of a science (Sturm , ch. ). With respect to MAN, so far we have discussed only aspects of (), but not of (). Kant indeed often speaks about (). Consider familiar examples from other areas of his oeuvre. Pure general logic ought not to be confused with psychology, since even though both deal with rules of thinking, they differ in their aims and in the epistemic status of their claims. Pure general logic details the necessary a priori laws of thought, whereas psychology provides the empirical, and hence continent and descriptive, laws of mental processes (KrV, A/B; Log, :–). Metaphysics and mathematics, though both sciences of reason, use different methods and thus ought to be kept apart (KrV, A/B; Log, :). Ethics and anthropology have human action as their subject matter, but only the former deals with categorical Oughts concerning action (GMS, :, –; MS, :–; for more, see Sturm , –, –). Kant thinks that sciences cannot make rational progress if we fail to separate them from one another. But while an “architectonic mind” brings such distinctions to light, such a mind also “methodically recognizes how all sciences are connected and how they mutually support one another” (Anth, :; cf. KrV, A/ B; Sturm , ch. ). Accordingly, while it could be profitable for both transcendental philosophy and the metaphysical foundations of natural science if we can distinguish between them in principled ways, we need not deny that Kant sees places or steps in his arguments where
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they connect with one another. But all this must be done in such a way that avoids confusion. Our question, therefore, must be: What justifies the distinction between transcendental principles of cognition u¨berhaupt and metaphysical foundations of natural science? For this, we need a general conceptual explanation. In Kant’s terminology, we must identify the elements or features of the “idea” of that doctrine or discipline. One achieves a clear (enough) distinction between different but neighboring disciplines by defining or explicating the idea of each; that allows one to avoid distortions and trespassing among different sciences. ..
Defining Sciences: Subject Matter, Method, and Aim
What elements make up such an idea? At the beginning of Prolegomena, Kant claims: If one wishes to present a body of cognition as science, then one must first be able to determine precisely the differentia it has in common with no other science, and which is therefore its distinguishing feature; otherwise the boundaries of all the sciences run together. (Prol, :)
The “distinguishing feature” of a science, he adds, may consist “in a difference of the object, or the source of cognition, or even of the type of cognition, or several if not all of these things together” (Prol, :). So, two distinguishing features of the idea of any given special science are its “object” or subject matter, or its proper ontological domain, and its “sources” or the “type” of cognition the science provides, that is, its epistemological characteristics. As Kant makes clear on other occasions, ends or aims – that is, axiological features – can also play a crucial role for defining sciences. In the Preface to MAN, he declares: For if it is permissible to draw the boundaries of a science, not simply according to the constitution of its object and its specific mode of cognition, but also according to the end that one has in mind for this science itself in uses elsewhere; and if one finds that metaphysics has busied so many heads until now, and will continue to do so, not in order thereby to extend natural cognitions (which takes place much more easily and surely through observation, experiment, and the application of mathematics to outer appearances), but rather so as to attain cognition of that which lies wholly beyond all boundaries of experience, of God, Freedom, and Immortality;
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This term is Laudan’s (, ): an axiology is a “set of cognitive goals or ideals.”
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then one gains in the advancement of this goal if one frees it from an offshoot that certainly springs from its root, but nonetheless only hinders its regular growth, and one plants this offshoot specially, yet without failing to appreciate the origin of [this offshoot] from it; and without omitting the mature plant from the system of general metaphysics. (MAN, :; emphasis in st edition, though not the nd and rd editions; cf. Refl , )
Thus, after repeating that the specification of the “object” as well as a suitable “source” and/or “type” of cognition are necessary parts of the definition of any science, Kant adds the role of its “end” (Zweck) too. Note how this passage also suggests that he has, up to then in the Preface, been occupied with determining the object and the type and sources of cognition of MAN and is only now turning to its end: natural science aims primarily at extending our “natural cognitions” (Naturerkenntnisse). Taken by itself, this claim appears uninformative or trivial; but we can see that Kant’s point is to make clear that the end of a special science need not be one concerning its “uses elsewhere,” or a practical one. Some philosophers, however, wish to put natural science to further use, namely – paradigmatically – to the ends of religion and morality. Such ends are not merely epistemic ones. Similarly, in the Architectonic of KrV, Kant declares that each doctrine that aims to be a science requires “a single supreme and inner end, which first makes possible the whole” (KrV, A/B). He distinguishes such an “inner end” – one that helps to define a given science – from so-called “arbitrary external ends” (KrV, A–/B–). The three – ontological, epistemological, and axiological – differentiating features of the idea of any special science are flexibly interrelated at various levels, can be rationally discussed and improved, and ideally make possible a well-ordered and principled system of all sciences (Sturm , ch. ). In this vein, Kant presents his own systematic classification of metaphysics, of its parts and subparts, and relates it to various special sciences within the Architectonic of KrV. Not all details of that system remain stable in his development, as the discussed shifts from the first edition of KrV to Prolegomena to MAN have made clear. That is as it should be: Kant believes that ideas of sciences, though rooted in reason, can be refined, reformed, or even replaced as research makes progress (KrV, A/B). ..
Defining the Metaphysical Foundations of Natural Science
Let us apply this account by returning to MAN’s Preface. Whatever else this text does, it must explain the aim, methods, and subject matter – using
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my terminology, the “external” systematicity – of MAN. Let us state each feature and compare it with parallel determinations of transcendental philosophy. Having explicated the notions of nature, science, and natural science in the early pages of the Preface (MAN, :–), Kant argues that neither empirical psychology nor chemistry satisfy (at least not sufficiently) the conditions for the status of natural science “properly so called” (:–). This leads him to the conclusion that only physics, or the “doctrine of body,” deserves this honorific title. This, then, provides the determination of the subject matter of MAN: it deals only with cognition of corporeal nature. In contrast, the KrV, in its constructive first half, is an inquiry that identifies and justifies the synthetic a priori principles of an “object of cognition in general” (Gegenstand der Erkenntnis u¨berhaupt). What is the point of this contrast? To answer this question, we must clarify what Kant thinks about the “type” or “sources” of cognition presented in MAN. In the first seven paragraphs (MAN, :–), Kant aims to define the concept of natural science in such a way that it becomes clear that physics is not a mere empirical science; it also needs a priori principles, and among these he is specifically concerned with metaphysical principles. The function of these principles of natural science can be stated by a comparison with the transcendental “logic of truth” of the first Critique (see Section .). The transcendental inquiry aims to state and to justify general synthetic a priori conditions for our judgments to be objective – to be determinately true or false of objects of experience, no matter what the specific ontological nature of these objects is (whether they be material or mental, what we mean by “material,” “mental,” and so on). The metaphysical principles of natural science, in turn, do not show that and why empirical judgments of physics are true, but show the conditions under which such judgments can be determinately true or false (Brittan , –; Friedman a, –). Put differently, the distinctive task of MAN is to explain the (real, not merely logical) possibility of empirical, and more specifically mathematical, natural science, such as measuring the distance between the Moon and the Earth (MAN, :), studying rectilinear and curved motions (:, ) as well as the attraction between planets and their satellites
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I add this qualification since () Kant revises his views about chemistry a few years later (Carrier ; McNulty ) and () his criticism of empirical psychology does not, as many interpreters claim, result in an unrestricted impossibility claim; instead, the claim is a restricted one (see Nayak and Sotnak ; Sturm ; Sturm , ch. ).
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(:), determining the axial rotation of the Earth relative to the stars (:), and perhaps even Kant’s own nebular hypothesis concerning the formation of the universe (albeit he never gives that hypothesis a precise mathematical form). The concept of matter is explicated using the categories such that relevant judgments obtain a determinate meaning, so that they can thereafter be confirmed or rejected empirically and/or mathematically. One consequence of this is that the term “Foundations” in the title of MAN has to be understood moderately, and perhaps it is not even correct to translate the German term Anfangsgru¨nde by “Foundations” in the first place: Kant presents principles from which research can begin, not a set of basic, intuitively known axioms from which one could already derive, by means of demonstration, all the knowledge contained in the relevant science. He does not follow an overly demanding form of rationalism, like that derived from the Aristotelian definition of episteme. Instead, he is trying to show the conditions under which quantitative empirical research of matter in motion is possible. However, a defense of the metaphysical principles as necessary conditions under which physical judgments can be determinately true or false may be too weak. After all, perhaps one can explain the possibility of empirical science, for instance, of Kepler’s laws or of other features of the solar system, by means of different principles? In this way, the classical objection against Kant’s transcendental principles (see Section .) resurfaces, once again, at the level of metaphysical foundations of natural science: they would lose their status as unchangeable truths. This objection leads us to more specific epistemological features of MAN. The principles developed in its four main chapters are, in Kant’s view, synthetic and a priori; and since they are a priori, they must be necessary in some sense. That is what needs to be justified. They are synthetic because they involve predicate concepts that are not contained in the subject concepts at issue in the respective principles. In other words, they are not analytic. The empirical concept of matter as the movable in space is determined completely through the four types of categories, thus creating synthetic judgments about physical objects (cf. also MAN, :, , , ). More importantly for countering the objection, these judgments are a priori, not only because they are presuppositions of existing cognition in
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I here broadly agree with Plaass () and disagree with Van den Berg (). In line with this, Kant explicitly rejects the view that the law of equality of action and reaction is empirical. He thereby distances himself from Newton, who “by no means dared to prove this law a priori, and therefore appealed rather to experience” (MAN, :).
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natural science (such as the empirical laws of planetary motion or the law of gravity), but because their apriority is justified by nonempirical arguments. The transcendental principle that “every alteration has a cause” becomes applied to material objects such that its metaphysical counterpart states that “every alteration in matter has an external cause” (MAN, :). But this principle is not simply derived from the transcendental one by means of a substitution of the subject term. Instead, it requires additional premises pertaining directly to the metaphysical nature of matter. Accordingly, in MAN Kant supports each metaphysical principle of physics by a non-empirical proof of its own. These proofs are what one needs to look at in order to assess the claims of apriority and necessity of the principles (Watkins a; cf. the contributions by Stan, Messina, and De Bianchi, this volume). Insofar as the proofs do not deliver, we must be prepared to revise the epistemic or modal status of these principles, if not the principles themselves. Conversely, the weight of the objection of revisability can be judged fairly only if one takes a closer look at Kant’s proofs. A mere pointing to historical changes in scientific theories will not do. The analysis I have given of the epistemological or methodological specificities of Kant’s idea of MAN is not complete. Among other things, a complete analysis would have to include his considerations concerning the distinction between two types of “rational” cognition, namely, metaphysical and mathematical cognition. It would have to explain Kant’s claims () that these are distinct types of cognition (MAN, :), () that any mathematical cognition of physical objects presupposes a metaphysical construction of the concept of these objects, and () that the widespread rejection of metaphysics by natural scientists should not be accepted (MAN, :–; cf., e.g., Plaass , ch. ; Pollok , –, –; Friedman , –). Simply noting these additional features of Kant’s conception of MAN shows, once again, how this work does not aim to provide mere applications of the transcendental principles, but also pursues an agenda of its own. Finally, we come to the end or ends of MAN. As shown above (Section .), Kant distinguishes between two types of ends: first, an end that is purely epistemic or internal to a given science, and second, an external end that “one has in mind for this science itself in uses elsewhere” (MAN, :). With respect to MAN, this difference must be understood as follows. On the one hand, the integration of cognitions in physics is not possible without an “inner end” of MAN. That end is to state and justify, in a systematic way, the a priori concepts and principles essential for the
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metaphysical determination of matter. This “inner end” is intelligible, at least given certain Kantian assumptions, and supported by intelligible notions of the method by which it can be achieved. Aim and method fit together. On the other hand, Kant delivers a clear plea to keep traditional preoccupations of metaphysics – with which he still deals, if critically, in the Dialectic of KrV – out of the inquiries into the metaphysical foundations of natural science. The “inner end” stands for itself; it need not and should not be burdened with the “external end” to discover “that which lies wholly beyond all boundaries of experience, of God, Freedom, and Immortality.” Thus, for instance, natural science should no longer be burdened with purposes of physicotheology. Kant also illustrates the relation between traditional, pre-critical metaphysics and the critically informed metaphysical foundation of natural science by an organismic metaphor: “one gains in the advancement of this goal” – namely, the traditional metaphysical interest in God, Freedom, and Immortality – “if one frees it from an offshoot that certainly springs from its root, but nonetheless only hinders its regular growth” (MAN, :). He is fully aware of historical dependencies and contingencies in the development of disciplines, but he encourages the reader to promote a division of cognitive labor, whenever possible for the rational progress of our scientific understanding of reality and our place in it. In sum, a close analysis of the conception of MAN shows that Kant explicates it in three basic dimensions – determination of subject matter (ontology), of type and source of knowledge (epistemology/methodology), and of ends (axiology). These form the coherent set of elements in his definition of pure natural science. Thereby, he conceptually distinguishes the metaphysical foundations of natural science from his transcendental philosophy. This is also one way of making sense of two somewhat obscure claims of Kant’s concerning his division of metaphysical disciplines: () that the KrV is engaged in “criticizing” the very faculty of reason or in a “self-cognition” of reason, including a critical discussion of rationalistic metaphysics, which implies that transcendental philosophy is not itself part of the “system of pure reason” (KrV, A/B); and () that MAN, in contrast, is a specific part of the system of pure reason: a metaphysics of the concept of matter. Thus, MAN depends on KrV, but not the other way around. We understand that now, in a more principled manner than through the
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The attempt to deal with physicotheology had occupied the young Kant in many sections of his early masterpiece, the Universal Natural History and Theory of the Heavens ().
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discussion of the relation between the Analogies of Experience and their counterparts in MAN, the Laws of Mechanics. While we should never overlook the connections between Kant’s two projects, we should also not ignore that he understood each resulting doctrine as a unified whole of cognitions. If we ignore disciplinary differences, it may be to the disadvantage of each system – potentially a step back on its way to becoming a successful science.
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For comments and criticisms, I am highly grateful to Angela Breitenbach, Gabriele Gava, Rudolf Meer, Michael Bennett McNulty, Bernhard Thöle, Eric Watkins, an anonymous referee, and Max Dresow, who also provided linguistic assistance. Research for this chapter was supported by the Ministry of Science and Higher Education of the Russian Federation, project Kantian Rationality and Its Impact in Contemporary Science, Technology, and Social Institutions (–, grant no. ---), Immanuel Kant Baltic Federal University (IKBFU), Kaliningrad.
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Kant’s Normative Conception of Natural Science Angela Breitenbach
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Introduction
Kant is well known for spelling out a remarkably strict conception of natural science, or “proper natural science,” as he calls it in the Preface of the Metaphysical Foundations of Natural Science (MAN, :). A proper natural science, he argues, is any body of cognition that is systematically unified, ordered by rational principles, and known with apodictic certainty. Kant’s interest in disciplines that have reached the rank of science is well motivated by his desire to circumscribe the most venerable form of knowledge. It offers the standard against which Kant measures the progress of philosophy when, in the Critique of Pure Reason, he famously asks whether metaphysics can find “the secure course of a science” (KrV, Bxix). But Kant’s account of proper natural science raises a problem. On Kant’s strict conception, there turn out to be very few such sciences: one, to be precise, namely, physics. At the same time, it is clear that Kant is deeply interested in a large variety of disciplines including chemistry, biology, and psychology that, today, would be classified as genuine sciences and that Kant himself regards as genuine forms of the study of nature. For instance, in the Critique of Pure Reason, Kant uses distinctively chemical examples to illustrate the way necessary ideas guide the classificatory efforts of “investigators of nature” (Naturforscher) (KrV, A/ B). Similarly, in the Critique of Judgment, Kant argues that, although biology does not belong to “(properly so-called) natural science” (KU, :–), its teleological heuristic contributes to the genuine study of nature. Moreover, even as he spells out his strict conception of natural science in the Preface to the Metaphysical Foundations, he seems to leave room for a more inclusive notion of science. In some passages he suggests
For more on the status of biology in Kant, see my essays (b, ).
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that chemistry and empirical psychology are doctrines of nature that, while apparently missing the mark of science proper, nevertheless qualify as a kind of science (see MAN, :–). How should we make sense of these assertions against the background of Kant’s strict conception of natural science? Can such disciplines as chemistry, biology, and empirical psychology be awarded scientific status if they fail to qualify as proper natural sciences? In the literature, Kant’s strict notion of proper natural science has been the focus of significant scrutiny. But commentators have paid much less attention to whether Kant may be assuming a more inclusive notion of science, and what that notion might consist in. I believe that this narrow focus makes it difficult to properly understand and situate Kant’s views on science. I argue that, instead, we must understand Kant’s strict conception of proper natural science in context. More specifically, Kant’s account in the Metaphysical Foundations must be construed against the background of a broad notion of science Kant introduces in the Critique of Pure Reason. According to this broad notion, both ‘proper’ and ‘improper’ natural science are forms of genuine science. But my proposal is not simply that Kant develops a broad notion that subsumes his conception of proper natural science. I argue, more specifically, that according to Kant’s broad notion all improper natural science is a form of genuine natural science because it has an important normative relation to natural science proper: not all natural science is proper natural science, but all natural science ought to be proper natural science. On my reading, to qualify as a science in general, a discipline must seek systematicity, order under rational principles, and apodictic certainty. My reading has an important advantage. It resolves the apparent tension between the strict conception Kant develops in the Metaphysical Foundations and his wider interests in those disciplines that do not meet this strict conception. It explains how Kant can hold that there is a broad domain of genuine science, while reserving a central place for proper natural science. My interpretation thereby shows that Kant’s conception of science is integral, and indeed central, to his investigation into the nature of human knowledge and cognition. Kant’s discussion in the Preface to the Metaphysical Foundations presents science not simply as a
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E.g., Plaass (), Parsons (), Friedman (b, ), Watkins (a), Pollok (), Van den Berg (). There are exceptions, including Sturm () and the contributors to Watkins (b), who signal the existence of a plurality of sciences, not all of which are proper natural sciences. They do not however address the question of what these sciences have in common. Gava () discusses Kant’s conception of science in the Architectonic of Pure Reason without, however, explaining its relation to the account of the Metaphysical Foundations.
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set of systematically ordered knowledge claims but as a human activity that is central to our cognitive endeavors. The plan for this chapter is as follows. In the next section, I begin with a sketch of Kant’s contrast between proper and improper natural science in the Preface to the Metaphysical Foundations. In Section ., I carve out and discuss two interpretations of this contrast, which I call the Demarcation Reading and the Classification Reading. The first construes Kant’s contrast as demarcating science from non-science. The second interprets the contrast as a distinction between two equally genuine kinds of science. I argue that neither is satisfactory. In Section ., I present my proposed interpretation, the Normative Reading, as an alternative. I argue that if we interpret the Preface in light of the conception of science Kant presents in the Critique of Pure Reason, we can overcome the difficulties that threaten the Demarcation and Classification readings. I conclude in Section . by ascribing to Kant a broad notion of science as a central human cognitive activity.
. Proper and Improper Natural Science Kant presents his distinction between proper and improper natural science in the Preface to the Metaphysical Foundations. There, the conception of proper natural science, and its contrast with improper natural science, emerges as the third of three increasingly demanding characterizations of science. In this section, I offer a sketch of this progressive characterization before highlighting tensions in the text, which any satisfactory interpretation needs to address. First, Kant begins by characterizing systematicity as the fundamental mark of science: “every doctrine, if it is supposed to be a system, that is, a whole of cognition ordered according to principles, is called a science” (MAN, :). It is striking that, in this passage, Kant does not characterize as science those doctrines, or bodies of cognition, that actually are systematic. He argues, instead, that the title ‘science’ applies to any doctrine that is supposed to be a system (“eine jede Lehre, wenn sie ein System . . . sein soll”; my emphasis). Why Kant uses this apparently normative formulation is left unexplained at this point. It is a thread I pick up in Section .. What is immediately clear, by contrast, is that Kant regards the tight link with systematicity as central to his conception of science. It is this link that distinguishes science from non-science or, as Kant had put it earlier, “scientific” from “common cognition” (KrV, A/B). The connection of science with systematicity is familiar from the Critique of Pure Reason. In the Doctrine of Method, Kant puts it
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succinctly: “systematic unity is that which first makes common cognition into science” (ibid.). And, as he explains in the Appendix to the Transcendental Dialectic, seeking the systematicity of cognition necessary for science is the task of reason: If we survey the cognitions of our understanding in their entire range, then we find that what reason quite uniquely prescribes and seeks to bring about concerning it is the systematic in cognition. (KrV, A/B)
By ‘reason’ Kant means the “faculty of principles” (KrV, A/B). Just like the understanding, it contributes to cognition. But while the understanding confers unity on the manifold of sensory intuitions by subsuming them under concepts, reason seeks unity, in turn, among the cognitions of the understanding. The unity reason seeks is a unity under principles. As the faculty of principles, reason works to bring systematic unity into the cognitions provided by the understanding by ordering them under higherlevel principles (see A/B). For instance, reason aims to systematize cognitions of the various different kinds of matter by postulating pure kinds, “pure earth, pure water, pure air, etc.” (KrV, A/B). According to the Preface to the Metaphysical Foundations, it is this systematic unity, or “interconnection based on one principle” (A/B), that constitutes the first distinguishing mark of science. Second, in the Preface, Kant goes on to suggest that we can distinguish different kinds of science, depending on the kinds of ordering principle the science employs in its systematizing endeavors. In the case of the cognitions of nature, Kant argues, we can distinguish between “historical” principles, which classify natural phenomena according to observed similarities, and “rational” principles, which explain these phenomena according to “an interconnection of grounds and consequences” (MAN, :). Correspondingly, it would seem that we could distinguish between two kinds of natural science, “historical or rational natural sciences” (ibid.), depending on which principles they employ. In one, the historical natural sciences, we would be concerned with systems ordered according to purely classificatory principles. For example, on Kant’s account in the Metaphysical Foundations, both natural description and natural history employ principles of the first kind. The principles of natural description classify “natural things in accordance with their similarities”; the
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In a smaller essay, Kant mentions Linnaean taxonomy as an example of this type of classification (see ÜGTP, :). Linnaeus’ classifications are based on structural similarities, for instance, the number of stamens and pistils of the flowers of plants.
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principles of natural history order “natural things at various times and places” (ibid.). Both generate an order in accordance with principles that emerge from contingent events, which occurred at particular moments in time (“Facta,” ibid.). By contrast, in the other, the rational natural sciences, we would be concerned with principles whose function is not simply to classify but to explain natural phenomena. For example, Newtonian physics explains the movement of matter by means of universal laws of motion. These laws are necessary principles, governing timeless characteristics that are not reducible to particular contingent occurrences. Cognitions ordered by the laws of Newtonian physics thus constitute “an interconnection of grounds and consequences” and, hence, a rational natural science (ibid.). Third, having introduced the distinction between historical and rational natural science, Kant comes to his final distinction, that between proper and improper natural science. As he points out, only sciences that are ordered exclusively according to a priori principles can be known with apodictic certainty, that is, with the “consciousness of their necessity” (ibid.). Only these natural sciences are natural sciences in the proper sense. By contrast, natural sciences that also contain empirical principles can only ever be known with empirical certainty. They are only natural sciences improperly so-called: Natural science would . . . be either properly or improperly so-called natural science, where the first treats its object wholly according to a priori principles, the second according to laws of experience. What can be called proper science is only that whose certainty is apodictic; cognition that can contain mere empirical certainty is only knowledge improperly so-called. (ibid.)
The a priori laws of mechanics furnish the theoretical basis for explaining the movements of the planets. And they do so in keeping with the requirements of apodictic certainty. This is why Newtonian physics is a proper natural science according to Kant. But the empirical laws of chemistry deliver explanations that may always be revised upon discovery
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It is not obvious that all natural history must be purely classificatory. Thus, in other places, Kant seems to think that Buffon’s natural history is more than classificatory. On Buffon’s theory, different races are ordered according to the principle of whether they developed from one another. And this principle of historical descent may well constitute a principle of ‘grounds and consequences’ (see VvRM, :). As Plaass points out (, ), Kant here follows Baumgarten and Meier in understanding Facta as past occurrences of which we can have empirical knowledge.
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of further evidence. Chemistry contains empirical principles and, for this reason, counts as an improper natural science (see MAN, :–). Kant’s conception of proper natural science further delimits his account of rational natural science. While the historical natural sciences must rely on empirical, and hence contingent, principles, the rational natural sciences may be grounded in either empirical or a priori principles of grounds and consequences. And only those rational principles that are known a priori can be the basis for proper science. All other principles – empirical, rational, or historical – result in ‘knowledge improperly so-called.’ In the Preface, Kant has thus laid out his characterization of proper natural science by gradually delineating a more and more restrictive notion of science and the criteria necessary for realizing it: systematicity, ordering according to rational principles, and apodictic certainty. Having presented his third and most restrictive conception of science as natural science ‘properly so-called,’ in the rest of the Preface Kant is then concerned to outline how, and to what extent, such proper natural science can be achieved. He argues that we can formulate a proper natural science, and know natural phenomena with apodictic certainty, only by investigating what we can know from reason alone. Proper natural science must contain a pure part. And since such a pure part is what Kant calls metaphysics, proper natural science requires metaphysics. But it requires a metaphysics of a particular sort. For natural science is concerned not with nature in general but with the special characteristics of specific kinds of natural objects. The metaphysics required for proper natural science must therefore be a “special metaphysics” (MAN, :), an a priori investigation into the features of a particular kind of natural phenomenon. In physics, for instance, it consists in an a priori investigation into the nature of matter. This requirement is a complex one. How can we have a priori knowledge of the particular nature of empirical phenomena? Kant’s answer is that we cannot achieve such a priori knowledge analytically, from mere concepts, but must proceed synthetically, by constructing the object of inquiry in a priori intuition. This answer is deeply rooted in his account of human cognition. Only by examining objects given in intuition can we learn anything beyond the concepts we already possess; but only by examining objects given in a priori intuition can what we learn qualify as a priori knowledge. Kant argues that this process consists in mathematical construction. In mathematical construction the imagination produces a representation of a concept in a priori intuition. While the result is an “individual object,” the construction nevertheless provides a “universal”
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representation of all those objects that fall under the concept (KrV, A/ B). It does so by representing the rule-governed “act of construction” (KrV, A/B), or “universal procedure,” of producing the object (KrV, A/B). In this way, the construction grounds the apodictic certainty, or “consciousness of [the] necessity” of the laws that govern material phenomena. On Kant’s account, special metaphysics, and with it proper science, thus requires mathematics. It is this insight that leads Kant to his famous claim that “in any special doctrine of nature there can be only as much proper science as there is mathematics therein” (MAN, :). These details about the requirement of mathematics for science become relevant in Section . below. In the Preface, Kant argues, moreover, that in order to apply mathematics to material objects we need to introduce the “principles for the construction of the concepts that belong to the possibility of matter in general” (MAN, :). The special metaphysical investigation into these principles that are required for proper natural science is the aim of the body of the Metaphysical Foundations of Natural Science. Whether the book does indeed succeed in laying the foundations for what Kant regards as a proper natural science, namely, physics, is a question I shall not pursue here. Instead, I highlight two important complications for Kant’s progressively more demanding characterizations of science outlined thus far. As I have shown, Kant develops three increasingly restrictive characterizations of science: proper natural science emerges as a specification of rational natural science, which in turn qualifies science in general. But the Preface contains problematic texts that shed doubt on the stability of Kant’s distinctions. Two are of specific relevance for my purposes. First, the taxonomy set out so far allows both historical, that is, classificatory, and rational, that is, explanatory, disciplines to count as genuine sciences. But Kant also claims that only rational disciplines count as genuine sciences of nature while historical ones constitute mere doctrines of nature. His proclaimed reason for not calling historical doctrines of nature ‘natural sciences’ is that a science of nature ought to be concerned with the cognition of natural phenomena “from their inner principle” (MAN, :), that is, from principles of the ‘interconnection of grounds and consequences,’ or necessary laws. Kant demands such a conception of natural science because he does not allow for a conception of nature that is independent of determination by inner principles. As he argues
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Why, exactly, Kant regards mathematics as necessary for special metaphysics is a question that is discussed controversially in the literature. I cannot go into the details here. See, e.g., Plaass (, ff.), McNulty (), Van den Berg ().
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throughout his work, nature is the sum total of the possible objects of experience under laws. And if nature is necessarily determined by laws, then natural science must consist in the study of such laws. It would follow that such historical doctrines as Linnean taxonomy do not constitute genuine sciences of nature because they are not grounded in laws, or inner principles, that govern their objects of study. The science of nature, it would therefore follow, reduces to rational natural science. Second, in some passages, Kant even refuses to give the title ‘natural science’ to any doctrine of nature, even a rational one, if it is not grounded in a priori principles, and thus is not a natural science proper. For instance, Kant claims that chemistry “should . . . be called a systematic art rather than a science,” since it is grounded in empirical laws and therefore fails to meet the definition of science proper (ibid.). Furthermore, he maintains that empirical psychology “must remain even further from the rank of a properly so-called natural science than chemistry,” since we cannot provide a priori constructions of the phenomena of inner sense (MAN, :). More generally, Kant states that “natural science must derive the legitimacy of this title only from its pure part – namely, that which contains the a priori principles of all other natural explanations” (MAN, :–). Kant’s reason appears to be that for something to be a natural science at all it must be grounded in principles that we can know a priori and with apodictic certainty. The science of nature, we must conclude, is coextensive with proper natural science. These complications introduce a serious tension into the Preface that any reading of Kant’s conception of science should address. What, exactly, does Kant intend to achieve with his distinction between proper and improper science? Does he seek to set up a contrast between real science and a form of cognition that is not really science? Or is he aiming at a contrast between distinct types of science, ‘proper’ and ‘improper,’ where the first qualifies as science in a stricter sense, the second only in a looser sense? In the next section, I argue that two existing interpretations along these lines fail to give satisfactory answers.
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A science of nature in the material sense, concerned with “the sum total of all things, insofar as they can be objects of our senses,” must therefore also be a science of nature in the formal sense, concerned with the inner laws of those objects, or “the first inner principle of all that belongs to [their] existence” (MAN, :). On Kant’s complicated and fraught distinction between nature (and natural science) in the material and formal sense, see Plaass (, –), Pollok (, –), McNulty and Stan (, –). Kant does not explain the terminology of his ‘proper’/’improper’ distinction, and eighteenthcentury German has both meanings: ‘proper’ as ‘true’ (“genau, der Sache völlig gemäß”; “der
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Two Problematic Interpretations
I identify two broad interpretive trends among philosophers who have commented on Kant’s distinction between proper and improper natural science. The first trend emerges from those readings that construe Kant’s definition of proper natural science as his conception of the science of nature in general. For example, in his discussion of Kant’s complex and changing views about chemistry, Michael Friedman reads Kant as identifying “proper” with “true” science (b, ). Similarly, John Zammito spells out what he regards as the immediate implications of Kant’s “highly restrictive characterization of Naturwissenschaft” in the Metaphysical Foundations (, ). According to Zammito, disciplines that fail to meet the three-fold definition of proper science such as biology cannot be reconciled “at all with Kant’s prescriptions for science” (p. ). Kant is engaged in “boundary maintenance,” as Jennifer Mensch puts it (, –n). He – strictly speaking – misspeaks when he attributes scientific status to disciplines that are based on empirical principles and have merely empirical certainty. On this reading, which I call the ‘Demarcation Reading’ because it demarcates scientific from common cognition, proper natural science is the only possible natural science. The Demarcation Reading has the great advantage that it ascribes to Kant a single conception of natural science, coherently and definitively characterized by the three criteria I outlined in the previous section. But the Demarcation Reading does not explain how to resolve the obvious tension between this single notion of natural science and Kant’s more inclusive conception that emerged from my discussion in Section .. For example, Kant claims that “any whole of cognition that is systematic can, for this reason, already be called science” (MAN, :). And he goes on to maintain that in some of these wholes of cognition “as in chemistry, for example . . . the laws from which the given facts are explained through reason are mere laws of experience” (ibid.). In these passages, Kant is clear that there are genuine sciences, and even genuine natural sciences, that do not fit the strict definition of proper natural science.
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Wahrheit nach”) and as ‘primarily’ or ‘strictly’ belonging to the meaning of a word (“einer Sache allein eigen, oder derselben doch vor vielen andern zukommend”). See Adelung (, –). See also Friedman (a, –; b, –). For a similar conclusion concerning the relationship between biology and science, see Richards () and Beiser ().
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A natural way to resolve the tension might consist in distinguishing between two senses of the term ‘natural science’: one broad, ranging over both proper and improper natural science (MAN, :), the other strict, limiting science to natural science proper. On this second reading, natural science in the broad sense includes, but is more than, proper natural science. The contrast between proper and improper science turns out not to demarcate the science of nature from non-science, but to classify natural science in the broad sense into distinct but equally genuine kinds, proper and improper. I refer to this second proposal as the ‘Classification Reading.’ It is often assumed, even if not always explicitly, by commentators who find in Kant a philosophy of the special sciences. For example, McNulty maintains that “in the preface to MAN, Kant claims that chemistry is a science but not a proper science, like physics” (, ). Similarly, Thomas Sturm contends that Kant’s criterion of aprioricity is not a “universal criterion of scientificity” but rather a criterion of natural science in a very specific sense – one which, presumably, can stand alongside sciences in other senses (, ). More broadly, Eric Watkins argues that Kant “does not let the special status of physics blind him to the fact that other sciences, such as chemistry, anthropology, and biology, can be scientific in different senses” (a, ). The Classification Reading has the advantage that it avoids charging Kant with a contradiction. Kant can, for example, exclude chemistry from the domain of natural science in the strict sense while also ascribing to chemistry scientific status in the broad sense. Indeed, Kant implies this dual conception when he claims that “the whole of [chemical] cognition does not deserve the name of science in the strict sense” (MAN, :; my emphasis), thereby suggesting that it may deserve the name of science in a broader and more inclusive sense. The Classification Reading thus helpfully accounts for a number of different ways Kant uses the term ‘science.’ Despite these advantages, I argue that the Classification Reading faces a considerable difficulty of its own. For, without further amendments, it is unclear how the Classification Reading can explain what makes improper natural science genuine natural science. The difficulty arises because, on the Classification Reading, improper natural science differs from proper natural science in not fulfilling the demands of natural science proper. A science is improper either because it is not grounded in any laws at all or, more specifically, because it is not grounded in a priori laws. But, according to the first complication spelled out in the previous section, Kant does not allow for a conception of genuine natural science that is independent of the study of laws. Kant does not allow for such a
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conception of natural science because, on his conception, nature is necessarily determined by inner principles, or principles of ground and consequence. And if nature is necessarily lawful, then the science of nature must consist in the study of those laws. Moreover, according to the second complication I mentioned in the previous section, Kant does not even allow for a conception of science that is independent of the study of laws that are known a priori and with apodictic certainty. There thus seems to be no room, in Kant, for a conception of natural science that is satisfied with purely empirical classifications, or even with empirical regularities, and the merely empirical certainty they provide. The science of nature, therefore, cannot be independent from the demand for a priori knowledge of the necessary laws of nature. But, unless it is developed further, the Classification Reading implies precisely such independence. To be fair, I do not take this shortcoming to threaten the readings of those proponents of the Classification Reading I have mentioned above. Insofar as they hold the Classification Reading as a background assumption, and not as a developed theory, they do not positively assert the independence of improper natural science from the study of necessary laws known with apodictic certainty. And, indeed, they might disagree over the nature of such independence. For example, they might argue over whether improper natural science is only contingently or necessarily lacking in a priori laws and, hence, only contingently or necessarily unlike proper natural science. They might thus disagree over whether the improper natural sciences can at least in principle achieve proper scientific status at some point in the future, or whether the metaphysical nature of their objects of study is such that they cannot even in principle be given an a priori grounding in necessary laws. I think however that either way of resolving the ambiguity would put pressure on the Classification Reading. Whether a natural science is improper because it has not yet achieved a priori knowledge, or because it can never in principle achieve it, the science would be sufficiently disconnected from the demand for a priori knowledge of necessary laws to count as a genuine natural science. On my reading, any attempt to classify the natural sciences into two distinct but equally genuine kinds would be wanting unless it could also account for
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See Watkins (a), Sturm (), and McNulty (), discussed above (p. ). Because the Classification Reading is not argued for explicitly, I do not intend to attribute to these authors any particular views on this issue. In Breitenbach () I have argued that, even though Kant does not think we know any biological laws, such laws are in principle possible on his account.
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the tight link between Kant’s conception of science and the demand for a priori knowledge of necessary laws. Both the Demarcation Reading and the Classification Reading must thus address serious concerns. The Demarcation Reading cannot explain why Kant ascribes scientific status to disciplines that lie outside the strict limits of proper science. It cannot account for the progressive picture of Kant’s increasingly demanding conceptions of science that emerges from the Preface to the Metaphysical Foundations as I have presented it in Section .. The Classification Reading fares better on this score and offers a way around the tension between some of Kant’s apparently contradictory views. But the Classification Reading fails to illuminate Kant’s professed claim that a science of nature must be concerned with necessary laws that can be known with apodictic certainty. Do these failings show that Kant’s own views about natural science are ultimately self-contradictory? Must we disregard part of Kant’s commitments if we want to extract a coherent account of natural science? In the following section, I argue for a third interpretation that avoids these conclusions.
. An Alternative Interpretation On my proposed Normative Reading, Kant construes both proper and improper natural science as genuine kinds of natural science. In this regard, my interpretation sides with the Classification Reading against the Demarcation Reading. However, by contrast with the Classification Reading, I argue that improper natural science is essentially linked to the criteria of proper natural science: systematicity, grounding in rational principles of ‘grounds and consequences,’ and apodictic certainty. I thus agree with the Demarcation Reading that these criteria are the hallmarks of genuine science. Importantly, on my reading, Kant’s apparent ambiguity in the Preface to the Metaphysical Foundations can be resolved if we see that, while Kant allows for two kinds of genuine natural science, proper and improper, these kinds are connected in an important way. Moreover, I argue that this connection is normative: while the improper natural sciences do not actually fulfil the criteria of proper natural science, they ought to fulfil them. Improper natural science is legitimately called ‘science’ because it is normatively guided by the criteria of proper natural science. I call this the ‘Normative Reading.’ My reading takes its cue from Kant’s surprising normative claim, which I flagged at the beginning of Section .. Kant maintains that “every doctrine, if it is supposed to be a system, that is, a whole of cognition ordered
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according to principles, is called a science” (MAN, :; my emphasis). In the Preface to the Metaphysical Foundations, Kant does not make it explicit why he regards this normative relation to systematicity as a criterion for scientificity. But this is to be expected once we see that Kant has already provided an explanation in the Critique of Pure Reason. There, as I argue in this section, Kant shows that the normative relation to systematicity implies a normative demand to fulfil the criteria of proper natural science. It is this first Critique conception, I argue furthermore, that Kant takes for granted in the Metaphysical Foundations. There is thus good reason why Kant begins his characterization of natural science in the Preface with a general account of science, as I have shown in Section .: he deliberately frames his argument in the Metaphysical Foundations within the first Critique conception of science. In the Critique of Pure Reason, Kant argues that science requires systematicity. It is the systematic unity of cognition, its ordering under principles, that “makes common cognition into science” (KrV, A/ B). Three aspects of this conception are important for my interpretation. The first is the insight that the systematicity required for science is brought about by reason and guided by reason’s idea of the systematic unity of all possible cognitions of the understanding under a highest principle. This is the idea of the “complete unity of the understanding’s cognition” (KrV, A/B; my emphasis). According to Kant, it is a “mere idea,” since it does not apply to anything in experience, and thus cannot in principle be realized in cognition (KrV, A/B). But it is also a “regulative” idea, since it sets a “goal” and guides the activities of reason (KrV, A/B). Even when reason has systematized a set of cognitions under a principle, it is thus guided by the demand to search for yet higher principles that subsume yet further cognitions. This demand is not satisfied by anything other than complete systematic unity. Thus, according to Kant’s first Critique account, while cognitions must have achieved some systematicity in order to count as scientific, any actual system of cognitions stands under the continued demand of reason for complete systematicity. Second, the idea of the complete systematic unity of cognition, applied to cognitions of nature, is the idea of unity under rational, or explanatory, principles. Although reason is also involved in classifying natural objects, it ultimately aims at uncovering the conditions of things; it is after
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I can only give a sketch of these three points here. For a more detailed account see my forthcoming essay.
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explanation. For instance, reason is involved in classifying materials into earths, salts, combustibles, water, and air (see KrV, A/B). However, in doing so, reason seeks systematicity under explanatory principles. It is involved in classifying materials into earths, salts, combustibles, water, and air “in order to explain the chemical effects of [these] materials” (KrV, A/B; my emphasis). The activities that constitute the historical sciences are thus aimed at those that constitute the rational sciences. In systematizing cognitions, reason is guided by an idea of the complete unity of cognitions ordered according to rational principles, or laws. As Kant puts it, “this idea [of reason] postulates complete unity of the understanding’s cognition, through which this cognition comes to be not merely a contingent aggregate but a system interconnected in accordance with necessary laws” (KrV, A/B). Third, the idea of reason is also the idea of the complete unity of cognitions under laws that can be known a priori and with apodictic certainty. This is because, on Kant’s account, in seeking systematicity under rational principles, we are searching for a unity of cognitions, a unity whose parts, the individual cognitions, and their ordering principles can be inferred a priori and, thus, with apodictic certainty. Reason is after certainty, and after the a priori insight that gives us this certainty. In the Architectonic of Pure Reason, Kant characterizes this unity as the unity of the manifold cognitions under one idea. This [idea] is the concept of reason of the form of a whole, insofar as through it the domain of the manifold as well as the position of the parts with respect to each other is determined a priori. (KrV, A/B)
In order to understand the significance of Kant’s third point, it is important to see that Kant construes the idea of the whole in teleological terms. The whole, or complete system, is construed as the end-directed realization of the idea of the whole. As Kant continues the passage from the Architectonic, “the scientific concept of reason,” that is, reason’s idea of a complete system under rational principles, “contains the end and the form of the whole” (KrV, A/B). As an end of reason, the idea guides the realization of the system. And it is because the idea of the whole contains an end in this teleological sense that, in a complete system, it
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Willaschek highlights this point (, –). Kant’s use of ‘end’ can here be understood according to the characterization Kant puts forward in the Critique of Judgment: an “end” is “the concept of an object insofar as it at the same time contains the ground of the reality of this object” (KU, :). See my book (a, ch. ) for a related account of Kant’s teleological conception of the systematic unity of reason.
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“precedes the determinate cognition of the parts” (KrV, A/B), as Kant puts it in the Appendix to the Transcendental Dialectic. The idea determines the realization of the system and thus precedes our cognition of its parts. It follows that, if we had fully determinate knowledge of the idea, we would be able to infer a priori not only the “domain of the manifold,” that is, all those cognitions that form part of the whole, but also “the position of the parts with respect to each other” and, hence, the rational principles that govern their relation (KrV, A/B). We would know the system and its ordering principles a priori and, therefore, with apodictic certainty. In seeking to uncover systematic unity under rational principles, reason is thus guided by an idea of systematic unity that can be known a priori and with apodictic certainty. My analysis highlights that, according to the Critique of Pure Reason, a manifold of cognition is scientific if it is systematized under principles, even if it is incompletely systematic. For to be incompletely systematic is to stand under reason’s demand for complete systematicity. And this, in turn, implies the demand for ordering under rational principles and for apodictic certainty. For cognition to be scientific, in other words, is for it to stand under the demand to fulfil the criteria of proper natural science: not only systematicity, but also grounding in rational principles, and apodictic certainty. All incompletely systematized cognition, even the ‘improper’ sciences such as the merely classificatory historical sciences and the merely empirical explanatory science, stand under this demand of reason. My analysis also shows that, according to the first Critique, the demand for proper scientific status is a normative one. It is a necessary end of the “regulative employment of reason” (KrV, A/B) to search for complete systematicity, and thereby for grounding in rational principles and apodictic certainty. It is an end reason “prescribes and seeks to bring about” (KrV, A/B), by regulating or “directing the understanding to a certain goal” (KrV, A/B), the complete systematicity of its cognitions. Kant’s conception of science in the first Critique thus implies my proposed Normative Reading. It shows why improper natural science is a genuine form of natural science: improper natural science is cognition
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My reading crucially differs from well-known best systems accounts of laws and science associated with Ph. Kitcher (e.g., ) and others such as Buchdahl (b, ), Brittan (), and Allison (). See my essay () for a criticism of the best systems account. On my account, our best scientific systems can count as science, but not by virtue of their actually grounding the necessity of laws. They can count as science by virtue of their aiming for complete systematicity and, thereby, for necessary laws.
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that is incompletely systematic and normatively guided by the criteria for proper science. Let us now turn back to the Metaphysical Foundations of Natural Science. There is clear evidence that there, too, Kant assumes the first Critique conception of science. After introducing the distinction between proper and improper science, Kant adds that one can easily see “that, in accordance with demands of reason, every doctrine of nature must finally lead to natural science and conclude there” (MAN, :). Kant makes this claim after arguing that natural science derives the legitimacy of this title from its pure part. In this context, Kant refers by ‘natural science’ to proper natural science. He suggests that any doctrine of nature, in accordance with demands of reason, must strive to become a proper science. Any study of nature must aim to achieve complete systematic unity under rational principles and apodictic certainty. The ‘must’ Kant refers to is the ‘must’ of the necessary demands of reason. In the Metaphysical Foundations, just as in the first Critique, he maintains that for something to count as science in the broad sense it must seek to realize what reason demands. When Kant states that “every doctrine, if it is supposed to be a system, that is, a whole of cognition ordered according to principles, is called a science” (MAN, :), he claims that something is a science if it stands under reason’s demands for complete systematicity and, hence, is normatively guided by the criteria of proper science. Improper science ‘must finally lead to’ proper science ‘in accordance with the demands of reason,’ because the practice of improper science is directed at the norm of the complete systematicity of cognitions, an ideal in which we would have apodictically certain a priori knowledge of the laws of nature. My Normative Reading thus receives explicit support from the Preface to the Metaphysical Foundations. My Normative Reading also solves the problems that beset the two prevalent alternative interpretations, the Demarcation and Classification Readings. The Normative Reading can account for a broad notion of science, which includes disciplines that do not satisfy the three criteria of proper science. And it makes sense of the tight link between natural science and the demand for a priori knowledge of necessary laws. Kant can consistently hold that there is a wide class of cognition that may legitimately be called ‘natural science’ even if it does not qualify as natural science proper, while also maintaining that these improper natural sciences are supposed to have proper scientific status. He can without contradiction claim that not all natural sciences do in fact explain the phenomena within their domain in terms of necessary laws, while also arguing that all natural
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sciences aim for such explanation. Kant’s broad notion of natural science has inherently normative character. One might wonder where my reading leaves physics, the one discipline that, on Kant’s account in the Metaphysical Foundations, is established as a proper science. How, if at all, can my reading account for the exceptional status of this discipline? As I have argued, Kant offers a broad account of natural science. According to this account, all natural science aims for proper scientific status by virtue of its aiming for complete systematicity of cognitions. On my reading, this general account does not conflict with Kant’s view that there are some sciences whose laws are actually known with apodictic certainty. Thus, according to Kant, we can know the fundamental laws of physics a priori. And, similarly, there might be other disciplines that may be grounded in a priori laws. However, on my Normative Reading, physics lacks the foundational role ascribed to it by some commentators. Striving for proper scientific status does not amount to striving for derivation from the laws of physics. On my reading, science is guided by the norm of a complete systematic unity of all cognitions. Integration in the complete system can secure a normative relation to a priori grounding and apodictic certainty. But it can do so not by aiming for derivation from a set of a priori laws of physics but by aiming for derivation from the idea of the whole. Physics may serve as a model of a discipline that fulfils the criteria of proper natural science. But even as a model, its status is limited. Even physics, just as any other science, has a further task to complete: ‘in accordance with demands of reason,’ it must be unified with other sciences in the complete systematicity of cognition. This construal of the place of physics in science also implies that we must read Kant’s famous statement about the need for mathematics with care. On my reading, when Kant suggests that science proper “is only possible by means of mathematics” (MAN, :), he is making a claim about the requirements for proper natural science insofar as such science is actually achievable by us. Kant is specifically concerned with proper science insofar as it “can be met with [angetroffen werden könne]” in the study of nature (ibid.). Proper natural science construed in this sense is proportional to the mathematics that, too, “is to be met with [anzutreffen ist]” in science (ibid.). But none of this rules out that, where mathematization is not achievable, we must aim for the apodictic certainty that comes
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For Friedman, for example, the laws of matter provide the “brilliantly successful Newtonian paradigm” of proper natural science (b, ). For related views, see Kreines () and Stang (, ch. ).
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with the idea of the whole. It is because of this normative demand that all natural sciences, whether proper or improper, are genuine sciences.
. Conclusion I have argued that Kant presents a well-defined broad conception of science that ranges over natural science ‘properly’ and ‘improperly socalled.’ A body of cognition qualifies as a natural science in this broad sense if it stands in a normative relation to science proper. And since aiming at proper natural science is construed, by Kant, in terms of the intellectual activities involved in systematizing cognitions, we can see now that science in the broad sense concerns the scientific enterprise broadly construed. It encompasses not only scientific claims such as theories and explanations but also scientific practices such as calculating and experimenting, modeling, representing and visualizing of data, and, more broadly, theorizing. Even in the text that is known for its strict, and even restrictive, notion of science, Kant is deeply committed to this broad, normative conception of science as a central human activity.
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For this reason I disagree with Van den Berg () and McNulty () that the mathematization requirement is equivalent to the aprioricity and apodictic certainty requirement. On my reading, apodictic certainty is the third requirement of proper science, and this requirement can but need not be met in the form of mathematization. Earlier versions of this chapter were presented at King’s College London, University College London, the University of Edinburgh, and (virtually) the University of Kaliningrad. I am grateful to the audiences at these events for their helpful comments and discussion. Special thanks go to John Callanan, Yoon Choi, Alix Cohen, Andrew Cooper, Bennett McNulty, Thomas Sturm, and an anonymous referee for Cambridge University Press for their invaluable feedback.
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The Applicability of Mathematics as a Metaphysical Problem Kant’s Principles for the Construction of Concepts Katherine Dunlop .
Introduction
In the seventh paragraph of the Preface to MAN, Kant contends that a discipline contains only as much “proper science” as mathematics is applicable within it (:). In the eighth and ninth paragraphs, he uses the examples of psychology and chemistry to illustrate how “improper” sciences can fail to meet this condition. The tenth through twelfth paragraphs aim to show that metaphysics is necessary for “proper” natural science (as Kant first asserted in the sixth paragraph), by arguing that metaphysical principles are needed specifically for the application of mathematics. This discussion spans the middle third of the (eighteen-paragraph-long) Preface, and its length and centrality indicate that securing the applicability of mathematics is crucial for Kant’s foundational project. Yet it is not clear how the foundations Kant proposes can fulfill this objective. The seventh paragraph of the Preface asserts that determinate natural things can be cognized in respect of their possibility, thus a priori (and with the apodictic necessity required for science), only by constructing their concepts. It is precisely because the concepts must be constructed that “a pure doctrine of nature concerning determinate natural things is possible only by means of mathematics” (MAN, :). Accordingly, Kant says (in the Preface’s tenth paragraph) that he will supply a “complete analysis of the concept of matter,” which is to underlie “principles for the construction of concepts that belong to the possibility of matter” (MAN, :; emphasis original); the twelfth paragraph makes clear that MAN is to include principles pertaining to the construction (Principien der Konstruktion) of concepts as well as the metaphysical ones that underlie them. This raises the immediate problem that construction, as Kant understands it elsewhere, is insufficient to establish the possibility of objects of
See McNulty () for an illuminating account of these paragraphs.
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natural science. In the first Critique, Kant claims that what construction produces is “only the form of an object” and “a product of the imagination, the possibility of whose object would still remain doubtful, as requiring something more.” Construction suffices to prove the possibility of geometrical figures, even if only as “forms,” because the requisite “something more” is just that they are thought in accordance with a formal condition on all (outer) experience, namely, space (KrV, A–/B). But the objects of natural science are conceived of as objects of experience themselves, thus as having (concrete) existence, which Kant insists is not at issue in the “mathematical problems” that construction solves. Indeed, Kant asserts in the Preface’s sixth paragraph that the laws investigated in natural science concern “the necessity of that which belongs to the existence of a thing,” and that no such concept can be constructed, “because existence cannot be presented a priori in any intuition” (MAN, :). For consistency’s sake, we must suppose that when Kant later claims concepts of determinate natural things are capable of construction, he is using “construction” with a different meaning than the familiar one presumed here. But he leaves the reader to work out what notion he has in mind. Kant’s suggestion that the application of mathematics is made possible by constructing “concepts belonging to the possibility of matter” (MAN, :) raises a further problem, as I explain in Section .: of the four properties into which he “analyzes” the concept of matter, Kant constructs only the first and third in the course of MAN’s main argument. Kant also outlines a construction of the second property, but says that metaphysics is “not responsible if the attempt to construct the concept of matter in this way should perhaps not succeed.” For, he claims, metaphysics “is responsible only for the correctness of the elements of the construction granted to our rational cognition, not for the insufficiency and limits of our reason in carrying it out” (MAN, :–). I think this limited conception of metaphysics’ responsibility gives us a good way to understand Kant’s phrase “principles for the construction of concepts” (“Principien der Konstruktion der Begriffe”). On the reading I propose in this chapter, these metaphysical principles vouchsafe the correctness of the “elements for construction” they deliver, without necessarily sufficing for the construction itself. In Section ., I show that the problem with “application of mathematics” that features most prominently in Kant’s writings is the incompatibility of Wolffian metaphysics with
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In mathematical problems, “the question is not about . . . existence as such at all [ist . . . u¨berhaupt von der Existenz gar nicht die Frage]” (KrV, A/B).
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mathematical doctrines. This indicates that the problem of mathematics’ applicability, as Kant conceives it, arises within metaphysics (rather than the philosophy of mathematics or science). That Kant also criticizes the opposed Newtonian metaphysics, although it succeeds in “opening the field of appearances for mathematical assertions” (KrV, A/B), indicates that “elements” for the application or construction of concepts must meet some standard of metaphysical “correctness,” regardless of how well they provide for “mathematical assertions” about objects of experience. So for such elements to be correct, it is neither necessary nor sufficient that they issue in actual constructions or applications of mathematical concepts. Section . presents examples of metaphysical views that Kant rejects, although they allow for the application of mathematics to nature, not only in the weak sense of being consistent with mathematical doctrines but even in the stronger sense that they “build” mathematical features into their conceptions of body and matter. These are the conceptions of body and of matter, respectively, articulated in the doctrine Kant calls physica generalis and by J. H. Lambert. I contend that Kant rejects them because they fail to relate mathematical concepts and properties to the conditions of possible experience.
. Construction in MAN’s Chapters .. Construction in Phoronomy Following the plan laid out in the Preface, each of MAN’s four numbered chapters begins with an “Explication” stating a fundamental property of matter, then adduces principles (“Propositions”) employing the property in question, as well as further Explications of related concepts. The property considered in the first main chapter, Phoronomy, is movability. In the Remark that concludes the chapter, Kant describes “phoronomy,” “in which matter is thought with respect to no other property than its mere movability,” as a “pure” doctrine (Lehre) of “the quantity of motion.” He then identifies the pure doctrine of motion’s quantity with “the doctrine of the composition of motion,” on the grounds that “the concept of quantity always contains that of the composition of the homogeneous” (MAN, :). Thus “construction of the quantity of a motion is the composition of many motions equivalent to one another” (MAN, :), and by
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This explicit formulation comes from the Mechanics chapter, where Kant elaborates that “the determinate concept of quantity is possible only through the construction of the quantum,” and “in regard to the concept of quantity, this is nothing but the composition of the equivalent” (MAN, :).
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exhibiting this quantity as a “composition of the homogeneous,” the construction serves to bring motion under the category of quantity. Kant’s account of how the composition of two motions is represented takes pride of place in Phoronomy, as the chapter’s single Proposition. The accompanying Remarks and Explications state requirements on geometrical construction in general and on the composition of motion, in particular. While Kant is particularly concerned to distinguish his procedure for combining the motions from “mechanical” alternatives (in which the quantities being combined function as causes of the resultant quantity, rather than standing in the relation of equality to it), his discussion also sheds light on how the constructions in MAN entitle us to regard what is constructed as a quantity. Kant’s Proposition asserts that to represent two equal speeds AB and ab in the same direction “as contained in one speed of motion,” that is, to represent this “one speed” as their sum, it is necessary to represent “the body A with speed AB as moved in absolute space” and to represent “the relative space,” in which this first component motion takes place, as moving with speed ab in the opposite direction (MAN, :–). In a Remark, Kant describes this as a “mediate composition,” contrasting it with the way spatial intervals are combined: If . . . one explicates a doubled speed by saying that it is a motion through which a doubled space is traversed in the same time, then something is assumed here that is not obvious in itself – namely, that two equal speeds can be combined in precisely the same way as two equal spaces – and it is not clear in itself that a given speed consists of smaller speeds, and of a rapidity of slownesses, in precisely the same way that a space consists of smaller spaces. For the parts of the speed are not external to each other like the parts of the space, and if the former is to be considered as a quantity, then the concept of its quantity, since this is intensive, must be constructed in a different way from that of the extensive quantity of space. But this construction is possible in no other way than through the mediate composition of two equal motions, such that one is the motion of the body, and
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Kant’s claim that “the concept of quantity always contains that of the composition of the homogeneous [Gleichartigen]” comes as part of his account of how the Proposition of the Phoronomy chapter is “connected” with “the schema of classification of all pure concepts of the understanding – namely, here that of the concept of quantity” (MAN, :). In the first Critique’s second edition, in a note discussing the principles governing the application of the categories, Kant glosses “composition” as “the synthesis of the homogeneous in everything that can be considered mathematically” (KrV, Bn). See Marius Stan’s contribution to this volume and Dunlop (forthcoming) for a different account of earlier treatments that Kant rejects as “mechanical.” This is, of course, only one of the three cases of composite motion that Kant considers.
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the other the motion of the relative space in the opposite direction. (MAN, :–)
In KrV, Kant characterizes “extensive” magnitude in terms of the priority of the parts over the whole (A/B) and “intensive” magnitude as that “which can only be apprehended as a unity” (A/B), thus not as made up of antecedently given parts. Daniel Warren explains how our inability to “take for granted the idea of distinguishable smaller parts that make up the whole magnitude,” in the case of intensive magnitude, leaves us without “a means of representing what we understand by the addition or subtraction of such intensive magnitudes” (which is “not a problem” for extensive magnitudes, since the sum of two homogeneous extensive magnitudes “is just an extensive magnitude with these two magnitudes as its parts”). [F]or Kant, homogeneous intensive magnitudes, as such, do admit of comparison as greater or less, or as equal, in their degrees of intensity. But that is not sufficient for the representation of addition or subtraction of such quantities. One can know that intensity a is greater than intensity b, without knowing how much greater a is than b, that is, without being able to represent that intensity c, such that a = b + c. (Warren , )
By showing how motions can be represented as sums of other motions, Kant solves this problem. It thus becomes possible to assign quantity to the differences between motions and order the “degrees” of motion on an interval (rather than merely an ordinal) scale. According to Warren, solving this problem allows us to represent “intensive magnitudes as quantities in the full sense” (ibid.). Similarly, Gordon Brittan claims that for Kant “the constructible is the addable” (, ), and Michael Friedman holds that “to conceptualize something as a magnitude or quantity is to exhibit or construct an appropriate operation of addition” (, ). ..
“All that metaphysics can ever achieve”: Construction in Dynamics
The forces and motions treated in the following chapters of MAN are also represented as intensive magnitudes, thus not in terms of antecedently given parts, and we face the same problem in extending quantitative consideration to them. So if the “principles for the construction of concepts” given in MAN are meant to suffice for constructing the concepts, we should expect that once the Phoronomy chapter clarifies what
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These categories of measurement scales were introduced in Stevens (). For helpful discussion, see Brittan ().
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construction accomplishes and provides an example, the Dynamics, Mechanics, and Phenomenology chapters will follow its model. In fact, only Mechanics culminates in a construction of the property it considers. In Mechanics matter is considered as “the movable insofar as it, as such, has moving force,” that is, as having the force to produce motion “by means of its [own] motion” (MAN, :; my emphasis). Kant refers to this sort of generation of motion as “communication” of motion (in contrast to the generation of motion by a body not “needing to be seen as itself moved” [ibid.]). The fourth and final Proposition gives what is described in an accompanying Remark as a “construction of the communication of motion” (MAN, :), such that the “action in the community of the two bodies,” reckoned as the product of mass and speed, is evenly divided between the “action or effect” (the motion received by the second body) and the “reaction” (the motion assigned to the first body) (MAN, :). So Kant again shows how to represent a whole as the sum of parts (although in this case, unlike the Phoronomy chapter, a condition of the problem is to represent the “action in the community” of the bodies as the sum of two equal parts). The second chapter, Dynamics, considers matter as filling space. Since MAN’s chapters correspond to the headings of the Table of Categories, Dynamics deals with the categories of quality, as does the Anticipations of Perception in KrV. The Principle of the Anticipations reads (in the second edition): “In all appearances the real,” or object of sensation, “has intensive magnitude, i.e., a degree” (KrV, B). In Prol, Kant describes this Principle as “the second application of mathematics (mathesis intensorum) to natural science” (:; the “first” application is presumably the Principle of the Axioms of Intuition, which asserts, “All intuitions are extensive magnitudes” [KrV, B]). Insofar as the application of mathesis
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To “fill” a certain space is to resist “every movable that strives through its motion to penetrate into” the space. Because filling space involves the power of resisting, it is “a more specific determination of the concept of occupying a space” (which signifies merely being “immediately present in all points of [the] space,” without the supposition that some “effect arises from this presence” [MAN, :]). It is easy to overlook the “second application of mathematics” codified in the Anticipations’ Principle. For Kant’s claim in the Axioms that its principle “alone makes pure mathematics in its complete precision applicable to objects of experience” (KrV, A/B) suggests that no other principle is needed (see Ph. Kitcher , ). On my reading, the principle that all intuitions are extensive magnitudes “alone makes possible” the application of mathematics just in the sense that it is a necessary condition: without this very principle, no such application would be possible. (That it is a necessary condition for Kant is clear from the fact that intensive magnitudes are constructed by exhibiting them in terms of extensive magnitudes.) But this is not to say it is also a sufficient condition. The expression “alone makes possible” seems to demand such a reading at A/ B–: “pure concepts a priori which represent objects prior to all experience, or rather which
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intensorum to natural science crucially involves the categories of quality, we might expect to find a general treatment of the construction of intensive magnitudes in Dynamics. But the most general finding concerning their construction is a limitative one, as I will now explain. While Dynamics includes a construction of the property of filling space, the construction does not feature prominently, in contrast to the Phoronomy and Mechanics chapters. The construction Kant outlines is based on his theory that matter fills space, “not through its mere existence,” but through the exercise of repulsive and attractive forces (MAN, :). Specifically, the “dynamical concept of matter, as that of the movable filling its space (to a determinate degree),” would be constructed by deriving the limitation of repulsive force by a universally penetrating attractive force, according to “a law of the ratio of both original attraction and repulsion at various distances of matter and its parts from one another” (MAN, :). As Warren has observed, however, the passage containing this construction occupies “a marginal position” in MAN: it does not occur in any Explication or Proposition, nor any proof of a Proposition, but “only in various Notes and Remarks appended to, and going well beyond the scope of, the last [Proposition]” in Dynamics (, ). Indeed, as Warren notes, Kant emphasizes that he “does not want the present exposition of the law of original repulsion to be viewed as necessarily belonging to the goals of my metaphysical treatment of matter,” for which “it is enough to have presented the filling of space as a dynamical property of matter” (MAN, :–). It appears that Kant frames his construction as dispensable because he does not find it wholly satisfactory. He describes his formulation of the law of the forces’ ratio (namely, that “the original attraction of matter would act in inverse ratio to the squares of the distance at all distances” and “the original repulsion in inverse ratio to the cubes of the infinitely small distances”) as “a small preliminary suggestion on behalf of the attempt at such a perhaps possible construction” (MAN, :), and follows it
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indicate the synthetic unity that alone makes possible an empirical cognition of objects.” Kant surely does not mean that these pure concepts (or the synthetic unity they indicate) alone suffice for empirical cognition of objects, without any contribution from sensibility. This does not quite follow the example of Kant’s constructions of composite motion and of action “in community,” inasmuch as it does not show that space-filling can be composed from lesser amounts of space-filling. But because the “law of the ratio” of the fundamental forces just asserts the “ratio” (i.e., the integral or fractional exponential power) with which each force acts on a point, as a function of the point’s distance from the force’s origin, this construction does relate the intensive magnitude of the filling of space to extensive (spatial) magnitude, and thus supplies a means to express the filling of space as a “quantity in the full sense.”
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immediately with the acknowledgment that he is “well aware of the difficulty in this mode of explaining [Erklärungsart] the possibility of a matter in general” (MAN, :). Kant’s reference to “a perhaps possible construction” is in keeping with his use of subjunctive mode throughout this passage, in contrast to the constructions in the Phoronomy and Mechanics chapters. Not only does Kant claim in Dynamics that metaphysics “is not responsible if the attempt to construct the concept of matter in this way should perhaps not succeed” (because metaphysics is “responsible only for the correctness of the elements of the construction granted to our rational cognition” [:]), but in the General Remark that concludes the chapter, he explicitly admits that his preferred dynamical concept of matter cannot be constructed, while a rival conception can be. The General Remark to Dynamics focuses on what Kant describes as “the most important of all tasks” of natural science, which is to explain “a potentially infinite specific variety of matters” (MAN, :), so Kant here contrasts the “mechanical” with the “dynamical” as “approaches” (Wege) to this problem. But it is clear that these contrasting “modes of explanation” (Erklärungsarten), as he more usually calls them, embody different conceptions of matter (see MAN, :). According to Kant, the mechanical mode of explanation presupposes an “absolutely impenetrable” and “absolutely homogeneous” primitive matter which coheres with “absolute insurmountability” into “fundamental particles,” and “the first and foremost authentication for this system rests on the apparently unavoidable necessity of using empty spaces on behalf of the specific difference in the density of matters,” that is, distributing empty spaces “within the matters, and between [the fundamental] particles, in any proportion found necessary” (MAN, :). This way of generating specific varieties of matter, “through the varying shape of the parts and the empty interstices interspersed among them,” confers on the mechanical mode an “advantage over the [dynamical mode] which cannot be wrested from it.”
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Discussing the acknowledged “difficulty” would take us too far afield. See Friedman (, ch. ), Warren (), and James Messina’s contribution to this volume. Kant writes at : that “the dynamical concept of matter, as that of the movable filling its space (to a determinate degree), would be constructed”; at :, that “the original attraction of matter would act in inverse ratio to the squares of the distances at all distances, the original repulsive force in inverse ratio to the cubes of the infinitely small distances, and . . . matter filling its space to a determinate degree would be possible”; and at :, that the “universal law of dynamics would [be that] the action of the moving force, exerted by a point on every other point external to it, stands in inverse ratio to the space into which the same quantum of moving force would need to have diffused, in order to act immediately on this point” (all emphases mine).
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For the possibility of both the shapes and the empty interstices can be verified with mathematical evidence. By contrast, if the material itself is transformed into fundamental forces (whose laws we cannot enumerate a priori, and are even less capable of enumerating reliably a manifold of such forces sufficient for explaining the specific variety of matter), we lack all means for constructing this concept of matter, and presenting what we thought universally as possible in intuition. (MAN, :)
Since one of Kant’s main points is that “one should guard against going beyond that which makes possible the general concept of a matter as such, and wishing to explain a priori” its specific varieties (MAN, :), we might take him to deny only that the dynamical approach to specific varieties of matter can be constructed, not that its “general concept of a matter as such” can be. But the claim that “we lack all means for constructing” the dynamical concept of matter seems just to make explicit what is already implied by Kant’s denial that the possibility of the “fundamental” (moving) forces to which matter is “reduced,” on the “dynamical” conception, can be “comprehended” (einsehen) (MAN, :; cf. :). To “comprehend” their possibility would be “to cognize [it] through reason,” that is, a priori (Log, :; cf. V-Lo/Dohna, :), and according to the Preface’s seventh paragraph, that would be to construct their concepts. Kant holds that despite our inability to construct the dynamical concept of matter, we must adopt it, precisely as a bulwark against the presupposition of absolute impenetrability and empty space. He contends that “a merely mathematical physics pays double” for the advantage of constructibility “on the other side”: First, it must take an empty concept (of absolute impenetrability) as basis; and second, it must give up all forces inherent in matter; and beyond this, further, with its original configurations of the fundamental material and its interspersing of empty spaces, as the need for explanation requires them, such a physics must allow more freedom, and indeed rightful claims, to the imagination in the field of philosophy than is truly consistent with the caution of the latter. (MAN, :)
In both KrV and the General Remark to Dynamics, Kant claims that while the dynamical approach is not sufficient to explain the specific varieties of
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It might be objected that Kant’s denial that we can cognize the possibility of the fundamental forces a priori implies only that they are not capable of “construction” as Kant understands it outside MAN, not that we cannot “construct” them in the sense of showing them to be additive. But I take it that no object of natural science is capable of “construction” in the former sense, so Kant could not be claiming that being constructible in this sense is an advantage of the mechanical approach.
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matter, the prospect of explanation that it offers counts decisively in its favor, because it “obviates the alleged necessity of the presupposition” of empty space (KrV, A/B). On his view “everything that relieves us of the need to resort to empty spaces is a real gain for natural science, for they give the imagination far too much freedom to make up by fabrication for” the knowledge it lacks (MAN, :). Kant’s attack on the mechanical approach in the General Remark – and with it the Dynamics chapter as a whole – concludes with the strong claim that the only purpose for which “metaphysical investigation behind that which lies at the basis of the empirical concept of matter is useful” is that of “guiding natural philosophy” toward “dynamical grounds of explanation,” and a parallel claim about the limited usefulness of metaphysics with respect to constructing concepts. This is now all that metaphysics can ever achieve towards the construction of the concept of matter, and thus to promote the application of mathematics to natural science, with respect to those properties whereby matter fills a space in a determinate measure – namely, to view these properties as dynamical, and not as unconditioned original positings, as a merely mathematical [sc. mechanical] treatment might postulate them. (MAN, :)
In claiming that metaphysics can achieve no more, in regard to the construction of the concept of matter, than to construe its space-filling property as dynamical, Kant appears to embrace the conclusion that the dynamical concept of matter cannot itself be constructed. It certainly appears paradoxical that “all that metaphysics can ever achieve” in promoting mathematics’ application to natural science, with respect to matter’s property of filling space, is to direct natural science away from the mechanical approach, which Kant himself describes as “the most tractable for mathematics” (MAN, :–). We could try to brush this odd-seeming result aside, as in some way peculiar to matter’s property of filling space, and take Kant’s treatments of the properties of movability and communicating motion as more representative of what principles “for the construction of concepts” can accomplish (cf. Friedman , ). But I will now argue that this limited conception of the principles’ role, as steering us away from a pernicious metaphysics, fits very well with Kant’s treatment of mathematics’ applicability (as a philosophical problem) from
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Cf. MAN, :: That matter fills space by repulsive force, which can differ in degree between “different matters,” is “the one and only assumption that we make, simply because it can be thought, but only to controvert a hypothesis (of empty spaces), which rests solely on the pretension that such a thing cannot be thought without empty spaces.”
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his pre-critical works through the first Critique. So the Dynamics can after all be taken to establish a general thesis about the construction of (intensive) magnitudes, as the correspondence with the Anticipations of Perception suggests.
. Metaphysical Obstacles to the Application of Mathematics To understand Kant’s concern to secure the applicability of mathematics, in laying the foundations of natural science, we must first distinguish the issue Kant is addressing from some more familiar ones. The problem of how the paradigmatically abstract science of mathematics can aid in the investigation of concrete natural things is pressing for some philosophies of mathematics, such as Platonism and formalism. But I take it that for Kant (at least in his critical period), the problem of the abstract-particular relationship already arises and is solved within pure mathematics. Kant takes pure mathematics to essentially consist of arithmetic and geometry, where the relevant abstracta are concepts or “general representations,” and the particulars are figures or sequences of units. These particular exemplifications of quantity can be drawn or merely imagined, but in either case they are exhibited “completely a priori, without having had to borrow the pattern . . . from any experience,” and as constructions of the concepts, they can “express in the representation universal validity for all possible intuitions that belong under the same concept” (KrV, A/B). I take it that because the concepts of pure mathematics already enjoy this sort of particular exemplification, there is no special or further problem about their application to concrete natural things (although the concreteness of natural things does preclude there being “constructions of the concepts” in the very same sense). A familiar set of problems concerning the use of mathematics in natural science are those of measurement: what empirical means are available, and how well they suffice, for assessing the quantitative features of objects. Kant makes some claims of a broadly conceptual nature about empirical
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As Charles Parsons succinctly explains, within pure mathematics pure intuition is “supposed somehow to get us across the divide between the fuzzy Lebenswelt with its everyday objects and the sharp, precise realm of the mathematical” (, ). Such a view of pure mathematics is implied by the argument Kant gives in Prol (:–) and the introduction to the B () edition of the Critique (KrV, B–) for the syntheticity of all judgments of pure mathematics. The argument aims to establish this conclusion by separately considering the two cases of arithmetic and geometry. As Lisa Shabel explains, this strategy “reflects [Kant’s] understanding of the elementary mathematics of his day, which took mathematics to be the science of discrete and continuous magnitudes (number and extension, respectively)” (, ).
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means of determining quantities, most notably, that quantity of matter “manifests itself in experience only by the quantity of motion at equal speed” (MAN, :; see also Warren ). But he does not seem very concerned about the reliability or precision of measurement techniques, or the related issue of how convincingly any given result can stand as a quantity “manifesting itself in experience” rather than an artifact of our activity of measuring. I think a better way to understand Kant’s concern for securing the applicability of mathematics is to consider how metaphysical views could raise obstacles for its applicability. For on my reading, Kant’s “principles for the construction of concepts” are meant mainly to forestall these obstacles (rather than to suffice for the construction themselves). For Kant, the most salient source of obstacles is the broadly Leibnizian view that dominated his philosophical context; I will now show that he speaks specifically of the “application of mathematics” (or equivalently of “opening the field of appearances for mathematical assertions”) in discussing its problems. I will then show that Kant thinks the Newtonian view to which the Leibnizian is opposed raises problems of its own. Throughout his published philosophical writings, Kant adverts to conflicts between currently prevailing metaphysical views and the claims of mathematics, in particular, geometry. He acknowledges that philosophers have sought to emulate the method of mathematics, which he considers a mistake (NG, :; UD, :–; KrV, A/B), but finds that philosophers have resisted “the genuine application of [mathematical] propositions to the objects of philosophy”: As for metaphysics, this science, instead of turning certain of the concepts or doctrines of mathematics to its own advantage, has, on the contrary, frequently turned itself against them. And where it might, perhaps, have
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In this respect my reading contrasts with Michael Friedman’s, which makes the argument of Book III of Newton’s Principia central to both the “balancing argument” of the Dynamics chapter and the contrast between “true motion” and “semblance” in the General Remark to the Phenomenology chapter. Briefly, Book III explains how the masses of heavenly bodies are determined from the observed motions of their satellites and, by finding the center of gravity of the solar system (from the ratios of the masses), decides in favor of the Copernican world-system over the Brahean. Friedman takes Kant to be particularly impressed by the quantitative precision with which Newton accounts for the planets’ deviations from Keplerian orbits: “To exactly the extent to which all perturbations of the initial (Keplerian) orbits can be successfully calculated . . . by reference to the universal inversesquare attractive forces in question, we are then justified in concluding that this force is indeed the ground of the truth or actuality of all the rotational motions” (, –). And to make this comparison, between the magnitude of the perturbing effect predicted by Newton’s theory and observed deviations from Keplerian orbits, we must first know that we have determined their trajectories with sufficient precision and that the deviations are genuine rather than spurious.
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been able to gain secure foundations on which to base its reflections, it is to be seen trying to turn mathematical concepts into subtle fictions, which have little truth to them outside the field of mathematics. (NG, :)
In this essay, Kant gives as an example metaphysics’ endeavor “to discover the nature of space and establish the ultimate principles, in terms of which its possibility can be understood.” He does not identify this metaphysics by name. But he indicates that metaphysics’ criticism of mathematics (that the latter’s “fundamental concepts have not been derived from the true nature of space at all, but arbitrarily invented”) is how metaphysics defends its conclusions if they should conflict with the “reliably established data” of geometry, such as “that space does not consist of simple parts” (NG, :). In UD, Kant explains how “the geometer” can “demonstrate that space is infinitely divisible” (:), summarizing an argument that appears in the textbooks of John Keill and Jacques Rohault (see also MonPh, :). Kant appears to allude to this same argument in his remarks on the Antithesis of the Second Antinomy, where he claims that the infinite divisibility of matter is proved on “merely mathematical” grounds. Here Kant identifies the refractory metaphysics as “monadism.” As the term suggests, monadism derives from Leibniz’ view of space and material things as phenomenal manifestations of simple, mind-like entities (“monads”), the only ultimately real things. But the monadists go further than Leibniz, holding that the ontologically basic simples are themselves situated in space. Michael Friedman () makes a compelling case that these remarks are crucial for understanding Kant’s charge, in the Transcendental Aesthetic, that philosophers of the Leibnizian school must “dispute the validity or at least the apodictic certainty of mathematical doctrines in regard to real things (e.g. in space)” (KrV, A/B); we should also note the similar claim in the “Axioms of Intuition” that “all objections” to the necessary validity of mathematical propositions with regard to objects of experience “are only the chicanery of a falsely instructed reason, which erroneously thinks of freeing the objects of the senses from the formal condition of our sensibility” (KrV, A/ B–). For the monadists, extended things “consist of simple parts”
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In other words, this “falsely instructed reason” regards spatiotemporal objects, and space and time themselves, as transcendentally real. As I am about to show (in the next section), it is specifically realists of the Leibnizian school who cannot account for the necessary validity of mathematics, in application to objects of experience. There is a more explicit allusion to monadism earlier in the paragraph: “Appearances are not things in themselves. Empirical intuition is possible only through the pure intuition (of space and time); what geometry says about the latter is therefore undeniably
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in the sense that they are composed of “physical points” (as at MAN, :). As Friedman explains, this lands the monadists with “the problem of the composition of the continuum and, more specifically . . . Zeno’s metrical paradox of extension,” according to which no extended region of space can be composed of “any number of unextended simple elements (points), not even an infinite number of such elements” (, ). The monadists’ only recourse is to understand these simple elements, the physical points, as having (very small amounts of ) extension and to deny the possibility of further division within these tiny regions. So they must deny the infinite divisibility of space, which counts as a mathematical doctrine for Kant (as we have just seen; cf. MAN, :). Kant accordingly makes clear that it is the monadists who “would not allow even the clearest mathematical proofs to count as insight into the constitution of space, insofar as it is in fact the formal condition of the possibility of all matter, but would rather regard these proofs only as inferences from abstract but arbitrary concepts which could not be related to real things” (KrV, A/ B). They thus erect an insurmountable obstacle to the application of mathematics.
. Wrong Ways to Explain the Applicability of Mathematics ..
Empty Space in Newton’s Metaphysics and in the Mechanical Theory of Matter
Kant’s charge that Leibnizian philosophers find themselves disagreeing with “mathematical doctrines” is part of a broader argument that philosophers who “assert the absolute reality of space and time, whether they assume it to be subsisting or only inhering” – inhering, that is, in “determinations and relations” which things have independently of being intuited (KrV, A/B) – must “come into conflict with the principles of experience” (KrV, A/B). Let us now consider how, according to Kant,
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valid of the former, and evasions, as if objects of the senses did not have to be in agreement with the rules of construction in space (e.g., the rules of the infinite divisibility of lines or angles), must cease” (KrV, A/B). On the doctrine of “physical points” in Wolff and Baumgarten, and how it differs from Leibniz’ view, see De Risi (, –) and Sarmiento (). Friedman (, ch. ; ) appeals to a number of texts, including the Dynamics chapter of MAN, to show that Kant was aware of this difference.
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the realist philosophers who have appropriate regard for mathematics’ validity and certainty are nonetheless in disaccord with “principles” governing experience. For if they decide in favor of [subsisting reality, i.e., Newtonian absolutes], then they must assume two eternal and infinite self-subsisting non-entities (space and time), which exist (yet without there being anything real) only in order to comprehend everything real within themselves. If they adopt [Leibnizian relationalism], and hold space and time to be relations of appearances that are abstracted from experience . . ., then they must dispute the validity or at least the apodictic certainty of a priori mathematical doctrines in regard to real things (e.g. in space), since [on this view the concepts of space and time are made from] abstracted relations [which] cannot occur without the restrictions that nature has attached to them. The first [sc. Newtonians] succeed in opening the field of appearances for mathematical assertions. However, they themselves become very confused through precisely these conditions if the understanding would go beyond this field. The second [sc. Leibnizians] succeed, to be sure, with respect to the latter, in that the representations of space and time do not stand in their way if they would judge of objects not as appearances but merely in relation to the understanding; but they [cannot] bring the propositions of experience into necessary accord with [a priori mathematical cognitions]. (KrV, A–/B–)
The objection to Newtonian philosophers is that they “become very confused” through the “restrictions that nature has attached” to the relations of appearances, if the understanding should go “beyond the field” of appearances. Kant doesn’t elaborate the objection, presumably because the Newtonian view of the ontological status of space and time (which, for both the Newtonians and Kant, in contrast to the Leibnizians, are prior to and in that sense “beyond” appearances) as “self-subsisting non-entities” is so evidently confused. Another obvious respect in which these “representations of space and time stand in the way,” in judging objects “not as appearances but merely in relation to the understanding,” is that the Newtonian view of time and space as “eternal and infinite” makes them conditions on things-in-themselves, in particular, God, which are properly understood as outside space and time. What is important for our purposes is that the Newtonians’ success in accounting for the application of mathematics (bringing the propositions of experience into “necessary accord” with mathematical propositions) in no way insulates their metaphysics from criticism. Rather, this success is outweighed by the metaphysics’ defects, just as the mechanical conception of matter “pays double on the other side” for the advantage of
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constructibility. That Kant takes the same view of these philosophies’ competitive advantage is suggested by the fact that they pay for it in the same coin, as it were: with a concept of empty space. To be sure, the Phenomenology chapter of MAN clearly distinguishes the “dynamical” concept of empty space as vacuum disseminatum (which “constitutes only a part of the volume of matter” and is used to derive specific differences in density) from the “phoronomical” concept of a space in which any “empirical space” can be thought as movable (MAN, :). But it is clear that empty space is not, on either conception, an object of possible experience. Now, there is an important difference between the dynamical concept of empty space (as part of the volume of matter) and Newton’s conception of absolute space, and a further important difference between Newton’s conception and Kant’s own phoronomical concept of empty space. Kant makes very clear that in contrast to the evident absurdity of Newtonian space, the impossibility of empty space in the dynamical sense cannot be proved “from its concept alone, in accordance with the principle of contradiction” (MAN, :; cf. :). Kant’s own phoronomical concept, as “only the idea of a space, in which I abstract from all particular matter that makes it an object of experience” and “nothing at all that belongs to the existence of things” (ibid.), is distinguished from Newton’s conception of “self-subsisting” absolute space, whose role it assumes (namely, as a reference frame in which any given “empirical” or “material space,” one fixed by the spatial relations of material objects, can be thought as movable). There is then a further difference between Kant’s phoronomical and dynamical concepts of empty space. Kant contends that the phoronomical concept is “necessary, not as a concept of an actual object, but rather as an idea, which is to serve as a rule for considering all motion therein merely as relative” (MAN, :). But the thrust of Kant’s attack on the mechanical conception of matter is that its assumption of empty space, in accounting for the specific varieties of matter, is unnecessary.
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It is a matter of dispute whether Kant takes Newton to subscribe to the mechanical conception of matter, as might be indicated by Newton’s speculation in Query of Opticks: “All these things being considered, it seems probable to me, that God in the beginning formed matter in solid, massy, hard, impenetrable, moveable particles, of such sizes and figures . . . and in such proportion to space, as most conduced to the end for which he formed them” (Newton , ). Kant asserts at : that “the pure space that is also called absolute space, in contrast to relative (empirical) space, is no object of experience”; regarding empty space in the dynamical sense, Kant says “all experience yields only comparatively empty spaces for our cognition,” not absolutely empty ones (MAN, :). See the passages quoted in note above and the paragraph to which it is appended.
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These comparisons indicate that something that is not an object of possible experience – in particular, a concept of empty space – can nonetheless be assumed as “necessary,” but not merely on the grounds that it effects the application of mathematics. So the project for which such concepts are necessary must be a distinctively philosophical one. According to Kant, the phoronomical concept of empty space must be assumed in order to arrive at a “concept of motion or rest valid for all appearance,” and thus to transform the appearance of motion or rest into “a determinate concept of experience (which unites all appearances)” (MAN, :–). Without going into this challenging reasoning, we can take it that the phoronomical concept must be assumed for the sake of articulating necessary conditions on experience. Its use is thus warranted by what Kant calls “true metaphysics,” the “a priori concepts and principles, which first bring the manifold of empirical representations into the law-governed connection through which it can become empirical cognition, that is, experience” (:). I now want to consider how Kant rejects certain metaphysical views which lend themselves to quantitative treatment of the objects of natural science, but not to this distinctive philosophical project. .. “Metaphysical Constructions” in Physica generalis Kant’s understanding of “construction” in the later paragraphs of the Preface to MAN is, as we have seen, broader than his use of it elsewhere – so much so that he speaks of “metaphysical constructions” in the Preface’s twelfth paragraph, although he argues strenuously in UD and the Discipline of Pure Reason chapter of KrV (A–/B–) that philosophy (in particular, metaphysics) cannot follow mathematics’ constructive method. Even in the fifth paragraph of the Preface (just as in the “Discipline” chapter), Kant sharply distinguishes “pure philosophy or metaphysics,” as “rational cognition from mere concepts,” from mathematics, which is grounded “only on the construction of concepts” (MAN, :). I will now show how Kant’s puzzling reference to “metaphysical constructions” can be explained, and thereby supply another example of a
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In UD, Kant does not in fact speak of constructing figures or concepts. But he does say, for example, that “in geometry, in order . . . to discover the properties of all circles, one circle is drawn” (:) and that “in mathematics the concept [of a cone] is the product of the arbitrary representation of a right-angled triangle that is rotated on one of its sides” (:). In contrast, in philosophy “one is constrained to represent the universal in abstracto” (:), rather than in a particular instance, and concepts are “given” prior to any such constructive procedures; see :.
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metaphysical view that does not conflict with mathematical doctrines, but which Kant finds incapable of explaining mathematics’ applicability. Having argued (in the Preface’s tenth paragraph) that mathematics’ application to “the doctrine of body” requires principles pertaining to the construction of the concepts that belong to the possibility of matter, and that these principles ultimately originate in the “metaphysics of corporeal nature,” Kant asserts (in the eleventh paragraph) that “all natural philosophers who have wished to proceed mathematically in their occupation” were compelled to employ “metaphysical principles (albeit unconsciously).” (Kant makes clear that these principles include “those that make the concept . . . of matter a priori suitable for application to outer experience, such as the concept of motion, the filling of space, inertia, and so on” (MAN, :) – that is, the properties attributed to matter in the Phoronomy, Dynamics, and Mechanics chapters.) In the twelfth paragraph, Kant explains why, given that such “metaphysical principles” inevitably enter into mathematical physics, it is nonetheless crucial to treat them separately: Yet it is of the greatest importance to separate heterogeneous principles from one another . . . and to place each in a special system so that it constitutes a science of its own kind.. . . For this purpose I have considered it necessary [to isolate] the former from the pure part of natural science (physica generalis), where metaphysical and mathematical constructions customarily run together, and to present them, together with [mit ihnen zugleich] principles of the construction of those [dieser] concepts . . . in a system. (MAN, :–)
In his elegant summary of the secondary literature, Konstantin Pollok distinguishes two ways to read the second sentence without committing Kant to a notion of “metaphysical construction” (which his sharp distinction between the methods of metaphysics and mathematics would not allow). The first option, due to the scholars Hansgeorg Hoppe and Karen Gloy, is to read the adjective “metaphysical” as modifying the noun “concepts,” so that the demonstrative adjective “dieser” refers back to already-mentioned metaphysical concepts. The second option is to understand the expression “metaphysical and mathematical constructions” “in the sense of the textual surround” (Pollok , ), as referring to “expositions [Erklärungen]” which need to be distinguished into
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See Pollok () for bibliographic information. On this reading, the adverb “zugleich” in this sentence exactly parallels the adverb “mithin” in the first sentence of the eleventh paragraph, as Pollok observes (, ). Both sentences then state that principles of the construction of such concepts as motion, filling space, etc. must be supplied in the course of presenting the concepts.
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metaphysical and mathematical varieties, as they are not in “physica generalis.” So, far from counting certain constructions as “metaphysical,” on this reading, Kant aims to correct errors that arise from overlooking the difference between genuine (mathematical) constructions and merely metaphysical expositions. Following Pollok’s lead, I will pursue this second option, and turn to Kant’s criticism of physica generalis to better understand the importance of distinguishing mathematical constructions from metaphysical expositions. My understanding of this criticism is, however, slightly different from Pollok’s, on which Newton’s Principia is “representative of ‘physica generalis,’ in which mathematics and metaphysics run together” ( , ). To be sure, Kant alludes to Newton and his followers in the (immediately preceding) eleventh paragraph, as natural philosophers who “made use of metaphysical principles (albeit unconsciously),” despite seeking to avoid metaphysics, construed as “the folly of contriving possibilities at will and playing with concepts” (MAN, :). This fits Newton’s own insistence that he “feigns” no hypotheses (, ) as well as the “common supposition” reported by Kant that “Newton did not find it necessary for his system to assume the immediate attraction of matter, but, with the most rigorous abstinence of pure mathematics, allowed the physicists full freedom to explain the possibility of attraction as they might see fit, without mixing his propositions with their play of hypotheses” (MAN, :). Kant contends, however, that Newton himself assumed that the planets “attracted other matter merely as matter, and thus according to a universal property of matter” (MAN, :), and it is because the Principia makes use of this obviously metaphysical principle – as well as the notions of absolute space and time – that I do not think it counts as physica generalis by Kant’s lights. For in a footnote to a discussion in KrV of the method and possibility of a “metaphysics of nature,” Kant describes “what is commonly called physica generalis” as “more mathematics than philosophy of nature” (KrV, A/Bn). Assuming that in MAN he also uses “physica generalis” for “what is commonly called” by that name,
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As Pollok notes, this second interpretative option does not exclude the first. We could understand Kant as saying that the expositions run together in physica generalis must be distinguished into metaphysical concepts and mathematical constructions. I admit that in a sense it would be correct to describe Newton’s Principia as “what is commonly called” physica generalis, given Kant’s account of what is “commonly supposed” about Newton, and the frequently leveled criticism that because the Principia fails to specify a (mechanical) cause for gravitation it counts as “applied mathematics” rather than “physics” (see Lind , ).
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Newton’s Principia falls squarely on the other side of the distinction between mathematics and “philosophy of nature.” Another possibility is that Kant uses “physica generalis” to refer to the branch of Wolffian rational cosmology that treats bodies in general, which Kant mentions by this name in his lectures on metaphysics (V-Met-L/ Pölitz, :; V-Met-L/Pölitz, :; V-Met/Mrong, :). But given Kant’s view that the monadists must dispute the validity of mathematics, it seems very unlikely that he would regard Wolffian physics as “more mathematics than philosophy of nature.” I think “physica generalis” most likely refers to the broadly Aristotelian theory – which also treats bodies in general – that still appeared in Latinlanguage textbooks entitled Physica generalis in the second half of the eighteenth century. These volumes were published and used for teaching mainly in Catholic regions. But it’s a fair bet that Kant would have been familiar with their organization, both because Aristotelianism also dominated the Königsberg university well into the eighteenth century and because they provided the model for German-language textbooks that were surely known to Kant (Eberhard ; Erxleben ). These textbooks begin by designating body as the object of physics and asserting its fundamental properties, which may include solidity (and fluidity), density (and rarity), and porosity, in addition to extension, divisibility, impenetrability, and mobility. Clearly, from Kant’s perspective such lists “run together” mathematical and metaphysical properties, and the justification for asserting the mathematical properties is not apparent. For instance, in the third of Biwald’s four paragraphs on “extension,” which he conceives as “the measurability of body in length, breadth, and depth” (, §, ), Biwald maintains that every body, no matter how small, has magnitude and is thus a quantum (§, ); this obviously begs whatever questions we may have about attributing quantity to bodies. Erxleben similarly asserts, with no apparent justification, that physics (Naturlehre) treats the properties and forces of bodies “according to magnitude,” even going so far as to claim that “applied mathematics actually consists only of particular parts of
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In addition to Biwald () and Hauser (), discussed below, examples include Beck () and Johann Baptist Horváth’s Physica generalis, quam in usum auditorium philosophiae, of which at least four editions appeared between and (in Trynau, Dillingen, Budapest, and Venice). See Kuehn (, ). Hauser () includes all of these properties, as well as elasticity; Biwald’s list is limited to solidity, extension, divisibility, and mobility. Erxleben’s first chapter, a “general inquiry [allegemeine Untersuchungen]” concerning “bodies in general [Körper u¨berhaupt],” discusses hardness and cohesion in addition to extension, matter (and with it impenetrability), density, porosity, rarity, and divisibility.
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physics [Naturlehre]” (, –). (In taking Erxleben’s text as an example of physica generalis, I follow Pollok [, ], who also counts such eighteenth-century textbooks as representative.) Although Erxleben does not make the converse claim that physics consists of applied mathematics, it is easy to see Kant would regard at least these “particular parts of physics” as “more mathematics than philosophy of nature.” In physica generalis, we have an example of a metaphysical view that Kant rejects, not because it is incompatible with mathematics (like monadism) or evidently absurd (like Newtonian absolute space), but because it accounts for mathematics’ applicability in the wrong way. Now it is obvious that in flatly asserting that mathematical or quantitative properties hold of physical objects, physica generalis fails to explain mathematics’ applicability. But I think Kant is making a more specific point when he objects that physica generalis mixes up mathematics and metaphysics: physica generalis does not respect the need for a separate, independent metaphysics to secure mathematics’ applicability. ..
Justifying a Quantitative Concept of Impenetrability
In Kant’s Remark to the first Proposition of Dynamics, he criticizes a proffered construction for falling short, and from his criticism we learn more about what a construction is supposed to accomplish. Proposition asserts that matter “fills a space, not through its mere existence, but through a particular moving force” (MAN, :). In the Remark, Kant explicates the opposing view, that matter fills space through its mere existence, as follows: “the presence of something real in space must already imply” resistance to penetration, “through its concept, and thus in accordance with the principle of noncontradiction.” But since, as Kant objects, “the principle of contradiction does not repel a matter advancing to penetrate into a space where another is found,” the only way to understand “how it contains a contradiction” for a body’s space to be penetrated by another is to “ascribe to that which occupies a space a force to repel every external movable that approaches” (MAN, :). Kant then remarks that here, the mathematician has assumed something, as a first datum for constructing the concept [or “of the construction of the concept”; Datum der Konstruktion des Begriffs] of a matter, which is itself incapable of further construction. Now he can indeed begin his construction of a concept from any chosen datum, without engaging in the explication of this datum in turn. But he is not therefore permitted to declare this to be something
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entirely incapable of any mathematical construction, so as thereby to obstruct us from going back to first principles in natural science. (MAN, :)
Proposition and the accompanying Remark set up a contrast between a concept of matter that includes resistance to penetration as a primitive property and Kant’s own dynamical concept, on which repulsive force is primitive and this resistance derives from it. It is thus natural to understand this passage as an opening salvo in Kant’s attack on the mechanical concept of matter. On Michael Friedman’s interpretation, to take a prominent example, “filling a space through mere existence is associated with the concept of solidity Kant is most concerned to reject – the concept of absolute impenetrability admitting no compression or penetration whatsoever,” and Kant’s “introduction of a repulsive force is . . . intended decisively to undermine this concept” (, ). Kant’s argument for Proposition appeals to the single Proposition of the Phoronomy chapter, and Friedman supplies an ingenious solution to the puzzle of how Kant could rest his attribution of repulsive force to matter on his procedure for constructing composite motion. On Friedman’s interpretation, if a body resists with “absolute” impenetrability, then the motion of a particle that “strives” to penetrate it (MAN, :) cannot be constructed, because it has no “well-defined velocity at the turn-around point” (, ) where the body’s resistance is exercised. But on Kant’s own conception of impenetrability (in terms of repulsive force), the motion of a particle “striving” to penetrate can indeed be constructed. On the assumption that Kant is already concerned with the mechanical concept of matter in Proposition , it appears puzzling that he continues to argue for the dynamical concept through the very end of the chapter. Friedman’s account elegantly solves this puzzle: “Rather than ruling out the merely mathematical concept of impenetrability all by itself, the first proposition is simply intended to demonstrate how Kant’s preferred dynamical concept is intimately connected, in turn, with his conception of the composition of motions already developed in the Phoronomy” (p. ). The problem I would raise for Friedman’s interpretation is that it seems at odds with Kant’s description of “the mathematician’s” procedure. Kant distinguishes between the construction “of the concept of a matter,” for the sake of which the mathematician assumes impenetrability “as a first datum,” and the “further construction” of which the mathematician’s assumption is incapable. So he does not claim that the mathematician is incapable of constructing the concept of matter (as filling space), as on
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Friedman’s account. Kant’s objection is, rather, that by taking impenetrability as primitive, the mathematician “declares” the datum from which he constructs the concept of matter “to be something entirely incapable of any mathematical construction” and thus prevents us “from going back to first principles in natural science” (MAN, :). I am not so much concerned to dispute the details of Friedman’s account as to question the assumption, which I think is widely held, that the concept of matter as filling space through its mere existence is just the mechanical concept of matter that comes in for criticism later in Dynamics. Now it is clear that Kant associates the “purely mathematical concept of impenetrability,” on which “matter as matter resists all penetration utterly [schlecterdings] and with absolute necessity” (MAN, :), with the mechanical concept of matter. On Kant’s dynamical concept, in contrast, impenetrability varies in degree according to the compression of matter. The issue is whether resisting penetration “utterly and with absolute necessity” is the same as resisting penetration “through mere existence,” or in accordance with the principle of contradiction. One reason to think it may not be is that J. H. Lambert is the only representative of “the mathematician’s” conception that Kant names in the Remark to Proposition , and while Lambert cleaves to the view that matter’s “solidity” (the property by which it fills a space) is primitive – and thus belongs to matter through its mere existence – he expressly leaves open the question of whether this property might vary in degree according to the compression of matter. Following Friedman, we may note that Lambert (who was deeply impressed by Euclid’s deductive method) supplies axioms and postulates for every “simple” concept of his metaphysics. The axioms governing the concept of solidity include that “the solid excludes other solids from the place where it is,” that “space can be no more than filled with solids,” and that “the solid has an absolute density and is therefore an unchangeable unity” (, §; translation Friedman , ). Lambert’s postulates “express the idea that there is a maximum density whereby a space is completely filled, without interspersed empty spaces, so that the only way in which a filled space can be diminished or compressed is by eliminating such empty spaces until it is completely filled” (Friedman , ). But Lambert nonetheless regards it as a genuine question “whether all solids are in themselves equally dense and are in this respect an absolute and
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In contrast, Euler explicitly claims that impenetrability is not capable of quantity in § of c; see Gaukroger (, ).
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unchangeable unity,” or whether “a completely filled space could not be still more filled intensively, [such that] the solid that fills it could be brought into a still smaller space.” According to Lambert, we acquire the concept of solidity “through touch [Gefu¨hl], and this does not provide us with the inner differences thereof.” In the concept that we have, there also seems to be no impossibility [in the idea] that the solid could have different degrees of inner density. The above axioms would thereby have to be changed so that solidity would not be an absolute and unchangeable unity, so that a completely filled space could be filled with more or less dense solids, and so on. (Lambert , §; translation Friedman , ).
It is striking that Lambert continues to acknowledge this possibility even with regard to “what the whole thing depends on”: whether we can go “through the entire realm of possibilities” by means of his axioms and postulates, so as to realize the “Ciceronian [ideal]: Si dederis, omnia danda sunt” (, §). Here Lambert notes that if “inner differences” in solidity are admitted, the resulting increase in “combinations and permutations” will greatly increase the multiplicity of (physical) possibilities (§). The example of Lambert seems to show that the conception of matter as filling space through its mere existence does not imply that its impenetrability is absolute (i.e., that its resistance to penetration cannot vary in degree). This raises the question of why Kant later characterizes the alternative “dynamical concept of matter” as “that of the movable filling its space (to a determinate degree)” (MAN, :). I do not think we should understand this claim as a definition, for then it would beg the question (and in any case the adjective “dynamical” pertains, in the first instance, to moving forces). Rather, I think that the initial contrast between matter as filling space through a moving force, and as doing so through its mere existence, leaves indeterminate which concept (if either, and not both) allows space-filling to vary in degree. The intervening argument (beginning, I take it, with Proposition ) is intended to show that while it might not be inconsistent with Lambert’s conception of matter to suppose impenetrability varies in degree, he is not entitled to this supposition. For he cannot give this quantitative property a metaphysical basis of the right kind, nor can anyone else who holds this conception of matter. The problem is that in directly linking impenetrability to the existence of matter, “the mathematician” has precluded any
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Cf. MAN (:): “The general principle of the dynamics of material nature is that everything real in the objects of the outer senses . . . must be viewed as moving force.”
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“further construction” of this property (which, as itself quantitative, may suffice to construct the concept of matter). We get insight into what such a “further construction” would be from Kant’s account of the advantages of his own conception of impenetrability, in terms of the fundamental force of repulsion. Even before introducing the fundamental force of attraction (and thus before presenting the unsatisfactory construction of matter’s property of filling space), Kant emphasizes that his concept of impenetrability “yields a concept of an acting cause, together with its laws, whereby the action, namely the resistance in the filled space, can be estimated in regard to its degrees” (MAN, :). We can now ask in which respects Kant’s account of impenetrability surpasses that of the concept of filling space through mere existence. If matter’s resistance to penetration arises through its mere existence, but comes in degrees, then presumably these degrees could be “estimated” according to a law – for instance, a law relating them to the matter’s compression. The decisive advantage of Kant’s account would seem to be that on it, this law can be one of an “acting cause,” whereas on the rival concept what is “estimated in regard to its degrees” is mere existence rather than action. But here Kant’s account faces the problem that, by his own admission, “no law of either attractive or repulsive force may be risked on a priori conjectures” (MAN, :): we cannot specify the content of any such law a priori, and it is not clear how much we can determine a priori about its form. It thus appears that either the law Kant associates with an “acting cause” must have an empirical basis (in which case it could not belong to a metaphysical account of matter) or else it is Kant himself who gives too much leeway to a priori speculation. I take it that Kant’s best hope for escaping this dilemma is to appeal to the role of repulsive force as a condition of possible experience (specifically, as explaining the possibility of determinate volumes of matter, and “furnish[ing] us with concepts of determinate objects in space” [MAN, :]). Only in a condition of possible experience do we have an a priori factor that cannot be arbitrarily assumed, and which at least might offer a contentful explanation of a law for “estimating degrees.” So this is the respect in which Kant’s account of impenetrability surpasses what Lambert has to offer.
. Conclusion In the Preface, Kant claims his “analysis of the concept of a matter in general” is to yield “principles for the construction of the concepts that belong to the possibility of matter in general” (MAN, :). Contrasting
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the moments of Kant’s analysis with Lambert’s treatment of the concept of matter and with physica generalis indicates that the concepts in question belong specifically to the possibility of matter as object of experience. This much could perhaps be gathered from Kant’s remarks in the Preface, but it is not explicitly stated there. Conversely, Kant’s rather obscure criticisms of physica generalis and of Lambert are clarified by his positive view of how metaphysics is to secure the applicability of mathematical concepts. It does not come as news to readers of the first Critique that the a priori treatment of concepts such as force must proceed in “relation to the form of an experience in general and the synthetic unity in which alone objects can be empirically cognized” (KrV, A/B) – that is, to conditions on the possibility of experience – if the concepts are to be more than “figments of the brain” and have objective reality. Nor is it news that for Kant, “only a metaphysical, or nonmathematical and nonexperimental, theory of matter can effectively restrain pure speculation in the philosophy of nature and thus provide a foundation for the development of a mathematical and experimental physics” (Pollok , ). But I think it is still worth reflecting on cases, such as Kant’s treatment of empty space and of the “mathematical” concept of impenetrability, in which Kant seems to want to restrain pure speculation precisely by demanding a relation to conditions on the possibility of experience. For one thing, Kant conceived his theory of matter as “nonmathematical and nonexperimental” as early as the mid-s; in fact, the remark of Pollok’s just quoted is about MonPh. But Kant’s appeal to conditions on experience’s possibility is a distinctively critical move, so highlighting it helps us to see how MAN is the culmination, as Pollok also says, of a three-decade-long “research program” (, ). For another, these examples show that an account of how mathematics can be true of objects of experience is not an eo ipso account of how experience itself is possible. They thus invite us to consider what more (than the application of mathematics) is involved in the “law-governed connection” that makes the manifold of empirical representations into experience (:).
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In particular, from Kant’s description of “true metaphysics” at : and from his claim that “the basic determination of something that is to be an object of the outer senses had to be motion, because only thereby can these senses be affected” (MAN, :). I worry that Brittan, for instance, may conflate these two explanatory projects; see Brittan (, –).
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Phoronomy Space, Construction, and Mathematizing Motion Marius Stan
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Introduction
With his Phoronomy chapter, Kant defies even the seasoned interpreter of his philosophy of physics. Exegetes have given it little attention, and understandably so: his aims are opaque, his turns in argument little motivated, and his context mysterious, which makes his project there look alienating. I seek to illuminate here some of the darker corners in that chapter. Specifically, I aim to clarify three notions in it: his concepts of velocity, of composite motion, and of the construction required to compose motions. I defend three theses about Kant: () His choice of velocity concept is ultimately insufficient. () He sided with the rationalist faction in the early-modern debate on directed quantities. () It remains an open question if his algebra of motion is a priori, though he believed it was. I begin in Section . by explaining Kant’s notion of phoronomy and its argument structure in his chapter. In Section ., I present four pictures of velocity current in Kant’s century, and I assess the one he chose. Section . is in three parts: a historical account of why algebra of motion became a topic of early modern debate, a synopsis of the two sides that emerged then, and a brief account of his contribution to the debate. Finally, Section . assesses how general his account of composite motion is, and if it counts as a priori knowledge. To achieve my aims, I use two methods. One is to translate his key terms into the language of modern kinematics. This approach lets us disambiguate his notions, clarify their explanatory connections, and better
Hereafter, by ‘Phoronomy’ I mean his chapter, “Metaphysical Foundations of Phoronomy,” and by ‘phoronomy’ the discipline that Kant so denoted. The same goes for his other chapters (Dynamics, Mechanics, etc.). Throughout, I use ‘Foundations’ as a convenient name for his book at issue in this volume, viz., Metaphysical Foundations of Natural Science. Unless noted otherwise, all translations are mine.
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grasp the scope and limits of his phoronomic foundation. The other method is to read the problems and theses of Phoronomy in the context of early-modern efforts to mathematize motion. I show that Kant was part of a long effort to secure algebraic structure for directed quantities – which the new science of motion needed, but classical mathematics could not provide. This casts fresh light on Kant’s phoronomy and on his place in the long history of foundational debates in mechanics. More broadly, grappling with Phoronomy is good training for a muchneeded examination of Kant’s notion of proof in mathematical physics: its epistemology, sources of evidence, and reliance on central distinctions in his thought, such as a priori/a posteriori, pure/empirical, and the like.
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The Subject Matter of Phoronomy
By his account, phoronomy studies matter regarded just as “the movable in space.” It “abstracts from,” or leaves out, all internal structure in matter, and so it considers just “motion, and its magnitude” – specifically, its speed and direction (MAN, :). The challenge is to explain these ideas without paraphrase, in clear concepts – preferably, our concepts. That task faces several obstacles, so I try here to remove them first. In particular, there are some red herrings that can lead astray even the wary reader. One is Kant’s very term. ‘Phoronomy’ had been a coinage used just twice before Foundations, and in contexts quite unrelated to Kant’s usage. Leibniz invented the word in the s to denote the doctrine of the “laws of nature,” whereby he meant the dynamical principles of collision theory. Then his disciple, Jakob Hermann, adopted it for his own project, a comprehensive treatise in particle dynamics. Kant does not mean his term in these senses. For the Leibnizians, phoronomic doctrine dealt indispensably with the causes of motion processes, whereas Kant is clear that his phoronomy is a pre-causal treatment. Another red herring is the modern notion of ‘kinematics.’ Just decades after Kant, Ampère and Poncelet invented it to denote the purely geometric, descriptive, and non-causal account of motion. That sounds much like
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By ‘algebraic structure’ I mean the concepts and principles that legitimize addition, subtraction, and related operations. In this regard, I also use ‘algebra of motions’ as an auxiliary synonym for the idea that motion-quantities can be added and subtracted. The term came from the Greek for local motion (phora) and law (nomos). For a sense of Leibniz’ meaning, cf. his dialogue Phoranomus, sive de potentia et legibus naturae. The book’s full title was Phoronomy: The Forces and Motions of Bodies, Solid and Fluid. Kant had it in his library (Warda , ).
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Kant’s objective in Phoronomy, and so we might think that his term is an exact synonym for ‘kinematics.’ But that would be a mistake. There is no single, all-encompassing discipline that covers descriptively all species of motion; there are just local theories, fit to describe some species but not others. Namely, each branch of classical physics has its own kinematics, with just enough mathematical structure for the needs of that branch, not generally. Hence saying that phoronomy is a kinematics leaves things fundamentally incomplete – we ought to also add what his kinematics is for. Finally, the third red herring is Kant’s announcing that phoronomy is a general account, qua part of the “general doctrine” of body. Recall his claim to have “completely exhausted this metaphysical doctrine of body, so far as it may extend” (MAN, :; my emphasis). That is unhelpful in two respects. Qua descriptive theory of motion quantities, phoronomy is not general – far from it, in fact. There are many motion species that his chapter does not cover and could not possibly cover, because the conceptual basis he offers is too weak for them. Moreover, his phoronomy does not really describe the motion of bodies. Namely, it is too weak to describe their motion as extended volumes of matter, which require stronger concepts (and mathematics, too) well beyond the basis of his chapter. Insofar as it applies to bodies, phoronomy is valid of them only under very narrow, restrictive assumptions that Kant unfortunately mischaracterized. I detail these charges in Section .. Then what is phoronomy? I propose here a construal of his movability doctrine in modern terms. To avoid the threat of incompleteness I signaled above, I specify these terms as best I can. I claim that Kant’s phoronomy is a kinematics for particle collision in a force-free vacuum. Just what grounds I have for my construal will become clearer in Section ..
. Kant’s Concept of Speed The keystone of Kant’s account of how we mathematize motion is a notion of speed at an instant. He was right to give it such attention,
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That is, there is a kinematics of wave motion; one for free-particle orbits; one for rigid-body motion; yet another one for continuous deformations; a kinematics for constrained-motion systems; one for continua with microstructure, e.g., liquid crystals; and so on. No single branch of mathematics has the conceptual resources to treat all of these motions within one framework. For example, for free particles and deformable continua, the most comprehensive kinematics requires as mathematics the differential geometry of skew curves and surfaces; whereas for rigid bodies we need to add tensor algebra, so as to describe the kind of rigid motion known as ‘change of attitude.’
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because it was a thorny problem. Between Galileo and Lagrange, mechanics relied on mathematical descriptions of motion that were often implicit and dependent on foundations shifting rapidly. Kant found himself amid this long age of drastic change. In particular, around the time of Foundations there were four notions of (instantaneous) speed available. Here I present those concepts; I use them to elucidate Kant’s preferred concept; and then I assess his choice, as befits this guide. To keep them apart, I give them below custom names; in every case, let P be a particle moving in some curve. Pre-classical. Speed is the ‘intension’ of a ‘form.’ Namely, it assumes that P has a ‘form of motion’ at every point of its trajectory. Imagine P to travel for some finite time T over some distance S while having the same ‘form’ at every instant, namely, ‘uni-formly.’ Instantaneous speed is the numeric ‘intension’ C of the form, and it equals S/T defined as above. Condensive. Speed at a location X is the ‘condensation point’ of a series of decreasing values. Namely, let XA, XB, XD, and so on be segments standing for future paths of the particle P currently passing through X. And let their lengths increase serially, that is, XA < XB < XD, and so on. The speed C is the value toward which these lengths tend, if taken smaller and smaller. In our terms, C is their ‘limit from above.’ Differential. Speed is a ratio of two infinitesimals. Let ds be the infinitely small path that P crosses in an instant, that is, an infinitely small time dt. P’s speed is the ratio ds/dt. Analytic. Speed is a part of an algebraic object. Let P’s motion be representable by an analytic function of time f(x, y, z, t). At any location X on its path, P’s speed equals the coefficient of the second term in the Taylor-series representation of the function f at X. Alternatively, at any instant, the speed of P is f’s derivative with respect to time. Now I give evidence for the concepts above. I called the first notion ‘preclassical’ because it shows up early, well before the law of inertia and the classical mechanics that it engendered. The notion received clear expression already in the fourteenth century from figures who reflected on mathematizing motion. At Paris in the s, Nicole Oresme explained how speed links up with extended distance: “We imagine punctual instantaneous speed by means of a straight line” (, ; my emphasis). The verb ‘to imagine’ was his way of conveying that in considering instantaneous speed we do not grasp a stretch of space actually crossed, but rather
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one that we may imagine the mobile to traverse (spatium quod ymaginatur pertransiri) counterfactually, if it crossed it at the same punctual speed (Oresme , ). This need (to rely counterfactually on stretches not actually traversed) is implicit in another, equivalent definition by Roger Swineshead at Oxford: “of local motions, one is swifter [velocior] when, by the intension of the former motion, one could cross [poterit pertransiri] a greater space, during some time, than by the latter motion’s intension.” The thought that we may use line segments to quantify punctual speeds is even older; we see it voiced by : “the proportion of the motions of points is as the proportion of straight lines described in the same time.” And the pre-classical notion of speed survived well into the early modernity. John Wallis relied on it: “Speed [celeritas] is an affection of motion, and it results from comparing distance [longitudo] and time; that is, from determining how much distance is crossed in how much time” (, ). But Wallis and his age made a mistake while adopting the medieval concept. Note that, as Oresme and Swineshead above knew, to define punctual speed we must resort to counterfactual distances; but the early moderns omit to specify this crucial point. In so doing, their failure (to mention that S and T denote non-actual stretches) becomes a crippling defect. In sum, it replaces the desired concept (i.e., instantaneous speed) with the wrong one, namely, average speed, which concept is not at issue here. The condensive notion of speed is not explicit in eighteenth-century works. Only its underlying concept, namely, of limit of a sequence, can be found. D’Alembert defined it as the fixed value toward which a convergent sequence tends: “a magnitude [grandeur] is the limit of another one when the latter can approach the former magnitude by more than any given, arbitrarily small magnitude, such that the difference (of the approaching magnitude) to its limit is unaccountably small” (, ). Elsewhere, he singled out certain sequences – for instance, a sequence of ever-decreasing distances, as Kant’s diagram (for his Parallelogram Rule) allows us to visualize – an especially easy way to test for convergence: “for a series to be as perfect as possible, it must be the case that ) its terms consecutively decrease, after the first one; and ) all of its terms have the same sign” (d’Alembert , ). But I cannot find direct evidence for the condensive speed concept in the s. Sutherland (, ff.) attributes it
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Translated from Roger’s unpublished Erfurt Manuscript, page retro, column a; excerpted in Sylla (, ). Translated from Gerard of Bruxelles, Liber de motu, excerpted in Clagett (, ).
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to Kant. But we must count it merely as a possible interpretation – namely, just compatible with the textual evidence – because Kant himself did not declare it overtly. The differential concept shows up as the official notion of speed in Euler’s Theoria motus, the first significant tract in rigid-body dynamics; Kant appears to have seen it, though perhaps too late. There Euler introduces uniform and non-uniform motion, declares that “they differ in their essence,” and goes on to treat non-uniform motion (inaequabilis) in a straight line first: [In problems of particle mechanics], the entire business reduces to finding the place where the moving point shall be at any arbitrary given time. Thus let AB be the straight line in which the point moves, starting from A. After a time ¼ t, the point will be at place S, and let AS ¼ s, the distance crossed in time t.. . . By differentiation, we obtain the element of distance ds that the particle crosses in an element of time dt. And the fraction ds=dt expresses the moving point’s speed at the place S. Evidently, this fraction is a finite quantity. Hence, if we let v denote the speed at S, we have v = ds/dt. Consequently, for particle motion we can assign a speed at any place or also at any instant. (Euler c, ; my emphasis)
Thus Euler taught lucidly the notion of instantaneous speed qua local magnitude defined at a point. And he warned that C ¼ S=T, the definition favored by Kant and others, is valid only in a very narrow context, namely, when a particle moves in uniform translation. Last, the analytic notion of speed was Lagrange’s singlehanded creation. He began by taking for granted that we may represent particle position by some coordinate function; indeed, that was established practice by then. Lagrange refined this assumption in two respects. First, he imposed the condition that any such coordinate mapping must be an analytic function. Second, he assumed that, for any such function, its value at a point ðx þ i Þ can be represented by a Taylor-series expansion (, ): f ðx þ i Þ ¼ f ðx Þ þ pði Þ þ q ði Þ2 þ rði Þ3 þ . . . :
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Euler’s book was reissued in , and Kant in the Opus postumum, written largely in the s, once refers to “Euler’s materia rigida” (OP, :). Start with the notions of a variable quantity x, y, and so on; of value of a variable, namely, a particular number; and of a constant quantity a, b, c, and so on. (The post-Leibnizians called them ‘symbols,’ as did Kant.) In the eighteenth century, an analytic function was any ‘expression,’ or syntactic string composed from variables, constants, and the common algebraic symbols þ, % , :, &, plus exponentiation. See the definition in Lagrange (, ). In the formula below, p, q, r are component functions of Lagrange’s “derived function.” See below.
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Against this backdrop, for any particle motion given by a function f of coordinates and time, its speed at point w is the second term in the Taylorseries representation of the value f ðwÞ that the function takes at w: The functions [that we use in mechanics] inevitably relate to time, which I will always designate by t. And so – since a point’s position in space depends on three rectangular coordinates x, y, and z – in all mechanical problems I will take these coordinates to be function of t.. . .
Generally then, in any rectilinear motion where the distance crossed is a given function of the time passed, the first function of this function represents the speed, and the second represents the accelerating force at some instant. (Lagrange , , ; my emphasis)
That was his way of saying that speed is the first derivative of the particle’s change of coordinates with respect to time. With this synopsis of kinematic concepts behind us, three questions need our attention now. Which concept of speed really was Kant’s notion? We have no direct evidence for an answer – he remained oracular about it. Exegetes have ascribed him two such concepts. Sutherland says Kant used the condensive notion, but Friedman gave a strong argument that it would not be the right account to attribute to Kant. Why did Kant choose his particular speed concept? There is an externalist explanation for his choice, but it is philosophically unsatisfying. Fortunately, Kant had an internal reason to prefer the pre-classical concept. Namely, the concept integrates very naturally with his theory of mathematical cognition in two respects. One, it allows us to show that speed can be constructed geometrically, that is, represented by the
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Lagrange’s phrasing is opaque, but easy to explain. For him, the Taylor expansion represents a function f by means of a “derived function” g, which is an (infinite) sum of other functions 00 0 00 f 0 , f , f , etc. Let u, v, w be consecutive values of some real variable, such that they differ from the next one by an infinitesimal amount du. (For example, they could be points in space lying next to each other; or consecutive instants of time.) To represent f ðv Þ, i.e., f ’s value at v, we use g ðvÞ, built 00 0 00 as follows: g ðv Þ ¼ f ðuÞ þ af 0 ðv þ duÞ þ bf ðv þ duÞ þ cf ðv þ duÞ, etc. In this infinite sum, 0 f ðuÞ is a fixed value, not a variable. Therefore, f counts as Lagrange’s “first function of the function” g. As is well known, f 0 is the first derivative of f , the represented function. See Friedman (, –) and Sutherland (, –). I am not sure what Sutherland’s evidence is for attributing the condensive speed-concept to Kant. Apart from brief interludes around mid-century, the German lands in the s were a backwater of mathematics; what cutting-edge research there was (by Euler, and then Lagrange) was confined to the Berlin Academy, from which Kant and his peers – campus metaphysicians, with few exceptions – were cut off. To compound his predicament, Kant’s mathematical training was always deficient, and his lack of French (the lingua franca for new science then, with Latin a distant second) prevented him from trying to keep up with the work of Euler and Lagrange. Cf. also Rusnock (, sec. ).
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picturable singulars that he considered essential to construction. Two, Kant thought that pre-classical speeds being constructable in pure intuition lets him argue that operations on speeds count as synthetic a priori knowledge. I turn to this point below, in Section .. Finally, was it a wise choice? On this point, I am not sanguine. Kant’s velocity concept has an unobvious but real shortcoming. Namely, it is not general. More exactly put, the defect is this. Systematic reasons drove him to adopt geometric representings of the quantities crucial to a modern theory of motion – position, velocity, and acceleration. But, qua descriptive language for those parameters, the geometric framework is too weak for modern theory as it had grown by his age. First, it lacks a general way to express accelerations, though it admittedly allows us local, configurationspecific ways of representing velocity difference or acceleration. Second, it cannot be used to write differential equations. And yet mechanics after had evolved into a stage that required the two capabilities above as sine qua non features of any mathematical language aspiring to be a representational vehicle for kinematics. In particular, consider how Euler stated his “new principle of mechanics,” that is, the law of motion that governed all the mechanical processes solved mathematically by then, including some that neither Newton nor Kant had analyzed. Euler wrote his law in component form, that is, stated relative to each of three orthogonal axes of an inertial frame external to the system to be described. And for each motion-component, he used a coordinate function to represent it: I : 2Mddx ¼ Pdt 2 ; II : 2Mddy ¼ Q dt 2 ; III : 2Mddz ¼ Rdt 2 :
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One local approach to acceleration was by means of ‘natural’ coordinates. Specifically, at a given point C of the particle’s motion (in a plane), one represented the induced acceleration as two speed increments, along two lines: one tangent to the trajectory at C – hence, locally – and one normal to the trajectory, likewise at C. The task was to infer the magnitude of these speed increments, from the known forces acting on the particle at that location. Once known, these increments would be ‘composed’ – via the Parallelogram Rule – with the particle’s (already known) velocity at C. The result of this composition was knowledge of the location D where the particle would be at the next instant, once it moved past C. At that stage, the task above had to be reprised: a new ‘composition of motions’ was needed, to find the location E at the following instant; and so on. A vivid illustration of this natural-coordinates approach is Newton’s proof of the Area Law in Principia, Book One, Proposition VI. P, Q, and R are the components of the total impressed force (on a point mass) along three orthogonal axes Ox, Oy, and Oz supposed immobile; x, y, and z are coordinate functions, while dx/dt, dy/dt, and dz/dt are the components of the resulting acceleration (induced by the total force). The factor is needed because M is not really a mass; it is a weight, with the value of g, the acceleration of gravity, set by convention at ½.
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Thus Euler makes clear that, in regard to mathematical form, all the laws of motion are differential equations. For the scientific elite, then, successfully mathematizing matter in motion amounted to deriving equations of kinematic change at a point, over an instant, for that particular type of matter. The import of these facts is: in Kant’s time consensus among theorists had coalesced around three convictions. First, the laws of physics had to be stated in coordinate functions. Second, quantifying the motions of particular bodies amounted to finding the derivatives of these functions. Third, inferring to those derivatives was through deductive reasoning from differential equations, not geometric construction – because those equations cannot be represented geometrically. To be sure, some role remained for the geometric approach that Kant favored: it was useful for motions tractable in ‘natural coordinates.’ The fact is, however, that such geometric approaches were not general. They worked just for particle motions under gravity-like forces as Newton had geometrized, with great success, in the previous century (see Fig. .).
Fig. . Newton’s diagram for his proof of Kepler’s Area Law. Bc is a counterfactual path that the planet-qua-particle would cross in an instant, if it moved inertially with the velocity it had at location B. And BV is another counterfactual path, which the planet would cross in an instant if it were sitting stationary at B (instead of moving, as it in fact does) while a gravity-like force (emanating from S) acted on it. Finally, BC is the effective path, which the particle actually crosses in an instant – as a result of arriving at B, where the force S acts on it. Note that BC results from applying the Parallelogram Rule to the two counterfactual motions above. Mutatis mutandis for the planet’s motion at the subsequent locations C, D, E, and so on.
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In sum, Kant’s concept of velocity appears seriously hampered, if the foundational intent was to ground a sufficiently general theory of kinematic experience. If I am right, then we should ask again, Just how much explanatory scope does his metaphysics of phoronomy have?
. Kant on Composite Motion The key result in Phoronomy is the Parallelogram Rule, a composition theorem for adding velocities. I begin with a baffling fact: Kant constructs the Rule – but why? Why is there a geometric figure (and diagrammatic reasoning on it) in a treatise of metaphysics? After all, Kant is famous for drawing a sharp line between the evidence-gathering methods of metaphysics and mathematics. One reason is that velocity crosses architectonic boundaries, so to speak. It counts as an intensive magnitude, but it is associated with extensive ones as well (the size of spaces crossed at various speeds). Thus, treating degrees of speed with the mathematics of extension seems to rest on certain grounding assumptions that Kant wishes to uncover and defend: “it is not clear by itself that a given speed consists of smaller speeds (and a rapidity consists of slownesses) in the way that a space consists of smaller ones” (MAN, :). Another rationale for him to discuss composition is for the sake of a philosophical explanation. In particular, Sutherland has argued that Kant needs his diagram so as to explain how we represent an identity: of parts and the whole they make up. The compounding motions are the parts, and the composite motion is the whole (Sutherland ). I suggest that Kant would have had yet another, third, philosophical aim with his diagrammatic construction and philosophical explication of it. Specifically, I read Kant’s diagram as an episode in the early-modern efforts to show that directed quantities have an algebra – they add and subtract – and to clarify the epistemology of their mathematization. Adding velocities (by the Parallelogram Rule) was the gateway insight of early-modern science, on a par with the Law of Inertia but far more useful than it. And yet, crucial and indispensable as it was, velocity addition was a foundational enigma for early-modern theorists. There were two sources
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Like any inertial dynamics, early-modern theory grappled with the generic task of mathematizing deflections from inertial paths. (Inert translation, if it ever occurs, is trivial to quantify.) But that process always requires adding velocities as the sine qua non operation. It is because, from one instant to the next, forced particles move along the resultant, or ‘composite’ motion, of two velocities: one that the particle has and keeps (by the Law of Inertia), and an acquired velocity
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Fig. . Pairs of velocities respectively equal in size. In each case, the heavy arrow represents their resultant, obtained in accordance with the Parallelogram Rule
of difficulty, and they required much skill and insight to navigate safely past them. Briefly, the difficulties were the following: • •
Velocities have directions too, not just sizes. And direction makes a difference to the size of the sum, or resultant, of their addition. But the algebraic framework of classical mathematics had no way to determine the result of adding velocities. It lacked rules for adding directions.
To grasp the first difficulty above, consider the two combinations in Fig. .. Each involves the addition of two velocities pairwise equal in size: their speeds are the same. But one velocity differs in direction from its sizetwin in the other pair. That difference alone is enough to make a difference to their addition – by affecting the size of the resultant. And so, a difficulty of composite motion was that, when it comes to adding velocities, knowing the size of the parts is not enough to infer the size of the whole. A confounding predicament, clearly. To grasp the second difficulty, recall a key fact. Before the s reforms in mathematics, the canon for adding magnitudes – more generally, the algebra of quantities, as we put it – was recorded halfway through Euclid’s Elements. It is the theory of proportions historically credited to Eudoxus of Knidos. Like everyone, then, Kant accepted it as the algebraic framework for mathematics qua general science of magnitudes. But that framework had a debilitating gap. Born well before early modern kinematics, Eudoxean proportion theory was not designed to
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increment, due to the net impressed force (acting on it at that point and instant). See, again, Newton’s above derivation of the Area Law. See the conclusive case for that in Sutherland ().
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Phoronomy: Space, Construction, and Mathematizing Motion
Fig. . Pairs of equal speeds. Which pair represents two equal motions, and why – on what grounds?
handle oriented magnitudes, namely, objects having a size and also a direction. The reason is that Euclid, Eudoxus, and their pre-modern descendants counted as magnitudes just two basic species of object: discretes, that is, integers and their fractions; and continuous magnitudes, that is, Euclidean multiples of straight-line segments. In consequence, this algebra had no explicit rules for taking direction into account when adding motion-magnitudes. As illustration, consider the question (Fig. .): What counts as two equal motions? This question is not well posed in Eudoxean proportion theory – it has no rule or algorithm whereby to answer it. But, then, without an answer to this question there is no way to decide whether one motion counts as twice as much as another – it is a “doubled celerity,” as Kant has it – or not; and so on, for every multiple. In sum: the classical theory of magnitude had no official rule for adding oriented magnitudes. The lacuna was felt painfully when Descartes and others began to theorize in optics, collision theory, and dynamics – the ‘mixed-mathematics’ areas in which resolving and adding motions is the key device. Lacking any guidance from Eudoxus and Aristotle, it took the seventeenth century many decades to understand how (and accept that) size is linked to direction inextricably, such that treating them separately would give the
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Suppose a line segment AB to be given, and let it be unit length, by convention. Starting with AB and using nothing but straightedge and compass, construct an arbitrary segment CD. A Euclidean multiple is any magnitude m such that m is to as CD is to AB. To be sure, embryonic forms of the Parallelogram Rule were known in Antiquity; for instance, in the Alexandrian tradition of statics. My point is that knowledge (of what results by applying the Rule) could not be fit into the official algebra, viz., Eudoxean proportion theory. As a result, knowers of the Rule would not have been able to explain why applying the Rule counted as an operation on magnitudes.
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wrong result. Then the early moderns faced up to another conundrum. Knowing that the Parallelogram Rule was true – but not in virtue of Eudoxean algebra – they had to answer the epistemological question, Why is the Rule true? That is, what is the evidence for its truth, and how strong is it? By the s, two approaches to answering this question had emerged, and they would compete into Kant’s age. Each approach had several species, but I can only very briefly mention them here; a real discussion of their subtleties belongs elsewhere. One approach was broadly empiricist and indirect. That is, the approach rested on empirical facts about forces and on conclusions about the motions – and their composition – associated with these forces. An example is Newton, who showed that his first two laws jointly entail that forces add by the Parallelogram Rule. In turn, single impressed forces generate accelerations, or velocity increments. It follows that a composite motion is proportional to (and inferable by the same Rule as) the composite force associated with it. Later, ’s Gravesande devised another proof procedure, even more decidedly empiricist. It relied on experiments with static forces on weights (, –). They showed that any two forces jointly acting on a body have a resultant equal to the diagonal of the parallelogram they form with each other (see Fig. ., top). The other approach was broadly rationalist. It relied on premises made true by some broad principle of rationality or on evidence accessible to rational intuition. The former was Daniel Bernoulli’s way, who relied on the Principle of Sufficient Reason: he started with three static forces f, g, h that add up to zero, (by the PSR, allegedly), and so they produce equilibrium in the body on which they act. Then he argued that any other three forces a, b, c obeying the Parallelogram Rule will be dynamically equivalent to f, g, h; that is, they too will cause equilibrium. The latter was d’Alembert’s procedure, who relied on a Gedankenexperiment with two parallel planes in uniform translation and a point moving relative to them – such that three relative velocities arise and can be added (d’Alembert , –). From this thought scenario, he concluded we can grasp rationally that his three velocities above obey the Parallelogram Rule (see Fig. ., bottom).
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These struggles are lucidly and rigorously explained in Miller (). See Corollaries II (on composition of forces) and I (on composite motions) to the section ‘Axioms, or Laws of Motion,’ in Newton, Principia, Book I (, –). Bernoulli’s starting premise posits three forces f, g, and h to be equal, at degrees with one another; he claims that PSR entails they are in equilibrium; cf. Bernoulli ().
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Phoronomy: Space, Construction, and Mathematizing Motion
Fig. . Top: ’s Gravesande’s experimental apparatus for confirming the Parallelogram Rule. The two outer weights produce a net resultant force R on the middle body. R is equal and opposite to the body’s weight (because the body remains at rest). The apparatus yields evidence that R is always equal and collinear with the diagonal of the parallelogram formed by the two outer-weight forces. Bottom: d’Alembert’s a priori proof of the Rule. A is a point particle, mobile in a plane BDCA that sits, and slides without friction across, plane HMLK underneath it; cgOa represents where plane BDCA arrives, relative to absolute space, after a finite time
We can put the early-moderns’ problem above in Kant’s own terms, to make vivid his predicament. Let a, b, c L be ‘motions,’ that is, instantaneous velocities qua directed speeds, and let ‘ L ’ denote addition with direction factored in. For him, statements like ‘a b ¼ c’ have truth values, and they are knowable a priori. But Eudoxean algebra had no way to secure
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non-empirical evidence for their truth or L falsity. Hence Kant had to expand that algebra, so as to make ‘a b ¼ c’ into meaningful, determinate judgments. In effect, he argued that L the Parallelogram Rule (for inferring the size and direction of c in ‘a b ¼ c’) is synthetic a priori. In Phoronomy, I suggest, he aimed to show that the Rule is true on non-empirical grounds, so it is a priori; and it is synthetic, not analytic, as the evidence for it comes from a synthesis, not from discursive reasoning with concepts. I assess his dual claim below. Hopefully, knowing the early-modern context I uncovered above might make it easier to understand why Kant decided to insert a diagrammatic proof in a chapter ostensibly on metaphysical foundations.
. Scope and Warrant This being a critical guide, I end my contribution by assessing phoronomy in regard to its scope, or descriptive reach; and in regard to Kant’s warrant for his Composition Rule. Generality. So far, I have hinted at how phoronomy lacks generality when examined from outside, as it were: in light of mechanics’ thengrowing need for a descriptive vocabulary (of motion quantities) that far exceeds geometry in representational content. But the question of generality arises internally as well, from within his doctrine. In particular, his Phoronomy has two concepts of motion: ‘change of outer relations in space’ and rectilinear pointmotion. Kant show that the latter motion is mathematizable: its size is C = S/T, and its directions are additive. Still, this concept is just a species of motion (and a narrow one, at that). It is the former motionconcept that is general. However, Kant left it fallow: he did not analyze the notion of outer relations in space, did not survey its scope, and did not try to show that it is mathematizable. He gave us no argument that “change of outer relations in space” belongs in the “pure mathematics of motion.” To restate my worry above, then, Phoronomy has shown just that there is a Kantian-proper science of linear velocity, nothing more. And yet. Phoronomy may not be general, but it is consilient – with his main agenda in Foundations. In Mechanics Kant’s epitome of interaction is the collision of bodies, which he treats under very restrictive conditions: he reduces each body to a ‘representative point’ and treats their collision. In effect, Kant disregards the bodies’ extension and the very complicated
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Phoronomy: Space, Construction, and Mathematizing Motion
motions that bodies undergo because they are extended. He just singles out their mass-centers and analyzes their impulsive motion (in impact) as mere points. Given that restrictive treatment, to represent their motion quantitatively he needs just a kinematics for particle collision in a force-free vacuum. Which is exactly what I claimed (in Section .) that his phoronomy amounts to. Specifically, it is a meager kinematics that includes a notion of velocity of translation and the Galilean transformation – that is what Kant’s ‘composite motion’ really amounts to – which he needs to describe the collision from two different frames: the observer’s perspective and the ‘absolute space’ in which the two bodies collide with equal forces of motion. Apriority. Now I examine the claim that his proof (of the Parallelogram Rule) is a priori. That he thinks so emerges from his objection to empiricist attempts to prove it: the Rule “must be constituted wholly a priori, and indeed intuitively, on behalf of applied mathematics” (; my emphasis). But is he right to think that his own proof counts as synthetic a priori? The Rule is synthetic in the strongest sense: he infers it by diagrammatic reasoning on the output of a synthesis, namely, a parallelogram constructed in intuition. Still, is it a priori? That is not clear yet. There is a chance that, in the final analysis, the Rule might count as a posteriori. I give here two signposts to guide future discussion on the real epistemic status of his Rule, because I consider it an open question. Recall that apriority is a dual notion in Kant: semantic and epistemological. On semantic criteria, the Rule would count as a priori if it constructs pure concepts in pure intuition. Epistemologically, the Rule counts as a priori if all the evidence for it comes from non-empirical sources. This clarification should help us see why we cannot yet grant that his Rule is apodictically true. Consider the following. Semantically, the Rule relies on the concept of a relative space: some of the motions it constructs are motions of relative spaces relative to one another. But it is unclear if ‹relative space› counts as an a priori concept. Kant’s own words suggest the opposite, when read naturally: In any experience, something must be sensed – that is the real of the sensible intuition. Hence, the space too (in which we set up our experience of motion) must be perceptible: we must designate it through what can be sensed. (MAN, :; my emphasis)
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Again, see Stan () for the full details.
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That is, ‹relative space› is an acquired representation, so it must count as an a posteriori concept. Also, ‘relative space’ is not a term in the inventory of basic concepts of the mathematical disciplines that Kant regards as pure, namely, elementary geometry and arithmetic. In sum, if future exegesis wishes to count ‹relative space› as a pure concept, it must make a case for it. More broadly, Kant admits that ‹motion› is an empirical concept. Then rules for composing motion would count as operations on a posteriori inputs. Epistemologically, things are just as ambiguous. Kant wishes to restrict the scope of his Rule to rectilinear motions alone. Namely, he infers the Rule from a ‘phoronomic principle,’ that is, that rest and straight-line motion are equivalent: “I assume here that all motions [subject to the Phoronomic Principle] are rectilinear.” But what justifies these restrictive premises? Why not let all motions be phoronomic-relative, that is, equivalent to rest? He gives us a hint: “for, in the case of curvilinear motion, it is not the same in all respects whether I regard the body as moving and the relative space as resting,” or vice versa (MAN, :). In plain English, he means to say that rest and straight-line motion are dynamically indistinguishable – they count as the same state of motion – because of Galilean relativity. (Or, even more strongly, because of Newton’s Corollary VI.) However, things are not the same – and so the Phoronomic Principle is not true – in the case of rest versus circular motion. But Galilean relativity is an empirical fact. It falls out of Newton’s Third Law, namely, that impressed forces come in pairs and balance each other. Newton regarded the law as true a posteriori. In sum, a key part of Kant’s warrant for his Rule is empirical. Then how can it count as a priori true?
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Plausibly, we obtain it by ‘abstraction.’ We see a few instances of containers (with edges and walls stable enough for some time) that enclose a volume with bodies free to move inside it. Mentally, we strip away irrelevant differences between these seen containers (e.g., color, texture, apparent size, ingredient materials), and we keep what they have in common (viz., the straightness and nearrigidity of their edges, and the free mobility inside their contained volume). Then we cluster these commonalities into a new representation: the concept ‘relative space.’ In this context, Galilean relativity means two things. () Rest and uniform straight-line motion are equivalent states: neither requires any forces, nor produces any effects, that the other one does not. () The laws of motion work just as well in a relative space at rest and one in uniform rectilinear motion. In contrast, circular motion is different: neither () nor () hold of it, as Kant knows well (see below). Further, in Kant’s doctrine – and also in Newton’s science – Galilean relativity obtains only because of their respective Third Law, not unconditionally. Newton’s Corollary VI is a stronger version of Galilean relativity. It asserts that rest and all rectilinear motion (uniform and accelerated) count as equivalent states. Friedman () argues that Kant meant his Phoronomic Principle to be compatible with Corollary VI. Based on evidence from collision experiments. After him, in the eighteenth century, new evidence for the Third Law came from the mutual perturbations that Saturn and Jupiter induce on each other gravitationally. Cf. Wilson ().
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Phoronomy: Space, Construction, and Mathematizing Motion
Objection: Kant has his own version of the Third Law; he argues that it is a priori; and it too entails Galilean relativity. Ergo, all of the evidence for his Rule amounts to a priori warrant, ultimately. Answer: the law is established late in Foundations, as a foundation for mechanics. So, it is not available as a premise in Phoronomy, where he would need it for the Parallelogram Rule. Plus, he derives his law of action-reaction from premises about force (mechanical and dynamical) and about ‘active relations of matters in space.’ But those notions do not count as pure concepts. This attempted defense thus must face up to Kant’s own injunction against ‘impure’ constructions and inference from them: For the construction of concepts, we require that the condition for presenting them not be borrowed from experience. So, their construction must not presuppose certain forces, whose existence can be inferred only from experience. Put more generally: the condition for constructing must not be itself a concept that cannot at all be given a priori in intuition. Such are the concepts of cause and effect, action and resistance, etc. (MAN, :–; my emphasis)
If some of the concepts required for a construction are empirical, and if some of its premises are justified empirically, does the purported conclusion still count as a priori knowledge? McNulty () likewise notes these delicate aspects of constructing motion. His efforts to elucidate these aspects, combined with my worries above, support the notion that we need more scholarly work to untangle this knot in Kant.
. Conclusions On the reading I have proposed here, phoronomy appears to be Kant’s cautious, conservative attempt to articulate a geometry of motion compatible with his overall framework, methodology, and foundational agenda for the science of nature. Read this way, his project succeeds. Phoronomy, I argued, is consilient with his broader picture of how we use geometry to describe nature, with his constructive methods for mathematics, and with the centrality of collision in his philosophy of mechanics.
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I have spelled out these conceptual assumptions and premises in Stan (). I thank Bennett McNulty for his trenchant comments and careful editorial advice, and an anonymous referee for their advice. For helpful discussion, I thank Ian Proops, Katherine Dunlop, James Messina, and Daniel Warren.
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Space, Pure Intuition, and Laws in the Metaphysical Foundations James Messina
.
Introduction
In the Metaphysical Foundations of Natural Science, Kant’s project is to provide the metaphysical basis for a proper science of matter. What he calls a special metaphysics of corporeal nature involves determining the powers, properties, and laws of matter. These include the fundamental forces of attraction and repulsion, along with their laws of diffusion (treated in the Dynamics chapter), and the three laws of mechanics (treated in the Mechanics chapter). In the preface, Kant indicates that the pure intuition of space (MAN, :) and the “form and principles of outer intuition” have a crucial (though not yet specified) role to play in his project (MAN, :). I am interested in the role of the pure intuition of space, and of properties and principles of space and spaces (i.e., figures, such as spheres), in the Metaphysical Foundations’ account of laws. Kant speaks of space as the “ground,” “condition,” and “basis” of various laws of matter, including the inverse-square and inverse-cube laws of attractive and repulsive force (Prol, :; MAN, :) and the Third Law of Mechanics (Br, :; KrV, B; cf. BDG, :). Moreover, in his proofs of all the laws just mentioned, the language of “construction” figures prominently, which suggests that Kant’s proofs (somehow) rest on or involve mathematical construction in his technical sense (MAN, :–, :, :; MS, :–). Such claims give rise to a number of questions. How do spatial
For Kant, mathematical construction of a concept proceeds by means of the “presentation of the object [corresponding to a concept] in an a priori intuition” (MAN, :), where the intuition in question has “universal validity for all possible intuitions that belong under the same concept” (KrV, A/B). Given this last point, Kant speaks at times of construction of synthetic a priori propositions (KrV, A/B), “principles” (MAN, :; KrV, A), and laws, like the law for the surface area of a sphere (Prol, :–). Construction proves in the first instance as it were formal truths or formal laws, pertaining to mathematical objects, which, as considered in pure mathematics, are not existing entities but rather possible forms for such entities (KrV, B, A/B, A/
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properties and principles serve to ground this particular set of laws? Which ones is Kant appealing to? What, if anything, does the spatial grounding of the inverse-square and inverse-cube laws of diffusion have in common with that of the Third Law? What role, if any, does mathematical construction play in Kant’s proofs of these laws? Finally, how, if at all, are Kant’s grounding claims consistent with his other commitments, for example, his notorious denial in Prolegomena § that there are any laws that “lie in space” (Prol, :)? I try to answer these questions. On my interpretation, Kant takes spatial properties and principles to be the grounds of the laws of diffusion and the Third Law insofar as they serve to explain the mathematical character of those laws. (Grounding is thus here an explanatory notion.) In the case of both types of laws (of diffusion, on the one hand, and of the Third Law, on the other), Kant’s explanation has two stages. The first stage involves the mathematical construction of a spatial (or spatiotemporal) model that is designed to exhibit certain formal, broadly geometrical laws. In the first stage of explanation, those laws are then used to explain why, given various factors, the corresponding “real” laws of both types have the mathematical content that they do. In the second stage of explanation, Kant uses global properties of space, including what I call below its efficient and equitable character, to explain why the phenomena behave in accordance with the relevant model. I show how this account can be squared with, among other things, Kant’s denial that matter and force can be mathematically constructed as well as the claim in the Prolegomena referred to above. My interpretation runs counter to interpretations that deny or downplay the role of mathematical construction after the Phoronomy chapter, and bears on the question of whether and in what sense Kant is mathematically
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B; MAN, :n; KU, :n) – mathematical construction shows us what is formally possible (e.g., what shapes could be shapes of objects of experience [KrV, A–/B–]) and what properties are necessary consequences of what shapes and properties; but we cannot construct the following related things: existence, reality (KrV, A/B), the matter of appearance (KrV, A/ B, A–/B–), and forces (MAN, :–; KrV, A–/B–). Geometry supplies paradigm examples of mathematical construction, but construction also occurs in other mathematical disciplines, such as phoronomy (MAN, :), For helpful discussion of Kant’s understanding of construction (and of interpretive debates regarding it, which I can’t settle here), see Shabel (). E.g., Friedman (, , –). Plaass (, ) actually denies at one point that there is any mathematical cognition in MAN (see Pollok , – for criticism). Brittan (, ) initially grants a role for construction in the Phoronomy and Dynamics chapters, though his analysis ultimately seems to undercut any sort of construction or mathematization of forces in the Dynamics (e.g., p. ).
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constructing the concept of matter. More generally, it serves as a corrective to interpretations that focus on the dependence of Kantian laws of nature on the categories and other non-spatial conditions of experience. In Section ., I consider the role that properties and principles of space play in grounding the laws of diffusion, as well as what he calls the “universal law of dynamics.” In Section ., I respond to some objections. In Section ., I explore the role of these things in grounding the Third Law. In Section ., I conclude by explaining how my interpretation fits with the Prolegomena passage.
. The Laws of Diffusion and the Universal Law of Dynamics ..
Preliminaries
Some of the ways in which Kant deploys properties and principles of space and spaces in his account of matter in the Metaphysical Foundations are well recognized. There is, for example, his appeal to the infinite divisibility of space in his proof of the infinite divisibility of matter (MAN, :), his appeal to the infinitude of space and to the fact that it alone cannot limit matter and its forces in the balancing argument (MAN, :ff.), and his appeal to the properties of a line in his explanation of why there can be at most two fundamental forces (MAN, :–). That Kant also makes use of spatial properties and principles in his account of laws is indicated, inter alia, by the following two parallel texts. The first occurs in the General Note to the System of Principles in the B-edition of the first Critique: “For [space] already contains in itself a priori formal outer relations as conditions of the possibility of the real (in effect and countereffect, thus in community)” (KrV, B; cf. MSI, :). The parenthetical remark contains an allusion to the Third Law (the law of the equality of action and reaction). The second passage occurs in the General Remark at the end of the Dynamics chapter when he says that “space is required for all forces of
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Washburn (), Förster (), and Heis (), among others, claim that Kant holds that the concept of matter needs to be mathematically constructed in pure intuition. Friedman (), among others, denies this. I have in mind, inter alia, the accounts of Friedman (), Stang (), and Watkins (). See, e.g., Guyer (, –). If McLear’s recent reconstruction of Kant’s “affection” argument () is correct, then this is yet another place where Kant appeals to a priori properties of space, in particular, to its essentially relational character.
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matter, and since it also contains the conditions of the laws of diffusion of these forces, it is necessarily presupposed prior to all matter” (MAN, :). By the “laws of diffusion,” Kant means the inverse-square and inverse-cube laws of, respectively, attractive and repulsive force, which he had discussed earlier in the Dynamics chapter. In language parallel to that used above, he claims that these laws depend on “conditions” (Bedingungen) contained in space. To see what this grounding relationship is, and to see what similarities there are in the spatial grounding of both types of laws, we need to turn to the details of Kant’s account of each type of law, starting with the laws of diffusion. .. The Inverse-Square and Inverse-Cube Laws I will concentrate on Kant’s treatment of the laws of diffusion in the Metaphysical Foundations, but it should be noted that it is anticipated by discussions in earlier texts. In the Physical Monadology (MonPh, :–), Kant offers a geometric derivation of these laws that resembles in a number of respects his later treatment in the Metaphysical Foundations. Moreover, just a few years prior to the publication of the Metaphysical Foundations, Kant says that the inverse-square law is “customarily presented as cognizable a priori” and says that the “sources of this law” are “simple” “in that they rest merely on the relation of spherical surfaces with different radii” (Prol, :). Kant introduces his treatment of these laws in the Metaphysical Foundations as follows. In Note to Proposition of the Dynamics chapter, he says that it should be possible to derive from the two fundamental forces of matter “the possibility of a space filled to a determinate degree” (MAN, :). But to complete this task, and in turn to construct the “dynamic concept of matter, as that of the movable filling its space (to a determinate degree),” it is necessary to determine the ratios of attractive and repulsive force. That is to say, it requires deriving the laws of diffusion of these forces. Kant describes this derivation as a “mathematical task, which no longer belongs to metaphysics” (MAN, :). Since mathematics is a matter of construction, Kant is saying that deriving the laws of diffusion requires mathematical construction. In Remark , which
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Friedman (b, ch. ) offers a reading of Prolegomena § according to which Kant is not in fact endorsing a geometric derivation of the inverse-square law. For a critique, see Messina (a). Friedman (, n, –) acknowledges that it is natural to read Kant as deriving the laws of diffusion from geometry and thereby constructing the concept of matter in pure intuition, but he
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is divided into four points, Kant offers “an attempt at such a perhaps possible construction.” In Remark , Kant mentions and responds to some difficulties for his construction. (I return to these in Section ...) In Remark , I take Kant be showing how it is possible to construct a geometric model of the diffusion of the action of attractive and repulsion forces. Kant’s constructed geometric models are designed to exhibit formal, geometric laws from which we can then derive the mathematical character of the corresponding laws of diffusion. The general idea is as follows: if we represent – that is, model – the action/effect (Wirkung) of an attractive force in such a way that it diffuses uniformly from a central point across concentric spherical surfaces so that the total quantum of action/effect is always the same (with the intensity diminishing uniformly at each given distance), then the intensity of the action of the force at a point on a given spherical surface will be inversely proportional to the square of the radius. As the areas of the concentric spherical surfaces grow in proportion to the square of the distance from the radius (this being a geometric law of the surface area of spheres), the intensity of the action of attractive force will diminish accordingly. So we get an inverse-square law of diffusion for attractive force. If we model the action/effect of a repulsive force in such a way that it diffuses uniformly from a central point across an (infinitesimally small) spherical volume, then the intensity of the action/effect will be inversely proportional to the cube of the (infinitely small) distance. As the volume grows in proportion to the cube of the radius (this being a geometric law of the volume of spheres), the intensity of the repulsive force will diminish accordingly. So we get an inverse-cube law of diffusion for repulsive force. In what sense do properties and laws of space and spaces ground the inverse-square and inverse-cube laws? And what is the role of construction in the derivations? As I understand it, grounding for Kant is, in this instance, a matter of explanation; this is in line with his typical usage of ‘ground’ and its cognates elsewhere. In this particular case, Kant takes the geometric laws that are exhibited along with his models to explain
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concludes this reading is untenable. I respond to some of Friedman’s objections to the natural reading in Section .. As Warren (, ) points out, the total quantum of action/effect is to be conceived “as the product of the intensive magnitude of the effects and the extensive magnitude of the space” (here the size of the two-dimensional spherical surface). For helpful accounts of Kant’s notion of a ground, see Hogan (), Smit (), Stratman (), Stang (, ). As Stang shows, Kant’s account is part of a larger German rationalist tradition, exemplified by Wolff’s definition of a ground as “that through which one can understand why something is” (Wolff , §; quoted in Stang , ).
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why – given that matter exists and has attractive and repulsive forces, and given that these forces follow the pattern expressed in the models – these forces of matter must diffuse according to inverse-square and inverse-cube laws. That’s because the necessary, formal mathematical relationships – the laws pertaining to spheres – exhibited along with the model constrain the mathematical content of the dynamical (real) laws in such a way that we can explain why they must have the mathematical character they do. (Again, this stage of explanation takes for granted that matter has attractive and repulsive forces and that they radiate uniformly outward according to the spatial pattern depicted in the model. We will return below to the question of whether the “fit” of forces with the model itself admits of an explanation.) Explanatory grounds come in various forms for Kant. For example, grounds can be logical (where the ground and consequent are mere concepts and the grounding occurs by means of the law of noncontradiction) or non-logical/real; grounds can be rationes essendi (grounds of the possible being of a thing) or rationes fiendi (grounds of becoming or actuality); and they can be causal or non-causal. I take it that, as explanatory grounds of the laws of diffusion, the properties and laws of space and spaces (in this case, spheres) are non-logical/real, non-causal, rationes essendi. They are non-logical grounds because the sort of grounding at issue is not a matter of a relation between mere concepts as in an analytic truth; instead, the connection is synthetic. They are non-causal, rationes essendi of the laws of diffusion because space, spatial figures, and their properties and laws, as object of pure mathematics, are merely formal entities. As such they lack causal efficacy and existence – as Kant says, figures have an essence but not a nature as existing things do (MAN, :n). Nevertheless, as rationes essendi, properties of space (and spaces) can be said to make possible certain geometric properties and also to constrain the space of possibilities (as it were). They rule out certain logically possible properties (e.g., properties of the whole of space rule out the geometric possibility of a two-sided, enclosed rectilinear figure [KrV, A–/B]) and (non-causally and non-logically) necessitate the presence of other geometric properties (e.g., the three-sides of a triangle necessitate its having three corners [V-Met/Mrong, :]). My suggestion is that the properties of space and spaces are, in the same way, rationes
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See Stang’s excellent discussion () for such distinctions. In the Metaphysics Mrongovius Kant describes geometric objects using the language of “ratio essendi.” He gives the following example: “the three sides in the triangle are the ground of the three corners” (V-Met/Mrong, :). See note .
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essendi of some real laws governing the interaction of material substances. For Kant, spatial properties not only make possible the interaction of material substances; they also constrain the mathematical content of various causal (real) laws. (Recall KrV, B, and MAN, :.) Spatial properties (and associated spatial laws) rule out some logically possible laws and properties of the real, while requiring others. The interpretation I am offering of the spatial grounding of the laws of diffusion allows us to make sense of the fact that, at least at times, Kant appears to regard the laws of diffusion (especially the inverse-square law) as a priori (Prol, :–; OP, :–). Consider the definition Kant gives of a priori cognition in the Preface: “Now to cognize something a priori means to cognize it from its mere possibility” (MAN, :). As rationes essendi, the formal properties and formal, geometric laws exhibited along with the constructed spatial models for the diffusion of attractive and repulsive force make possible, and constrain the possibility of, the mathematical character of the laws of diffusion. We see how the given forces could have the mathematical character they do by constructing the spherical model(s), and we also see why, assuming the forces operate in accordance with the respective spherical models and their respective geometric laws (of the surface area and volume of a sphere, respectively), they must have the character they do. Because any forces that diffuse according to the model must obey the formal laws (which themselves are constructed a priori), a priori knowledge of the formal, geometric laws yields a priori knowledge in the Preface’s technical sense of the mathematical character of the dynamic (real) laws. For this reason, cognition of the inverse-square and inverse-cube laws based on mathematical construction is also a priori in the sense of involving cognition of necessity (see, e.g., Prol, :; KrV, B); the construction enables us to see why, given the various assumptions mentioned, their laws must be this way. Finally, it is a priori in the traditional rationalist sense of a priori knowledge as knowledge from (explanatory) grounds. My interpretation of the role of mathematical construction in Kant’s derivation of the laws of diffusion is further supported by other significant
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Some commentators have been reluctant to concede the a priori character of these laws, such as Brittan (, ), Friedman (b, ), Laywine (, ff.), and Stang (, n). For some important (and importantly different) takes on this tradition and Kant’s appropriation of it, see Hogan (), Smit (), Hebbeler (), and Stang (). Hebbeler’s account is particularly germane, since he is concerned in part to show how Kant’s views on laws, geometry, explanation, necessity, and a priori knowledge fit with this tradition. I return to the idea that properties of space (and not just properties of spaces) are rationes essendi of the mathematical content of laws below.
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passages in which Kant brings together the themes of construction and laws. In Kant’s explanation in the Preface of the Metaphysical Foundations of why chemistry does not yet meet the conditions for a natural science, he points to the fact that it has been unable to mathematically construct a spatial model of the chemical actions of matter and thus has been unable to exhibit the spatial principles or laws that make possible and explain the mathematical character of the chemical “laws of approach and withdrawal” (MAN, :–). Thus, what is missing from chemistry is precisely what can be achieved in the case of the laws of diffusion of attractive and repulsive force. Another thing to note about the chemistry passage is how closely Kant links all of the following together: () construction of formal geometrical concepts and formal principles/laws, () construction of principles/laws of chemical action, and () construction of a corresponding concept of (a type of ) matter. In fact, we have also seen Kant link together (); (0 ) construction of (the mathematical content of ) the laws of diffusion; and (0 ) construction of “the dynamical concept of matter, as that of the movable filling its space to a determinate degree.” This occurred in Note of Proposition . Especially interesting for our purposes is that in an intriguing passage in, of all places, the Doctrine of Right in the Metaphysics of Morals, Kant links together (), (00 ) construction of the Third Law, and (00 ) construction of an unspecified “dynamical concept”: The law of a reciprocal coercion necessarily in accord with the freedom of everyone under the principle of universal freedom is, as it were, the construction of that concept, that is, the presentation of it in pure intuition a priori, by analogy with presenting the possibility of bodies moving freely under the law of the equality of action and reaction. In pure mathematics we cannot derive the properties of its objects immediately from concepts but we can discover them only by constructing concepts. Similarly, it is not so much the concept of right as rather a fully reciprocal and equal coercion brought under a universal law and consistent with it, that makes the presentation of that concept possible. Moreover, just as a purely formal concept of pure mathematics (e.g. of geometry) underlies this dynamical concept, reason has taken care to furnish the understanding as far as possible with a priori intuitions for constructing the concept of right. (MS, :–)
Kant says here that a “fully reciprocal and equal coercion brought under a universal law” (what he earlier calls the “law of a reciprocal coercion necessarily in accord with the freedom of everyone”) makes possible
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That mathematical rather than metaphysical construction is at issue in this passage is emphasized by Pollok (, –n) and McNulty ().
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“as it were” a construction of the concept of right. He draws an analogy between the way that the former law enables a (quasi) construction of the concept of right and the way that the law of the equality of action and reaction (a fairly clear reference to his own Third Law of Mechanics) enables a construction of an unspecified dynamical concept. Kant’s specific views on the concept of right need not concern us here (this includes how literally to take his talk of construction of the concept of right). What I want to emphasize is that Kant speaks of presenting in a priori intuition the Third Law of Mechanics and he describes this presentation (i.e., this construction of the law) as enabling the construction of a corresponding dynamical concept. Also of keen relevance is his claim that “a purely formal concept of pure mathematics (e.g. of geometry) underlies this dynamical concept.” In making this claim, Kant appears to be suggesting that a kind of geometric construction (of some sort of formal, geometric concept) is at work in the construction of the law of the equality of action and reaction – the construction of which in turn enables the construction of the corresponding dynamical concept. Kant doesn’t fill in the details here; one will have to look to his various formulations of the proof of the Third Law to determine, for example, what formal concept and formal, broadly geometric law is being constructed. Given, however, that the Mechanics chapter is devoted to the concept of matter as the “movable, insofar as it, as such a thing, has moving force” (MAN, :), it is plausible that this is the dynamical concept he is referring to, the one whose construction is enabled by the construction of the Third Law. The upshot here is that in all three of the passages discussed Kant indicates that by mathematically constructing a formal concept of space and/or formal spatial/principles we are able to construct in a fashion a real law (or at least its mathematical content) and exhibit its real possibility. This, in turn, enables a sort of construction of a concept of matter. (I return to this point in Section ...) .. Universal Law of Dynamics In our treatment of Kant’s geometric derivation of the laws of attraction and repulsion, we saw that in each case Kant constructs a model – a figure – whose formal properties and laws are used to demonstrate the
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Note that Kant distinguishes between a mechanical and a dynamical law of the equality of action and reaction (MAN, :).
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possibility of the mathematical character of the corresponding law of diffusion and moreover to explain why they are as they are – given a number of things, including that there are forces, and that the forces follow the pattern of the model. Now consider the following question: Does the fact that the forces obey the pattern (as opposed to, say, behaving like shrapnel from an exploding grenade, to use the example in Laywine , ) admit of even a partial explanation, from Kant’s standpoint, or is it a purely contingent fact? I will approach this question by considering Kant’s account of the grounding of what he calls the “universal law of dynamics.” The action of the moving force, exerted by a point on every other point external to it, stands in inverse ratio to the space in which the same quantum of moving force would need to have diffused, in order to act immediately on this point at the determinate distance. (MAN, :)
According to this more general law, every action of a fundamental force that acts immediately on others is inversely proportional to the distance from the center of force (F ! /rx). What is the ground of this more general dynamical law, of which the inverse-cube and inverse-square laws are species? Warren () argues that it is grounded in Kant’s broader account of intensive magnitudes. I think, though, that Kant takes it to be grounded in features of space (indeed, Warren himself also ends up appealing to features of space, including its inertness, in his account of the law [p. ]). While the whole of space, as Kant conceives it, is causally inert, it also has various global properties that not only serve to constrain the spaces that are possible within it but serve to constrain the possible properties of its material occupants, along with their forces, motions, and their laws. These properties include space’s continuity, unity, threedimensionality, uniformity, orientability, and isotropy. The inverse-square and inverse-cube laws would not obtain if at least some of the global spatial properties just mentioned were changed (e.g., if space were two- or four
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For other treatments of this law, see Pollok (, ) and Warren (). Warren (, –) plausibly takes the immediacy and fundamentality conditions to be crucial for ruling out additional causal factors that would, as it were, disturb the diffusion. One difference between Warren’s treatment of these issues (from which I have learned a great deal) and my own is that he does not appeal to anything like what I am calling the efficient and equitable character of space. As a result, it remains unclear (at least to me) whether and how his Kant could explain why fundamental, immediate forces must diffuse evenly from a central point. A relatively trivial example is the infinite divisibility of material substances, which is grounded in the continuity of space.
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dimensional), and neither would the universal law of dynamics. So, I take it to be non-controversial that some global features of space are at least a necessary condition of the universal law of dynamics. But do these or any other global features of space require attractive and repulsive force to conform to the spherical models and thereby to obey the universal law of dynamics? I want to suggest that Kant takes the whole of space to have a structure that not only makes possible but also requires diffusions of force and communications of motion that are maximally efficient and equitable or balanced. In addition to the features of space already mentioned, space for Kant has an efficient and equitable character that constrains attractive and repulsive forces to adopt spherical as opposed to other patterns. For, in these models, force is distributed equally and efficiently in all directions in three-dimensional space in a shape that maximizes area (as opposed to, say, behaving according to the grenade model). Kant’s musings about space in the Only Possible Argument support the ascription of such a view to him. Throughout the second part of this text, Kant emphasizes the harmony, regularity, and beauty of space and spatial relations (BDG, :–, ). He also claims that “spatial relations can also enable us to recognize, from the simplest and most universal grounds, the rules of perfection present in naturally necessary causal laws, insofar as they depend upon relations” (BDG, :). He specifically discusses Maupertuis’ Principle of Least Action and the Law of the Equality of Action and Reaction in this regard. He proposes that these laws derive from facts about geometrical relationships, such as the fact that figures that are more equal and regular (with the circle being the limit case) are the most economical (they enclose the most space with the least outlay of perimeter). Kant seems to draw the conclusion that it is because of the privileged character of equal spatial relationships that the motions and actions of interacting bodies are equal and opposite. (I will return to this point below, in my discussion of the Third Law.) Though Kant doesn’t specifically discuss the inverse-square and inverse-cube laws in the Only Possible Argument, he does have much to say about the special properties of spheres, writing that the sphere is “a form which subsequently harmonizes with the other purposes of the universe better than any other possible form, a spherical surface being capable, for example, of the most uniform dispersion of light” (BDG, :).
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Kant’s view as expressed in this passage can be fruitfully compared with that of Leibniz, who similarly finds in various laws (such as those of optics) confirmation “of a principle of determination
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In the Only Possible Argument, then, Kant thinks that properties of the space that matter operates in constrain it to act and interact in certain ways. Certain patterns of action across space are more efficient and equitable than others. The whole of space has a structure that is in a sense globally what the sphere is locally: as the arena for diffusions of force and communication of motion, it constrains matter to act in ways that maximize equality and efficiency. In virtue of the equal and efficient character of space, which I think Kant regards as a corollary of its overall “harmony,” the default pattern of force diffusion (the one that will occur in the absence of any confounding factors) will be a spherical pattern. This view of space cannot easily be dismissed as a mere relic of the precritical period. There are critical texts that echo examples and claims from the Only Possible Argument, such as Prolegomena §, where Kant relates various harmonious, purposive-seeming geometric facts to the inversesquare law of gravitation (which he takes to itself be uniquely purposive [Prol, :]) and the Critique of Judgment §, where Kant speaks of the harmony and purposiveness of space and of spaces (especially the circle). Nor does the claim that space has an efficient and equitable character conflict with Kant’s denial that space is causally efficacious. Claiming that space has global features that constrain the causal activity (and the laws) of the material occupants of space is not the same as saying that space is a cause in the sense of the Second Analogy; space doesn’t accelerate bodies, it is not a force. Nor, for that matter, does space bring anything into being. But this doesn’t mean its features do not have implications for the causal order. Consider, for example, the orientable character of space (the fact that it permits incongruent counterparts). This property of space non-controversially imposes constraints on causal possibility – preventing, for example, the explosion that would otherwise result from inserting a piece of fissile material into the shape of its incongruent counterpart. Similarly, I take it to be non-controversial that the three-dimensionality of space has implications for the causal order and for causal laws. So, it simply cannot be the case that Kant’s denial of causal efficacy to space is meant to
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in nature which must be sought by maxima and minima; namely, that a maximum effect should be achieved with a minimum outlay, so to speak” (L, /AG, ). In addition to the optical laws of reflection and refraction, Leibniz cites, inter alia, the fact that “liquids placed in a different medium compose themselves in the most spacious figure, a sphere” (L, /AG, ). A key difference, though, at least on my reading, is that Kant takes this principle of least action to be as it were geometrically necessary and to obtain in virtue of the nature of space. For Leibniz (and Maupertuis), by contrast, the principle is contingent. I have been urged by an anonymous reader to address this worry. See Hogan () for this example and for discussion.
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include a denial of a constraining role by global features of space. I am merely suggesting additional properties of space that play a constraining role. The efficient and equitable character of space is a ground in the sense of a non-causal ratio essendi (as explained in Section ..). I have been trying to show that and how, for Kant, spatial properties and geometric laws ground the laws of diffusion, as well as the universal law of dynamics. In the case of the universal law of dynamics, he appeals to global features of space, including its efficient and equitable character, to explain why these forces diffuse in an even, spherical pattern, rather than in a scattershot sort of way. In light of this analysis, we can discern two distinct stages in Kant’s explanation of the laws of diffusion (though admittedly Kant himself does not sharply separate them). In the initial stage, we construct spatial, geometric models (which exhibit formal, geometric laws) for the diffusion of attractive force and repulsive force. At this stage of explanation, not only do we take for granted that there are attractive and repulsive forces that are fundamental and essential to matter and act immediately, but we also take as given that they behave in accordance with the model. Given these things, the formal laws constructed along with the models serve to explain why the laws of diffusion are inverse-square and inverse-cube laws. At the second stage, we appeal to global properties of space (in particular, its three-dimensionality and its efficient and equitable character) to explain why forces that are fundamental and act immediately in all directions conform to the spherical models. In this way, we see, then, that their conformity with the spherical models is not contingent.
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That said, it is not clear that spatial considerations can explain why attractive force must be modelled in terms of spherical surfaces. Warren (, –) suggests that experience plays a key role in showing that attractive force obeys that particular pattern; by contrast, the need for repulsive force to be modelled in terms of a volume would follow simply from its character as a space-filling force. I take it that Kant thinks that these features of the forces rule out their being influenced by an outside causal factor which would lead them to depart from, as it were, the default pattern that space privileges (see note ). This allows that other kinds of (non-fundamental) forces can be anisotropic. Thanks to Bennett McNulty for pushing me on this. Notice that the constructed models/laws and spatial properties only serve to explain and derive the laws when they are, as it were, discursively thought in conjunction with the various assumptions at the different stages. So while Kant can be said to prove (and explain) these real laws by means of construction, and while he speaks of constructing the (mathematical content of ) such laws, one also needs to discursively reason through the results of the construction together with assumptions and concepts that cannot themselves be constructed. Thanks to Houston Smit for prompting clarification here.
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. Some Objections .. First Objection: Kant’s Ambivalence One concern has to do with the apparent “hesitation” and “ambivalence” (to use Friedman’s terms [, –]) with which Kant offers these derivations. Consider that Kant offers the derivations as something of an aside, describing them as a “purely mathematical task, which no longer belongs to metaphysics – nor is metaphysics responsible if the attempt to construct the concept of matter in this way should perhaps not succeed” (MAN, :–). Consider, too, the way he appears to distance himself especially from the derivation of the inverse-cube law of repulsion in Remark (MAN, :). Reply: I do not think Kant’s cautious language is inconsistent with his believing in his constructions, taking them to be crucial to understanding the mathematical content of the laws, and seeing mathematical construction as essential to the project of MAN. With regard to the last two points, Kant indeed emphasizes in numerous texts that metaphysics is not itself mathematical construction (MAN, :, :; KrV, A/B). But this claim does not preclude mathematical construction being not only relevant to but indeed essential to the aims of special metaphysics. Kant says of special metaphysics that it is concerned with the application of mathematics to body, that it contains “principles of the construction” (MAN, :–) of concepts (note the plural) belonging to matter and its possibility, and that it is “responsible . . . for the correctness of the elements of construction granted to our rational cognition” (MAN, :). As I understand this, special metaphysics provides principles and elements of construction by analyzing the concept of matter into concepts of matter and then showing how those partial concepts admit of a sort of mathematical construction in principle. The partial concepts into which matter is analyzed are, for example, “the movable in space” (MAN, :), “the movable insofar as it fills space” (MAN, :), and “the movable insofar as it, as such a thing, has a moving force” (MAN, :). In the case of the movable in space, Kant shows in the Phoronomy chapter how this concept can be in principle constructed by showing how composite motion admits of construction in pure intuition. In the case of the movable insofar as it
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For some historical background, see Shabel (). I am in agreement with Heis (, ) and McNulty (, ) that mathematical construction is at issue here. Cf. Stang (, ).
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fills space, I think Kant is trying to show how it is in principle possible to construct the concept, insofar as it is in principle possible to construct a model and formal laws that explain how the mathematical content of the dynamical laws is both possible and necessary. Kant is plausibly read as doing something very similar for the Third Law and the corresponding concept of matter treated in the Mechanics chapter (as the passage from the Metaphysics of Morals quoted above suggests). The point of the “in principle” qualification is that special metaphysics qua metaphysics does not concern itself overly with messy technical details of actual constructions – the success of Kant’s metaphysical-dynamical analysis of matter does not stand or fall with such details. What matters is that he has done enough to show constructibility along the lines of the account above. Kant is particularly keen to emphasize this in the lead-up to the actual constructions that he cannot “forebear” giving because of some technical issues having to do with his derivation of the inverse-cube law. A key “difficulty” (MAN, :) that Kant refers to in Remark to Proposition is that matter is in fact a continuum – there are no discrete centers of force, as in his pre-Critical Physical Monadology. This generates a problem, because the sort of model we construct for repulsive force is naturally taken to involve point-centers that diffuse across a finite volume (and are separated from others by a finite distance). Nevertheless, Kant suggests that the gap between the idealized model we construct and the physical reality can be bridged by taking the distances and volumes in the model to be infinitely small. Since infinitely small distance/volume is “not different from contact,” according to Kant, the idealized aspects of our model do not in fact present an insuperable difficulty (MAN, :). Now, evidently Kant thinks that readers will be skeptical about this point, and he doesn’t want their skepticism to extend to the larger project of the Metaphysical Foundations. But it is far from obvious that Kant takes this technical, mathematical worry, or the empirical worry he raises immediately after about Mariotte’s law, to seriously call into question that these forces obey these laws and that they are in principle constructible and
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My position on laws and construction has some affinities with McNulty (). That the philosopher must establish constructibility is a point also made by Brittan (, ) and Förster (, ). As Warren (, n) notes, Kant does not seem to have the sort of concerns about the inversesquare law he has about the inverse-cube law. For similar descriptions of the problem (though differing in their construal of its severity for establishing mathematical constructibility), see Förster (, –), Pollok (, –), and Friedman (, –).
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explicable from properties and laws of space and spaces (something Kant had claimed as early as the Physical Monadology). .. Second Objection: The Non-Constructibility of Force and Matter Another worry relates to Kant’s claim that force cannot be constructed in pure intuition (KrV, A/B; MAN, :), nor relatedly can matter as it is conceived through his own metaphysical-dynamical approach (MAN, :). Reply: Our inability to construct the forces doesn’t obviously entail our inability to construct the laws that they stand in, or at least the mathematical content of the laws. Moreover, Kant’s denials about the constructibility of matter are consistent with his holding () that the partial concepts into which the metaphysician analyzes matter themselves admit of a sort of construction and () that the sort of construction these partial concepts admit of is very different from the kind of construction that mathematical-mechanical investigators of nature can undertake on behalf of their preferred concept of matter. The latter do not have forces to worry about, and can, working with their concepts of solidity and empty space, more directly and straightforwardly construct in pure intuition properties of matter, like density (MAN, :). By contrast, for Kant, the only sort of construction that partial concepts of matter admit of occurs in the manner I have described above. .. Third Objection: The Need for Experience A final worry concerns Kant’s claims that we need experience to find out these laws: “For, aside from this, no law of either attractive or repulsive force may be risked on a priori conjectures” (MAN, :; cf. :). Reply: We need experience to acquire concepts of matter and force. Moreover, Kant thinks one could not hope to initially discover and justify the inverse-square law from the armchair without any empirical data, just by considering geometric relationships in pure intuition. However, this doesn’t preclude him from thinking that there is a two-stage procedure involving construction of the sort we have described that explains why and
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Friedman () emphasizes such passages. Friedman (, –) also emphasizes the distinction between the concept ‹matter› and these partial concepts (Theilbegriffe), though he denies that, apart from the Phoronomy, they admit of construction in pure intuition.
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how the inverse-square law holds. The properties and laws of space and spaces described above are not the ratio cognoscendi of the inverse-square law. However, if I am right, they are the ratio essendi of this law (as well as the inverse-cube law).
. The Third Law We’ve now encountered three passages in which Kant suggests that the Third Law is grounded in properties of space (KrV, B; BDG, :; MS, :–). But Kant makes his view particularly explicit in a letter to Christoph Friedrich Hellwag, a German doctor and physicist. Kant’s letter includes a response to a question from Hellwag about the basis of the laws of inertia and the equality of action and reaction (Kant’s Second and Third Laws, respectively). Hellwag proposes that these laws have their source in some kind of inertial force that is in space but not identical with it (Br, :). In his response, Kant reformulates Hellwag’s question as follows: “What is the ground of the law that matter, in all its changes, is dependent on outer causes and also the law that requires the equality of action and reaction in these changes occasioned by outer causes?” (Br, :). Against Hellwag’s proposal, Kant argues that “the general and sufficient ground of these laws lies in the character of space” (Br, :). Before we turn to the account in Kant’s letter, it will be helpful to first review the official proof in MAN. Kant’s Third Law states, “In all communication of motion, action and reaction are always equal to one another” (MAN, :). In his proof, Kant “borrows” from the Third Analogy the idea that communication of motion involves a causal community (a community of moving forces), so that when A communicates motion to B, both A and B must act (MAN, :, :). Kant then claims that we must “represent” communication of motion in such a way that when there is a communication of motion between A and B, considered in absolute space “the change of relation (and thus the motion) between the two is completely mutual” (MAN, :). That is, we should “construct the action in the community of the bodies” (MAN, :–) in a case of impact in such a way that (a) leading up to the impact both are moving toward each other with equal and opposite momenta and (b) in the impact, the momenta transferred or communicated are equal and
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Hebbeler (, –) and Warren (, –) make similar claims. There may be an asymmetry in this regard between the inverse-square and inverse-cube law. See note above.
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opposite. We do this by viewing the motions of the impacting bodies “in absolute space,” that is, relative to their common center of mass. Kant’s “construction” of such a community of motion, which I take to be a mathematical construction in the technical sense, involves a spatial diagram – a model – with A and B, whose respective masses are depicted by circles of differing sizes, connected by a line, and with a point C on the line depicting absolute space (their center of mass) with respect to which they move. I take it the constructed model is supposed to be one where it is intuitively obvious that and how conditions (a) and (b) would be met. In that respect, the constructed model exhibits a formal law of the equality of momenta and transfer of momenta in cases of communication of motion. What does any of this have to do with the equality of action and reaction (the Third Law)? Kant says in Remark that the Third Law is a “necessary condition” of this construction. I take him to mean by this that if we apportion motion in cases of communication of motion in accordance with the model (i.e., we assume actual cases of communication of motion work in accordance with the model), and if we assume further that () both A and B are causally active in the communication of motion, with each acting on the other rather than itself (Kant here draws on the Third Analogy, and perhaps also the Second Analogy and Second Law), and () the causal action of each is measured by the change of momentum in the other it effects, then the equality of action and reaction in the communication of motion follows. One thing that is not particularly clear is why we are entitled to assume, and what would account for the fact, that actual motions of interacting bodies conform to the constructed model? Why are we required, once we are talking about communication of motion, to apportion the motion according to the equality of momenta law exhibited in the model, and what would explain this fact? The letter to Hellwag sheds light on this issue: As for the second law it is based on the relationship of active forces in space in general, a relationship that must necessarily be one of reciprocal opposition and must always be equal (actio est aequalis reactioni), for space makes
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Kant also identifies a further condition, namely, that the motion of A and B in absolute space is cancelled through their impact (the impact would thus be a perfectly inelastic one). This appears to be a consequence of the other two conditions. For further details, see Friedman (), Stan (), and Watkins () Following Watkins (), I take this step of the argument to establish what Kant (in Note ) calls the “mechanical law” of equality and reaction; the next establishes the “dynamical law” of action and reaction (MAN, :). Cf. Watkins (, ).
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possible only reciprocal relationships such as these, precluding any unilateral relationships. Consequently it makes possible change in those spatial relationships, that is, motion and the action of bodies in producing motion in other bodies, requiring nothing but reciprocal and equal [gleiche] motions. I cannot conceive of a line drawn from body A to every point of body B without drawing equally as many lines [so viele gleiche Linien] in the opposite direction, so that I conceive the change of relationship in which body B is moved by the thrust of body A as a reciprocal and equal change. Here, too, there is no need for a special positive cause of reaction in the moved body, just as there was no such need in the case of the law of inertia, which I mentioned above. The general and sufficient ground of these laws lies in the character of space, viz., that spatial relationships are reciprocal and equal [zugleich] (which is not true of the relations between successive positions in time). (Br, :–)
Kant claims that space makes possible only reciprocal relationships, a point which he illustrates (as he did in the official proof in MAN) with a line model. As he notes, when there are two points A and B on a line, A’s changing distance to B always requires B’s changing distance to A. This constructed model allows us to see how it is possible to distribute motions in the way Kant’s proof of the Third Law requires – and in that sense (given the role that this way of apportioning motion plays in Kant’s proof ) also allows us to see how it is at least possible for the Third Law to have the equal and opposite character it does. (Recall in this regard MS, :–.) But again: Why must actual cases of communication of motion occur in accordance with the model – couldn’t they instead involve just one body moving, or bodies moving in something other than an equal and opposite way? This, I take it, is where Kant’s appeal to the “character of space, viz., that spatial relationships are reciprocal and equal” (and, relatedly, that space requires nothing but equal and opposite motions) is relevant. I take Kant to be alluding to what I referred to as the efficient and equitable character of space. The efficient and equitable character grounds – in the sense of explaining – why, given that there are bodies that causally communicate motion to each other, they do so in accordance with the constructed spatial (or spatiotemporal) model, in which momenta are equal and opposite.
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I follow the Cambridge translation of the Correspondence here. An anonymous reader insists that “zugleich” should instead be translated as “simultaneous.” Given the repeated use of “gleich” and its cognates in the earlier lines, however, I think the translator’s choice is understandable. In any case, even if Kant did intend “simultaneous” here, there’s plenty of other indications in the passage that he holds that equality of spatial relationships is part of the explanation for why the law is as it is.
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Space, Pure Intuition, Laws in the Metaphysical Foundations
This reading allows us to understand the significance of Kant’s claim in the official proof in the Metaphysical Foundations that “there is no more reason to ascribe more of the motion to one than to the other” (MAN, :). Kant is not appealing to the principle of sufficient reason, as Marius Stan and various other commentators have suggested (, ) – or, more carefully, he is not appealing to this principle rather than considerations about space. The reason the quoted remark is relevant is that, due to the efficient and equitable character of space, where there are no special, additional causes for deviation, the default distributions of motion and force are maximally efficient and equitable. If this correct, then we can distinguish two stages of explanation in Kant’s construction-cum-derivation of the Third Law, just as we did with the laws of diffusion. In the first stage we construct a spatial model (with a temporal element) – in this case a line with geometric points taken to be changing their distances to each other and to the point standing in for their center of mass. In doing so, we simultaneously construct a formal, broadly geometric law, in this case a law of the equality of momenta. By means of this model and the law expressed through it, we can explain why, given that bodies communicating motion to each other conform to the model, and given the other assumptions laid out above, it is not only possible but also necessary that “in all communication of motion, action and reaction are always equal to one another” (MAN, :). In the second stage, we appeal to properties of space, namely, its efficient and equitable character, to explain why, given that there are bodies that communicate motion by means of force, the motion must conform to the model – that is, why real bodies must act in these ways.
. Conclusion How, if at all, does the view I have ascribed to Kant fit with his claim in Prolegomena § that “space is something so uniform and so indeterminate
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See, e.g, Watkins (, n) and Adickes (–, :–). The latter mentions Kant’s appeal to space in the letter to Hellwag (:n) but dismisses it quickly as a “mere assertion without any proof.” This leaves it open that the structure of space might be itself a result or manifestation of the principle of sufficient reason (as, e.g., Schopenhauer holds). Thanks to Colin McLear for pushing me here. In the case of the laws of diffusion, the absence of these is secured by the fundamentality and immediacy of the force. It is less clear what in the present case secures their absence. Notice that, while there appears to be such a temporal (and phoronomical) element, Kant consistently highlights the spatial, geometric aspects of the model. I take it that the line model corresponds to the formal, geometric concept that Kant spoke of in the Metaphysics of Morals passage.
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with respect to all specific properties that certainly no one will look for a stock of natural laws within it” (Prol, :)? I take Kant to be rejecting here a kind of view – to which he thinks some transcendental realists are committed – that takes space all on its own, apart from the other conditions that make experience possible, to be sufficient for the existence of laws, including geometric laws, transcendental laws of nature, laws of diffusion, and Laws of Mechanics. On my interpretation, Kant denies the sufficiency claims. The space that we are given in pure intuition does not alone suffice for any laws – and this is related to the fact that it doesn’t, as it were, produce the objects (whether real or merely formal/geometrical) that are subject to the law. In the case of geometric laws (such as the law of the surface area of spheres), Kant thinks they depend not just on pure space but also on constructive acts of the geometer (which themselves involve a synthesis in accordance with the mathematical categories) by which the objects bound by the laws are generated. In the case of the transcendental laws (like the Third Analogy), Kant thinks they depend not just on pure space but also on the dynamical categorial synthesis of empirical intuitions in space and time in virtue of which experience of a unitary, objective space and time (and of real, causally interacting objects within it) is possible. As for the laws of diffusion and the Third Law, they depend not just on pure space but also on constructed object-spaces and their formal geometric laws – which means they also depend on the constructive acts and categorial synthesis they presuppose. They depend further on non-spatial conditions of the experience of matter; these include intellectual conditions (e.g., the categories of reality, and also the relational categories and Third Analogy in the case of the Third Law) as well as empirically given ones, like the presence of forces. All of this, I think, is compatible with holding that properties and laws of space and spaces explain in the two stages (each stage of which involves certain “givens” associated with various intellectual and sensible conditions of matter/the experience of matter) the mathematical character of the laws of diffusion and the Third Law.
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Cf. Messina (a). I am grateful to participants in the UT Austin-Workshop on the Metaphysical Foundations and to participants in the Biennial NAKS for feedback. Special thanks to Colin McLear, Katherine Dunlop, Anton Kabeshkin, Clinton Tolley, and Houston Smit for detailed comments.
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Finitism in the Metaphysical Foundations Lydia Patton
. Introduction Recent readings of Kant’s Metaphysical Foundations of Natural Science begin with the claim that Kant presents a theory of matter, and of the interactions between material substances, in that work. Emphasizing this point raises a question that comes up in interpretations of the Metaphysical Foundations more generally: What is more fundamental to Kant’s account of matter: A description of how law-governed interactions are explained as arising from the essential properties and powers of objects (the Necessitation Account)? Or an account of how we come to know universal laws of nature via formal inferences regarding the a priori foundations of particular empirical laws (the Derivation Account)? Or a Best System Interpretation (BSI), on which “the particular laws of nature are those empirical generalizations that would figure in the best systematization of the empirical data at the ideal end of inquiry” (Breitenbach , )? I have argued (Patton ) that Kant espouses an “essentialist” view and that a complete exposition of the “real essences” of material bodies is necessary to Kant’s argument in the Metaphysical Foundations of Natural Science. Thus I have defended a version of the Necessitation Account (NA), which is the first view above: that “empirical laws are necessary governing principles that obtain by virtue of the particular natures of things” (Breitenbach , –). In this chapter, building on recent and long-standing work (Warren ; Friedman ; Glezer ), I investigate how the account of the essences or natures of material substances in the Metaphysical Foundations is related to Kant’s demand for the completeness of the system of nature. We must ascribe causal powers to material substances for the properties of those substances to be observable and knowable. But defining those causal
See Kreines () for the original classification of Kantian accounts in this way.
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powers requires admitting laws of nature, taken as axioms or principles of natural science, which govern anything that can be constructed mathematically or “given as an object of experience” (MAN, :–). Presenting a complete system of nature requires adumbrating the properties of material substances that can be objects of experience. But it is not possible to show how those properties are involved in explanations of the phenomena without involving the laws that govern interactions between material substances. And the enumeration of the comparatively inner properties of material substances requires appeal to how they would interact with other substances. For Kant, or so I will argue, it is not possible to account for the natures of things, or for the features of laws of nature, independently of each other. The account presented here thus may seem at first to push in the direction of a Best System Interpretation, on which “the systematic unity of our cognitions of nature . . . confers the status of a necessary law on empirical regularities” (Breitenbach , ). But instead, my account is intended to substantiate a fourth reading of Kant on laws: the Finitist Account (FA). The Finitist Account has it that modal judgments about the possible proofs that can be made, or interactions that can be explained, are () based on concrete intuitive reasoning and motivated by the desire to avoid appeal to the actual infinite and () grounded in finite decision procedures, which are () based on reliable systems of axioms or rules of inference. The difference between the Best System Interpretation and the Finitist Account may seem small at first. According to the BSI, “systematic unity is . . . constitutive of the necessity [of] and our knowledge of the laws” (Breitenbach , ). Thus, for the BSI, systematic unity has an independent role to play in Kant’s justification of the laws of nature. The Finitist Account links Kant’s “ontological commitments” regarding matter to his account of the system of nature that can be constructed in concert with an account of matter as phenomenal substance. The role of laws of nature in finitist accounts is regulative, and it is relative to our
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This terminology is due to Warren (, ). The BSI was inaugurated by David Lewis in general, and by Philip Kitcher in Kant’s case. It is notable that a version of the BSI has been defended by Michela Massimi as her own position on the laws of nature (a), while in Kant’s case in particular Massimi defends a version of ‘dispositional essentialism’ (a). For a reading of Kant as a finitist about arithmetic, see Tait () and the response from Sieg (). The phrase “ontological commitments” and this general characterization are drawn from Detlefsen (, –), in his description of Hilbert’s finitist methods.
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capacity to analyze substances using mathematical principles, and to demonstrate that such substances can be objects of a possible experience. Matter can be defined as a phenomenal substance with certain properties and powers only insofar as two conditions are met jointly: () that we can show how those powers and properties interact with the formal requirements of our system of laws and () that we can show that all possible events that can be objects of experience will fall under that system. Kant does not appeal to a “systematic unity” governed by a priori laws at the ideal end of inquiry, but rather to our ability to integrate any proposed feature of matter into the system of nature, considered as a nexus of phenomenal substances. In Section . of the chapter, I will explain how the exposition of the concept of matter features in Kant’s argument for the physical completeness of the system of nature. Kant’s system of nature can be built, as Section . will discuss, without appeal to actual infinity or to divine intervention. Section . deals with two key positions of the MAN: () that judgments about the possible motions and interactions of material substances based on the a priori concept of matter are synthetic and () that we may argue for the observability and reality of matter based on its causal powers. Section . lays out the case that Kant’s reasoning in the MAN is finitist and argues that the NA can be consistent with the FA.
. Completeness and the Exposition of Concepts Part of the goal of the Metaphysical Foundations of Natural Science is to demonstrate the completeness of the metaphysics of corporeal nature. In order to do so, it is necessary to show that “all that may be either thought a priori in this concept, or presented in mathematical construction, or given as a determinate object of experience” must be brought under the categories (MAN, :–). Kant makes the following remark about this: But in order to make possible the application of mathematics to the doctrine of body, which only through this can become natural science, principles for the construction of the concepts that belong to the possibility of matter in general must be introduced first. Therefore, a complete analysis of the concept of matter in general will have to be taken as the basis, and this is a task for pure philosophy – which, for this purpose, makes use of no particular experiences, but only that which it finds in the isolated (although intrinsically empirical) concept itself, in relation to the pure intuitions in space and time, and in accordance with laws that already essentially attach to the concept of nature in general, and is therefore a genuine metaphysics of corporeal nature. (MAN, :)
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Kant refers to this shortly thereafter (MAN, :) as “detaching” the “metaphysical foundations of the doctrine of body” from the “general system of metaphysics,” as well as from mathematical physics. For any science to be complete requires, first, a critique of (pure) reason (Lu-Adler , , –). Kant’s notion of ‘completeness’ involves not just the idea that a presentation is exhaustive but the claim that the system in question is ordered by a principle or principles (MAN, Preface). That ordering allows for a decision on whether the system provides an exhaustive account of the concepts in use in the system, which then allows for an a priori account of how those concepts function in the system. In the case of MAN, the parts of the book lay out the specifications of the concept of matter. For Kant, conceptual exposition grounds the process of showing how a concept makes a difference to systematic thinking in some domain. Concepts must first be ‘analyzed’ – that is, a complete list of their characteristic marks must be made and presented in an ‘exposition’ – before they can be employed systematically. But that exposition does not exhaust the possible a priori knowledge about that concept within a given system. In particular, Kant argues that “principles for the construction of the concepts that belong to the possibility of matter in general” must be provided if natural science is to be possible. These principles involve the mathematical and dynamical properties of matter that can be derived from a theory of matter, which involves, as a priori principles, space, time, and the categories. In the MAN, Kant analyzes the concept of ‘matter’ and provides an exposition of the concept via four of its characteristic marks: Phoronomy. “Matter is the movable in space.” (MAN, :) Dynamics. “Matter is the movable, insofar as it fills a space. To fill a space means to resist any movable, which attempts to penetrate a certain space through its movement. A space that is not filled is an empty space.” (MAN, :)
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I am grateful to a reviewer for raising this point. Here, I am following the persuasive account of Messina () regarding the concept of “exposition,” and I am using the term in a substantive sense here, as “a metaphysical exposition, for Kant, requires the analysis of a ‘given concept,’ an analysis that justifies propositions describing the marks of the ‘given concept’ being analyzed”; and moreover, on Messina’s novel account, “at least some of the marks uncovered in the analysis of a given concept correspond to essential features of the objects (or object) in the extension of the concept” (Messina , ). For Kant’s account along these lines, Messina cites KrV (Bff.) and Log (:), among other texts. Messina () and Stang () emphasize the modal reasoning that metaphysically substantive conceptual determinations of space and time can afford.
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Mechanics. “Matter is the movable, insofar as it has motive force in itself.” (MAN, :) Phenomenology. “Matter is the movable, insofar as it as such can be an object of experience.” (MAN, :)
Listing the characteristic marks of central concepts is the first step to showing how those concepts function in one account of how we obtain knowledge a priori: by means of demonstrations, using those concepts. The second idea of ‘completeness’ here is that, via the Methodenlehre, every judgment can be referred back to a concept that has been characterized, in its essential features, in the Elementenlehre. Hence, ‘completeness’ is not restricted to the idea that we have exhausted the essential features of concepts in the Elementenlehre. It is defined in terms of the claim that the nexus of knowledge gained via those concepts can be traced back to its sources or grounds, in a priori principles of construction. In the texts for which Kant distinguishes between an Elementenlehre and a Methodenlehre, it is crucial to distinguish between metaphysical expositions of concepts and classification of elements as they work within a system. Kant’s four propositions that make up the parts of the Metaphysical Foundations collectively provide an exhaustive list of the characteristic marks of the concepts of “matter” and “material body” (Patton , ). In the MAN, Kant explains how the concept of matter as object of experience can fit into a system complete in itself, without additional ontological commitments (e.g., to monads, God, or actual infinities). The central problem of the MAN is to show that the concept of matter, with the four characteristics Kant ascribes to it, can be integrated into a system, based on the categories, which allows for judgments to be made about observable material bodies. But it turns out that for material bodies to be observable, for Kant, they must be ascribed certain causal powers. Then Kant must show not only that material bodies can be observed in space and time but that their interactions with other bodies are measurable using finite methods available to subjects of experience.
. Matter as Phenomenal Substance A central question of this chapter is how Kant is able to argue that the a priori classification and elucidation of concepts such as “material body” can promote the completeness of the system of nature, including
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For an explanation of how this works in the MAN, see Hyder (, ch. ).
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knowledge of the interaction between and motion of objects, which – as Kant argues himself – cannot be known a priori. Instead, if we are to account for matter as a possible object of experience, we must ascribe certain properties to material bodies. These include causal powers that material bodies must have if they are to be experienced at all, and even if we are to experience temporal succession. Kant’s system of nature is constructed on the basis of his theory of matter as phenomenal substance. The laws he proposes are valid universally, but that ‘universality’ is founded on Kant’s matter theory. For Kant, a theory of matter as substantia phaenomenon is necessary to solve fundamental problems in natural philosophy (Pollok , –). Such a theory requires determining what Warren (, –) dubs the ‘comparatively inner’ properties of matter. These are the properties elucidated by the relationships between the characteristics of material bodies considered independently of relation to other objects, with respect to how these features ground perceptible phenomena when material bodies do interact. In the Physical Monadology of , Kant had argued for an ontology of ‘physical monads’ with a sphere of activity (via forces of attraction and repulsion), but which were not observable. In his view developed after the Physical Monadology, he argues that monads are not necessary to explaining the observed phenomena and that material bodies can be known (only) through their causal interactions. However, Kant maintains the idea that the ‘comparatively inner’ properties of bodies are the basis for explaining the observable, measurable phenomena in which those bodies are involved, for example, their interactions with each other. In demonstrating how inner properties of matter can be made sensible and measurable, Kant now rejects the idea that monads are only intelligible and do not interact with other objects (Glezer , –; Pollok , ). However, as Warren notes, that does not mean that Kant abandons the idea that the measurable, sensible properties of matter are knowable or that they cannot be traced back to the inner properties that ground them (Warren ; Pollok , n). For instance, a material body’s taking up space is a result of its impenetrability (Warren , ch. ). That is not
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See the (in)famous footnote to B, in which Kant argues, “Motion of an object does not belong in pure science and hence not in geometry, because it cannot be known a priori that something is moveable, but only in experience. But motion as the describing of a space is a pure Actus of the successive synthesis of the manifold in outer intuition as such through productive imagination and belongs not only to geometry, but also to transcendental philosophy.” See Friedman (, ff.) and Laywine (, n) for recent discussion.
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just because we can determine the mechanical properties of the body quantitatively but also because we can determine the dynamical properties with which bodies must be endowed, or else they would never become objects of sensible experience in the first place – for instance, a force of repulsion, which grounds impenetrability. It is correct to do as many commentators do, and to emphasize that Kant’s path from the Physical Monadology was to develop a theory of matter that no longer depended on the idea of intelligible monads that do not interact with other substances. However, it is important to note that Kant does not reject the idea of giving a complete determination of a concept a priori by giving its characteristic marks. That is, in fact, his task in the Metaphysical Foundations of Natural Science: to provide an a priori exposition of the concept of matter (Patton ). Only if the concept of matter is given a complete exposition can we decide, for instance, whether a given judgment about an observable material body is analytic or synthetic (KrV, A–/B). In the case of matter, the complete determination of what Warren () calls the “inner relations” of a material body provides a ground for judgments about the possible relations of material bodies to perceiving subjects. These relations depend on the a priori features of space and time, as well as on the categories (Friedman , ). It is well known that Kant argues that material bodies must be impenetrable if they are to be observable (Warren ). In addition, Kant argues that there are forces we must assume to exist if we are not to endow material substances with what Warren () calls ‘absolutely inner’ properties, that is, if we are to consider matter as a phenomenon, not as a thing in itself. The relationship between the properties of a concept (elucidated a priori) and the possible relations and interactions of observables that can be synthesized using that concept is the following: (A) Synthetic (Warren ; Friedman , –) (B) An argument for the observability and reality of matter based on its causal powers (Warren , and passim).
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See Warren (, –) and Pollok (, ); see McNulty () for Kant’s rejection of absolute impenetrability. The model for this is the classification of the concepts of space and time in the Critique of Pure Reason (Messina , ).
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(A) The Principle of Succession as a Synthetic A Priori Principle The relationship between the properties of a concept (elucidated a priori) and the possible relations and interactions of observables that can be synthesized using that concept is not analytic, but synthetic, because you cannot derive from the ‘mere existence’ or ‘mere concept’ of a thing that it will have particular relations to other objects. For Kant, we cannot know whether matter fills space by virtue of its ‘mere existence’ (Warren , ). More generally, it is not possible to derive from a description of matter or of a material body, hypothetically and a priori, an account of its actual relations with other objects. And Kant is clear, even as early as the MonPh, that a body cannot be known completely in isolation from all other material bodies. Even in the last proposition of the Nova dilucidatio, Kant notes that “finite substances by their mere existence are unrelated” (PND, :–; cited in Warren , ). However, Kant can allow a theory of how the phenomena that can be unified under a concept would interact with other things, if we are to have knowledge of those interactions and of the properties that ground them. Kant’s account of the possible interactions of material bodies is given as part of his clarification of the a priori foundations of Newtonian natural science, from his very earliest writings. Kant makes a number of such arguments during the pre-critical period. In the Metaphysical Foundations, Kant will use Newtonian laws as the basis of his reasoning. As Laywine () and Schönfeld () remark, Kant’s method in the Metaphysical Foundations is foreshadowed by another of Kant’s early works: his doctoral dissertation, the New Elucidation of the First Principles of Metaphysical Cognition (), sometimes called the Nova dilucidatio. Newton’s laws in the Principia are the following: . Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed. [law of inertia] . A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed. [law of acceleration] . To any action there is always an opposite and equal reaction; in other words, the actions of two bodies upon each other are always equal and
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As Pollok () describes, beginning with the MonPh and continuing to the MAN, Kant develops “an original theory of matter” (p. ). Such a theory is necessary to solve problems in “geometry and metaphysics,” such as whether space is divisible to infinity or not (ibid.). Pollok calls Kant’s progress between and the “phenomenalization of nature.”
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always opposite in direction. [law of interaction] (Newton , –) In the Nova dilucidatio, Kant proposes as an axiom the “principle of succession”: No change can happen to substances except in so far as they are connected with other substances; their reciprocal dependency on each other determines their reciprocal changes of state. (PND, :)
There can be no change of state in a substance if it is not connected to some other substance. As we saw above, this claim is central to Kant’s move away from his more Leibnizian position in the Physical Monadology. Along similar lines, Laywine (, –) observes that the “principle of succession” in Kant seems to “stand in a peculiar relation” to Newton’s laws of motion, because it seems to contain version of all three laws within itself, which Schönfeld summarizes as the following: . . .
no substance has the power to affect change in itself; all change in a substance must be the effect of a connection with, or the action of, some other substance; change of state in substances is mutual; that is, equal and opposite. (, )
In the Nova dilucidatio, Kant proposes the principles of succession and coexistence as axioms. As Laywine () notes, Kant’s arguments for the principle of succession appear weak prima facie: at best “contrived,” at worst “question-begging” (p. ). As Laywine summarizes one argument: If we deny real interaction among substances, we must conclude that every substance has within itself the sufficient reason of every change it will ever undergo. Kant’s point is that these changes will unfold instantaneously unless there is some sufficient reason to delay them. The sufficient reason of this delay has to be outside of the substance. For so long as the substance has within itself the sufficient reason of all its future changes, it will not be reluctant – as it were – to bring these about. But if the substance presents all its future states all at once, it would not really undergo any change. So Kant
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Discussion in Schönfeld (, ). Stan () argues for Kant’s continuing allegiance to Leibnizian mechanics, examining Kant’s Principle of Action and Reaction as a foundation for Leibnizian mechanics, not a version of Newton’s Third Law. I do not see my account here as in conflict with Stan’s, since Stan argues – and I agree – that Kant’s move away from Leibniz and toward Newton is found in the dynamics, not the mechanics.
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concludes that something must act on the substance from without, otherwise no real change will take place in it. (p. )
Kant seems to argue from whether change is conceivable in an isolated substance to whether change is actually possible, which appears to beg the question: Why can’t we conceive of change in such a substance? If the ground of conceivability is intelligible and not physical, then it seems that Kant is missing an argument that physical change in an isolated substance must rest on the internal, intelligible principles of that substance. Laywine argues, however, that Kant’s arguments for the principle of succession are stronger if they are understood to center on a more general question: “the possibility of any kind of temporal order in the world” (p. ). If changes in a substance take place instantaneously and result only from the internal powers of that substance rather than from real interaction, then no temporal order would result from the observable “order of succession” of changes in nature. For the changes will all take place instantaneously, and in accordance with the (possibly unobservable) internal principles of substances. But Kant argues, as Laywine points out, that time is that order of succession. If this is right, then, as early as the Nova dilucidatio, we find Kant arguing that the principles or laws of nature are bound up with the status of space and time as principles of order (though not necessarily, yet, as a priori principles). One might extend this argument, in the MAN, to Kant’s analysis of continuity and change. McNulty () analyzes Kant’s commitment to a dynamical theory of matter as resting on his “endorsement of Leibniz’s law of continuity” (p. ). McNulty argues that this endorsement is on metaphysical, not just mathematical grounds and that it is fundamental to a number of Kantian texts (pp. –). For instance, “In the General Remark to the Mechanics, Kant endorses a particular instantiation of the law of continuity, according to which mechanical communication of motion occurs bit-by-bit through an extended duration and not at an instant ([MAN,] :f.)” (p. ). Kant’s insistence that change, causality, and interaction be represented as taking place continuously over time is, of course, fundamental to his analysis of causality (Watkins ). It is also linked to his overall method
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See PND, :. If we understand Kant to be working in a more Leibnizian framework in the Nova dilucidatio than is sometimes assumed, this difficulty may be resolved to a degree (only to introduce difficulties associated with that framework, of course). For a substantial account of how time, especially, is involved in the argument and structure of the MAN, see Hyder ().
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in the MAN. In the Preface, Kant makes an explicit distinction between the a priori foundations of natural science and the ‘transcendental underpinning’ found in Newtonian physics and related systems of natural philosophy (MAN, :). The “natural philosophers” assume, according to Kant, that metaphysics could consist only in concepts and principles that are employed speculatively or inventively and argue that the better method of natural philosophy is to employ empirical principles that are “borrowed from experience.” Kant argues that this is a false dichotomy, and that a third route is available: All true metaphysics is drawn from the essence of the faculty of thinking itself . . . it contains the pure actions of thought, and thus a priori concepts and principles, which first bring the manifold of empirical representations into the law-governed connection through which it can become empirical knowledge [Erkenntnis], that is, experience. (MAN, :; translation emended)
Kantian critical metaphysics involves making explicit the principles originating in a priori thinking that first makes experience itself possible, and, through it, empirical knowledge of objects and phenomena. Kant emphasizes the need to investigate the a priori sources of the metaphysical principles bound up with mathematical physics, which “make the concept of their proper object, namely, matter, a priori suitable for application to outer experience, such as the concept of motion, the filling of space, inertia, and so on” (MAN, :). These principles cannot be postulated: we cannot assume that matter is the movable in space, nor can we state that if matter is given, then it follows without any argument that it is movable, fills space, has an inertial force, and so on. Instead, we must demonstrate that the concept of matter has these features. That is the purpose of the Metaphysical Foundations of Natural Science: to demonstrate the completeness of the “metaphysics of corporeal nature” (MAN, :) by showing that “All determinations of the general concept of a matter in general must be able to be brought under the four classes of [pure concepts of the understanding], those of quantity, of quality, of relation, and finally of modality – and so, too, must all that may be either thought a priori in this concept, or presented in mathematical construction, or given as a determinate object of experience” (MAN, :–). Doing this requires not only listing the properties of matter, but also specifying how those features fit in an exposition of matter as a possible object of experience. Showing how matter can be a possible object of experience requires demonstrating that material bodies can be observed in space and time.
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Fitting a theory of the interactions of material bodies into a system of nature requires ascribing continuous forces to those bodies, which requires, for instance, our ability to apply infinite division to bodies and their motions. Those ‘requirements’ may not be logically necessary, but they can be physically necessary, if we are to show how matter can fit into a causal, temporal nexus. Thus, while Kant argues that “true metaphysics” must be drawn from “pure actions of thought,” he also argues that among those pure actions are proofs concerning how matter, its motion, and the actions of material bodies can be observable and measurable. (B) Motion, Reality, and Causal Powers In Kant’s mature theory of matter, found in the MAN, we must ascribe causal efficacy to material bodies in order to explain, given the a priori exposition of the concept of ‘material body,’ how we can come to have sensible experience of phenomena unified by that concept. As Warren, Glezer, and Friedman analyze this, in order to be subsumed under the category of reality, matter must be understood as possessing causal powers. The Metaphysical Foundations begins with the proposition that “Matter is the movable in space.” Each of the chapters adds a determination to that proposition. Given that matter is the movable (in space), it is not possible to determine a priori that a particular material body has actually moved, according to the following reasoning, which is my own synthesis and reconstruction of recent work on this subject. . According to Kant’s principle of relativity, it is one and the same a priori whether an object is seen as taking a particular path with respect to the subject, or taking another, opposite path with respect to another frame of reference. . An object cannot be determined to move in isolation from all other objects (the principle of succession). In particular, if we cannot show that an object has made a measurable difference to the motion of another object, then there is no real phenomenon of motion to be observed, and there is no determinably real alteration to the state of
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This may seem to cut against a finitist reading of Kant – but see Section . below. See Massimi (a) for the idea of a causal nexus in Kant. A main thesis of Warren () and Glezer (). See Patton (), including the references there. McLear () provides a relevant analysis of motion as the fundamental determination of matter. Friedman ().
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. .
any object. One reason for that is because the possible motions the object could take with respect to other objects, including the observing subject, will sum to zero. Thus, to observe or to know an object’s real motion requires being able to determine an object’s causal interactions with other objects. Finally, therefore, material objects must be considered to be endowed with causal powers if their motion is to be measurable or quantifiable, and if they are to fall under the category of reality.
It is entirely possible that bodies may have properties, including causal properties, that are “absolutely inner” in Warren’s terms; that is, the properties are determinable in isolation from any possible relation to another object. For Kant, we cannot know those properties, for these would be properties of things in themselves (Warren , ch. , §). We can know, however, bodies’ “comparatively inner” determinations: specifications that are grounded in perceptible internal relations of the body, which describe possible relations with other objects, including causal powers to affect them (Warren , ). Analysis of comparatively inner properties a priori yields only potential effects, including potential alterations to a body’s state of motion. However, if something is a material body, we know a priori that it is movable and endowed with its own comparatively inner properties, which ground the possible interactions it may have with the original object in question. With Glezer, Warren, and Friedman, we can conclude that, in the MAN, matter is real to us insofar as we can measure its “force or power” to alter the motion or state of other objects. In Leibniz, substantial forms
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Glezer (, ), Warren (, ), Friedman (, ). Patton (). Warren (, ch. , §, and passim); Glezer (, chs. and ). One might object to my reading (here and in Patton ) that it seems to ascribe a version of the old substance-accident metaphysics to Kant. Pollok (, n) has raised a similar objection to this kind of reading. Without giving a full response, it is worth noting that Glezer () defends a cogent picture where Kant’s category of reality “descends from Early Modern – especially, Leibnizian – versions of the Scholastic concept of realitas, often identified with that of substantial form . . . [R]eality’s central role in Kant’s thought is analogous to the central role of substantial forms in seventeenth-century debates over the nature of physical bodies and physical explanation” (). For instance, in Leibniz’s letter to de Beauval, we find the claim that what is real in motion “is force or power” (G :; L ; cited in Glezer , ch. ). The question of Leibniz’s – as opposed to Wolff’s – influence on Kant is complex. For skepticism about how much Leibniz Kant knew firsthand, see Garber (). For the view that Kant could have read much of Leibniz’s works and correspondence, see Storrie () and McNulty ().
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are intelligible. They ground, in turn, the intelligibility of our ascription of properties to objects. In moving away from the Physical Monadology and toward the Metaphysical Foundations of Natural Science, Kant’s challenge is to show that “substantial form,” that is, “the grounding of properties in the individual essences of substance,” can be shown to function in a framework in which all of the properties of those substances are phenomenal. And he needs to do this instead of showing that observable properties are grounded in purely intelligible essences. Kant provides an account of how properties are grounded in the ‘essences’ of substances, but where that essence is defined entirely in terms of phenomenal properties. This is done by showing that a physically complete system of forces in nature is grounded in an account of the essence of matter as substantia phaenomenon. That account of the essence or nature of matter does put conditions on the elaboration of a system of nature, consistent with the Necessitation Account. But, in turn, the requirements for enumerating a system of nature, in terms of a nexus of finite, observable substances and their interactions, set conditions for the exposition of the concept of matter.
. Finitism and Kant’s Matter Theory It is odd that finitist readings of Kant have not had more purchase to date. It is well known by now, almost a philosophical cliché, that among Kant’s motivations for the critical turn was to rein in metaphysical speculation about God and the infinite by limiting knowledge claims to what can be proven of objects of a possible experience. And, later, David Hilbert and Paul Bernays “justify finitary knowledge in broadly Kantian terms (without however going so far as to provide a transcendental deduction), characterizing finitary reasoning as the kind of reasoning that underlies all mathematical, and indeed, scientific, thinking, and without which such thought would be impossible” (Zach , §.). Tait () has argued recently that Kant’s theory of arithmetic is “finitist” in Tait’s own sense, that it “refers to the mathematics that is
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“Even if it is impossible fully to articulate the relation between the essences of substances and their phenomenal manifestation as quantifiable physical forces, Leibniz can justify their intelligibility to the extent that he can provide a metaphysical picture in which each substance is characterized by an individual essence that determines all its features, and so functions as the basis for the entire series of its phenomenal states. This idea of a thoroughly determined substance, represented by a complete concept, is the idea of the Leibnizian monad, the centerpiece of Leibniz’s mature metaphysics” (Glezer , ).
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capable of proving general propositions about the natural numbers without presupposing any infinite objects (actual infinities) – such as the totality of natural numbers” (p. ). Clearly, in proposing a Finitist Account of Kant on the laws of nature, I am not arguing that Kant’s matter theory is finitist in this exact sense. But I do want to argue that it is finitist in a broader sense. The key characteristics of finitism for my purposes are the following: . . .
Finitism does not appeal to actually infinite objects or magnitudes (an infinite universe, an infinity of monads, the infinite sensorium of God, the totality of the natural numbers, and the like). Finitism appeals to finite, concrete, usually intuitive construction procedures in generating a domain. Finitism appeals to explanations using systems of axioms or principles to demonstrate that the construction procedures in item are finite (and will not need to be extended to the infinite) in the domain of interest.
In these three ways, Kant uses finitist methods in the MAN. Kant needs a theory of matter as phenomenal substance. This theory must demonstrate how law-governed observable interactions are grounded in the comparatively inner properties of material bodies. The physical system of nature that Kant constructs cannot be metaphysically complete in the sense that Leibniz’s system is, because Kant rules out the appeal to absolutely inner properties of simple substances. According to his own principles, Kant cannot argue, for instance, that the apparent infinite divisibility of substances is resolved by arguing that substances are composed of metaphysically simple monads. Nor can Kant’s system be given the “final vehicles” (e.g., God) with which Newton endows his natural philosophy. Instead, the exposition of the concept of matter engages with the theory of matter as observable substantial phenomenon in the MAN. In The Philosophy of the Young Kant, Martin Schönfeld charts the wandering path of the young Kant between the metaphysics of the
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One might object, on grounds of anachronism, that one should not apply this term to Kant. I might agree, and then note that, on the same grounds, one shouldn’t say Kant has a Best System Interpretation. Insofar as both are being used as evaluative, rather than straightforwardly descriptive, categories, some leeway can perhaps be given. Moreover, there is an argument – of course, not given in this chapter – that Hilbert’s finitism is derived partly from Kant’s, and thus even the historical category may be appropriate. On Leibniz’s changing views on the “substantial bond” theory, see Garber (, –, passim) and Storrie (, §).
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Leibniz-Wolff school and the Newtonian method of mathematical natural science (Schönfeld ). In Universal Natural History and Theory of the Heavens (), Kant conceives of a teleology that rests on the Newtonian forces of attraction and repulsion. Kant’s use of immanent forces to explain what Newton (supposedly) had explained using extrinsic agency means that Kant already is trying to show that the dynamical framework of nature is a self-sufficient system. Kant uses the Newtonian laws to explain how nature could sustain itself without the teleological background and could consist of a system of interacting forces that contribute to the continuation of that system. Kant’s arguments are motivated by the desire to avoid an appeal to actual infinity – Leibniz’s infinity of monads, for instance – and to avoid Newton’s and Descartes’ accounts of God as law-giver, which provide an extrinsically grounded, rather than an immanent, law-governed framework to nature. Both of these desires are deeply finitist in nature and exemplify characteristic number above. The second characteristic of finitism above is: “Finitism appeals to finite, concrete, usually intuitive construction procedures in generating a domain.” Here, there may seem to be a snag for interpreting Kant as a finitist. Kant endorsed Leibniz’s law of continuity throughout his career (McNulty ), and, arguably, this law requires being able to divide matter to infinity. But, in fact, Kant makes characteristically finitist arguments in laying out his dynamical theory of matter. He argues that we must show that the principles of mathematical construction apply to material bodies and their motions. Those principles may themselves appeal to infinite processes (division to infinity, for instance). But these regulative, mathematical rules do not require that we engage in constitutive physical reasoning using actual infinities. Kant is definite in not ascribing actual infinite extension to the universe, or actual infinite divisibility to any substance. As McRobert () analyzes the role of mathematical construction in the MAN, “Kant’s argument in the Dynamics makes clear that construction is a procedure which is necessary to exhibit the mathematical part of a concept. The problem, as Kant sees it, is neither mathematicians nor metaphysicians had shown why mathematics applies to matter” (p. ). As McRobert cites Kant in the Dynamics section,
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Schönfeld identifies Kant’s selection of “the Newtonian forces of attraction and repulsion as the vehicles for nature’s unfolding toward perfection” as, “paradoxically, his point of departure from Newton” (Schönfeld , §.).
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For it does not necessarily follow that matter is physically divisible to infinity, even if it is so from a mathematical point of view, even if every part of space is a space in turn, and thus always contains [more] parts external to one another. For so far it cannot be proved that in each of the possible parts of this filled space there is also substance, which therefore also exists in separation from all else as movable in itself. (MAN, :–)
McRobert continues: This ‘something,’ Kant maintains, is a constructive procedure which can be infinitely continued, and in terms of which matter can be represented as infinitely divisible. In the Dynamics, Kant’s preference for a dynamical concept of matter is explained in light of the constructive procedure [–], which requires that the iteration of an operation be possible indefinitely. (p. )
Reason demands the infinite divisibility of matter as a regulative principle. This demand does not promote the completeness of the system of nature, if we consider that system to consist of observable phenomena. In fact, it detracts from it, by showing that any actual, observable division of matter is incomplete and can be taken further. But if we can show that no physical division is a division into real parts, because geometrical divisions applied to physical bodies are divisions in thought but not in actuality, then we can ground the (geometrical) division of matter to infinity without requiring that matter be an aggregate of an infinite number of (real, physical) parts. And that is exactly what Kant goes on to say (MAN, :–). Reason can make a demand for infinite division. The understanding can show that no infinite aggregate of parts can be brought under the conditions of the possibility of actual experience. No rule for the connection of representations can be given such that an infinite aggregate can be constructed. And this fact, that no rule can be given, can be demonstrated a priori, even though it involves an empirical concept (matter or material body). Geometrical division is not real division. Kant can demonstrate this, because he can show that, despite reason’s demand for infinite division, we cannot have experience of a material body with infinite parts. We cannot provide any principle for the law-governed connection of representations that would be the basis of such an experience.
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The passage quoted from Kant picks up on many of the preoccupations of Émilie du Châtelet in her Institutions de Physique (Foundations of Physics), an important source of context for Kant’s arguments in this regard (see du Châtelet ). Holden () does not discuss du Châtelet, but investigates the debates over the metaphysical, geometrical, and mathematical concepts of matter. See Holden () for detailed discussion.
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Kant’s account of mathematical construction and its application to the concept of matter in the Dynamics thus exhibits the crucial characteristics of finitist theories. First, it requires what’s sometimes called the “surveyability” of objects in the domain of interest (Sieg , ). Second, it relies on the conditions for producing demonstrations via finite construction and derivation procedures. Of course, in the Antinomies, Kant argues that one could build a system using actual infinities, if one chooses. However, a key task of the MAN is to show that one can also build a complete system of nature without appeal to actual infinities. A final objection may occur to insightful readers. After all, Kant does appeal to two infinite given magnitudes in his proofs: space and time. And so one might object that Kant is not a finitist (noting that this objection would apply to Tait as well). I cannot resolve this complex question here but will point toward an answer. It depends very much on what Kant means when he says that space and time are “infinite given magnitudes.” On my own reading, the statement that space and time are infinite given magnitudes applies only to Kantian metaphysical space and time, as opposed to particular, constructed, empirical spaces and times. Metaphysical space and time contain not just the actual constructed measures of space and time, but any possible measures, and this is why Kant argues that the entire measure of metaphysical space and time is zero (Patton , ). The set of all such measures contains all the possible magnitudes that could be generated from a subject’s position. That set is ‘infinite’ in Galileo’s sense, but is not an ‘actual infinity’ in any sense that conflicts with a finitist reading of Kant.
. Conclusion The reconstruction above focuses on several interlocking features of Kant’s view: •
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Kant appeals to the ‘real essences’ or natures of material bodies (attainable by a priori exposition) in proofs, following a “mathematical For a proposed distinction between “original,” “geometrical,” and “metaphysical” space in Kant, and for a somewhat different reading of Kant on space and time, see Tolley (). For the purposes of evaluating Kant’s finitism, nothing hangs on whether we call the a priori space and time discussed here “original” or “metaphysical.” See Patton (, ff.). Kant argues that any actual measure of geometrical magnitudes must take place in time, an argument that dovetails well with the arguments above from Laywine () on the principle of succession and McNulty () on the law of continuity.
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method,” of how material bodies and their motions and interactions can be o Observable, and o Part of a complete system of nature. • Kant’s “completeness” in the MAN has two key elements: o It is derivable from the natures of material bodies, working with the a priori principles of space, time, and the categories, along with certain methodological principles. These include that bodies must be objects of a possible experience to form part of a system of knowledge. o It is “complete” in the sense that any possible experience of an epistemic subject must be classifiable as part of the nexus of relations it specifies. Kant’s comprehensive specification of the features (“real essences”) of material bodies is the basis of his argument that we can construct a complete system of nature without appeal to actual infinities. Kant’s finitism works hand in hand with his necessitarianism, and each puts limits on the other. For Kant, necessitarianism may not extend to positing a monadology when specifying the natures of objects: those natures may not include positing actual infinities or purely intelligible, unobservable substances. So Kant’s finitism puts limits on any Necessitarian reading of Kant. On the other hand, Kant’s finitist proofs are limited to those that can be elaborated in terms of observable material bodies and their phenomenal interactions. In general, I do not think one needs to choose only one of Kreines’ readings of Kant on the laws of nature. Kant’s system, in my view, is a blend of Necessitarianism and Finitism. The modal judgments that constitute the ‘edges’ of the system of corporeal nature in the Metaphysical Foundations of Natural Science do not ground limits in principle, or absolute limits. They are determined by the limits of the constructibility of proofs on the basis of the empirical concept of observable material body. They are explanations and descriptions of what it would mean for a system of nature to be constituted without outside intervention and without the need to make metaphysical postulates of God, Freedom, or Infinity: of what a science of nature that is complete in itself would be.
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Michael Bennett McNulty extended a gracious invitation to contribute to this volume and has provided substantial comments on drafts of the chapter. I presented an early version remotely at a workshop hosted by McNulty and by Katherine Dunlop, at which I received insightful suggestions and remarks from Dunlop, McNulty, Daniel Warren, Silvia di Bianchi, and James Messina.
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The Construction of the Concept of Space-Filling Kant’s Approach and Intentions in the Dynamics Chapter of the Metaphysical Foundations Daniel Warren .
Introduction
What does Kant intend the Dynamics chapter (chapter ) of the Metaphysical Foundations of Natural Science (MAN) to accomplish? Is there a goal specific to the Dynamics chapter? How does this aim fit into those of MAN as a whole, as described in the Preface to that work? At a first pass, we can say that the goal of MAN as a whole is the construction of the concept of matter and that the specific goal Kant is concerned with in the Dynamics chapter is the construction of the concept of matter as filling its space. But it is particularly difficult to determine what this amounts to, and how it is related to “construction,” in Kant’s technical sense. Moreover, there is a distinction between the goal that is specific to the Dynamics chapter, that is, the goal the chapter is concerned with, on the one hand, and what Kant says the chapter is actually meant to accomplish with regard to that goal, that is, its intended contribution toward that goal, on the other. And in order to understand this distinction, we need to ask what general claims Kant thinks he can make use of in that chapter. In this chapter I will approach these questions in three stages, which correspond to the division of the paper into Sections .–.. First, I will discuss what Kant says in the Dynamics chapter itself about the construction under consideration there (viz., the degree of space-filling). Second, I will consider what Kant says in the Preface to MAN about the construction of the concept of matter more generally, and I will discuss how its implications (with respect to the centrality of the concept of motion) are to be
Kant typically speaks of “construction of the concept of __.” Nevertheless, I take it that this essentially involves the intuitive elements of representation just as it does in geometrical examples, whether such construction is performed on a blackboard or just in the imagination. I say “at a first pass” because part of the aim this chapter is to point out the ways this claim requires qualification if it is not to be misleading. This is discussed at greatest length in the “Discipline of Pure Reason” (KrV, A–/B–).
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The Dynamics Chapter of the Metaphysical Foundations
understood in relation to the Dynamics chapter specifically. In the final section, which is the longest, I will examine a further law (which Kant calls the “universal law of dynamics”), which governs several crucial steps in the argumentation of the Dynamics. This law concerns the mathematical treatment of intensive magnitudes (such as the degree of space-filling) and allows us to see what Kant actually has in mind when he refers to construction in that chapter. This incorporates what he says in the Preface about construction of the concept of matter, but goes beyond anything suggested in the programmatic characterization of it given there.
. The Dynamics Chapter: Forces Leading to the Expansion or Compression of Matter ..
Filling Space (to a Determinate Degree) and Construction
MAN is concerned with the possibility of a mathematical physics, that is, the possibility of mathematizing certain basic concepts that figure in natural science, and with the role this plays in making laws of nature possible. In the first chapter (Phoronomy) of the work, Kant is concerned to give an account of motion considered as a “pure quantum.” In the second (Dynamics), he gives an account of the filling of space as a matter of degree, rather than all-or-none. That is, the property by which matter is distinguished from empty space, according to Kant, has a degree, that is, what he calls an “intensive” magnitude. In chapter (Mechanics) of MAN, Kant gives us a way of representing what was then called the “quantity of motion,” that is, the product of the quantity of matter (inertial mass) and the speed. Moreover, in each of these chapters, Kant is concerned with the “construction” of some feature of the concept of matter, which is in line with his view that in mathematical reasoning, the construction of concepts is essential. Kant writes in the Preface to MAN
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Intensive magnitudes are contrasted to extensive magnitudes. An extensive magnitude is made up of parts distinguishable from one another, and are in some sense outside one another; with an intensive magnitude, the parts are not outside one another (if in this case there is talk of parts at all) (KrV, A/B; V-Met/Dohna, :). Moreover, extensive magnitudes, e.g., a (finite) region of space or extended piece of matter, are made up of smaller distinguishable regions of space or pieces of matter; an intensive magnitude, e.g., a degree of heat, by contrast, is not made up of distinguishable parts with lesser degrees of heat (Refl , :). See MAN (:). Chapter (Phenomenology) is different in an important way. This chapter involves critical reflection on what had been done in the previous three chapters, which is particularly illuminating philosophically, but does not involve some fundamentally new construction.
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that “in order to make possible the application of mathematics to the doctrine of body, which only through this can become natural science, principles for the construction of the concepts that belong to the possibility of matter in general must be introduced first” (:). In the Phoronomy and Mechanics chapters, Kant actually exhibits the corresponding constructions. The Dynamics chapter, by contrast, proceeds differently in this regard. Kant explains at one point in the chapter why he thinks that the corresponding construction cannot be carried out within the confines of the chapter itself, and indeed why it cannot be carried out within the confines of a metaphysical work like MAN. But, as I indicated earlier, that doesn’t mean that the Dynamics chapter cannot make significant contributions to carrying out this construction. Here it will be important that the Preface passage only referred to the introduction of “the principles” for the construction. We will return to this point in Section ... What is the construction that Kant is concerned with in the Dynamics chapter? (We can ask this question even if Kant thinks the construction can’t be carried out within the confines of MAN.) His focus here is on the property by which matter “fills” its space (also called “solidity”: cf. : and :). Space that is said to be filled is contrasted to space that is empty. Space-filling is, or is the ground of, impenetrability. This is the property by which matter resists compression; it is not the same as cohesion or rigidity. It involves a resistance to a decrease in volume, not, for example, to a change in surface area or in the relation of parts. Kant refers to the corresponding construction right after he proves the last Proposition in the chapter: From this original attractive force . . . it should now be possible, in combination with the force counteracting it, namely, repulsive force, to derive the limitation of the latter, and thus the possibility of a space filled to a determinate degree. And thus the dynamical concept of matter, as that of the movable filling its space (to a determinate degree), would be constructed. (MAN, :)
Much of Section . of this chapter will concern the interpretation of this quotation and the sentences which directly follow it. This passage suggests that, in the Dynamics chapter, construction is concerned with the
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The concepts Kant is referring to here probably correspond to the concepts of “motion, the filling of space, inertia” (MAN, :). According to Kant, the impenetrability of matter, is, or is grounded in, its property of filling space. For to fill a space is to exert a force of repulsion, and the parts of a piece of matter in turn exert repulsive forces on one another; this is the basis for that matter’s impenetrability (conceived as a resistance to compression).
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The Dynamics Chapter of the Metaphysical Foundations
representation of space-filling as having an intensive magnitude (“a determinate degree”). After elaborating on the significance of this idea that filling space is a matter of degree, I will discuss two further issues before concluding Section .. The first concerns the fact that Kant suggests that it is only when attractive force is brought into the picture, along with repulsive force, that we can carry out the construction. The second concerns the fact that Kant says, “thus the dynamical concept of matter. . . would be constructed,” rather than saying “is [or will be] constructed.” The idea that space-filling is a matter of degree, and the connection between this and the constructability of the concept of space-filling, comes up at the earliest stages of the Dynamics chapter, significantly before Kant argues that an attractive force must be posited. The passage just cited, which came after the last Proposition of the chapter, picks up on Kant’s characterization of his project in the Remark accompanying the first Proposition. There he opposes his view of space-filling (as a matter of degree) with the all-or-none conception, the “purely mathematical concept of impenetrability” (MAN, :), which Kant associates with what he calls (:–) the “mathematical-mechanical” approach to explaining the concept of matter: Here the mathematician has assumed something, as a first datum, for constructing the concept of a matter, which is itself incapable of further construction. Now he can indeed begin his construction of a concept from any chosen datum, without engaging in the explication [zu erklären] of this datum in turn. But he is not therefore permitted to declare [zu erklären] this to be something entirely incapable of any mathematical construction, so as to obstruct us from going back to first principles in natural science (MAN, :).
Kant is criticizing the “mathematicians” for blocking a deeper understanding of matter; they do so by starting with something said to be incapable of mathematical construction, namely, existence in space, the all-or-none conception of “absolute” solidity or impenetrability they think this entails. Kant thinks that his own view of space-filling is not subject to this criticism. I take it that Kant thinks his own conception of space-filling as matter of degree, in contrast to that of the mathematicians he is criticizing, is capable of mathematical construction. However, the “deeper understanding of matter,” which it allows for, though significant, is strictly limited. Kant contrasts the mathematician’s absolute
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Kant is here opposing his own conception of space-filling to a conception of matter that fills space “by its mere existence” (MAN, :).
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impenetrability with his own dynamical conception, a bit later in the Dynamics chapter, as follows: Absolute impenetrability is in fact nothing more nor less than an occult quality [qualitas occulta]. For one asks what the cause is for the inability of matters to penetrate one another in their motion, and one receives the answer: because they are impenetrable. The appeal to repulsive force is not subject to this reproach. For, although this force cannot be further explicated in regard to its possibility, and therefore must count as a fundamental force, it does yield a concept of an acting cause, together with its laws, whereby the action, namely, the resistance in the filled space, can be estimated in regard to its degrees. (MAN, :)
So, although space-filling as a matter of degree, understood in terms of repulsive force, is explanatorily basic and incapable of being explained by something more fundamental, it is not an occult quality; it is not an “empty concept,” as Kant puts it nearer the end of the Dynamics chapter (MAN, : and :). The ability to construct the degree of spacefilling is not a matter of construction out of something more basic than repulsive force. Rather, it involves the appeal to laws of force (e.g., laws governing the degree of resistance as a function of the compression of a given piece of matter). And for this reason Kant thinks that positing impenetrability, conceived dynamically, is not explanatorily empty in the way that positing an un-constructible absolute impenetrability is. The idea that the all-or-none conception of space-filling blocks a certain sort of deeper scientific understanding is tied, in Kant’s view, to the fact that it is incapable of mathematical construction. By contrast, the conception of space-filling as having a magnitude (an intensive magnitude) allows for the application of mathematical reasoning along with its characteristic methods, as Kant understands them. Thus the idea of space-filling as a matter of degree, from the first Proposition of the chapter, fits with the idea from the last Proposition, discussed earlier, that it is the concept of matter as filling-space-to-a-certain-degree that is being constructed in the Dynamics chapter. It is by reference to the constructability of the concept of space-filling that Kant distinguishes his own dynamical conception of matter from the mathematical-mechanistic conception he is rejects. However, the picture of space-filling Kant is proposing has a further feature which has not yet been brought out. In the second Proposition of the Dynamics chapter, Kant is already claiming that “[m]atter fills space through the repulsive force of all its parts, that is, through an expansive force of its own, having a determinate degree, such that smaller or larger degrees can be thought to infinity” (MAN, :). And correspondingly,
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we can say that the degree of resistance to compression has “a determinate degree, such that smaller or larger degrees can be thought to infinity.” But one thing that has not yet been shown is that the repulsive forces by means of which matter fills its space are sufficient to account for its impenetrability. For even if it has been shown that space-filling (resistance to compression) is a matter of degree, it still needs to be shown that the degree of resistance can match that of any compressive force no matter how strong. That is required even by Kant’s dynamical conception of impenetrability. But unless the degree of resistance to compression increases without limit, as it is further compressed, there will be some degree of external compressive force which will be able to reduce the extension of the body to zero. And this, according to Kant, would constitute the penetration of matter. That smaller or larger degrees of space-filling by matter can be thought to infinity, as Kant put it in the second Proposition, or even that such larger degrees are really possible, does not amount to showing that the repulsive force between the parts, and, therefore, the degree of resistance to compression that is mounted, is such that it will be stronger and stronger without limit as the matter is compressed more and more. This is what the next Proposition, the third, is meant to prove: the repulsive force among the parts of a piece of matter, which it must have just insofar as it fills its space, is sufficient to check any compressive force, no matter how strong. This is the further feature that needed to be established. And it is only after that (viz., Proposition ), and in the succeeding Explication, that Kant will allow that he has shown that his notion of space-filling understood in terms of repulsive force is sufficient to account for the impenetrability of matter. But, as I understand it, the grounds Kant appeals to in establishing this in Proposition are of an entirely different sort from those that had come up earlier in the Dynamics chapter. The grounds concern a principle Kant calls “the universal law of dynamics” (MAN, :). This will receive more detailed attention in Section . of this chapter. ..
A Force That Checks Matter’s Expansive Tendency: Attraction
Now, as I noted earlier, it is only after Proposition (the last in the Dynamics chapter), after having established the existence of an attractive force, that Kant says we should be able to derive “the possibility of a space filled to a determinate degree.” And in this way, “the dynamical concept of matter, as that of the movable filling its space (to a determinate degree), would be constructed.” What explanatory role is Kant assigning to attractive force and how is it meant to play this role?
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In the first Proposition of the Dynamics chapter, Kant says that “matter fills space . . . through a particular moving force,” namely, repulsive force. He goes on to argue (in the subsequent Proposition) that this space-filling property is due to the repulsive forces between the parts of the matter. This, however, leads to a tendency toward expansion of the matter, and constitutes what Kant calls matter’s “expansive force.” Moreover, as the matter expands (i.e., as its volume increases), it becomes more and more rarified, and so the degree of space-filling decreases. For this reason, Kant thinks that some further force – a force which tends toward the compression of the matter, specifically, an attractive force – must be brought in to oppose this expansive tendency. Otherwise, according to Kant, it will not fill its space to a determinate degree. Doubts can be raised about whether Kant is justified in making this claim and in the reasoning which leads to it. I will not address these doubts in this chapter. More precisely, I will bracket the doubts about the supposed need to posit a compressive force of some sort in order to counter the tendency toward expansion. What I will examine in the remainder of this subsection and the next (albeit briefly) is why Kant thinks this compressive force must be grounded in a force of attraction, specifically, as well as how, in his view, this force of compression results from that attraction. Earlier in the Dynamics chapter (namely, in Note to Proposition , and in Explication , Proposition , and Explication , along with the associated Remarks), Kant had already brought in a force that opposes the tendency toward expansion which was due to the mutual repulsion of parts, though it is not yet assumed to be a force of attraction. Kant is arguing in this stretch of text that this expansive tendency is the ground of the impenetrability of matter. Instead of thinking of the expansive force as producing an actual expansion, we think of it as countered by a force of compression. And if, in this way, we also think of the expansive force as countering the force of compression, then we are thinking of the former as the ground of impenetrability. The concept of impenetrability is the concept of resistance to something (namely, compression). And the degree of impenetrability is conceived of as the degree of compression it can (just exactly) resist. (This will in turn be equal to the degree of the expansive force.) However, here, what counters the expansive tendency is not an
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This is not meant to suggest that these parts are ultimate (indivisible) parts. I take it that, in MAN, Kant’s view is that the parts can themselves be divided into smaller parts, and those, in turn, into further parts. I explore these doubts in Warren ().
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attractive force; rather, it is an externally applied compressive force, as might be applied to a piston acting on matter in a cylinder, rather than an attractive force exerted by that piece of matter itself. The idea that the expansive tendency must be countered by attraction comes in only later. ..
Two Roles for Attractive Force
It is notable that the Dynamics chapter brings in attractive force to counter the expansive tendency in two distinct stages. This in turn gives us a framework for understanding the overall progression of the Explications and Propositions in that chapter and their role in advancing its central theme, namely, matter’s property of filling space to a determinate degree. Kant’s announcement after the last Proposition of that chapter, that it should now be possible to “derive . . . the possibility of space filled to a determinate degree” (MAN, :), comes after the second of the two stages. The first stage in considering a force that can counteract the expansive tendency is found in Proposition , where Kant first brings in attractive force. Kant explains: All matter requires for its existence forces that are opposed to the expansive forces, that is, compressing forces. But these in turn cannot originally be sought in the contrary striving of another matter, for this matter itself requires a compressive force in order to be matter. Hence there must somewhere be assumed an original force of matter acting in the opposite direction to the repulsive force, and thus to produce approach, that is, an attractive force. (MAN, :–)
Here, rather than appealing to a counteracting force exerted by something else (an external force of compression), he appeals to a force dependent only on that piece of matter itself. The expansive tendency, which is due to the mutual repulsion between the parts of a piece of matter, is countered by the mutual attraction of these same parts. In the above passage, Kant makes this move in order to forestall what he apparently regards as the threat of an endless explanatory regress that would arise if the existence of one piece of matter depended on that of another, and that, on another in turn. This is an essential part of what motivates the introduction of a
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Kant is here concerned with what he thinks “matter requires for its existence” – more specifically, as the argument makes clear, what matter requires for the independent existence of matter. This idea of the existence independent of the existence of others, as a traditional criterion of substance, thematically links this argument to the discussion of the concept of substance, specifically, material substance, in Explication and its associated Remark (MAN, :–).
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fundamental attractive force in the proof of Proposition . Thus, the appeal to an external compressive force seen in the early stages of the Dynamics is now shown to be insufficient for Kant’s purposes (as indicated in Proposition ), and so, ultimately, an appeal to attractive force must play the role of balancing matter’s repulsion. In Proposition , attractive force between the parts of a piece of matter is introduced because “matter requires [it] for its existence.” However there is a shift of focus beginning in Explication , which continues through the last Proposition of the Dynamics chapter. Kant is now turning his attention to “the action of one matter on another” (MAN, :–). He is now concerned not just with the forces at play within a piece of matter, but rather with the forces exerted between different pieces. This is not to say that different kinds of forces are in play. But different considerations are at work when we consider the forces between “one matter” and “another.” It is in this context (specifically, Propositions and ) that that we see the second stage in which a role for attractive force is brought in. We may well ask what attraction’s immediate action at a distance (Proposition ) and its infinite reach (Proposition ) have to do with the central theme of the Dynamics chapter, namely, matter’s filling its space (its resistance to penetration). I believe that Kant tells us the answer to this question in the two “Notes” immediately following the proof of Proposition . Kant begins the first note with the sentence we considered earlier. Here I quote it more fully: From this original attractive force as a penetrating force [i.e., a force acting immediately at a distance] and extending its action to all matter at all possible distances, it should now be possible, in combination with the force counteracting it, namely, repulsive force, to derive the limitation of the latter, and thus the possibility of a space filled to a determinate degree. (MAN, :)
As indicated by the parts of the sentence I have italicized (which were omitted when I earlier quoted the passage), Kant is saying that what he established in Propositions and helps us to understand how attractive force limits the repulsive forces responsible for matter’s tendency toward expansion. Kant explains this more fully in the second note: Since every given matter must fill its space with a determinate degree of repulsive force, in order to constitute a determinate material thing, only an original attraction in conflict with the original repulsion can make possible
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For example, in Proposition , the idea that one matter exerts forces on every other matter in the universe.
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a determinate degree of the filling of space, and thus matter. Now it may be that the former flows from the individual attraction of the parts of the compressed matter among one another, or from the uniting of this attraction with that of all matter in the universe. The original attraction is proportional to the quantity of matter and extends to infinity. Therefore, the determinate filling, in accordance with its measure, of a space by matter, can in the end be effected only by the attraction of matter extending to infinity, and imparted to each matter in accordance with the measure of its repulsive force. (MAN, :)
Each “determinate material thing” exerts an attractive force on every piece of matter outside it. Thus every material thing, through its attractive force, counters the expansive tendency due to the repulsion between its parts. And it does so not only through the mutual attraction of these parts themselves, as in the first stage, but also by drawing toward itself all the matter outside it (including ambient fluids like the ether). In this way, it is itself responsible for a compressive force acting on it from the outside, through the “forces of pressure and impact” (MAN, :) of the surrounding matter. This is the second stage in which the attractive force of a given matter is brought in to counter its tendency to expand and, thus, on Kant’s account, to provide for “the possibility of a space filled to a determinate degree” (MAN, :). In the first stage, a piece of matter is responsible for limiting its expansion, in a way that is independent of anything external to it. In the second stage, the piece of matter limits its own expansion through its interaction with all the matter in the world of which it is a part. ..
What the Dynamics Chapter Is Not Meant to Accomplish
Before moving on to the third section of this chapter, I wish to consider a further issue which is important in interpreting the passage we’ve been examining, namely, the first note to Proposition . Kant says that once we bring in attractive force, in addition to the repulsive, “it should now be possible . . . to derive the limitation of the latter . . . And thus the dynamical concept of matter, as that of the movable filling its space (to a determinate degree) would be constructed.” This passage might lead the reader to think that nothing further is needed to actually carry out the construction. But if a reader notices that Kant uses the subjunctive “would
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In conversation, Bernhard Thöle has been particularly helpful in drawing my attention to the significance of this passage.
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be constructed [wu¨rde . . . konstruiert werden],” rather than the future indicative “will be constructed” or even “can be constructed,” they will be a little less surprised by what comes immediately afterward: But for this one needs a law of the ratio of both original attraction and repulsion at various distances of matter and its parts from one another, which, since it now rests simply on the difference in direction of these two forces (where a point is driven either to approach others or to move away from them), and on the magnitude of the space into which each of these forces diffuses at various distances, is a purely mathematical task, which no longer belongs to metaphysics – nor is metaphysics responsible if the attempt to construct the concept of matter in this way should perhaps not succeed. For it is responsible only for the correctness of the elements of the construction granted to our rational cognition, not for the insufficiency and limits of our reason in carrying it out. (MAN, :–)
What might seem surprising is what Kant says can and cannot be guaranteed by metaphysics (the metaphysics of body). Metaphysics can vouch only for what he calls the “elements” of the construction. This corresponds, perhaps, to Kant’s idea seen in a passage from the Preface quoted earlier, that MAN is meant to vouch for the principles of the construction. But however that may be, the point here is that the construction of space-filling can actually be carried out only if certain mathematical problems can be solved, and these lie outside the province of metaphysics and their solution is not its responsibility. The reason Kant uses the subjunctive here is to suggest not that this construction is not possible, but rather that it is not possible until some further (mathematical) considerations are successfully addressed. I will go on to examine what Kant thinks this construction is, the execution of which is a task not lying wholly within the jurisdiction of metaphysics. And then I will try to determine what, if anything, Kant thinks the Propositions and Explications of the Dynamics chapter (insofar as they constitute a part of the metaphysics of body) must contribute to the task of this construction.
. The Preface: The Account of Motion and Its Applications ..
Pure Intuitions, Intuition Given A Priori, and Construction
There is a further set of questions one might have about the idea that what is being constructed is the concept of matter as something filling its space
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The subjunctive is also used when Kant writes, “it should now be possible . . . to derive [mu¨ßte . . . abgeleitet werden können].”
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to a certain degree. This seems in a crucial respect far removed from the paradigmatic examples of construction, for example, those which according to Kant feature prominently in geometry. For the intuitions involved in representing space-filling or impenetrability are empirical. There is no pure intuition of the intensive magnitude of an empirically given quality. Yet it is the purity of the intuitions involved in geometrical construction that is, on Kant’s view, essential to their role in geometrical construction. So, one might ask, how could the concept of space-filling be susceptible of construction? Of course, not every case of what Kant calls “construction” needs to fit these paradigmatic, narrowly geometrical cases. There may be a range of related senses of the term. But in using the term, and evaluating its relation to the paradigmatic cases, what must be kept in mind is the epistemic role that construction is meant to play for Kant. For it is meant to explain, in certain cases (like geometry), how we can have knowledge of necessary truths, where this goes beyond what we know merely from the analysis of the concepts involved. Is construction playing this role when it is the concept of space-filling that is said to be constructed? And how could it play this role given the differences between this concept and the paradigmatic cases? Moreover, if, according to Kant, the construction of concepts is able to play this role in the paradigmatic cases only because the intuitions corresponding to such concepts are pure intuitions (of space and of time), then what plays the analogous role in the “construction” of the concept of space-filling? Why is it that in this construction, a role in underwriting necessary claims is not blocked by the empirical character of the intuition corresponding to the concept of space-filling? I’ll begin with what Kant says in the Preface to MAN about the general purpose and method of the work. Then I will apply this to the Dynamics chapter more specifically. One of the best known claims of the Preface is that “in any special doctrine of nature there can be only as much proper science as there is mathematics therein” (MAN, :). This really sets the agenda for MAN. Kant supports the claim with the following line of thought: Now to cognize something a priori means to cognize it from its mere possibility. But the possibility of determinate natural things cannot be cognized from their mere concepts; for from these the possibility of the thought (that it does not contradict itself ) can certainly be cognized, but not the possibility of the object, as a natural thing that can be given outside the thought (as existing). Hence, in order to cognize the possibility of determinate natural things, and thus to cognize them a priori, it is still
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required that the intuition corresponding to the concept be given a priori, that is, that the concept be constructed. Now rational cognition through construction of concepts is mathematical. Hence, although a pure philosophy of nature in general, that is, that which investigates only what constitutes the concept of a nature in general, may indeed be possible even without mathematics, a pure doctrine of nature concerning determinate natural things (doctrine of body or doctrine of soul) is only possible by means of mathematics. (MAN, :)
When Kant speaks of a “special doctrine of nature,” he is making explicit that this is a doctrine concerning a determinate kind of thing, namely, matter, or corporeal nature. Our thought about nature in general, about objects of experience, abstracting from whether it is specifically an object of outer sense or of inner sense, is governed by what Kant calls a “general metaphysics,” of the sort presented in the discussions of the categories found in the first Critique. If a concept is to figure in cognition, then as Kant puts it, we must know “the possibility of the object.” However that phrase is to be understood, Kant means it to indicate that intuition must accompany the concept. General metaphysics involves intuition in that the categories must be schematized if they are to be applied to objects, and if they are to figure in the Principles. The representation of time (and in a way, space) places conditions on the application of the categories. However, for Kant, the role of intuition in representing a determinate kind of natural thing, for example, matter, goes beyond this. Here, the intuition corresponding to the concept allows us to represent something falling under the concept, an instance. If our concept of material nature is to amount to cognition, then it is required that “the intuition corresponding to the concept be given.” Yet if Kant’s line of thought ended there, we would not yet have made the connection to mathematics. What he says is that “it is [still] required that the intuition corresponding to the concept be given a priori, that is, that the concept be constructed.” It is the need for intuition given a priori that is crucial here, but it also brings to the foreground questions, raised earlier, about the sense in which the concept of matter is something that can be constructed. But what is clear from this passage is that, as in the paradigmatic cases of construction (like geometry), intuition given a priori will play a central role. Why does Kant think that it is intuition given a priori that is needed? It is because, as he emphasized earlier in the Preface, he is concerned with our knowledge of the natures of specific kinds of things (here, objects of
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I take it that MAN is concerned with both what Kant calls, in the Preface, nature “in its formal meaning” and “in its material meaning” (:). It concerns the laws governing a determinate kind
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outer sense, i.e., corporeal nature, or matter). And, for Kant, that involves knowledge of necessary claims (natural laws), which are specific to natures of the kind of thing in question. These are not just those claims that hold of the species (objects of outer sense) merely in virtue of their falling under the genus (objects of the senses in general). If we are to go further than what we already know from the Critique about objects of the senses in general, the intuitions we draw on must themselves be capable of underwriting these further necessary claims. This is the role that pure intuitions are intended to play; Kant takes it that empirical intuitions, no matter how many of them are amassed, could not. ..
The Representation of Motion
What is puzzling here, when Kant talks of cognizing “the possibility of determinate natural things,” is that the concepts that Kant is talking about, concepts of a specific kind of object, are empirical concepts. A metaphysics of nature can be general or special. As metaphysics (whether general or special), Kant says, it “must always contain solely principles that are not empirical” (MAN, :). And although the concept of nature in general is pure, a special metaphysics concerns “a particular nature of this or that kind of thing, for which an empirical concept is given,” for example, “the empirical concept of matter” (MAN, :–). And if the intuition corresponding to an empirical concept, like the concept of matter, is itself empirical, we are faced with the same problem we discussed earlier, namely, in what sense can an empirical concept be said to be “constructed”? In what sense can “the intuition corresponding to the concept be given a priori, that is, . . . the concept be constructed”? In fact, not only are the concept of matter and the concept of space-filling (discussed earlier) empirical, even the concept of motion is empirical, on Kant’s account. But presumably, empirical concepts depend on empirical intuition to establish their objective reality and, in part, to give them content. How can the “intuition corresponding to the concept be given a priori,” if the concept is empirical? In this most general form, the worry is easily allayed. Even if an empirical concept has an empirical intuition corresponding to it, there could still be a component of that intuition which is not empirical. There could be a of object, thus, nature in the formal meaning. But it is not the nature of things independent of their relation to us; rather, the determinate kind is picked out by its relation to our cognitive faculties, as objects of outer sense, and thus as belonging to nature in the material meaning.
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spatiotemporal component to this empirical intuition, which itself allows of construction a priori, and, on account of which, further synthetic claims can be known to hold necessarily. The concept of an orange sphere is empirical; the intuition corresponding to it is empirical, for it involves sensation. But there is also involved the representation of a sphere. And what we know a priori about the geometry of spheres will also carry over to the empirical concept of an orange sphere. Thus, even if an empirical intuition is needed to give content to the concept and to show its objective reality, we can by a process of abstraction obtain a non-empirical component of the intuition. However, this straightforward sort of example does not tell us how a spatiotemporal component allows for a priori knowledge in the case of the concepts under consideration in MAN. Although it will turn out that, in the construction of space-filling, the representation of the spatial extension (size) will in fact play an important role, this comes in only at a later stage. Kant is not at this point especially concerned with the fact that, insofar as these objects are spatial (and temporal), we can apply, for example, geometrical claims (and “principles of the relations of time, or axioms of time in general” [KrV, A/B]) to them. MAN is an application of the Transcendental Analytic, not the Transcendental Aesthetic. The concepts Kant says he is concerned to construct are those concepts “that belong to the possibility of matter in general.” These are, as I indicated earlier, probably to be identified with “the concept of motion, the filling of space, inertia, and so on,” which Kant lists in the Preface (MAN, :), and they clearly correspond to the Phoronomy, Dynamics, and Mechanics chapters, respectively. Moreover, insofar as they figure in our knowledge of (sensibly knowable) nature, they are and have traditionally been meant to play a role in explaining alteration or change. And for this reason we can’t just rely on what we know from the pure intuitive representation of regions of space and what we know correspondingly with respect to stretches of time. Neither of these affords us a representation of alteration. For, according to Kant, the representation of alteration requires empirical intuition (cf. KrV, A/B; also cf. B), and in an important sense the same will hold of motion. This is a point I will return to shortly. So the question remains. In the case of an intuition corresponding to the empirical concept of matter, what is the spatiotemporal component “given a priori” that allows for knowledge of laws of outer nature? Kant does not give any clear, general answer to this question, but we can at least attempt to identify the way he puts his proposal in the Preface to MAN. There he tells us,
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[pure philosophy] makes use of no particular experiences, but only that which it finds in the isolated (although intrinsically empirical) concept [of matter] itself, in relation to the pure intuitions in space and time (in accordance with laws that already essentially attach to the concept of nature in general). (MAN, :)
For our purposes, what still seems unclear in this formulation is the idea of making use only of that which is found in the concept “in relation to the pure intuitions of space and time.” In spite of this unclarity we can be fairly confident of what Kant has in mind by looking at what he says almost immediately after this passage: The basic determination of anything that is to be an object of outer sense had to be motion, because only thereby can these senses be affected. The understanding traces back all other predicates of matter belonging to its nature to this, and natural science, therefore, is either a pure or applied doctrine of motion. (MAN, :–)
Thus, when Kant talks about making use of the empirical content of the concept of matter “in relation to the pure intuitions in space and time,” he likely has in mind that we are to relate it to the representation of motion in some way. Kant then goes on to say how each of the four chapters of MAN corresponds to a distinct way of considering motion: The first considers motion as a pure quantum.. . . The second takes into consideration motion as belonging to the quality of matter, under the name of an original moving force.. . . The third considers matter with this quality as in relation to another through its own inherent motion.. . . The fourth chapter, however, determines matter’s motion or rest merely in relation to the mode of representation. (MAN, :)
As I mentioned earlier, Kant thinks that even the intuition corresponding to the concept of motion is empirical. Kant says that beyond “these two elements, namely space and time . . . all other concepts belonging to sensibility, even that of motion, which unites both elements, presuppose something empirical. For this presupposes the perception of something movable” (KrV, A/B). So, I take it, Kant’s point is that the representation of motion involves the representation of space and the
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I have changed Friedman’s translation slightly by putting the last phrase in parentheses, as it appears in Kant’s text. The interpretation of Kant’s claim that motion is the basic determination “because only thereby can these senses be affected” is extremely controversial. I will not address it in this chapter. What is important for my purposes is what Kant goes on to do with the idea that motion is this “basic determination” and is specially fitted to serve certain epistemic functions.
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representation of time, both of which are pure intuitions. But that doesn’t mean that the representation of this particular way of uniting them can be represented a priori. According to Kant, an empirical intuition is required to establish that such a combination is (“really”) possible. Kant’s reasoning in this passage is that “[i]n space, considered in itself, there is nothing movable; hence, the movable must be something that is found in space only through experience, thus an empirical datum.” In the same paragraph, Kant gives a parallel argument that the concept of alteration presupposes something empirical. The pure representation of time does not suffice. He writes that “for time itself does not alter, but only something that is within time.” Similarly, the pure intuition of space does not represent anything as being in space. But that is precisely what is needed to represent motion. Parts of space don’t move; only something that is in space can move. An intuition cannot be “of motion” without undermining precisely that condition on which the purity of spatial intuition rests. An intuition of motion necessarily involves a sensational component. However, the concept of motion has an ambiguous status for Kant. Even if it is empirical in the sense that it needs an empirical intuition to vouch for its objective reality, there is a special sense in which we can form the concept and an intuition corresponding to it by abstracting from the empirical component. In the Preface Kant says: “The first [chapter of MAN] considers motion as a pure quantum in accordance with its composition without any quality of the movable” (MAN, :). In the case of the orange sphere, mentioned earlier, I can abstract from the quality (orange), and then consider only its extension, with respect to both its size and its shape. In the case of the motion of a piece of matter, I can consider motions abstractly, as “pure quanta,” as I do in the Phoronomy, and I can then “[determine] these motions a priori solely as quantities, with respect to both their speed and direction” (MAN, :). In phoronomy, “motion can be considered only as the describing of a space – in such a way, however, that I attend, not solely, as in geometry, to the space described, but also to the time in which, and thus to the speed with which, a point describes the space” (MAN, :). We consider motion abstractly, as a quantum, by representing the motion of a point through
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The quality Kant is referring to is space-filling or impenetrability, and I understand this talk of considering motion “without any quality of the movable” as a way of talking about abstracting from this quality. The other thing that the Phoronomy abstracts from is the “quantity of the movable” (MAN, :), by which I take it that Kant has in mind the consideration of matter as an aggregate of smaller parts, which he goes on to treat in the Mechanics chapter when discussing quantity of matter.
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a space, which that point can be said to “describe.” According to Kant, this will count as an intuitive representation of motion in which we have abstracted from what is empirical in the intuition. What we do not abstract from are the spatial and temporal aspects of the intuition. Moreover, from the related discussions in the B-deduction, it is also clear that we are not here abstracting from the fact that “[m]otion, as description of a space, is a pure act of the successive synthesis of the manifold in outer intuition in general” (KrV, Bn), which in turn involved attending to “the action in accordance with which we determine the form of inner sense” (KrV, B). However, in an important sense, this still does not mean that we have a pure intuition of motion. The only pure intuitions we have are those of space and those of time. Recall that beyond “these two elements, namely space and time . . . all other concepts belonging to sensibility, even that of motion, which unites both elements, presuppose something empirical” (KrV, A/B). The representation of motion draws on two pure intuitions (of space and of time), but it “unites” them in a certain way. However, it is in establishing the possibility of something (viz., motion), in which space and time are thus united, that we must “presuppose something empirical.” In the very abstract representation of motion as a point describing a space, “a pure act of the successive synthesis of the manifold in outer intuition in general,” we represent this way of unifying space and time, but we do not thereby settle the question about whether motion, as “motion of an object in space” (KrV, Bn), is possible. And if the representation of this unity “presuppose[s] something empirical,” for its objective reality, then even the very abstract representation of motion will, in exactly the same sense and for exactly the same reason, “presuppose something empirical.” What, then, are we to think of Kant’s claim that in the Phoronomy he considers motion as a “pure quantum” (MAN, :)? Is there any sense in which the representation of this unity can be said to be formed from what is available to us a priori, and which is, in that sense, pure? As I’ve said, according to Kant, the concept of motion “unites” the pure representation of space and the pure representation of time. A pure concept of motion is possible, drawing, in part, on the “two elements, namely space and time” that are said to be “united” in it (KrV, A/B), only if we can represent a priori the way in which they are “united.” By abstracting from the empirical properties of what is moved, we formed a representation of
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That is, “really possible,” in Kant’s technical sense.
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motion merely as motion of a point in space. But unlike the case of the concept of an orange sphere, if we are to have a pure concept of motion after abstracting from what is specifically empirical in it, something must be added as well. What is also involved in the representation of motion, as I mentioned earlier, is “a pure act of the successive synthesis of the manifold in outer intuition in general” (KrV, Bn). Thus, what is added is itself something pure, something that does not depend on an empirical element. And if this “pure act of synthesis” is taken to be what is involved in the representation of motion, even the motion of a mere point in space, it will involve the use, even if only in a very abstract way, of certain pure concepts, namely, certain of the schematized categories. For I take it that the categories are rich enough to give us the concepts of being numerically the same object over time, and the concept of the same object having incompatible predicates at different times. In the case of motion, we need the concept of two objects at one distance from one another at one time and the same two being at another distance at some other time. That is, the same objects bear incompatible distance relations to one another, though at different times. The objects may be abstractly represented as mere points. But these further concepts are indispensable to the representation of motion, and thus to the representation of the way space and time are here “united.” This will be enough to give us a core component of the concept of motion, without drawing on anything available only through experience, which is sufficient to allow for a priori reasoning that draws on the two pure intuitions it “unites,” even if it is not enough to allow us to establish the objective reality of the concept. This is not to make the concept of motion a category; it is to make it a “predicable.” A predicable is a pure concept that falls under the categories, but has greater specificity than them, because of the particular way it draws on the pure representations of space and time. Kant characterizes them as “derivative a priori concepts” (A/B). And although Kant takes it that the objectivity of the categories can be established a priori, that does not mean that the objectivity of concepts falling under them, like the predicables, can be. I take it that the “construction” of the concept of motion is not itself
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Again, this is not to say that this secures the real possibility of the objects corresponding to concepts. The incompatibility of these two distance relations is presumably known from the pure intuition of space alone. In this paragraph I draw heavily on Plaass, especially his idea of treating the “pure” concept of motion as a predicable, along with the idea that there may be (empirical) conditions on establishing the objective reality of the concept of the species that do not apply to the concept of the genus. See Plaass (, ch. ).
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intended to establish its objective reality. That, as Kant says, requires experience. But just as the predicables are pure concepts that can be formed without any reliance on empirical input, so too the concept of motion as a “pure quantum” can be formed without relying on experience. What the construction of this concept can establish are certain necessary conditions on any motion that is or can be given to us, and this is sufficient to constitute a “pure doctrine of the quantity of motion.” But this is a task which is distinct from that of establishing the “real possibility” of motion. These features are, I believe, common to the construction of all “the concepts belonging to the possibility of matter in general” (MAN, :), with which MAN is concerned. None of these constructions is meant to establish real possibility. As in the case of motion, that is something that only experience can provide. So, according to Kant, in the Phoronomy chapter, motion is considered as “a pure quantum.” And phoronomy is regarded as “a pure doctrine of the quantity of motion” (MAN, :), which centrally involves the representation of the composition (adding, subtracting) of motions. The subsequent chapters of MAN will draw on this “pure doctrine of the quantity of motion.” The focus in this essay is on the second chapter, the Dynamics. Thus, as Kant puts it in the Preface, the focus is on “motion as belonging to the quality of matter” (MAN, :). It is not obvious what this phrase means. We get some indication of what Kant has in mind from the gloss “under the name of an original moving force.” However, in order to understand the point of this, it is necessary to look at how motion actually figures in the Dynamics chapter – although, even then, questions and obscurities remain. There, motion is always considered as an effect, that is, as something produced. In the Dynamics, motion is considered as something produced in virtue of matter’s having a certain quality (presumably, the property of filling a space or impenetrability). Perhaps this is all Kant means by “motion belonging to the quality of matter.” But the idea of something “belonging to” a quality might more naturally suggest cases in which we say that further determinations belong to a quality (as when we say that the quality has a certain degree of intensity, or, if it is a color quality, a certain hue, etc.). And so it is not clear that all Kant has in mind here is that motion is produced by (the possession of ) the quality.
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In his groundbreaking work, Kant’s Construction of Nature, Michael Friedman (, –) argues against an account of the role of construction in MAN presented by Eckart Förster (), which is in certain important respects like the picture I propose. However, the idea that these constructions are meant to establish real possibility, which most of Friedman’s criticisms of Förster focus on, is explicitly denied in the account I am presenting.
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In the Phoronomy chapter, Kant considers motion as a “pure quantum.” In the Dynamics chapter, as we just said, motion is considered as something produced. In the Mechanics chapter, Kant considers matter’s ability to produce motion in others “by means of its [own] motion” (MAN, :), as occurs in interactions between pieces of matter, in what is called the “communication of motion.” Without going into this in any detail, the quantity of matter will also be an essential component in evaluating what a body (an inertial mass) can communicate to others in virtue of having a given speed. But the point is that a body, in virtue of having a given speed, can be the ground (or part of the ground) of its effects on others. In a rough and ready way, we can say that in the Dynamics chapter, motion is considered as effect; in the Mechanics chapter it is considered as cause. And in both these chapters Kant relies on the representation of the addition and subtraction of motions that were discussed in the Phoronomy. ..
Application to the Dynamics Chapter
I return now to the Dynamics chapter. Kant begins the Dynamics by analyzing what it is to fill a space: “To fill a space is to resist every movable that strives through its motion to penetrate into a certain space” (MAN, :). He then starts off the proof of Proposition with two claims, which I take to be premises. In a somewhat simplified form they are () penetration into a space is a motion, and () resistance is the cause of the diminution of this motion. I take it that premise () can be defended only if we can defend the claim that something causes the diminution of this motion, which itself is an instance of the second analogy (given that this diminution constitutes an alteration). More important in the present context is the fact that premise () is what allows us to apply the results of the Phoronomy chapter to the concept of penetration; namely, if something is a motion, the Phoronomy will show us how to represent it as a pure quantum, where this means, in particular, how to represent the addition or subtraction of motions. According to Kant, part of treating motion as a quantum is that if two quanta, homogenous to one another, differ in their magnitude, the second is equivalent to the sum of the first and something (a motion) homogeneous to it. More specifically, when a motion increases in magnitude, this is equivalent to the addition of a motion in the same direction as the first; and when a motion diminishes this is equivalent to the addition of a motion in the opposite direction. In each case the Phoronomy tells us how to construct a representation of the
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added motion. So when a body intrudes into the space of another, and it slows down, this is equivalent to the addition (to the motion of the intruding body) of a constructible, outward-directed motion. As Kant puts it, “Now nothing can be combined with a motion which diminishes it or destroys it, except another motion of precisely the same movable in the opposite direction (Phoron. Prop.)” (MAN, :). Kant is asking, In virtue of what does matter fill a space; that is, in virtue of what does it resist the penetration of others? He analyzes the filling of space in terms of the motion added to the intruding matter, a motion whose magnitude and direction can be represented through a construction, given the initial inward motion and the final motion (perhaps rest). He then concludes that matter fills space in virtue of possessing a power to produce such motion, that is, in virtue of possessing a kind of moving force. Kant begins by engaging in consideration about motions. And he refers us to the previous chapter, the Phoronomy, the “pure doctrine of the quantity of motion” (MAN, :). It is the Phoronomy that tells us that a decrease in the quantity of motion is equivalent to an addition of an oppositely directed motion. And this is what tells us that the tendency to resist is a tendency to add a motion (which is the core of Kant’s argument at this stage, in opposition to the mechanist’s position). This may seem do no more than belabor something obvious, but it is really a matter of spelling out the consequences of treating both motion and rest as being quantities, rather than as the presence of a property and its mere privation. Resistance to intrusion is regarded as a matter of adding a motion, not simply a matter of taking away or canceling a property that something had. Kant takes it that what he has shown about combining motions in the pure doctrine of the quantity of motion (Phoronomy chapter) can be applied to thinking about motion as something produced, that is, about motion as effect (Dynamics chapter). Once he does this, the argument proceeds from the character of the effect to the character of the cause. And based on the direction of the produced motion, Kant is able to characterize the force as a repulsive force. Thus, the argument is from the observed effects to the existence of a power to produce these effects. In general, the interest of an argument of this kind really only emerges when we say what more can be explained by the possession of the same causal
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Near the end of the Second Analogy Kant writes of our “acquaintance with moving forces, or, what comes to the same thing, with certain successive appearances (as motions) which indicate such forces” (KrV, A/B). This suggests the very close connection between our knowledge of moving forces and our knowledge of the motions they produce.
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power, or force, and what cannot – and this is precisely how Kant continues. Having considered motions added to other (intruding) pieces of matter, Kant will now move on to consider motions of the parts of a piece of matter relative to one another. The repulsive force between these parts, as Kant sees it, can explain matter’s expansive force (a tendency to expand), its resistance to compression, and ultimately its impenetrability. Two kinds of change in spatial configuration are considered here: expansion and contraction. Later, Kant argues that there must be a tendency to resist expansion (a tendency toward contraction) which cannot be explained by the repulsive force. And this is a part of his argument for positing an attractive force. Expansion and contraction (increase and diminution in volume) are changes in spatial determinations. I will call them “motions.” In the context of the Dynamics, they can be considered motionsproduced, that is, motions-as-effects. Can they also be considered merely phoronomically, susceptible of mathematical addition and subtraction? Not directly: the Phoronomy chapter doesn’t deal directly with changes in spatial determinations like expansion and contraction in volume. But expansion and contraction of a piece of matter can be analyzed in terms of motions of its parts relative to one another, their mutual withdrawal and mutual approach. And these are motions that the Phoronomy does directly apply to. Kant writes: All motion that one matter can impress on another, since in this regard each of them is considered only as a point, must always be viewed as imparted in the straight line between these points. But in this straight line there are only two possible motions: the one through which the two points remove themselves from one another, the second through which they approach one another. (MAN, :–)
Mutual withdrawal and mutual approach are motions that pertain to matter even if we abstractly consider this matter as a mere point which can move in space, and thus are precisely the sort of motion that the Phoronomy considers. In the Dynamics’ Proposition , the contribution of Phoronomy was that resistance (to intruding matter) was the same as the addition (to that matter) of a motion – an oppositely directed motion. In the same way, by analyzing expansion and contraction in terms of the mutual withdrawal and approach of parts, the consideration advanced in the Phoronomy will lead us to the idea that a tendency to resist compression is the same as a tendency toward expansion. In this way, what we know about the motions, as pure quanta that can be added and subtracted by means of corresponding constructions,
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contributes to the consideration of motions as motions-produced in the Dynamics chapter. And it allows us to see how the considerations put forward in the Phoronomy, which concerned the motion of a point, can be applied to increases and decreases in volume, which are the kinds of change (motions in a broad sense) that are most immediately central to Kant’s treatment of matter-as-filling-its-space-to-a-determinate-degree. However, this might seem a rather meager yield for a chapter as long and elaborate as the Dynamics. But it is clearly only a first step in the direction of mathematizing the concept of space-filling. It satisfies the requirement announced in the Preface “that the intuition corresponding to the concept be given a priori,” a treatment of the “concept [of matter] itself, in relation to the pure intuitions in space and time” in that we conceive of motion as something-produced in a body, and we construct a representation of a composite motion as a quantum in the manner described in the Phoronomy chapter. But this does not account for much of what goes on in the Dynamics chapter in the service of constructing the concept of matter as filling space to a determinate degree. And in particular, no connection has been made between the intensive magnitude of space-filling and what we’ve just been focusing on, namely, the extensive magnitude of the space filled. Moreover, as I mentioned at the end of Section .., something new is already being brought into the proof of Proposition , where Kant is considering the effects of an external compressive force acting on a piece of matter, and in particular, where he considers which effects of compression are possible and which are not. So there is a further consideration, going beyond the mere application of what was inherited from the Phoronomy chapter, which concerns the filling of space “to a determinate degree,” and which is specific to the Dynamics chapter.
. Kant’s Universal Law of Dynamics ..
The Characterization of This Law and Its Role in the Dynamics Chapter
We can get a better sense of this “further consideration,” which is involved in the construction of the concept of space-filling, if we look again at the passage from Note to Proposition , discussed earlier in this chapter. Kant is talking about what one needs in order to be able to construct a determinate degree of space-filling: But for this one needs a law of the ratio of both original attraction and repulsion at various distances of matter and its parts from one another,
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which, since it now rests simply on the difference in direction of these two forces (where a point is driven either to approach others or to move away from them), and on the magnitude of the space into which each of these forces diffuses at various distances, is a purely mathematical task, which no longer belongs to metaphysics. (MAN, :)
It is worth examining more closely why Kant says that this problem – that of establishing “a law of the ratio of both original attraction and repulsion at various distances of matter and its parts from one another” – is (as he puts it) “a purely mathematical task.” It is mathematical, he says, “since it now rests simply on the difference in direction of these two forces . . . and on the magnitude of the space into which each of these forces diffuses at various distances.” Kant, in speaking of the force laws governing attraction and repulsion, is drawing our attention to the role of direction in space and magnitude of space. These are taken to be purely mathematical properties, and they indicate a role for pure intuition in characterizing these laws. But if the solution to the “purely mathematical task” is to be applied to the metaphysics of body, we need to keep in mind how, according to Kant, these mathematical properties are being applied. The direction in space part is easier, and has already been discussed. It concerns the contrast between attraction and repulsion, or between the motions they produce, which will be important when we are representing the one as canceling out the effects of the other. At this point, what is more important to note is how Kant applies the idea of the magnitude of space. He speaks of “the magnitude of the space into which each of these forces diffuses at various distances.” This talk of “the magnitude of the space” here is not simply a way of referring to the distance between two points, the distance across which the attractive or repulsive force acts. Rather, the idea of a force “diffusing” into a space is part of a somewhat elaborate picture of regions of influence, which Kant employs in thinking about the action of fundamental forces. And in order to understand the role that “the magnitude of the space” is playing in this context, and thus to see how Kant interprets the construction of spacefilling, we need to examine this view he has of the action of forces in space, along with the principle that he says governs the view. In a pair of Remarks associated with Proposition , Kant begins: “Yet I cannot forbear adding a small preliminary suggestion on behalf of the attempt at such a perhaps possible construction” (MAN, :). Here he is referring back to the passage from Note we’ve just been discussing. This is especially useful to us since it gives an indication of the construction the Dynamics is concerned with – even if its execution lies outside the purely metaphysical tasks of MAN. In the first of the two remarks, Kant
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introduces a number of considerations in favor of an inverse-square law of attraction and an inverse-cube law of repulsion, along with some cautionary notes regarding their proper representation. In the second of these Remarks, he refers to “the mathematical presentation of the proportion in accordance with which attraction takes place at various distances” and “that [mathematical presentation of the proportion] whereby every point in an expanding or contracting whole of matter immediately repels the others” (MAN, :). He then goes on to cite the law on which these mathematical presentations depend: the action [Wirkung] of the moving force, exerted by a point on every other point external to it, stands in inverse ratio to the space into which the same quantum of moving force would need to have diffused [sich ausbreiten], in order to act immediately on this point at the determinate distance. (MAN, :)
The idea, for Kant, is that, if the spaces through which the force would “need to have diffused” are surfaces, more specifically, concentric spherical surfaces, the force will obey an inverse-square law. And if the spaces through which the force would need to have diffused are closed volumes, for example, spherical volumes, then the force will vary as the inverse cube of the distance. Kant calls this “the universal law of dynamics” (MAN, :), and it covers both cases. In thinking about this law, it is helpful to distinguish two senses of force. In most cases, when Kant refers to a force (Kraft) in the Dynamics chapter, he is talking about a causal power, something an object could be endowed with. This may be considered as operating from an extended region of space (where the body possessing the power is located), in which case the power is regarded as having both an intensity (which may vary from place to place in the region) and an extensive magnitude; but it can, in many contexts, be regarded as operating from a point, in which case it is regarded as having intensive magnitude only. Force, in this sense, is a causal power something has, not its influence, actions, or effects. The latter, which is closer to the Newtonian notion of impressed force, is said to be exerted or produced by the former. Appeal to the former notion is meant to be explanatory with respect to the latter. When Kant speaks of a quantum of force diffused through a space, he has in mind the diffusion of its influence (Einfluß) or actions/effects (Wirkungen), where these are exerted or produced by a force (in the first sense). The diffusion of a force
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If what is diffused are the effects/actions [(Wirkungen] ) of the force (or its influence [Einfluβ]), it is hard to work out a precise positive characterization of the ontological status of what diffuses. For, it is often only potential or possible actions/effects [(Wirkungen] ) that are said to diffuse, rather than
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through a space does not mean that the causal power is located in a space outside the body that possesses that power; rather, the idea is that it acts or can act in that space; that is, it has or can have effects in that space. Yet in the end, not much hangs on deciding whether it is the force in the first sense or the actions/effects that are said to diffuse. Kant is going to work with a notion of the total quantum of action/effects in a given space (a surface or a volume), when he applies the universal law of dynamics. The total quantum of action or effects gives us a (conceptually privileged) measure of the quantum of force in the first (causal power) sense. So sameness of quantum of force always goes along with sameness of the total quantum of effects. And in drawing consequences from applying the universal law of dynamics, it makes no difference whether one says it is the same quantum of force that diffuses into different spaces or the same total quantum of actions/effects. It is not at all a straightforward matter to give a general definition of this notion of the “total quantum” of actions or effects. But I take it that the applicability of the general law of dynamics depends on there being something answering to this notion, a notion of something the same in magnitude, across different applications of the law to a given object exerting these actions/effects. The total quantum of actions/effects is not their intensive magnitude, nor is it the extensive magnitude of the space in which it is diffused. Rather, it is the product of the two. And this goes along with the idea that if the total magnitude is the same, then the intensive magnitude of the actions/effects will be inversely proportional to the extensive magnitude of the space in which they are diffused. The name “universal law of dynamics” might lead a reader to expect that this law would occupy a prominent place in the presentation of the Dynamics chapter. But it doesn’t – at least not as an explicitly enunciated claim. The single place it is stated explicitly and named a “universal law” only comes after the main line of argument of the Dynamics chapter has been completed, that is, in the Remarks we’ve been discussing, which were
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actual actions/effects on pieces of matter. But I don’t think that question needs to be addressed for my purposes here. It should also be noted that Kant’s focus on the notion of the total quantum of actions/effects is not, from a scientific point of view, misguided or capricious. The notion will become increasingly important with the development of field theory in electromagnetism. Its descendent is concept of flux, e.g., the magnetic flux, where the corresponding intensive magnitude is the flux density (symbolized as B in the case of magnetic flux density; corresponding notions are developed in the case of an electrostatic field). But the claim that Kant has focused on a scientifically respectable notion is distinct from the claim that he is fully entitled, by his own lights, to use it for the metaphysical purposes he has in mind, which seems to me significantly harder to defend.
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attached to the last Proposition of the chapter. Nevertheless, it is a principle that is crucial to the way the Dynamics chapter had already proceeded. It may not be explicitly stated in the main thread of the Dynamics chapter’s Explications and Propositions. It is, however, essential to these arguments at two pivotal stages: first, in the argument that matter cannot be penetrated by a force of compression, no matter how strong (Proposition ), as was mentioned at the end of Section .. of this chapter; and second, in the argument that attractive force is exercised on all bodies no matter how far away they may be (Proposition ). In both places, “the universal law of dynamics” is merely implicit, but that this is the principle being appealed to is, I think, clear and beyond question. Moreover, in the central proposition of the chapter, where Kant argues that, insofar as matter is endowed with a repulsive force, it must also possess an attractive force (Proposition ), Kant is appealing to reasoning that is exactly parallel to that seen in Proposition , and it is extremely likely that, like this later Proposition, it too is ultimately based on the “universal law of dynamics.” So, as I understand it, Kant is already relying on this principle in carrying out what he takes to be the essential metaphysical tasks the Dynamics chapter is meant to accomplish. The attempt to solve the mathematical problems which arise in the further application of this same principle, problems on behalf of which Kant “cannot forbear adding a small preliminary suggestion,” can be relegated to the appended Remarks, which the main line of the argument in the Dynamics chapter – the sequence of Explications and Propositions – doesn’t depend on. The actual mathematical form of the force laws (with respect to dependence of force on distance) may be needed for the construction of “the dynamical concept of matter, as that of the movable filling its space (to a determinate degree).” But the application of the principle to obtain more “qualitative” conclusions about the end point behavior of these functions, that is, what happens to the force as the distance approaches zero and as the distance approaches infinity, is regarded by Kant as essential to the metaphysics of body. One can legitimately ask why the line between these two kinds of application (getting the precise form of the function vs. merely getting the end-point behavior of the function) has the significance Kant seems to
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The account of the Dynamics Propositions and presented by Michael Friedman () differs significantly from the one I give. The contrast, I believe, can be seen most sharply on pp. – (for Proposition ) and p. (for Proposition ). Limitations of space do not allow me to pursue these points of difference here; they will be taken up in future work.
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think it does. But it seems to me that some progress has been made, even if we do no more than understand the context in which that question can arise. .. Diffusion of Force In his statement of what he calls the universal law of dynamics and in its applications, Kant talks about the “diffusion” (Ausbreitung) of the force, or of the quantum of force in a space. In his Physical Monadology of , in a passage (MonPh, :) concerning the derivation of force laws that closely parallels the passages in MAN we’ve been looking at, Kant indicates two sources for this idea of diffusion: () the theory of degrees of illumination at various distances from a source of light and () the seemingly aprioristic attempts to derive the inverse-square law of attraction by John Keill (a second-generation Newtonian). These sources have in common a focus on the intensive magnitudes of the actions or effects. And they also have in common a focus on the inverse relation between these intensities and the extensive magnitude of the region of a space (a surface) in which these actions or effects occur. In the theory of light, we distinguish a light source (a flame, a bright surface, etc.) and its action or effect, the illumination at a location some distance from the source. And here, what corresponds to force in the sense of causal power is the illuminating strength of the light source. However, when Kant extends the light model to the case of fundamental forces, he will need to take into account a basic difference in the character of the relation between source and action/effect. In the case of light, it is possible to propose some kind of mechanism to explain this relation, that is, to explain the transmission of light. However, in the case of fundamental force, we have reached the rock-bottom of explanation. The question about how the source produces actions or effects simply has no deeper answer.
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In these passages from the Dynamics chapter, Kant seems to use interchangeably both the noun forms “Verbreitung” and “Ausbreitung,” as well as both the verb forms “verbreiten” and “ausbreiten” (in both reflexive and irreflexive forms), all of which are translated into English as “diffuses” and its cognates. In a discussion many years ago, Michael Friedman suggested that we think about a role for geometrical optics, when Kant says we could “construct the degree of the sensation of sunlight out of about , illuminations from the moon” (KrV, A/B). In a way, this chapter is an attempt to work out that suggestion, though it may go in a different direction from what was intended.
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In Keill’s discussion of gravitational attraction, the relevant force law is presented as a particular instance of what he regards as an absolutely general rule governing any quality a body has. Keill writes: We have demonstrated this Theorem universally, whatever is the Nature of the Quality, so that it acts in straight Lines; and it hence follows that the Intensions [i.e., intensities] of Light, Heat, Cold, Perfumes, and the like Qualities, will be reciprocally as the Squares of their Distances from the Point whence they proceed. Hence also may be compared amongst themselves the Actions of the Sun on different Planets. (Keill , –)
Unlike Keill, Kant will not claim that this holds for every “Quality or Virtue” (ibid., –). As we will see, Kant only endorses a qualified and restricted version of the general claim. Kant only intends his universal law of dynamics to cover a special subset of these “Qualities and Virtues.” And part of the function of the Dynamics chapter, I claim, is to establish that the repulsive and attractive forces do in fact belong to this subset, in other words, that these forces satisfy the conditions for applying the universal law of dynamics to them. I want to make two points about this talk of diffusion in Kant. First, what is diffused is not necessarily some kind of substance, some kind of material or fluid, though it can seem natural to picture what is diffused in that way. Even if the transmission of fluids plays a role in explaining some cases of causal influence, it cannot, on Kant’s view, be an account of the action of fundamental forces. Nor can Kant be thinking that what is diffused could be like a pattern (of motion, or of any accidental property), which is propagated over a distance, for example, as a wave moves in a medium. It must be kept in mind that Kant takes his account of moving force to vindicate the idea of immediate (i.e., unmediated) action at a distance, and this kind of substance-picture or waves-in-a-medium-picture of what is diffused would, for such a force, render this vindication pointless. This is not to deny that in some cases, for example, the propagation of light, there is an underlying account of particles or waves that backs up the notion of diffusion. But my claim here is that the notion Kant is using does not require this. Although the theory of light serves as a model in thinking about diffusion, what Kant has in mind is something more abstract and metaphysically more fundamental. It is tied to an account of transeunt causal relations, the idea that a (finite) substance can be present, that is, active, outside itself, and that when a substance acts on a distinct substance, it is, in some sense, immediately present where that other substance is, at least if this action is immediate (unmediated).
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This is called virtual presence (“virtual” from vis, i.e., power or force), in contrast to what is called substantial or local presence. This notion of “external presence” figures in the early essay, the Physical Monadology, but this notion also appears in MAN. Second, diffusion is not necessarily to be regarded as a temporal process. We should not imagine that, in the course of diffusing, the activity of a force shifts or extends from one place to another. The representation of the diffusion is just the representation of the spatial distribution of something’s activity, which Kant also calls its sphere of action or influence. When Kant speaks of a body’s attractive force diffusing through a space, he does not mean that its activity is first closer to the body and then, later, further away. The diffusion through a space of a body’s essential repulsive forces is its “filling” that space, but this diffusion is no more a temporal process than its filling the space is. On the other hand, this doesn’t mean that the distribution of the activity of a force is unchanging. If an attracting body changes its shape from a sphere to a flattened ellipsoid, this will be accompanied by a change in the distribution of its activity in space. The point is that diffusion is not itself a temporal process, not that the diffusion of activity can never change. Indeed, the representation of changes in the distribution of this activity is central to the way the universal law of dynamics governs the construction of the degree of space-filling. We said earlier in the chapter that, for Kant, construction in the metaphysics of corporeal nature involved the representation of motion. This is relatively straightforward in the Phoronomy and in the Mechanics chapters. It is somewhat more complicated in the Dynamics chapter. To be sure, as we have already noted, motion is brought in as something-produced-in-a-body through the action of some moving force. But the motions produced that are relevant to the construction of the degree of space filling are a very narrow subset of such motions. The magnitude of the space into which a given force diffuses can change, and when it does change this can be considered a species of motion – expansion or contraction – which is reducible to the basic
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See MonPh (:; also cf. :). See :, where Kant brings out the close relation between being present in a space and occupying a space. See MAN, : and :, where this is discussed in relation to attractive force. Also cp. :, where Kant argues that even repulsive force acting by contact is an instance of the generalization that “every thing in space acts on another only at a place where the acting thing is not.” This is especially important for understanding the way Kant thinks about the diffusion of repulsive force, which is always confined to the region the matter fills.
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motions in the Dynamics chapter, namely, the withdrawal or approach of the parts. In constructing the degree of space-filling of a piece of matter – the degree at which the balance of forces has occurred – we are focusing on that magnitude of its extension for which the motions due to the opposed forces exactly cancel out one another: the motion of contraction cancels out the motion of expansion. The universal law of dynamics concerns the magnitude of a space; it does not itself concern change. But it can certainly be applied to cases of change, specifically, to changes of the magnitude of the relevant space. The motions we are considering – namely, changes in the “magnitude of the space into which a given force diffuses” – are ones to which the universal law of dynamics can be applied. And it is this law that allows us, with the aid of geometrical methods for representing inverse proportions, to move from the change in the “magnitude of the space” to a change in the intensive magnitude of the “action of the force” that has diffused into that space. It is in this way that the intensive magnitude of the action, given by a force law, can be said to be constructed. And the degree of space-filling (the intensive magnitude of the expansive tendency that the compressive forces had to cancel) is then constructed, by determining where the force laws will give equal and opposite magnitudes of action. Strictly speaking, it is not these force laws that are constructed. These force laws are special cases (species) of the universal law of dynamics (the genus), and as such they are what enable us to construct representations of the intensive magnitudes of the effects from the extensive magnitudes of the space into which the force diffuses. Although the universal law of dynamics brings in pure intuition when it mentions “the space into which a given force diffuses,” this does not in itself bring in the idea of motion or change. But in the construction of the degree of space-filling, this law must be applied to cases where the size of that space changes, and thus to cases of motion (in a broad sense). Insofar as this law contributes to the task of the Dynamics chapter, the role of pure intuition is not just to support the application of geometrical theorems to spaces of various shapes and sizes. The role of pure intuition is to afford us a representation of a species of motion (expansion or contraction of a volume). And this is in line with the idea, discussed earlier, that the required “relation to the pure intuitions of space and time” (MAN, :), which makes possible “a genuine metaphysics of corporeal substance”
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As in the case of the spatial component of the intuition of an orange sphere, mentioned in Section ...
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(ibid.) always involves making natural science “a pure or applied doctrine of motion” (MAN, :–). Kant takes himself to be entitled, in the theory of matter as filling-aspace, to make use of the claim he calls the universal law of dynamics, which is that a force or its activity, or more properly, its intensity, is inversely proportional to the space into which it must diffuse in order to act immediately at a given point. But what is behind this claim? What, in Kant’s view, grounds it? Is it an empirical claim or does Kant take it to be a priori? I suspect that Kant does consider it to be a priori, given the role he assigns it in metaphysics (although what it can be applied to might be, in part, an empirical question). To better understand why, I will consider in the final section of this chapter one objection that can be raised against treating it as a priori. Although I think this objection is particularly illuminating, I don’t claim to have ruled out other possible objections to treating the “universal law of dynamics” as a priori. It remains unclear what Kant would say about a deeper basis for this law. But it is significant that there is a long-standing metaphysical commitment on Kant’s part which concerns the idea – which the universal law of dynamics depends on – that the total quantum of actions/effects is always the same. In his early essay, the Nova dilucidatio of , Kant presents “certain genuine corollaries of the principle of determining ground” (PND, :). These corollaries include “There is nothing in the grounded which was not in the grounded,” that is, “There is no more in that which is grounded than there is in the ground itself” (PND, :–). Both formulations allow that that there might be something (more) in the ground than in what is grounded. That is because they are general enough to cover the case of God’s creation of the natural world, and in that case it would be granted that there is something in the cause that is not in the effect. But if we just focus on the relation between ground and grounded within nature, then a stronger principle holds, and that is that there is exactly as much in that which is grounded as there is in the ground itself. This gives us a conservation law, but it is not in general concerned with the conservation of the quantity of substance. As Kant puts it: “The quantity of absolute reality in the world does not change naturally, neither increasing nor decreasing” (PND, :), and by “quantity of absolute reality,” he does not mean the quantity of substances. His examples are forces and motions. But it is not clear on what grounds Kant could hold on to this principle in the critical period, even if its application is limited to experience. I argue elsewhere that the universal law of dynamics should be seen as a condition or presupposition (at least, in a specifiable subclass of cases) on applying
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the notion of intensive magnitude to force, as causal power, or ground. Kant’s view is that extensive magnitude is the magnitude of a thing as aggregate. Intensive magnitude, however, is said to be the magnitude of a thing as ground, at least if we consider intensive magnitude as a quantity in the full sense, meaning that we can represent the ratios, and the sums or differences, of such quantities (not merely the relations of greater than, less than, or the same). It is by restricting attention to spatial regions (e.g., the parts of a spherical surface) with the same intensity of actions/effects as one another that we can determine the total action/effect in the region (the whole spherical surface). By comparing this sphere with a sphere of a different size, we can then assign a ratio to the intensities in the two regions. At least we can do so if we can presuppose the general law of dynamics or something analogous to it. Assigning a magnitude to the force, as causal power, always involves assigning an intensive magnitude to it (the magnitude of the causal power “as ground”). And, as I said earlier, the magnitude of the total action/effect gives a measure of the magnitude of the force (considered as causal power). It is the universal law of dynamics, and the notion of the total quantity of action/effects that it relies on, that guarantees that the measure of the quantity assigned to the ground is univocal. In the case of light, for example, the spherical surfaces centered on the source each has a constant intensity of illumination. Within a given sphere, the sizes of sub-regions (parts of that sphere) may be added or subtracted, and in this way ratios, and the sums or differences of the total amount of light (the total action/effect) within those regions can be added, and so on. The total amount of light, which gives the measure of the magnitude of the source, is given as the amount light over the whole spherical surface. And so, if the total amount is the same no matter which sphere is chosen, then the light intensity will be inversely proportional to the square of the distance. .. Conditions on Applying the Universal Law of Dynamics This last point brings out an assumption – a qualification or restriction – which I have alluded to, but have not yet discussed. In photometry, there is an inverse relation between the intensity of illumination and the size of
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See Warren (). “The magnitude of a thing as aggregate is extensive, as ground, intensive” (Refl , :; see also V-Met/Volckmann, :; and V-Met/Dohna, :).
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the sphere illuminated. This kind of case is a model for applying the universal law of dynamics. But it may be objected, what if the air through which the light travels is smoky? What if the ethereal medium through which light waves move is slightly viscous, resulting in frictional losses as the underlying medium vibrates? In other words, what if there are causal processes which result in the dissipation of the light as it travels away from the source? In that case, the light intensity will not obey the inverse-square law, for the total quantum of light in the concentric spheres will not be constant – presumably, it will decrease as a function of the sphere’s radius. In that case we cannot suppose that the “total amount of action” is defined, and thus, that the universal law of dynamics can be applied. In this kind of case, we will tell a causal story about the process by which the dissipation of light occurs. And for Kant, such an account will certainly depend, at least in part, on what we can know only from experience. In the photometric case, we had simply assumed that the effects of the light source are not dissipated as they propagate outward, and that the empirical claims about the process in which the light dissipates could be set aside. Let’s now shift our attention from photometry to the case of attractive and repulsive forces. We can then raise the objection, as we did when we were considering the case of light: How do we know that there isn’t some dissipation of the action of, say, the attracting body as we get further away from it? For Kant, this is exactly where we must appeal to the fact, which he has established, that attraction acts immediately at a distance. And because it acts immediately at a distance, he says, “no intervening matter sets limits to the action of an attractive force” (MAN, :). For this reason, Kant thinks that it doesn’t make any difference if there is matter of some sort between the attracting body and the place it acts – it will act just as it would if the space in between were absolutely empty. This point is crucial when Kant presents his proof of Proposition , the claim that attractive force extends to infinity: Because the original attractive force belongs to the essence of matter, it also pertains to every part of matter to act immediately at a distance as well. But suppose there were a distance beyond which it did not extend. Then this limiting of the sphere of its activity would rest either on the matter lying within this sphere, or merely on the magnitude of the space in which it diffuses this influence. The first [case] does not hold; for this attraction is a penetrating force and acts immediately at a distance through that space, as
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See also the discussions at : and especially at :, where Kant considers the idea that matter, through its attractive force, “occupies a space without filling it.”
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an empty space, regardless of any matter lying in between. The second [case] likewise does not hold.. . . Thus, since there is nothing that has anywhere limited the sphere of activity of the original attraction of every part of matter, it extends beyond all specified limits to every other matter, and thus throughout the universe to infinity. (MAN, :–)
The “second [case]” was referred to earlier; it is the first that is relevant at this point. So, assume that a body endowed with a causal power (an attractive force) produces effects which are limited in their intensive magnitude as we get further from that body. Kant’s point is that this diminution of the body’s action can have two very different kinds of ground. It could be due to the intervening matter. And I take it that, in that case, the diminution has some kind of causal basis; it is something that can be traced to causal powers and susceptibilities of the intervening matter. But, according to Kant, in the case of a force which acts immediately at a distance, the intervening matter is, so to speak, circumvented, along with any possible role this matter could play in diminishing the effects the body can exert. The second kind of ground for the diminution of the effects produced by a body, according to Kant, is just the magnitude of the space itself. But this diminution by the magnitude of space, unlike a diminution due to some intervening matter, is not itself a causal process. Empty space, Kant assumes, is causally inert. It neither produces effects nor is it affected by anything. Nevertheless, it can be cited as a reason for the diminution of the intensity of the action or effect of a force. Thus Kant claims that “a greater distance would indeed be a reason for the degree of attraction to diminish in inverse ratio, in accordance with the measure of the diffusion of this force” (MAN, :; my emphasis). However, it would not be a reason for any diminution of the total quantity of action or effects. To account for that kind of dissipation, on Kant’s view, would require an appeal to some causal process. Differences in the magnitude of the space can be a reason for the effects to be more spread out. It is a reason for a change in the spatial distribution of the effects. But, as I understand the picture here, if all that happens is a redistribution in space of the effects, the idea is that the total magnitude of the effects is unchanged. The application of what Kant calls “the universal law of dynamics,” the law that the intensity of the effect is in inverse proportion to the space in which the force diffuses, depends on the contrast between cases in which there is dissipation with distance (as, e.g., in the smoky room) and the cases in which there is not. The former involve a diminution of the total effect, but the latter assume this to be unchanged. The latter can only involve a difference in the spatial distribution of that effect, that is, a difference in its
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extensive magnitude is a ground for a corresponding difference in the intensity of that effect. Cases of the first sort can be explained only by the appeal to causal laws, which cannot be known wholly a priori. Cases of the second sort involve a quite different form of explanation. And regarding the absence of the kind of explanatory grounds that make the first so clearly empirical, this absence may lead Kant to think he is justified in treating the law governing the second as a priori. But whether such an inference could ultimately be justified is less important to my purposes here than to have made out () the distinction between the two kinds of reason that could be cited for the diminution of the intensity of the effects and () the role of that distinction in Kant’s argumentation. We can apply the universal law of dynamics to a force only if it satisfies a certain condition. It must be characterized in terms of its immediate effects. “Immediate” is here contrasted to “mediated.” If there are intermediate steps by means of which it produces its effects – that is, if these effects are produced by a chain of causes and effects – then we must allow for the kind of causal influences that will lead us to worry about dissipation of the effects. And we won’t be able to apply the universal law of dynamics to generate a law governing these effects. The force laws Kant is attempting to ground in this way are laws governing the immediate effects of the force. Kant writes: Of any force that acts immediately at various distances, and is limited, as to the degree with which it exerts moving force on any given point at a certain distance, only by the magnitude of the space into which it must diffuse so as to act on this point, one can say that in all the spaces, large or small, into which it diffuses, it always constitutes an equal quantum, but [also] that the degree of its action on that point in this space is always in inverse ratio to the space, into which it has had to diffuse, so that it could act on this point. (MAN, :–; my emphasis)
In applying the universal law of dynamics, Kant is concerned with cases in which the degree of the effect is limited “only by the magnitude of the space into which it must diffuse.” But I suspect that Kant takes the immediacy of the effects to be part of a sufficient condition for this. For after having argued that matter’s essential attractive force acts immediately at a distance in Proposition , he makes essential use in the proof of Proposition of the claim that this immediacy of action (at a distance)
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Possibly, this is sufficient only when this condition is conjoined with the claim that attraction is a force “having a degree below which ever smaller degrees can always be thought to infinity” (MAN, :), which Kant also brings in as a premise in the proof of Proposition .
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justifies us in saying that even if matter is present in the space between the attracting body and the location of its effect, that body acts just as it would if that intervening space was completely empty. In any case, Kant seems to think that establishing the immediacy of the attractive force is a necessary condition of applying the universal law of dynamics to it. Kant is equally concerned to consider the immediate effects of matter’s repulsive force. He considers this to be a “surface force,” meaning that it acts only by contact (MAN, :). And for this reason he treats the repulsive force of a piece of matter as having its immediate effect only in the immediate (i.e., vanishingly close) neighborhood of that matter. Thus, when he applies the universal law of dynamics to repulsive force, he says that the force between mutually repelling parts of matter diminishes “in inverse cubic ratio to their infinitely small distances” (MAN, :; my emphasis). Moreover, in Proposition , which was mentioned earlier in the chapter, where Kant argues not for the inverse-cube law specifically, but just for the claim that “matter can be compressed to infinity, but can never be penetrated by a matter, no matter how great the compressing force of the matter may be” (MAN, :), I said that he makes use of the universal law of dynamics. It is this law that guarantees that, as the volume of a piece of matter is decreased more and more, “the same quantum of extending forces, when brought into a smaller space, must repel all the more strongly at every point, the smaller the space in which this quantum diffuses its activity” (ibid.). And for this reason, the intensity of the resistance to compression increases ad infinitum as the space filled by the matter decreases. But again, there are conditions on using the universal law of dynamics in Proposition . There he says in the attached Remark: In this proof I have assumed from the very beginning that an expanding force must counteract all the more strongly, the more it is driven into a smaller space. But this would not in fact hold for every kind of merely derivative elastic forces. However, it can be postulated in matter, insofar as essential elasticity belongs to it, as matter in general filling a space. For expansive force, exerted from every point, and in every direction, actually constitutes this concept. (MAN, :)
This “elasticity” is a property of a piece of matter based on the mutual repulsion of its parts. If this elasticity were “derivative,” according to Kant, the universal law of dynamics could not be applied to it, and there would be no reason to think its intensity would increase without limit as the space in which the force diffuses is compressed. It is not clear exactly what the
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relation is between something’s being “essential,” on the one hand, and its being characterized in terms of immediate effects, on the other. Each is meant to distinguish the case as one in which the universal law of dynamics can be applied. Kant typically contrasts “derivative” forces to “fundamental” ones. The effects of derivative forces are said to be explainable in terms of the effects of more basic ones. And the fundamental forces are explanatorily most basic. If an elastic force is not derivative, I assume it has this kind of fundamental explanatory role. But it is hard to see how the fundamental character of a force can be made out, within a Kantian framework, without thinking of these forces in terms of their immediate effects. However that may be, the main point I want to draw out is this. The construction of matter insofar as it fills space to a determinate degree is the ultimate aim of the Dynamics chapter. As part of a metaphysics of body, it can itself make only a limited, but nevertheless essential, contribution toward that goal. But whether we are talking about the completion of the task or just the part that metaphysics can accomplish in advancing it, what will be crucial to the construction is the application of the universal law of dynamics. And so part of what the Dynamics chapter must do is to establish that the repulsive and attractive forces essential to matter satisfy the conditions on applying this law. It is perhaps not so important whether those conditions are framed, specifically in terms of the immediacy of the effects or, for some purposes, in terms of other related notions. This goes some way toward explaining why Kant thought that it was necessary to include Propositions in the Dynamics concerning the action of the attractive force, and did not stop when he established, in Proposition , that it was essential to matter, as a necessary condition of there being a degree of space-filling where the expansive tendency of the repulsive force would be checked by a force of mutual attraction. Kant needed to establish that its action at a distance was immediate, if he was to apply the universal law of dynamics, as he does in the proof of Proposition , and thereby, to show what metaphysics could contribute to the construction of the degree of space-filling. This fits well with the idea that the chief reason Kant
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When Kant calls the elasticity “essential,” as compared with “derivative,” he might mean not only that it is in some way fundamental, as I’ve just pointed out, but also that it is based on forces that belong to matter as such. That is, this elasticity cannot be based on its containing a special kind of matter. This also accounts for the inclusion of a detailed analysis of action by contact, and thus the caveats on the proper interpretation of the immediate action of a contact force, like matter’s essential repulsive force.
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brings in attractive force into the chapter is that there must be some check on the action of the repulsive forces, and he needs to understand how this check or limitation works. This is confirmed by the brief “General Note” appearing in the Dynamics chapter after its Propositions and Explication have been presented, where Kant summarizes what he has done and connects it to the table of categories in the first Critique. There he emphasizes the fact that “the determination of the degree of filling of a space” rests on the category of limitation, namely, “the limitation of the first force [repulsion] by the second [attraction]” (MAN, :). In the course of the Propositions and Explications of the Dynamics chapter, there may be discussions of the quantitative treatment of a number of concepts. But even if there are some discussions that do not fit cleanly into the idea that Kant’s goal in the chapter is construction of something “filling its space (to a determinate degree),” it seems to me that, by Kant’s lights, this can be said to be the chief and guiding aim of the chapter. As Kant sees it, metaphysics has an essential contribution to make to that construction, even if its completion lies outside its proper sphere.
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I want to acknowledge the helpful and generous comments provided on an earlier draft of this chapter by the editor of this volume, Michael Bennett McNulty; the valuable feedback from the participants in a conference on MAN held at the University of Texas, Austin, in March ; and the useful remarks provided by an anonymous review of the volume. In addition, I wish to thank Bernhard Thöle for tremendously useful discussions, as well as the participants of a graduate seminars I taught on MAN in and . I also thank the Max-Planck-Institut for the History of Science in Berlin for support provided during the – academic year. Most of all, I want to express my gratitude to Hannah Ginsborg for extremely helpful discussions, for her support, and for her invaluable advice.
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Beyond the Metaphysical Foundations of Natural Science Kant’s Empirical Physics and the General Remark to the Dynamics Michael Bennett McNulty .
Introduction
The General Remark to the Dynamics (hereafter, “the Remark”), appended to the second chapter of Kant’s MAN (:–), is a perplexing tract. Therein, Kant offers a quadripartite characterization of the “specific variety of matter” – that is, those properties that vary among bodies, such as density, cohesion, aggregative state, friction, elasticity, and chemical affinity – that is bookended by reflections on his preference for a force-based methodology for explaining natural phenomena, which he dubs the “metaphysical-dynamical mode of explanation” (:). These considerations, particularly the discussion of the specific variety of matter, appear prima facie divorced from priorities of MAN. In MAN, Kant’s primary intention is to found the a priori part of the natural science of matter, or what he elsewhere calls the “metaphysics of corporeal nature” (:, , ) and “rational physics” (KrV, A/B). In the body of the book, he thus proffers purportedly a priori clarifications, propositions, and laws that hold universally of all matter. To wit, among other propositions, he demonstrates that matter possesses two fundamental forces, the repulsive (MAN, :) and attractive forces (:–); that it is infinitely divisible (:–); and that action and reaction are equal and opposite in the communication of motion (:–). The concerns of the Remark apparently fall outside the confines of this project. The aforementioned phenomena canvassed in the middle sections of the Remark are by their very nature variable among different kinds of matter and hence do not appear to belong among the proper topics of the metaphysics of body in general. Furthermore, the characterizations of these
The sense in which the judgments of MAN are a priori is a matter of some obscurity and controversy; see McNulty and Stan (, –).
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Kant’s Empirical Physics and the General Remark to the Dynamics natural phenomena lack the apodictic certainty and demonstrative character of the topics in the body of MAN. Whereas, for example, in MAN Kant offers “proofs” demonstrating the “propositions” of rational physics – including those mentioned above – his explanations of the specific variety of matter are less definitive and appear more speculative, or even tentative. Beyond these difficulties, Kant’s discussions of the moments of the specific variety of matter are regrettably brief and obscure. Coupled together, these difficulties make the Remark one of the more enigmatic passages in Kant’s notoriously difficult corpus. While some commentators have dismissed the Remark as concerning “a number of singular questions” (Adickes –, :) or as consisting of speculations on “marginal problems” (Tuschling , ), others have recently reoriented our understanding of the Remark. Michael Friedman () argues that the Remark plays an important role in demonstrating the present and potential utility of Kant’s metaphysicaldynamical, force-based approach. Another set of interpreters, including Dina Emundts (, –), Hein van den Berg (, ), and Oliver Thorndike (, –), explain that the Remark relates to the basis for empirical physics, the a posteriori counterpart to the rational physics developed in the body of MAN. Although these interpretations are undoubtedly edifying, crucial issues remain unclarified, chief among them: What, precisely, is the relationship that Kant envisioned between the topics of the body of MAN and those of the Remark? Or, to put it another way, what is the relationship between the rational and empirical parts of physics for Kant? A tricky question, no doubt, and one whose difficulty is compounded by Kant’s relative reticence on the topic. In this chapter, I clarify the relationship between rational and empirical physics to which Kant aspired in MAN and argue for the following theses. First, rational physics grounds empirical physics insofar as the forces or phenomena of empirical physics are derived from the fundamental forces of rational physics (attraction and repulsion). Second, the relation of derivation between these forces is a real relation; that is, the derivative forces are not simply logically defined in terms of the fundamental forces, but rather emerge from a complex interplay between the fundamental forces. Finally, I contend that additional posits are necessary to derive
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For more on the negative, received view of the Remark, see Thorndike (, –). Van den Berg (, , , ) prefers to claim that the Remark concerns ‘special’ physics. Thorndike (, ) conceives of the Remark as a precursor to Kant’s Übergangsprojekt of OP. Emundts (, –) describes a series of difficulties facing the thesis that the Übergangsprojekt of OP is a direct continuation of that of the Remark, though she notes a close kinship.
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empirical, physical phenomena from the interactions of fundamental forces. In particular, explanation of the specific variety of matter requires the assumption of different sorts of matter that exhibit different degrees of the fundamental forces. Among such requisite posits is an omnipresent dynamical ether, which exhibits a unique manifestation of the fundamental forces and is necessary to account for the elasticity and cohesion of bodies.
. “The Noblest of All the Tasks” of Natural Science: Explaining the Specific Variety of Matter As mentioned above, looming over the Remark is the division in physics (Physik), or the doctrine of nature (Naturlehre), between its rational and empirical subtypes (KrV, A–/B–; Prol, :; MAN, :; OP, :–). For Kant, rational physics is the a priori doctrine of matter, while empirical physics is its a posteriori counterpart. Rational physics is equivalent to the metaphysical foundations of natural science and consists of those a priori laws that derive from the categorical determination of the concept of matter (:–). Thus, in each chapter of MAN, Kant discusses a different categorial determination of matter: its quantity in Phoronomy, its quality in Dynamics, its relations in Mechanics, and its modality in Phenomenology. Among the fruit of rational physics are the aforementioned propositions – for example, that matter is the source of the fundamental forces and that it is infinitely divisible. Empirical physics is the doctrine of the contingent properties and relations of particular matters (GMS, :–; KrV, A/B; see Van den Berg , –), and it is grounded in rational physics (MAN, :; KrV, Bn.; Prol :). Empirical physics, for Kant, also has a wide breadth: it concerns not only motions of matter and their transfer in contingent settings, but additionally chemical and organic actions of matter (V-Phys/Mron, :). Throughout the Remark, Kant returns frequently to the ‘specific variety of matter.’ By this he means those properties and relations that constitute
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Eberhard (, –) and Erxleben (, –), whose textbooks Kant used in various of his physics lectures, distinguish between general and special features of bodies that are, respectively, objects of distinct doctrines: general and special physics. For more on this distinction, see Van den Berg (, –). It is fruitful to recall that physics, or the doctrine of nature (Naturlehre), is a broad doctrine, comprising not only laws of motion, but also all laws governing natural behavior (see McNulty , –, ).
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Kant’s Empirical Physics and the General Remark to the Dynamics the differences among matter, that is, those such as density, state of aggregation, friability, cohesion, and chemical affinity. Insofar as these phenomena are variable and are not universal of matter, as such, they belong to the empirical counterpart of physics. Although commentators classically focus on Kant’s aprioristic inclinations with respect to natural science – that is, the grounding of the possibility of natural science in the principles of the understanding and the categories – he nevertheless appreciates the empirical modalities of the science of nature. Indeed, in the Remark, Kant claims the “clarification” (Erklärung) of the “potentially infinite specific variety of matter” to be “the noblest of all the tasks [of natural science]” (MAN, :). In the Remark, he states that the fourpart discussion of the variable properties of matter is meant to constitute the “moments to which [matter’s] specific variety must collectively be a priori reducible” (MAN, :). A critical dimension of relation between rational and empirical physics consists in the distinction between “fundamental” (Grund-) (or “original” [urspru¨nglich]) and “derivative” (abgeleitet) forces. The fundamental forces are those that enable matter’s filling of space and thus make it a possible object of experience (:, ). They are essential to matter as such and therefore are exhibited by all matter in virtue of being matter. These forces are the fundamental repulsive force, whereby matter resists the penetration of its space by other matter, and the fundamental attractive force, which is the gravitational force and whereby matter does not, under the influence of the repulsive force, disperse itself infinitely throughout the universe. Although Kant says little explicitly about the general characterization of derivative forces, he mentions them throughout MAN (:, , , –). They are, at least, not essential to matter as such, in contrast to their fundamental counterparts. Further, in the Remark, Kant claims that cohesion (:), attractive elasticity (the disposition to regain a smaller volume after rarefaction) (:), and expansive elasticity (the propensity to regain a larger volume after compression) (:) are all derivative. So in the Remark Kant aims to account for the specific variety of matter, that is, to provide the conceptual underpinnings for empirical physics. A major part of this account involves a transition from the fundamental forces of rational physics to their derivative counterparts in empirical physics. Thus, in the quest to illuminate the connection between the Remark’s description of the specific variety of matter and the body of
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On Kant’s arguments for the joint necessity of the forces for the possibility of matter’s filling of space, see Warren () and Smith ().
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MAN, the first task ought to be to clarify the relation between these different sorts of forces. Commentators have observed the Leibnizian heritage of Kant’s division of fundamental and derivative forces (e.g., Thorndike , –). Primitive forces, for Leibniz, are fundamental, universal causal powers possessed by all substances. Derivative forces, by contrast, vary among substances and are grounded in the primitive forces as “modifications” of them. So, for Leibniz, the conative powers of monads are primitive, whereas the derivative forces are physical forces belonging to phenomenal, aggregative bodies (AG, –). However, merely observing this lineage for the distinction sheds little light on the relation between rational and empirical physics in Kant’s system. This is primarily due to the relation between primitive and derivative forces being notoriously obscure even in Leibniz’s thought. Robert Adams, for instance, writes that the relation is “probably the largest obstacle to understanding the relationship between Leibniz’s physics and Leibnizian metaphysics” (, ). Indeed, in a famous correspondence with Leibniz in –, Christian Wolff presses on the distinction, suggesting its unintelligibility (see Adams , –). Recently, Stephen Howard () has argued that up to his death Leibniz himself struggled unsuccessfully with the recalcitrant problem of developing his dynamics along these lines. Although it is right to say that Kant appropriates this distinction, such appropriation on its own fails to clarify the transition from empirical to rational physics. Thus, in the remainder of this chapter, I seek to clear up the connection between rational and empirical physics, particularly by interrogating the dependence of derivative forces and phenomena on the fundamental forces.
. Derivation and Reduction At the center of Kant’s comments on the relation between rational and empirical physics are two concepts: derivation and reduction. He regularly refers to “derivative” (abgeleitet) forces as well as to “deriving” (ableiten) forces and phenomena in general. For example, in a note to Proposition of the Dynamics, Kant writes that the fundamental expansive force “must therefore be called original, because it can be derived [abgeleitet] from no other property of matter” (:, see also :, , ). Kant sometimes writes that the specific variety of matter may be derived from
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Kant also occasionally writes of particular forces being derived from experience (see MAN, :–, :).
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Kant’s Empirical Physics and the General Remark to the Dynamics some other grounds: to wit, from atom and void, according to the mathematical-mechanist, and from moving forces, according to the metaphysical-dynamist (:). In the context of the relation between empirical and rational physics, Kant also occasionally mentions reduction (Reduction, reduzieren, or zuru¨ckfu¨hren), where, again, the term applies to forces and the specific variety of matter. So he writes that “only these two kinds of forces [sc. attractive and repulsive] can be thought, as forces to which all moving forces in material nature must be reduced [zuru¨ckgefu¨hrt]” (:) and that “[a]ll natural philosophy consists, rather, in the reduction [Zuru¨ckfu¨hrung] of given, apparently different forces to a smaller number of forces and powers that explain the actions of the former, although this reduction [Reduction] proceeds only up to fundamental forces, beyond which our reason cannot go” (:). Thus, the envisaged relationship between empirical and rational physics is one of derivation or reduction. The specific variety of matter, in general, and the forces of empirical physics, in particular, are supposed to be derivable from or to reduce to the fundamental forces of rational physics. These reflections, however, give rise to the next set of crucial interpretative questions, surrounding the notions of derivation and reduction. To put the central query succinctly: What does it mean to derive or, alternatively, to reduce a force for Kant? I first focus on the Kant’s conception of derivation. “Ableiten” has a variety of meanings, both in German generally and in Kant’s system specifically. These distinct usages are well noted by Corey Dyck (a) in an essay review of Dennis Schulting’s Kant’s Deduction and Apperception (), which makes crucial use of the notion of derivation. Dyck notes that “derivation” may refer to the process of logical deduction. Thus one may say that a conclusion is derived (wird abgeleitet) from premises in a syllogism. It is, apparently, such a logical notion of derivation that Kant discusses in his logic lectures, wherein derivation is defined in terms of marks of concepts. A mark of a concept is a partial concept “insofar as it is considered a ground of cognition of [a] whole representation” (Log, :). So, the partial concepts that constitute another concept are marks; the marks of ‹human› are the concepts ‹rational› and ‹animal›. Marks may then be used in two ways as grounds of cognition according to Kant: internally
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Kant also mentions “derivations” of the law of action and reaction (:). In the Remark, Kant also writes of the concept of matter being “reduced [zuru¨ckgefu¨hrt] to nothing but moving forces” according to the metaphysical-dynamical mode of explanation (:).
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and externally (see also V-Lo/Blomberg, :; V-Lo/Wiener, :). Marks are externally used to compare one thing with another, whereas “the internal use consists in derivation [Ableitung], in order to cognize the thing itself [sc. the concept] through marks as its grounds of cognition” (Log, :). Thus, in my example, to cognize ‹human› though its marks – ‹rational› and ‹animal› – is to derive the former from the latter. But, as Dyck rightly notes, elsewhere in Kant’s writings, “derivation” takes on a looser meaning, where it refers simply to a general sort of dependence. That is, a representation, A, may be derived from another, B, in the case that A depends on B. Such dependence is real, not logical, though there is some wiggle room in conceptualizing the precise relation. For example, in the case of A depending on B, we may think of B as being a condition of the possibility or existence of A, or of B as being the source for A’s content (Dyck a, ). It is this looser sense of derivation, Dyck contends, Kant has in mind, for instance, in the Second Analogy, where he writes that “I must therefore derive the subjective sequence of apprehension from the objective sequence of appearances” (A/B, italic emphasis mine) and in the Transcendental Deduction, where he writes that “the empirical unity of apperception . . . is derived only from the [transcendental unity of apperception], under given conditions in concreto” (B, emphasis mine). “Reduction” similarly admits of multiple interpretations. “Zuru¨ckfu¨hren” literally means “to lead back” or “to trace back,” although in some contexts it rather bears the same meaning as “to reduce.” Just as “ableiten” can be understood logically and non-logically, so too can “zuru¨ckfu¨hren.” Indeed, we find Kant leveraging both meanings of reduction in a few instances. For instance, in KrV, Kant mentions the logical sense of reduction in the appendix to the Transcendental Dialectic, writing that chemists “were able to reduce all salts to two main genera, acidic and alkaline” (A/B). In this sense, ‹acidic salt› and ‹alkaline salt› are two marks, through which all the different, specific sorts of salts can be thought. One can think of vinegar (or acetic acid, specifically) as a particular acidic salt, one produced through the fermentation of particular sugars or alcohols. Conceiving of ‹vinegar› through its marks, that is, the concepts contained in it, such as that of ‹acidic salt›, reduces ‹vinegar› to its marks. Equivalently, such a conception derives ‹vinegar› from said marks.
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In FM (:) Kant also distinguishes the two senses of derivation. There he states that attributes can be derived from essences in two ways: either analytically through the principle of contradiction or synthetically, in which case another principle is necessary.
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Kant’s Empirical Physics and the General Remark to the Dynamics In this context, the reducibility of salts to the acidic and alkaline genera is a logical relation: particular salts can be explicated as species of acidic and alkaline salts. For an example of a non-logical use of “reduction,” take the famous claim from the Leitfaden of KrV that “[w]e can, however, trace all actions of the understanding back to judgments, so that the understanding in general can be represented as a faculty for judging” (A/B), where “trace . . . back” translates “zuru¨ckfu¨hren” in the original. Although the particular relation that holds among the actions of the understanding is not immediately transparent in this context (and is naturally a matter of some controversy), Kant is not claiming that judgment is a mark – a constituent concept – of the other activities of the understanding (such as the production of concepts). In this passage, Kant clearly means to call attention to a tight interconnection among such activities of the understanding, but it is importantly a non-logical connection. Distinguishing these meanings of “derivation” and “reduction” helps to crystallize the pivotal questions about empirical and rational physics. First, is the derivativeness of the forces of empirical physics understood as logical or as real? Second, if the relation is non-logical (as I will argue), then can we say anything more specific about the particular relation of derivativeness between the fundamental and derivative forces? Third, are the fundamental forces collectively sufficient for the derivation (in whatever sense) of the derivative forces, or merely necessary? And fourth, if the fundamental forces are not collectively sufficient (as I will argue), then what, in addition, is required for the derivation of the derivative forces? First, I argue that the derivative forces cannot be derivative in the sense that they can be logically derived from the fundamental forces (after first clarifying the sense of logical derivation of a force). Second, consideration of the details of Kant’s accounts of the specific variety of matter reveals the relation of derivativeness to be best thought of in terms of explanation. Third, I argue that there is strong evidence that the fundamental forces, though conditions of the possibility of derivative forces, are insufficient on
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See also KrV (A/B). Notably, these reductions are merely logical in the sense of being definitional. Once we lay down our explications, it is simply an analytic truth that vinegar is an acidic salt. However, based on the pressure of experience, we may tweak or change our definitions (see A–/B–). So we might add a new mark to the concept of vinegar or distinguish different sorts thereof. But in all cases, once the explications are fixed, the logical reductions follow necessarily. The claim of exhaustivity in the above passage – that all salts reduce to these two genera – is distinct from the bare relation of reduction, which, in this case, is purely logical. Glezer intimates in the direction of a logical relation, suggesting that the reduction of derivative forces to fundamental forces in the Remark “parallels the subordination of species concepts under genus concepts” (, ).
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their own for the (non-logical) derivation of the derivative forces. Fourth, Kant transparently postulates different sorts of matters, which vary in their relative expression of the fundamental forces, in order to explain the phenomena belonging to the specific variety of matter. In particular, Kant speculates on the existence of a unique, omnipresent matter – the ether – which exhibits a unique combination of the fundamental forces and thereby makes possible contingent phenomena like cohesion and elasticity.
. No Logical Derivation of Forces As I discussed above, a concept may be logically derived from others when its meaning is analytically reducible to the meaning of the others. As in the above example, the concept ‹human› is derivable from the combination of ‹rational› and ‹animal›. This sort of logical derivation could not be what Kant had in mind with respect to the derivative forces. Take cohesion, which is supposed by Kant to be a derivative force. He repeatedly emphasizes that it is a superficial attractive force, so, at the very least, ‹cohesion› contains ‹attractive›, ‹superficial›, and ‹force›. In contrast, the fundamental forces are, respectively, attractive and penetrative, on one hand, and repulsive and superficial, on the other. So we might propose that ‹fundamental repulsive force› be defined as ‹repulsive›, ‹superficial›, and ‹force›, and ‹fundamental attractive force› as ‹attractive›, ‹penetrative›, and ‹force›. At the outset, it is clear that ‹repulsive force› and ‹attractive force› are not marks of ‹cohesion› so the latter cannot be derived in a straightforward sense from the pair of fundamental forces. That said, through analyzing the concepts of the fundamental forces, we come upon constituent concepts, by means of the combining of which we can formulate the concept of ‹cohesion›. But such a relation gets us nowhere useful. Even if we can explicate cohesion, or the derivative forces, in general, in this manner, such an explication tells us nothing substantive about the phenomenon. For instance, using marks of the fundamental forces to define cohesion provides no clue as to whether cohesion is even a really possible force. Relatedly, such a definition fails to determine whether cohesion belongs to empirical or rational physics – that is, whether it belongs to matter’s possibility. Another way of seeing this point: in the preface to MAN, Kant explains natural science concerns nature, that is, the “first inner principle that belongs to the existence of a thing” (MAN, :). Rational physics, he explains, concerns the nature of things falling under the concept of ‹matter›. Defining a derivative force in terms of other concepts analytic to fundamental forces does not determine whether we
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Kant’s Empirical Physics and the General Remark to the Dynamics are still investigating the nature of ‹matter›, or whether the derivative force belongs to the nature of a thing. The basic, logical understanding of derivativeness is simply a nonstarter.
. “Hidden Play of Moving Forces”: Explaining Derivative Forces via the Fundamental Forces For these reasons, I contend that the relation of derivativeness or reduction between the fundamental and derivative forces must be understood more loosely, or non-logically. In particular, in this section, I argue that the fundamental forces are understood by Kant to be an explanatory basis for the derivative forces. That is, when a derivative force – say, cohesion – is instantiated in a body, this force is explained by an interaction between the original attractive and repulsive forces of matter. Subsequently in Section ., I detail Kant’s accounts of the sense of derivation in the cases of the Remark and argue that the complex interactions of the fundamental forces that explain the phenomena of empirical physics require the postulation of distinct sorts of matter. In the first Remark to Proposition of the Dynamics, Kant offers his most explicit characterization of the distinction between fundamental and derivative forces: That the possibility of the fundamental forces should be made conceivable is a completely impossible demand; for they are called fundamental forces precisely because they cannot be derived [abgeleitet] from any other, that is, they can in no way be conceived [begriffen].. . . Thus because it [sc. the fundamental attractive force] is not felt, but is only to be inferred, it has so far the appearance of a derived force, exactly if it were only a hidden play of moving forces through repulsion. On closer consideration, we see that it can in no way be further derived from anywhere else, least of all from the moving force of matter through their impenetrability, since its action is precisely the reverse of the latter. (:, emphasis mine)
The attractive force is therefore fundamental, and not derivative, because it is no mere “hidden play” of the repulsive force. The figurative language of “hidden play” aside, Kant’s central claim is that the attractive force is no merely apparent manifestation of the impacts of impenetrable pieces of matter.
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Along different, though complementary, lines – especially by situating Kant’s account of forces with respect to his historical context – Howard () also supports this thesis and denies that the relation between fundamental and derivative forces is a logical one.
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An example helps to clarify what sort of account Kant is denying. During this period, there were various mechanical theories of gravitation, according to which gravitational attraction is due to impacts or pressure of material bodies. Descartes’ well-known vortex theory and its later rearticulations in the hands of the Cartesians fall under this characterization. According to Descartes, the heavens are filled with a particular, ethereal sort of matter, which is extremely subtle and flows in vortices. As it moves, it carries heavenly bodies along with it, producing the observed revolutionary behavior of, for example, the planets (see AT, :– [part III of the Principia philosophiæ]). For another example, take Georges-Louis Le Sage’s () account of gravitation. According to Le Sage, corpuscles strike bodies from all sides, yielding a uniform compressive force. If two bodies are beside one another, they will shield each other from the particles that would strike each other on the interior sides, meaning that a less intense force pushes the bodies apart than pushes them together. Thus, there is net force bringing the bodies together via the impact of corpuscles. For Le Sage, the gravitational attraction of bodies is hence merely an effect of the impact of fundamentally repulsive particles. Accounts like Descartes’ or Le Sage’s thereby conceive of gravitation as an appearance of the activity of impenetrability or repulsion; it is thereby understood as the hidden play of particles. In the above passage, Kant means to deny just this sort of account of gravitation that makes it a derivative force by reducing it to repulsion. The attractive force, for Kant, is rather fundamental,
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Kant was, at the very least, exposed to mechanical theories of gravitation through the texts that he used for his physics lectures, such as Erxleben’s (, ). Erxleben criticizes mechanical theories of gravitation, such as Descartes’ as well as Huygens’ and Bu¨lfinger’s improvements (, §§–). He goes on to criticize material theories of gravitation, which postulate an especial, gravitational substance that accumulates in heavy bodies (§§–). After doing so, Erxleben reports that that gravity must have another basis because it is “almost totally impossible that it is derived from a material or pressure” (§). He prefers the idea that it rests on an attractive force in bodies (§). In his expanded version of , Lichtenberg, a fan of mechanical theories of gravitation, takes editorial license and adds an incredulous “(?L)” to Erxleben’s claim that he has disproven the mechanical theory of gravitation (in the section now labeled §a). Lichtenberg subsequently adds in his commentary that it is best to admit that we do not yet know the cause of gravitation but that he holds hope for understanding it (§b). That said, he reports that he eagerly awaits Le Sage’s theory of gravitation (, unpublished at the time), which he expects to pick up where Newton left off and explain the mechanisms of the laws of nature. Interestingly enough, Lichtenberg later took Kant’s conception of the fundamental forces of matter as necessary to matter’s possibility as a strong objection to Le Sage’s mechanical theory of gravitation (see Lichtenberg , ). Kant states that centrifugal force is another example of a derivative force and explains that, in general, contingent motions “should be derived from the forces which are inherent in matter, even when it is at rest” (BDG, :).
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Kant’s Empirical Physics and the General Remark to the Dynamics irreducible, and essential to matter’s very existence. Thus, according to the understanding suggested in this passage, a derivative phenomenon or force is one that is apparent and results from the interaction among other, fundamental forces of various pieces of matter. A similar understanding of the difference between fundamental and derivative forces is can also be found in a Reflexion from the s: Moving force occurs every time by means of [vermittelst] the original (or the first forces, which act at rest) (therefore those [forces] are every time derivative), because either repulsive or attractive force acts: the latter, when bodies should penetrate, the former, when they should be removed from one another. (Refl , S. II [–], :–)
In this passage, Kant asserts that all moving forces occur by means of, or via, the original (fundamental) forces, and, for that reason, all such forces are derivative. This characterization of the relationship between the fundamental and derivative forces broadly agrees with the interpretation on offer: the derivative forces occur “by means of” the original forces insofar as they emerge from the interactions of the latter forces. Kant’s comments on the respective mechanical and metaphysical approaches to explanation that bookend the Remark (:–, –) support the idea that the interactive emergence of derivative forces is additionally explanatory. In these passages, Kant opposes two approaches to explaining natural phenomena (Erklärungsarten), as mentioned above. The first, “mathematical-mechanical” mode of explanation takes as basic the shape, size, and impact of absolutely impenetrable particles in the void, whereas the latter, “metaphysical-dynamical” approach, bases explanations on the activity of moving forces. Kant begins the Remark by ruminating on the respective benefits of the two approaches to explanation – unsurprisingly, the mathematical-mechanical mode is more mathematically adequate, whereas the metaphysical-dynamical approach is metaphysically acceptable – though he ultimately sides with the metaphysical-dynamical approach. The Remark then concludes with another discussion of the two approaches to natural explanation that connects more explicitly with
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See also Refl , S. I (–), where Kant claims that gravitation must be a fundamental force insofar as it cannot be explained via impact (:; see also DfS :). Kant appears to define derivative as that which depends upon a principle (Refl :). Throughout his Reflexionen on metaphysics as well as in his texts on physics, Kant opposes derived (abgeleitet) from original (urspru¨nglich). For different takes on Kant’s views on the mechanical and dynamical modes of explanation and his preference for the dynamical approach, see Brittan (), Warren (), Friedman (, –), and McNulty ().
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the specific variety of matter. There he writes that the task of “explaining [Erklärung] a potentially infinite specific variety of matters” can take either the mechanical or dynamical approach. According to the mechanical approach, one explains this variety “by combination of the absolutely full with the absolutely empty,” whereas, according to the dynamical approach, one does so “by the mere variety in combining of the original forces of repulsion and attraction” (:). This approach to explanation is “that which derives this specific variety of matters . . . from the moving forces of attraction and repulsion originally inherent in them, [and] can be called the dynamical natural philosophy” (ibid., first emphasis mine). These passages from the Remark hence demonstrate that, according to Kant’s metaphysical-dynamism, the derivative forces result from the interaction of the fundamental forces and, additionally, that this relation explains the derivative through the fundamental.
. The Variety of Matters and Their Expression of Fundamental Forces While it is relatively simple to conceptualize natural phenomena in terms of the various imaginable shapes, sizes, and impacts of impenetrable pieces of matter (as the mathematical-mechanist would have it), it is more difficult to conceive the specific variety of matter as resulting from the hidden play of the fundamental forces of attraction and repulsion. In the case of the metaphysical-dynamical mode of explanation, it is not transparent what the explanatory “combination” of fundamental forces looks like. In this section, I argue that, for Kant, explanation or derivation of the derivative forces requires the postulation of fundamentally different kinds of matter that express the fundamental forces to unique degrees. The unique expression of these forces and the consequent interactions – the
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Consider, as well, the first note to Proposition of the Dynamics, where Kant claims that, given the proof of the existence of the fundamental attractive force, “the possibility of a space filled to a determinate degree” may be “derive[d]” “[f]rom this original attractive force . . . in combination with the force counteracting it, namely, repulsive force” (MAN, :). I thus agree with Van den Berg (, , ) and Emundts (, –, –) that the fundamental principles are “explanatory principles” for the phenomena of empirical physics. Indeed, I argue that this is the salient aspect of the derivative phenomena and forces of empirical physics. I add, however, that additional explanatory resources are necessary for the phenomena of the Remark (see below). I hence concur with Friedman’s claim that the account of the specific variety of matter “presupposes only the fundamental forces” (, ). Indeed, these forces are the basis for the descriptions given in the Remark. However, as I will argue, fundamentally distinct sorts of matter expressing these forces differently must also be posited.
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Kant’s Empirical Physics and the General Remark to the Dynamics aforementioned “combinations” – among them explain derivative phenomena and forces. One material, in particular, is especially important to a dynamical account of the derivative forces, namely, the omnipresent ether, which uniquely expresses extreme repulsion and negligible attraction. I detail Kant’s account by proceeding through his discussions of the moments of the specific variety of matter in the body of the Remark, noting in each case how Kant incorporates the assumption of fundamentally distinct sorts of matter in each case. Density. For Kant, density results from the interplay between the fundamental forces varying among different sorts of matter and is thus a derivative phenomenon. In the Remark, Kant states, “The degree of the filling of a space with determinate content is called density” (MAN, :). Kant is clear in the second Note to Proposition of the Dynamics that the variability in degrees of space-filling comes down to the interaction between the fundamental forces of matter: Since every given matter must fill its space with a determinate degree of repulsive force, in order to constitute a determinate material thing, only an original attraction in conflict with the original repulsion can make possible a determinate degree of the filling of space, and thus matter. (MAN, :)
According to Kant’s account, the postulation of fundamentally distinct degrees of repulsion expressed by different kinds of matter is necessary to account for differences in density. Kant actually states his MAN account of density most clearly in an infamous letter to Johann Sigismund Beck, wherein he suggests that the account of density in MAN is circular. (This criticism is neither here nor there for my purposes, insofar as I am interested simply in recovering – not evaluating – Kant’s account of density in MAN.) In the letter, Kant summarizes his theory of density as follows: “(the universal, Newtonian) attraction is originally equal in all matter; it is only the repulsive force that varies in different kinds of matter, and this is what determines differences in density” (Br, :–). For Kant, the density of a body comes down to the balance between its
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A more radical thesis is expressed in the Berliner Physik lecture notes: “The supreme [oberste] cause of all derivative forces is the ether” (V-Phys/Berliner, :). As may be expected from a short appendix, the accounts of the derivative forces in the Remark are unfortunately brief and incomplete. Nevertheless, my aim in the present context is only to note the necessary assumption of distinct sorts of matter for the purpose of explaining the derivative forces of matter, which can be achieved without speculating on the complete accounts of said forces. For more on Kant’s account of density, see Kahn (), who also defends the account from the accusation of circularity, and Westphal (b), who argues that this circularity and other problems torpedo the project of MAN.
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attractive and repulsive forces. Take a unit of matter, which expresses a particular degree of the fundamental attractive force based on the quantity of matter. Since this quantity is fixed, the density of the body hence depends on the volume that it inhabits, which volume is, however, determined by the repulsive force by means of which the parts of the body repel each other. The more intense the repulsive force among these parts, the greater the volume and, derivatively, the lesser the density. Now different samples of one kind of matter may vary in density due to state of compression or rarefaction, but to account for the specific variety of matter – different sorts that have distinct innate densities – one must assume different kinds of matter that differ in their expression of the repulsive force. Cohesion. Throughout the s, Kant reiterates that the impact of external ether explains the cohesion of bodies. The everywheredistributed ether, according to Kant, is essentially expansive and repulses all matter. It is therefore in an original state of compression, exerting repulsive force always in all directions; all matter in the universe is thereby compressed by the ether. By means of this ethereal compression a body will cohere, that is, resist the separation of its parts. This is a dynamical explanation of cohesion, which derives the force from the interplay of fundamental forces. Although the ether is a unique substance – it is subtle, omnipresent, and repulses all matter to a high degree that totally outstrips its attraction – it expresses its effects by means of the fundamental forces. In particular, the ether impresses fundamental force of repulsion on the surface of cohesive bodies. Cohesive force is, according to Kant, simply the consequence of complex interactions between the repulsive force of the ether and gross bodies. So, again, Kant’s explanation requires the
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Friedman concurs that matters of different sorts have “specifically different expansive forces, by which they differentially resist the corresponding compressive forces” (, ). Kant offers a different conception of the ground of density in MonPh (:), albeit one that also presupposes a variety in the kinds of matter. See V-Phys/Berliner, :; Refl , S. I (–), : , –; Refl , S. I (–), :; Refl (–), :; and Refl (ca. mid-s to s), :–. Edwards (, –) and Friedman (, –, –) both highlight the role of the ether in the Remark’s explanation of cohesion. The story is, of course, more complicated, especially insofar as the ether insinuates in the interstitial space of bodies. In order for the ether to express a net compressive force on a body, the force of its external compression must outstrip its internal expansion. Newton, for one, explained the situation as follows: when ether penetrates a body, it attenuates; in virtue of being less dense inside bodies, it expresses an internal expansive force that is weaker than the external compressive force. The ether thereby expresses a net compressive force on bodies that explains cohesion (Heilbron , –).
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Kant’s Empirical Physics and the General Remark to the Dynamics postulation of a particular sort of material – the ether – exhibiting the fundamental forces to characteristic degrees. Elasticity. Although Kant’s account of attractive elasticity – whereby a body returns to its earlier figure after being bent or rarefied – is particularly sketchy, it similarly makes reference to the ether. For example, he claims, Attractive elasticity is obviously derivative, as the term already shows. An iron wire, stretched by a hanging weight, springs back into its volume when the band is cut. In virtue of the same attraction that is the cause of its cohesion, or, in the case of fluid matters, if heat were suddenly to be extracted from mercury, the matter would quickly reassume the previously smaller volume. (:)
Given the above-described arguments regarding the ether as the cause of cohesion, he thereby is attributing attractive elasticity to the ether as well. It is, however, a bit obscure how Kant envisages an account of attractive elasticity proceeding by means of the ether. In the case Kant imagines, where a spring is in a state of tension due to a weight hanging from it, the spring is subject to a downward force due to the gravity of the weight. When the band is cut, the spring then recompresses due to the expansive force of the external ether, which pushes the spring back up to its original figure. Why, however, the ether would push disproportionately on the bottom of the spring until it reaches an equilibrium at its original volume and position is left unexplained. Nonetheless, such an account, warts and all, would make the cause of attractive elasticity the same as that of cohesion, as Kant suggests in the passage from the Remark. Kant also attributes expansive elasticity – the propensity of a fluid, when compressed, to regain its original, greater extension – to the ether: “Expansive elasticity, however, can be either original or derivative. Thus air has a derivative elasticity in virtue of the matter of heat, which is most intimately united with it, and whose own elasticity is perhaps original” (MAN, :–). This passage connects with an earlier one, wherein Kant suggests that the expansive force of air may rest on heat, which causes the parts of air to rarefy due to its vibrations (:). Kant thus attributes the expansive elasticity of empirical matters to their containing the matter of heat, the originally elastic matter, which is identical to the ether (see also DI, :–). Kant’s idea appears to be that the ether is originally expansive, and when it is bound in a body, it similarly expresses its expansiveness on the parts of the body. After air has been compressed, it
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Alternatively, it could be that Kant holds fast to his pre-critical view that the parts of bodies attract each other by means of an interstitial elastic ether (DI, :), but this option faces similar puzzles.
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re-expands to its original volume due to the influence of the ether bound therein, whose own expansive force increased as the air was compressed. Thus, the derivative forces of cohesion and elasticity both depend upon the ether and are explained by the complex interplay of the fundamental forces. It is through the ether’s unique expression of the fundamental forces – its extreme fundamental force that outstrips its negligible attractive force – that it interacts with bodies in the right way to make possible the derivative forces. Namely, because the ether is originally extraordinarily expansive, it compresses bodies in such a way so as to make possible their cohesion and attractive elasticity, and it expresses its expansivity inside a body to explain its expansive elasticity. Furthermore, as Kant notes at the end of the Remark, the metaphysical-dynamical approach to explanation legitimates our thought of such an ether, a point in favor of his preferred mode of physical explanation. Chemical Activity. Regarding chemical forces, discussed in the final numbered section of the Remark, Kant thinks that there is no such transition from the fundamental forces to those of chemistry, which govern the dissolution and decomposition of substances (MAN, :–). In the Remark, Kant goes to lengths to emphasize that chemical forces are of their own unique genus. What is unique about them is that they effect an “absolute dissolution” or “chemical penetration” of matters. That is, by means of a chemical force of, say, dissolution, two matters are combined with one another such that every space within the solution contains an equal proportion of both solvent and solute. This means that the pre-dissolution spaces of the solvent and solution have both been reduced to zero, which is what Kant calls the “penetration of matter” in the body of the Dynamics (MAN, :–). In that context, Kant explicitly states that the mechanical penetration of matter – that is, communicated by the fundamental forces of matter – is impossible. Thus, one of the main points of this final numbered section of the Remark is that
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Similarly to the case of attractive elasticity, Kant’s account of expansive elasticity is preliminary and faces difficulties (e.g., How can the uniform ether explain the disproportionate expansion of a decompressing spring?). Kant returns to the ether at the end of the Remark, I suggest, to allude to the mode of progression in the four moments of its body. There he notes that the ether is thereby intelligible only in the metaphysical-dynamical framework. Kant does so, I suggest, because the ether is intimately involved in the explanation of cohesion and elasticity. The ether referenced is a mere “example,” but one that is absolutely crucial to the comprehension of the specific variety according to the metaphysicaldynamic approach and the transition to empirical physics portended in the Remark. For a more in-depth account of Kant’s views of chemical actions and a reading of this part of the Remark, see McNulty ().
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Kant’s Empirical Physics and the General Remark to the Dynamics chemical dissolutions and decompositions cannot be reduced to mechanical actions. That is, chemical forces cannot be derived from the fundamental forces. Indeed, when he mentions chemical forces throughout this passage, he notably calls them “inherent” and never “derivative” forces. Now Kant does ultimately conclude that the actuality of chemical forces is unknown and unknowable; however, his claim is that if genuine chemical dissolutions are possible, they can occur only by means of special, inherent forces and not the fundamental forces discussed in the body of the mechanics. Thus, in the context of the Remark, the chemical forces are unique. Chemical phenomena are not, like density, cohesion, and elasticity, derivative phenomena in the sense of being explained by the fundamental forces of matter. Yet, nonetheless, Kant recognizes that chemical phenomena require a place in the general theory of nature, meaning that some irreducible posits – the inherent forces of dissolution and decomposition – are necessary for the complete account of the specific variety of matter. Moreover, specifically different behavior of particular kinds of substances will need to be postulated, that is, different sorts and degrees of inherent chemical forces, insofar as chemical affinities are elective.
. Reduction to Fundamental Forces, and Beyond The phenomena canvassed in the Remark well substantiate Kant’s account of the method of natural philosophy near the conclusion of the Remark: “All natural philosophy consists, rather, in the reduction of given, apparently different forces to a smaller number of forces and powers that explain the actions of the former, although this reduction proceeds only up to fundamental forces, beyond which our reason cannot go” (:). In some cases, reduction to the fundamental forces is successful, as with density, cohesion, and elasticity. Other times this attempt to reduce fails, as when we try to derive the chemical forces from their fundamental counterparts. Yet, in each case, something more than the bare fundamental forces as described in the body of the Dynamics is required. For density, it is the original, essential variability of the repulsive force in different kinds of matter. For cohesion and elasticity, we require the postulation of the ether. For chemical actions, we need to assume chemical forces that vary among different sorts of matter and that cannot be reduced to or derived from the
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I should emphasize, again, that Kant gives us only an outline of the dynamical, reductive account of cohesion and elasticity. The reduction is potentially successful, given further articulation.
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fundamental forces of matter. All cases, however, involve the picture of natural philosophy Kant presents in the Remark and connect with the broad picture of the metaphysical-dynamical approach to matter he espoused. Even in the case of chemistry, wherein the reduction is impossible, we nonetheless best think of material happenings as effects of forces, not as resulting from mechanical interactions.
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Thanks to the audiences at the biennial meeting of the International Society for History of Philosophy of Science and the Central Division meeting of the American Philosophical Association for their feedback on earlier versions of this chapter. I also benefited greatly from the comments and questions of the participants in a workshop on MAN hosted by Katherine Dunlop as well as from thorough discussions with Daniel Warren and Andrew Janiak. Finally, I thank Stephen Howard for his extensive comments on the chapter and allowing me to read an early draft of his work. This work was supported by a Research Fellowship from the Alexander von Humboldt Foundation.
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How Do We Transform Appearance into Experience? Kant’s Metaphysical Foundations of Phenomenology Silvia De Bianchi
. Introduction to the Phenomenology: Beyond Newton’s Principia The Phenomenology has not received much consideration from commentators, at least compared with the Dynamics and the Mechanics. For instance, Pollok’s () commentary to Kant’s MAN devotes only around thirty-five pages (of a roughly -page book) to the Phenomenology. That said, Friedman’s book Kant’s Construction of Nature () represents a recent exception to this trend, offering the most complete account that extensively discusses this chapter. Nonetheless, the Phenomenology is the most intriguing, yet underexplored part of MAN, not only with respect to its relationship with other parts of Kant’s system but also with respect to the natural science of Kant’s time. This contribution builds upon and elaborates Friedman’s reading. More specifically, it facilitates the comparison of the Phenomenology with both other parts of Kant’s system and the science of his time, in particular, with Euler’s equation of motion for rigid bodies. In Kant’s Construction of Nature (), Michael Friedman proposed to read MAN as the result of Kant’s attempt to find the a priori principles underlying Newton’s laws and their application. Friedman argues that Kant’s work provides “Leibnizean” metaphysical foundations for Newtonian physics (, ). This inspired criticism from the side of historians of philosophy (Stan , a) who emphasize the limits of defining as ‘Leibnizean’ the metaphysics devised in MAN. Friedman’s work also stimulated further studies in the history and philosophy of science, such as Patton’s () work, and in the history of space-time theories (Hyder , ). In this contribution, () I extend Friedman’s
In Massimi and De Bianchi (), for instance, the legacy of Cartesians, such as De Mairan, is emphasized in order to show how the metaphysical concept of force in MAN cannot be defined either as “Leibnizean” nor as “Newtonian.”
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reading of the Phenomenology by including Euler in the picture and () I propose a more radical thesis to the Phenomenology than Friedman. In particular, I offer a deeper analysis of time representation and emphasize the function of “reduction” (Reduktion) to be ascribed to the idea of absolute space in order to highlight the possible link of the Phenomenology to the development of Kant’s thought, including his doctrine of measurement in KU. This move allows us to consider unexplored paths in Kant studies and contributes to a deeper understanding of the impact of the MAN on the development of Kant’s system itself. The rationale behind my reading is that in MAN Kant is not just assuming the perspective of transcendental philosophy, or that of determining the possibility of an object of experience in general. The perspective that Kant wants to develop is that of his metaphysics of nature, a metaphysics purified from errors and dogmatism. I therefore depart from Friedman’s reading and attribute a central role to the faculty of reason in the Phenomenology. In my view, Kant’s metaphysical foundations are not aimed at the conceptual foundation of Newton’s physics only, but at any development of Newtonian mechanics. As shown in the next sections, Kant tried to attain this goal by means of the assumption of the law of antagonism in all community of matter through motion based on the principle of reciprocal interaction (Wechselwirkung), which corresponds to the function of reducing and unifying all appearances under an idea of reason. I do not contend that Friedman overlooks this; instead, he offers a very detailed account of the Phenomenology and its connection to the first Kritik and Kant’s cosmology (Friedman , –). However, Friedman looks at the correlative function of the Phenomenology as a link between KrV and MAN. With respect to Friedman’s reading, I want to make a further step and frame the Phenomenology within a broader picture that includes not only KrV but also the KU. In other words, my claim is that in the Phenomenology Kant had to face fundamental questions related to his system and world view, including the metaphysical foundations of any scientific knowledge, be it a priori or empirical, as well as the a priori foundations of observational cosmology. Indeed, the conclusions of Phenomenology are related to the application of Newton’s mechanics beyond the solar system and are in agreement with Kant’s cosmology. They are related as such to the idea of the cosmos (Weltganze) as a whole (see Friedman , –). In this case, however, the object under discussion is not the whole of rational cosmology, that is, the world (die Welt), but rather the physical universe. Thus, it seems to me crucial to read Kant’s Phenomenology within the construction of a metaphysics of
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nature, which includes the problem of harmonizing the mathematical and philosophical representation of (ideal) time, (material) space, and motion with both the foundations of cosmology and the end of natural science in general (see MAN, :).
. Structure of the Phenomenology At the very beginning of the chapter, the Explication and its Remark give the definition of motion as appearance and show in which sense one needs to objectively represent motion in order to transform appearance into experience: Hence the movable, as such a thing, becomes an object of experience, when a certain object (here a material thing) is thought of as determined with respect to the predicate of motion. But motion is change of relation in space. There are thus always two correlates here, such that either, first, the change can be attributed in the appearance to one just as well as to the other, and either the one or the other can be said to be moved, because the two cases are equivalent; or, second, one must be thought in experience as moved to the exclusion of the other; or, third, both must be necessarily represented through reason as equally moved. In the appearance, which contains nothing but the relation in the motion (with respect to its change), none of these determinations are contained. But if the movable, as such a thing, namely with respect to its motion, is to be thought of as determined for the sake of a possible experience, it is necessary to indicate the conditions under which the object (matter) must be determined in one way or another by the predicate of motion. At issue here is not the transformation of semblance into truth, but of appearance into experience. (MAN, :–)
Proposition is based on the definition of motion and change offered in the Remark to the Explication. It states that the rectilinear motion of matter with respect to an empirical space is a merely possible predicate. The same rectilinear motion, when thought in no relation at all to matter external to it, that is, as absolute motion, is impossible. The Proof (:) goes as follows. Relative motion, be it of a body or a space that moves with same speed in opposite direction while the other is at rest, is always determinable with respect to its relation to the representation of the subject. Therefore, relative motion is always appearance and never experience. This is the case because either () the subject collocates himself in the space at rest and the body appears in motion to him or () if he locates in another space comprehending the first one relative to which the body is likewise at rest, then that relative space counts as moving. More
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specifically, in case () an external framework or space is needed in order to establish whether a space is in relative motion to a body, but this always implies a standpoint external to the system, for example, the body at rest and the relative space in motion, in such a way that the objective determination of the predicates of matter (as being at rest or moving) cannot be actually assessed but is a mere possibility. Subsequently comes the interesting part of Kant’s argument insofar as he gives a definition and a mixed proof to support this position. Kant writes: Thus in experience (a cognition that determines the object validly for all appearances) there is no difference at all between the motion of the body in the relative space, and the body being at rest in absolute space, together with an equal and opposite motion of the relative space. (MAN, :–).
In this mixed proof Kant appeals to principles that do not strictly pertain to metaphysics but that serve to help metaphysics to ground natural science. Namely, he refers to logical rules that make possible the determination of the object for all appearances. Whereas in the proof of the second Proposition Kant is more explicit and explicitly defines this principle, here we can extrapolate a negative rule, namely, that relative motion always brings with it the concept of underdetermination with respect to which of two opposed predicates that can be attributed to it. In particular, appearance gives rise to judgments that use one of the predicates that differ only in regard to the subject and its mode of representation. These are alternative judgments, not disjunctive judgments proper. Alternative judgments allow an arbitrary choice between the two predicates. Both can be applied depending on where the subject is located in the system, and therefore we can infer only the possibility of rectilinear motion. The chapter then proceeds to Proposition , which derives from Lambert’s treatment of motion in the Phenomenology chapter of his Neues Organon (). Kant reinterprets it as follows: The circular motion of a matter, as distinct from the opposite motion of the space, is an actual predicate of this matter; by contrast, the opposite motion of a relative space, assumed instead of the motion of the body, is no actual motion of the latter, but if taken to be such, is mere semblance [Schein]. (MAN, :–)
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Friedman (, –), among others, rightly put it in connection to Lambert. In his Neues Organon, indeed, the fourth section is called “Phenomenology,” in a quite similar way to Kant’s work. However, Kant investigates the difference between true and apparent motion, and, in a sense, he is echoing Lambert’s view of Phenomenology in Proposition where the distinction between apparent and true motion is associated with a disjunctive judgment.
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In the Proof of Proposition , Kant employs the conceptual dichotomy between internal and external and deploys again a mixed proof. The proof displays, I suggest, his adoption of Euler’s laws of motion, because he portrays circular motion as a change of a change in external relations in space and as continuous arising of new motions. This definition is compatible with Euler’s laws of motion. Indeed, Kant grounds this definition of circular motion on the law of inertia and a concept of moving force that is heavily linked with his metaphysics of causality. He then appeals again to general logic, arguing that in the case of circular motion we are appropriately, that is, universally and necessarily, determining the object when we are effectively using a disjunctive judgment in which one of the predicates that we attribute to matter is actual, whereas the one attributed to space is excluded. This logical underpinning helps in identifying any source of error, at least in Kant’s view, when attributing circular motion to space, because phoronomically it has no moving force; therefore, this predicate is not determining the object validly for all appearances. Only a physical body can possess circular motion. I shall come back to this point later, but for the time being we should notice that from the strict physical standpoint, Kant is here following upon Euler, who also attributed circular motion to the body rather than to space. In his work on the dynamics of rigid bodies, Euler did not endorse the necessity of determining through an external moving force circular motion as true motion; rather, he pointed out how the inner state of the body and the law of inertia could explain this thanks to the introduction of external frames of reference (Bartoloni-Meli ; Maronne and Panza ). Just as Euler did, Kant appeals to the conceptual distinction between internal state of the body and external force. However, the difference between Euler’s and Kant’s respective takes on force must be highlighted. Kant believes that force is in need of a definition, a metaphysical one, because it is a fundamental concept of natural science, whereas Euler thought that
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For a clearer account of the relationship between semblance (Schein), appearance (Erscheinung) and causality in connection with transcendental idealism of space and time, see FM (:). I am very thankful to Claudia Laos for pointing out this passage to me. For this reason, I think that one should compare Euler’s equations of motion with the Phenomenology rather than the Phoronomy, which deals with the kinematics of points rather than physical bodies. About this point see also Section .. below. As Maronne and Panza underscored: “A new fundamental change occurs when extrinsic reference frames, typically constituted by triplets of orthogonal fixed Cartesian coordinates, are introduced and when the relativity of motion is conceived to be the invariance of its laws with respect to different frames submitted to uniform retailer motions” (, ).
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impenetrability was the fundamental idea from which the concept of force could be derived. Also consider the following passage: Thus the circular motion of a body, as distinct from that of space is an actual motion so that the latter, even though it agrees with the former according to the appearance, nevertheless contradicts it in the context of all appearances, that is of a possible experience, and so is nothing but mere semblance. (MAN, :).
Here Kant remarks on the complex character of possible experience: we never transform appearance (in singular) into experience, but the latter always presupposes a collective interaction of a plurality of appearances. The fundamental concept of force serves as a bridging concept between metaphysics, mathematics, and physics and grounds the mutual interaction among bodies. In this respect, impenetrability is just a consequence of repulsive force. This passage also marks the transition to the third Proposition: “In every motion of a body, whereby it is moving relative to another, an opposite and equal motion of the latter is necessary” (:). Its Proof is based on the third Law of Mechanics: The communication of motion of bodies is possible only by the community of their original moving forces, and the latter only by mutually opposite and equal motion. The motion of both is therefore actual. (MAN, :)
Thus Kant proceeds to the conclusion that since this motion does not rest on the influence of external forces, as was in the case of circular motion, this motion is necessary. This is the case because the concept of the relation of the moved in space to anything else movable determines the object validly for all appearances and in an unavoidable way. However, also in this case, Kant provides a mixed proof, but the argument is stated in the General Remark to the Phenomenology. In particular, Kant there adds that Proposition is clearly a distributive judgment, in which the two bodies equally share the mutual attribution of motion. The General Remark clarifies that Proposition determines the modality of motion with respect to the Mechanics and adds that all three Propositions of the Phenomenology determine the motion of matter with regard to its possibility, actuality, and necessity; that is, they determine it with respect to all three categories of modality.
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For studies on Kant’s notion of repulsive force, see Warren (), Friedman (), and Massimi and De Bianchi ().
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Kant’s Metaphysical Foundations of Phenomenology
. The General Remark to Phenomenology and the Role of Reason In the General Remark to Phenomenology Kant asserts that this Chapter provides the general use of three concepts in natural science: . . .
Motion in relative (movable) space Motion in absolute (immovable) space Relative motion in general as excluding absolute motion
The basis (Grund) of these three concepts is the idea of absolute space. The relationship between these concepts of motion and the idea of absolute space should be understood as a ground-consequence (GrundFolge) relation. That is, absolute space as an idea of reason is necessary to ground the three kinds of motion: Absolute space is therefore necessary, not as a concept of an actual object, but rather as an idea, which is to serve as a rule for considering all motion therein as merely as relative; and all motion and rest must be reduced [reducirt] to absolute space, if the appearance thereof is to be transformed into a determinate concept of experience (which unite all appearances). (MAN, :)
We are facing here a new positive function of the idea of reason, that of reduction (Reduktion). How do we have to interpret this function? Should it be distinguished from the mere unifying capacity of the ideas of reason, as such? How does reduction contribute to the objective determination in metaphysics and the possibility of transforming appearance into experience? I offer an answer to these questions in the remaining part of this section. The general Proof relies on a hidden premise, namely, that external relative space can be generated ad infinitum. From this follows the general rule that all motion or rest can be relative only and not absolute; that is, matter can be thought as moved or as at rest only in relation to matter and never with respect to mere empty space. Therefore, absolute motion is impossible without any relation of one matter to another. In other words, this implies that no concept of motion or rest valid for all appearance is possible in relative space:
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For a recent study on absolute space and rotational motion, see Stan (). Studies on reduction and unity of science in Kant’s system and beyond include Buchdahl (), Morrison (), and Stan (b).
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Rather, one must think a space in which the latter can itself be thought as moved, but which depends for its determination on no further empirical space, and thus is not conditioned in turn – that is, an absolute space to which all relative motions can be referred, in which everything empirical is movable, precisely so that in it all motion of material things may count as merely relative with respect to one another, as alternatively mutual, but none as absolute motion or rest (where, while one is said to be moved, the other, in relation to which it is moved, is nonetheless represented as absolutely at rest). (MAN, :–)
Thus, the unconditioned idea of absolute space becomes the reference frame through which not only all relative motions, but also reciprocal empirical interactions are determined. We have therefore to conclude that the function of reduction attributed to the idea of absolute space is intimately connected to the operations of unifying all relative motions and of ensuring the objective determination of phenomena. At this stage, it is important to verify whether the idea of absolute space introduced in this chapter is different from Newton’s. Kant made a distinction between absolute and relative space which clearly refers to Newton’s Scholium on space, time, and motion at the beginning of the Principia: Absolute space, of its own nature without reference to anything external, always remains homogeneous and immovable. Relative space is any movable measure or dimension of this absolute space. (Newton , –)
In MAN, Kant claims that “absolute space is in itself nothing and no object at all,” but refers to an indefinite process of considering ever more extended relative spaces (:–). Moreover, Kant states in the Phenomenology that absolute space is not an existing external object but rather is to serve as the rule for considering all motion and rest therein merely as relative (:). This conception resembles Euler’s (, ) and once again points to the necessity of distinguishing when Kant is referring to Newton’s work and when he is instead trying to harmonize his system more broadly with the physics and the astronomy of his time. Nevertheless, the first difference to notice with respect to Newton consists in the application of the concept of central force to the whole hierarchical structure of the universe; second the transcendental status of absolute space; third the fact that according to Kant gravitation is only one case of the concept of central force and the universal law that Kant has in mind is the one from which Newton’s third law is derived, i.e. “the law of antagonism in all community of matter through motion.” (MAN, :)
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This means that we must be cautious before accepting a perfect overlap between the principles of MAN and Newton’s laws. Kant was far more ambitious: he wanted to explore the foundations of the generalizations of Newton’s physics, including Euler’s, and was aware of the limits of Newton’s physics to explain some astronomical observations, such as those made by William Herschel or Johann Hieronymus Schröter. We should also note the use that Kant makes of measurement in order to support his view. In the General Remark to Phenomenology (MAN, :), for instance, in order to show that true circular motion is not semblance and that circular motion is a continuous dynamical change in the relation of matter within space, Kant claims that we must be able to measure acceleration. In other words, we must measure how the direction of bodies’ motion deviates within the same reference system that is comoving. In order to obtain the objective determination of a moving body, we have to include its internal state in this determination, not only the external forces or states acting upon it. This means that to prove that the Earth is revolving upon its axis or completing its revolution around the Sun we do not have to observe the starry heavens; rather, the measurement apparatus must be posited within the system of which we want to determine the motion. For example, to prove the axial rotation of the Earth Kant references the example of throwing a stone within a deep hole descending to the center of the planet and observing the continuous deviation of its perpendicular trajectory. Likewise to prove the Earth’s revolution about the Sun, one has to consider the Sun-Earth system as two-body problem. Friedman understands these passages to imply that the law of universal gravitation falls under the category of necessity. According to this reading, Kant’s reconstruction of Newton’s “deduction” of the law of universal gravitation from the initial Keplerian “analogies” provides a perfect illustration of the three-step procedure, described in the Postulates of Empirical Thought, by which a mere “empirical rule” is transformed into a “necessary and universally valid” objective law. However, I contend that Friedman’s interpretation overlooks the role of measurement (and therefore the link to KU) as well as Kant’s attempt to ground Euler’s physics from a metaphysical standpoint (see Section ..).
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For more details on this topic, see De Bianchi (, ). Recall what Kant writes in the MAN: “Hence, the movable, as such a thing, becomes an object of experience, when a certain object (here a material thing) is though as determined with respect to the predicate of motion” (:). In other words, the equations must contain the internal state of motion of the body à la Euler.
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For the time being, let me return to my analysis of the General Remark to Phenomenology, which is fundamental for two reasons. First, it illuminates how Kant connects the application of formal conditions to the concept of “the reciprocal interaction in all community of matter through motion,” rather than to the object of the possible experience in general. Second, the General Remark implies the necessity of determining this reciprocal interaction by means of the application of mathematics to physics, that is, measurement. This enables us to establish a connection between this part of Kant’s system and the third Kritik where Kant makes explicit his doctrine of measurement (see Section ..). Second, the General Remark to Phenomenology is fundamental for Kant’s cosmology. It is at this point of the text that Kant denies the possibility of the universe moving or expanding, for this would presuppose a space external to it, which is impossible. However, Kant was aware that, according to his cosmology, it could have been perfectly possible for the universe to expand in all directions as a sphere. Nonetheless, at this stage Kant still seems to be prudently avoiding any explicit reference to his cosmological system, even if we know that Kant republished his cosmology at least partially in and that he discusses it again in the Opus postumum (see De Bianchi ).
. Phenomenology: Measurement and Foundations of Mathematical Physics In this section I both underscore Kant’s debt to Euler and advance a new reading of the Phenomenology as being meant to ground the equations of Euler’s mechanics of solid bodies. This shows that to transform appearance into experience by using absolute space as an idea of reason means to connect and measure the plurality of comoving bodies, thereby embodying in practice the law of antagonism in all community of matter through motion. In the second subsection, I show the consequences of such a view and connect the Phenomenology with Kant’s view of measurement as developed in the third Kritik. .. Phenomenology and the Foundations of Euler’s Equations The Phenomenology is the laboratory in which Kant not only tests the third law of mechanics in its capacity to ground the concept of actual
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See Sutherland () for Kant’s view of arithmetic and proportions in KrV.
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motion but also deploys the conceptual apparatus for the foundations of Euler’s equations of solid bodies. In a sense, the Phenomenology also results in a conceptual reconstruction of Newton’s a posteriori derivation of the law of universal gravitation from the observable phenomena described by Kepler’s “rules” or analogies. These phenomena are considered as appearances (Erscheinungen), which are to be transformed into experience (Erfahrung). This is attained by using the law of inertia to mathematically derive inverse-square accelerations of satellites directed toward every primary body in the solar system. Further, by identifying the accelerations in question as effects of a central force, it follows that these accelerations must hold immediately between each part of matter and every other part of matter and are also directly proportional to the mass. However, in order to transform appearance into experience, that is, in order to obtain experience of matter through the empirical concept of motion, we need to measure acceleration. This provides the ground for determining the other two elements which give us a necessary universal determination of bodily motion, that is, the translation of the center of mass and the rotation about the center of mass. Both requirements are present in Euler’s equations of motion of rigid bodies. Furthermore, in Kant’s view, to metaphysically construct matter as object of experience also means to exhibit the relationship between the idea of absolute space and the three types of motion, which are possible, actual, or necessary objects not with respect to the understanding (as would be the case if only the postulates of empirical thought in general were at stake) but with respect to reason. Now, in reviewing Friedman’s work, Stan correctly noticed that when talking about Newton we have carefully to distinguish it from Newtonianism. In turn, I want to stress a similar point that Massimi and De Bianchi () made about Descartes and Cartesianism in the eighteenth century; namely, one has to distinguish Newton or Descartes from Newtonianism or Cartesianism given the quasi-syncretistic nature of these positions in the first half of the eighteenth century. In other words, when approaching the MAN, we have to take into account those physicists and mathematicians who confronted the most fundamental and still-open questions raised by Newton; his predecessors, such as Descartes and Galilei; and his contemporaries, like Leibniz and Huygens. More emphasis should additionally be placed on the representatives of the Berlin Academy of Sciences, such as Euler and d’Alembert, as they inherited the burden of extending and formalizing Newton’s physics. Newton’s Principia indeed required much more than a reformulation in analytic terms, if it was to provide the basis for the comprehensive
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mechanics that was developed in the eighteenth century. According to Stephen Gaukroger (), there were also a number of areas crucial to the unification of mechanics that Newton did not deal with at all in the Principia, particularly, the dynamics of rigid, flexible, and elastic bodies and the dynamics of several bodies with mutual interactions. Euler’s contribution to the development of these topics, and hence to the unification of mechanics, was immense. Building upon Buchdahl’s (b) interpretation of MAN, Harman () focused on the link between Euler’s physics and Kant’s notion of force, but he did not attempt to demonstrate that in the Phenomenology Kant meant to refer to the advancements of Newtonian physics embodied by Euler’s equations. This cost him the sharp criticism of Okruhlik (), who rightly suggested that one must be careful in establishing an overlap between Kant’s and Euler’s notion of force. However, in what follows, I shall show how Kant’s Propositions of the Phenomenology are meant to explore the metaphysical underpinnings of Euler’s equations of the dynamics of rigid bodies, even if the meaning that they attributed to “force” is not the same. Leonhard Euler gave a formalization of the extension of Newton’s laws to rigid bodies, considered from the standpoint of continuous mechanics. Today this extension is designated through two equations or axioms that are complemented by a third one, the Newton-Euler law. Historically, Euler achieved the result of extending Newton’s laws of motion for point particles to rigid bodies in a series of papers, by further developing the law of inertia and the definition of impenetrability and abandoning the idea of conceptually defining force. In , Euler published Theoria motus corporum solidorum seu rigidorum. This work was improved by Euler’s son Johann Albrecht in a new edition in . In the introduction to the original work Euler confirmed the principles stated in his Mechanics (). He defines the main characteristic of a rigid body by the invariability of distances between any two points belonging to a body. In addition, Euler defined a “mass centrum” or an “inertiae centrum” for every and each body. In this work, Euler points out that the “center of gravity” of a rigid body implies a more restricted concept than a “mass center” or an “inertia center.” The last two concepts are better defined by the inertia itself when the system of forces acting on the rigid body is neglected. In addition, he adopted a reference frame attached to the rigid body and identified the principal axes of inertia.
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See Suisky (, , , ff.). The works are Euler (, , a, b, c, ).
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Euler’s study further included the rigid body dynamics by means of the decomposition of motion in two elements: the translation of the center of mass and the rotation about the center of mass. In this context he considered what we now call “Euler angles” and studied rotational problems motivated by the precession of the equinoxes. In particular, Euler distinguished between internal and external states of bodies; the internal state is described in terms of relations between different parts of the bodies, whereas the external state is obtained from the relation of the whole body (including its internal parts) to other bodies. In my view, Kant’s emphasis regarding the fundamental law of mutual interaction among bodies and his appeal in Proposition to the conceptual dichotomy of internal and external state of a body are meant to provide the metaphysical foundation of Euler’s equations. Furthermore, the definition of the states of rest or uniform motion was presented in Reflexions sur l’espace et le tems and is based on the external relations to other bodies and on the relations between the bodies and the space (Euler ). Euler claimed that the preservation of direction of a uniformly moving body cannot be explained by Leibniz’ relational theory of space and time (, –). Therefore, Euler accepted absolute time and space but rejected absolute motion. These sources shed considerable light on Kant’s Phenomenology and his rejection of absolute motion stated in Proposition and in the General Remark. In Découverte d’un nouveau principe de la mécanique, Euler states: Among the infinity of motions to which a solid body is susceptible, the first thing to consider is that in which all parts persist constantly directed toward the same points of absolute space. In other words, if we conceive of a straight line passing through any of two points of a body, this line will always conserve the same direction or, which is the same, it will perpetually persist parallel to itself. This sort of motion is called purely progressive. (Euler , )
This passage helps to clarify Kant’s statement that all three types of motion analyzed in the three Propositions of the Phenomenology presuppose the
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For more details on Euler’s external frames of reference, see Maronne and Panza (). See Suisky (, ff.). Kant rejected the idea of absolute time but accepted that of absolute space in the Phenomenology, a move that is not in contradiction with the doctrine of transcendental idealism as long as absolute space is a regulative idea of reason. The original text reads: “Entre l’infinité des mouvements, dont un corps solide est susceptible, le premier, qu’il faut considérer, est celui, ou toutes les parties demeurent constamment dirigées vers les mêmes points de l’espace absolu. C’est-à-dire, si nous concevons une ligne droite tirée par deux points quelconques du corps, cette ligne conservera toujours la même direction, ou ce qui revient au même, elle demeurera perpétuellement parallèle a elle-même. Un tel mouvement est nommé purement progressif.” English translation is mine.
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concept of absolute space. Nevertheless, Kant distances himself from Euler insofar as he assumes repulsion and attraction as fundamental properties of matter in dynamical terms. That said, Kant completely follows upon the mathematician when stating: There is thus no absolute motion, even when a body in empty space is thought as moved with respect to another; their motion here is not considered relative to the space surrounding them, but only to the space between them, which considered as absolute space, alone determines their external relations to one another and is in turn only relative. (MAN, :).
Euler’s explanation for the assumption of absolute space is grounded on mathematical considerations by means of the representation of projected trajectories, whereas Kant aims to harmonize Euler’s equations with transcendental philosophy. Thus, he defines absolute space as an idea of reason with a necessary use, capable of encompassing all possible (relative) motions and therefore all appearances according to the universal and necessary law of antagonism in all community of matter through motion. There are now two questions I would like to mention before concluding. First, I strongly agree with Friedman () that the notion of reference frame is fundamental in order to understand the Propositions of the Phenomenology and more generally his concepts of relative motion and rest. And precisely the notion of the coordinate system (x, x, x) as an inertial frame of reference is what Euler presupposed for his equations. Second, it is worth mentioning that the Newton-Euler equation fits nicely with the observations made by Kant in the General Remark to Phenomenology. In classical mechanics, the Newton-Euler equations describe the combined translational and rotational dynamics of a rigid body. This formalism relates the motion of the center of gravity of a rigid body with the sum of forces and torques (or moments) acting on the rigid body. In other words, it embodies in a very sophisticated way the scope of objectively determining material bodies through actual relative motions, by transforming the latter from appearances into experience; namely, actual relative motions extend Newton’s laws to all appearances,
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Stan () underscores the necessity for Kant to ground Euler’s equations and identifies in his early physical monadology the key to make transcendental philosophy compatible with Euler’s mechanics. In my view, there is no need to include Kant’s monadological theory of matter with the Phenomenology to show the latter’s compatibility with Euler’s mechanics. Indeed, what is at stake in both cases is the discussion of the torque law applied to comoving bodies. Of course, Stan’s observation on the necessary presupposition of a certain theory of matter excluding its infinite divisibility holds if one focuses on the kinematics discussed in the Phoronomy, because in that chapter Kant is considering points rather than bodies.
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such as comoving bodies, be they elastic, rigid, or fluid. Therefore, Kant’s Phenomenology aimed at encompassing the scope of Euler’s Mechanics. In distinction to Euler, however, Kant dismisses the idea of absolute time and removes it from the picture, even if he clearly thinks that the objective representation of time as a formal intuition is fundamental to ground measurement and therefore to transform appearance into experience, by connecting the plurality of comoving bodies. ..
Measurement and the Asymmetry of Space and Time: Phenomenology as the Laboratory of Pure Theoretical Reason
This subsection is devoted to showing in which sense we could interpret the Phenomenology as a laboratory for the empirical use of theoretical reason. In order to substantiate my claim, it is necessary to investigate the connection between the conception of time as formal intuition and Kant’s view of measurement. In KrV we find the distinction between space as formal intuition and as form of intuition: Space, represented as object (as is really required in geometry), contains more than the mere form of intuition, namely the comprehension of the manifold given in accordance with the form of sensibility in an intuitive representation, so that the form of intuition merely gives the manifold, but the formal intuition gives unity of the representation. In the Aesthetic I ascribed this unity merely to sensibility, only in order to note that it precedes all concepts, though to be sure, it presupposes a synthesis. (KrV, B–n)
This passage concerns the objective representation of space, but it is silent with respect to the results of the objective representation of time. Is there a science in which time is represented as object? Even if it is not possible to give a final answer to this question, because there could be more than one science apt to do that, it is remarkable that Phoronomy seems to be a candidate for this scope. The mathematical construction of motion in the Phoronomy accomplishes the task of measuring the object at the same
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It is worth noticing the relevance of investigating in a deeper fashion the relationship between the Phenomenology and the Phoronomy with respect to Kant’s view of measurement and the foundations of mathematics. The debate on this topic is rich and marked by conflicting interpretation on the role of schematism for the objective representation of time. See, e.g., Longuenesse (, ) and the debate with Friedman (a, b).
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time in which it is drawn; however, what is drawn, technically speaking, is velocity, therefore what is objectively represented is a certain time taken by a point to move from A to B. In the Phoronomy thus we find a sort of objective representation of time. Contrary to the geometrical representation of space, time must undergo an indirect objective representation through motion (see KrV, B–). In other words, only by means of measurement time can acquire meaning. However, things become subtler and more intriguing when comparing the objective representation of time of the Phoronomy with that of the Phenomenology. The connection between these two sections is stated by Kant himself in the text (MAN, :). The major difference between the Phoronomy and the Phenomenology is that the former concerns only the motion of space and is devoid of any reference to a moving force, that is, of a dynamic account. How does the Phenomenology represent time objectively? And why does Kant eventually need a second way in which time can be objectively represented through motion? Part of the answer lies in the previous passage: the Phenomenology shows us the dynamics underlying the reciprocal relationship among bodies and encodes within itself all three modes of judging motion as presented in the previous chapters of the MAN. Within this enlarged picture, time is represented as object through comoving reference frames and not by means of an abstract motion of a space. However, if comoving reference frames are at stake, it means that they can be measured relatively to one another and that their relative magnitude can be established. In other words, time is objectively represented, but not, for instance, as a single cosmic time. Time is, rather, parametrized and represented as object only relative to each reference frame. Now, this position is not simply intriguing with respect to the science of Kant’s time. What is extremely interesting is to compare this conclusion of the Phenomenology with Kant’s view of measurement as portrayed in KU, in particular, in the passages devoted to the sublime and the foundations of measurement (see KU, :). In the third Kritik the foundations of measurement are to be found in the capacity of reason to provide ideals to intuitions, rather than in the principle of the pure understanding. Once again, this operation, albeit in a different fashion, brings ideas and their faculty back in view. In the case of MAN, the idea of absolute space is the actor that can make sense of magnitudes. And what is the objective
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Dunlop () offers an interpretation of Kant’s passages in the B-Deduction (Bff.) that I find convincing and able to reconcile Friedman’s and Longuenesse’s views.
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representation of time, if not that of an infinite given magnitude, a continuous succession? And now comes the critical part of my reflection and perhaps a puzzling aspect not only of Kant’s philosophy but of philosophy of time in general. Whereas the idea of absolute space is able to fully justify the structure and aim of the Phenomenology, the concept of absolute time is not. And indeed it is not mentioned in the last chapter. What is even more striking is what we read in the KU regarding time and measurement: The measurement of a space (as apprehension) is at the same time the description of it, thus an objective movement in the imagination and a progression; by contrast, the comprehension of multiplicity in the unity not of thought but of intuition, hence the comprehension in one moment of that which is successively apprehended, is a regression, which in turn cancels the time-condition in the progression of the imagination and makes simultaneity intuitable. It is thus (since temporal succession is a condition of inner sense and of an intuition) a subjective movement of the imagination, by which it does violence to the inner sense, which must be all the more marked the greater the quantum is which the imagination comprehends in one intuition. Thus the effort to take up in a single intuition a measure for magnitudes, which requires an appreciable time for its apprehension, is a kind of apprehension which, subjectively considered, is contrapurposive, but which objectively, for the estimation of magnitude, is necessary, hence purposive; in this way, however, the very same violence that is inflicted on the subject by the imagination is judged as purposive for the whole vocation of the mind. (KU, :–)
The asymmetric treatment of space and time in Kant’s philosophy is not something new, but here we see that in order to obtain a temporal formal intuition, reason (and neither the imagination nor the understanding) must exert violence on the manifold of representation in intuition. In the case of the Phenomenology, something quite similar to what is described in KU is outlined, although with more accuracy and by means of the idea of absolute space. This move conforms Kant’s theory of matter to Euler’s equations but creates an asymmetry between space and time. This is due to the fact that an idea of absolute time would lead to a concept of absolute acceleration and the impossibility of comparing reference frames. In connection to the results of the Phenomenology regarding the three types of motion, this would also lead to consequences contrary both to the universal law of antagonism in all community of matter through motion and to Kant’s own cosmology. Indeed, to accord absolute time a role in the last chapter of the MAN would have meant to admit that there
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are times to be calculated in a different manner with respect or relative to various star systems. A fascinating conclusion, but it was not Kant’s.
. Conclusion In the previous sections, I highlighted the importance of considering the Phenomenology in connection with the advancements of Newtonian physics and with Euler’s equations of motion of rigid bodies in particular. I emphasized the relevance of the function of reduction attributed to the idea of absolute space and how it articulates in a more precise way the transcendental function of unity attributed to the ideas of reason in the first Kritik. Another crucial point highlighted in this contribution is the fundamental role played by the connotation of time as forms of intuition and formal intuition to determine acceleration and make it compatible with Euler’s mathematical and conceptual standpoint. The most relevant point that I would like to convey to the reader is that the Phenomenology represents Kant’s attempt to embody the empirical use of reason, which is identified with “transforming appearance into experience,” and to enable the application of mathematics to natural science through measurement. In other words, the last Chapter of MAN constitutes a unique laboratory to test the necessary empirical use of pure theoretical reason.
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Kant conserved the structure of his cosmology when publishing an excerpt again in . In the NTH he talks about spaces in plural, but not about times in the universe. The problem of absolute time is still present and object of inquiry in the manuscript of the Opus postumum, when Kant has to resort the idea of ether as the all-penetrating and oscillating cosmic matter to account for the measurable acceleration of the universe expansion, but to my knowledge he never refers to the idea of absolute time to make sense of both Herschel’s observations and the applications of Newton’s and Euler’s equations in the late writings. I am very thankful to Michael Bennett McNulty, James Messina, and Daniel Warren for their comments and suggestions on earlier drafts of this contribution. This result is part of the PROTEUS project that has received funding from the European Research Council (ERC) under the Horizon Research and Innovation Programme (grant agreement no. ).
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Absolute Space as a Necessary Idea Reading Kant’s Phenomenology through Perspectival Lenses Michela Massimi
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Introduction
Kant’s engagement with Newton’s notion of ‘absolute space’ is fascinating, complex, and spans over both the pre-critical and the critical period. The received view has it that in the pre-critical period Kant shifted from an originally Leibnizian view of space (still visible in Physical Monadology [MonPh, ], and New Doctrine of Motion and Rest [NLBR, ]) to a proper Newtonian view of absolute space via the incongruent counterparts argument in Directions in Space (GUGR, ), for then abandoning absolute space in the Inaugural Dissertation (MSI, ). Indeed, the same argument from incongruent counterparts was later employed in the Prolegomena () as an argument for space as “the form of outer intuition of . . . sensibility” (Prol, :). In the first Critique, Transcendental Aesthetic, the Newtonians are praised for succeeding in making mathematical knowledge of nature possible by “opening the field of appearances for mathematical assertions” (KrV, A/B), but criticized for making the mistake of assuming “two eternal and infinite self-subsisting non-entities (space and time)” (ibid.). And in a passage of the Critique of Practical Reason discussing the relation between freedom and natural necessity, Kant returns to Newton’s absolute space this time by offering an ‘argument from Spinozism’ against it (KpV, :–). Given the unequivocal criticism of absolute space in the two Critiques, it might seem prima facie surprising to see Kant returning to the topic in the Metaphysical Foundations of Phenomenology, the final chapter of the Metaphysical Foundations of Natural Science (MAN, ), in a seemingly more conciliatory tone. For there we are told, “Absolute space is therefore
See Brewer and Watkins () for the threat of theological determinism, and its relation to both Leibniz and Spinoza. And Massimi (b) for a reading of this passage in the light of Kant’s defense of idealism about space.
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necessary, not as a concept of an actual object, but rather as an idea, which is to serve as a rule for considering all motion therein merely as relative; and all motion and rest must be reduced to absolute space, if the appearance thereof is to be transformed into a determinate concept of experience” (MAN, :). Absolute space is said to be a “peculiar concept” that acts as the “basis” for three other concepts: “motion in relative (movable) space”; “motion in absolute (immovable) space” and “relative motion in general” (MAN, :; emphases in the original). What is surprising about this passage of MAN is not the fact that absolute space reappears or features prominently in the chapter on Phenomenology. For it is very clear from the passage that Kant neither is harking back to absolute space as an “essential determination of the original being itself” (which he criticizes adamantly in KpV, :–) nor is he retracting his earlier rebuttal (KrV, A/B) of absolute space qua an absolute (transcendental) reality of space. Thus, there is no inconsistency as such between the analysis of absolute space in MAN and the two Critiques. Back to absolute space nonetheless Kant goes, this time defended as “a necessary concept of reason, and thus as nothing more than a mere idea” (MAN, :). What is then surprising (and worth exploring in more detail) are the reasons why absolute space as an idea of reason (i.e., neither a determination of God’s omnipresence nor an eternal and self-subsisting entity) is said to be necessary to determine the empirical concept of matter (qua the movable in space) “with respect to the predicate of motion” according to the category of modality. In other words, why is absolute space – qua a “mere idea” (MAN, :) – necessary to determine the motion of matter as possible, actual, or necessary? How can absolute space as an idea transform “appearance into experience” (MAN, :)? Friedman (, –) has persuasively explained this passage of MAN by observing that “Kant’s notion of what he calls ‘absolute space’ is more like Newtonian absolute space than the modern conception of inertial frame. Yet Kantian absolute space differs from Newton’s in being what Kant considers a limiting idea of reason rather than the concept of an actual object” (p. ). In particular, Kant’s absolute space as an idea of reason is said to be connected with Kant’s argument for the first antinomy whereby “we can neither consider the sequence of (material) relative spaces
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Kant clearly says that absolute space qua empty space in the phoronomical sense “is therefore nothing at all that belongs to the existence of things, but merely to the determination of concepts, and to this extent no empty space exists” (MAN, :).
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Reading Kant’s Phenomenology through Perspectival Lenses
as a completed infinite totality of cumulative rotating structures nor as a finite such totality terminating in empty space” (p. ). Friedman’s (, ) detailed analysis of absolute space as a purely regulative idea provides the springboard for my discussion in what follows. For while I entirely agree with Friedman’s analysis of the regulative role of absolute space, I’d like to explore in more detail what is at stake in such a regulative idea and why it is said to be necessary to determine the motion of matter according to modality. If absolute space as a “mere idea” has only a regulative and not a constitutive function, why is its use deemed nonetheless necessary for matter (the movable) to become an object of experience? In other words, how can a purely regulative idea (such as absolute space) determine the empirical concept of matter with respect to the predicate of motion so as transform appearances into objects of experience? These questions take us right to the heart of one of the most important aspects of Kant’s Critical philosophy: namely, the regulative role of ideas of reason. I offer a ‘perspectivalist’ reading of the regulative role of ideas of reason that I hope will shed some light on Kant’s stance on absolute space in the Metaphysical Foundations of Natural Science. The chapter proceeds as follows. In Section ., I review some of the milestones in Kant’s lifelong engagement with the topic of absolute space, from the pre-critical to the critical period. In Section ., I analyze some of the main theses at play in Kant’s critical reassessment of Newton’s absolute space and the regulative role of the idea of absolute space in the Metaphysical Foundations of Natural Science. In Section ., I return to the Critique of Pure Reason, Appendix to the Transcendental Dialectic, to clarify my perspectivalist reading of the regulative role of ideas of reason as foci imaginarii, that is, ‘imaginary standpoints’ as I’d like to call them. And I explain how this reading can shed light on the stark assertions that Kant makes in the Metaphysical Foundations of Natural Science about the necessity of absolute space.
. Kant’s Lifelong Engagement with Absolute Space: A Primer Kant’s philosophy of nature evolved and developed from his very early contributions in the s to his mature critical period. And the presence of distinctive Newtonian elements in Kant’s writings on natural science can be traced back to what some scholars have portrayed as Kant’s ‘conversion to Newton’ as early as in Universal Natural History and Theory of the Heavens (NTH, ). But it was not just Newton’s mechanics and his principles of attraction and repulsion that the young
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Kant received and engaged with in his original and often idiosyncratic ways. Kant’s lifelong engagement with Newton’s view of absolute space deserves special attention. Friedman (b) and Schönfeld (), among others, have documented how in the pre-critical period Kant shifted from an originally Leibnizian/relationalist view of space (still evident in Physical Monadology and New Doctrine of Motion and Rest) to a proper, though short-lived, Newtonian view of absolute space via the incongruent counterparts argument in Directions in Space (). By the time of the Critique of Pure Reason, Kant famously criticized Newton’s conception of the absolute reality of space (as well as Leibniz’s relationalism) in the name of transcendental idealism about space and time (KrV, A/B). Did the young Kant convert to Newton’s absolute space (ca. ), short-lived as the conversion proved to be? In Massimi (a) I answered this question in a more nuanced way by showing that in the relevant period around –, Kant was in fact working with a thoroughgoing relational view of space ensuing from Kant’s dynamical matter theory. While sufficiently distant from both Leibniz’s and Wolff’s relationalism, Kant’s view of space was nonetheless elaborated primarily against the backdrop of the Leibnizian-Wolffian tradition and imbued with Kant’s new dynamical theory of matter, which was in turn inspired by speculative Newtonian experimentalism (as I clarified in Massimi ). Kant’s pre-critical view of space – from the original True Estimation (GSK) to the Directions in Space – betrays his idiosyncratic blend of the Wolffian and the Newtonian traditions. For example, the relationalism about space advocated in Physical Monadology is at some distance from both Leibniz and Wolff and betrays Kant’s debt to speculative Newtonian experimentalism in the treatment of physical monads as spheres of activity (with repulsion understood as a contact force and exemplified by the “ether, that is to say, the matter of fire” [MonPh, :]). And, vice versa, Kant’s enthusiasm for Newton’s program in Directions in Space requires a few caveats and, in my view, should be read as containing some of the seminal seeds of Kant’s later critical treatment of space. Let me briefly substantiate these two claims.
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In what follows, I draw on (and summarize the main points from) Massimi (c). For an excellent reconstruction of how Kant’s matter theory of this period, especially evident in Physical Monadology, borrows from and, at the same time, distances itself from Leibniz, Wolff, and also Baumgarten, see Watkins ().
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Reading Kant’s Phenomenology through Perspectival Lenses
In the Physical Monadology, Kant described physical monads as reciprocally acting substances, and responded to the proof of the Newtonian John Keill (), who in An Introduction to Natural Philosophy had offered a proof for the infinite divisibility of space. Kant argued that the infinite divisibility of space was compatible with the assumption that physical bodies are not themselves infinitely divisible on the ground that “space, which is entirely free from substantiality and which is the appearance of the external relations of unitary monads, will not at all be exhausted by division continued to infinity” (MonPh, :). Kant argued that Newtonians erred in wielding Keill’s proof against the “metaphysicians,” as much as the “metaphysicians” erred in maintaining against the “geometers” that the properties of space were imaginary. Key to this idiosyncratic blend of Leibnizian metaphysics and Newtonian geometry was Kant’s view of space understood as “a certain appearance of the external relations of substances.” But how can physical monads fill the space via their spheres of activity, and with space being in turn infinitely divisible without either jeopardizing the simplicity and unity of the monads (pace monadology) or reaching the contradictory conclusion that the divisibility of space (qua external relations among substances) can only be finite (i.e., pace Keill’s proof )? As I argued in Massimi (a), this conundrum could be solved by bringing in considerations from Newtonian experimentalism. Against the standard Newtonian view of specific density of bodies as the ratio between mass and volume (whereby bodies with the same volume may nonetheless possess different specific densities because of the different amount of interstitial vacua), on Kant’s view (MonPh, :–), specific densities were due to a perfectly elastic force “which is different in different things” and which constituted “a medium which is, in itself and without the admixture of a vacuum, primitively elastic” (Proposition XIII. Theorem, emphasis added). As a primary example of elastic bodies, Kant mentioned the ether, “that is to say, the matter of fire,” with an unequivocal homage to Boerhaave’s material fire, in continuity with his text On Fire (), where the ether was presented as both the matter of fire and the matter of light. Kant’s engagement with speculative Newtonian experimentalism (from Newton’s optical ether to Boerhaave’s material fire) made possible the understanding of the impenetrability of bodies in terms of a perfectly elastic repulsive force emanating from physical monads. In this way Kant could at once
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For an analysis of On Fire, see Massimi ().
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() ()
safeguard the simplicity and unity of physical monads by thinking of space along relationalist lines (i.e., as a bunch of external relations holding among monads’ respective spheres of activity) and guarantee the infinite divisibility of space by thinking of impenetrability as a perfectly elastic repulsive force that fills the space (e.g., the ether).
However, Kant’s view on space seems to have undergone a major overhaul in the following decade, as several scholars have noted and commented upon. In the text Concerning the Ultimate Ground of the Differentiation of Directions in Space, Kant seemed to side unequivocally with Newton and the Newtonians in rejecting this time the opinions of the “German philosophers,” who claimed that space consisted solely in the external relations of the parts of matter. Twelve years after Physical Monadology, Kant defended the view that directions in space could not be reduced to mutual relations among objects but should be explained “in relation of the system of these positions to the absolute space of the universe” (GUGR, :). This statement has often been read as marking a significant departure from Kant’s earlier flirtation with the LeibnizianWolffian dynamical metaphysics, and as him taking a more decisive stance in defense of Newton’s view of absolute space. Indeed, the declared goal of the essay was to offer a proof intended for “geometers” of the claim that absolute space has a reality of its own. Kant lamented that no metaphysical argument had been successful in establishing this claim. Nor were a posteriori proofs for absolute space available, Kant claimed, apart from Euler’s attempt to provide one for the prize essay of the Berlin Royal Academy of Sciences in , which Kant discarded as a proof intended for engineers, and not for “geometers.” A charitable reading of Kant’s above claim would be that Euler’s proof for absolute space could cut no ice with the German metaphysicians because it presupposed precisely what the metaphysicians would question, namely, the reality of absolute space. Kant then goes on to defend absolute space by using what might be called arguments from indexicality (I) and from incongruent counterparts (II): (I) In Cartesian coordinates, Kant notes that if we take our body as the origin of the three axes, we can establish the distinction between above and below; left and right; in front of and behind. Similar
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See Stan (, ).
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reference to our situated body is inevitably presupposed in the indexical use of geographical maps and of the compass. (II) Features found in some animal species (e.g., snails’ shells) and vegetable ones (e.g., the growth of beans and hops) reveal incongruent counterparts, despite the objects having same size, same proportion, and even same relative arrangements of their parts. These arguments are – to say the least – peculiar if intended as evidence for absolute space in Newton’s own sense (or, for that matter, absolute space as intended by the “geometers” more broadly). If anything, considerations about indexicality and incongruent counterparts may work at best as arguments against the German philosophers (i.e., Leibniz), who by appealing to relations among parts were unable to account for handedness and chirality. But it is unclear how (I) and (II) could function as arguments for Newton’s absolute space. Indeed, it would be a non sequitur to conclude from (I) and (II) that one must embrace Newton’s absolute space. In Massimi (a), I therefore argued that Kant’s text should not be read as an endorsement of Newton’s absolute space. Indeed, Kant begins the essay in a rather non-Newtonian sounding tone, by identifying absolute space with the “ultimate foundation of the possibility of the compound character of matter” where once again the whole problem about the compositionality of matter seems to belong more to metaphysics than to Newton’s physics. And Kant referred to absolute space not as a real object but as a Grundbegriff, that is, “a fundamental concept which first of all makes possible all such outer sensation” (GUGR, :). Two years later, in the Inaugural Dissertation, Kant presents the concept of space as “not abstracted from outer sensation” (MSI, :) and as presupposed for the possibility of outer perceptions – a view that anticipates some key features of Kant’s mature critical treatment. A decade later, in the first Critique – Transcendental Aesthetic, Metaphysical Exposition, and Transcendental Exposition (KrV, A/ B) – Kant returned to the topic and famously defended the apriority, necessity, and ideality of space. Space is there said to be a “necessary representation, a priori, that is the ground of all outer intuitions,” “the condition of the possibility of appearances, not as a determination dependent on them, as is an a priori representation that necessarily grounds outer appearances.” Hence, the “ideality of space in regard to things when they are considered in themselves through reason, i.e. without taking account of the constitution of our sensibility” (KrV, A/B).
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And this time, the Newtonians (described as “the mathematical investigators of nature”) are praised for succeeding in making mathematical knowledge of nature possible by “opening the field of appearances for mathematical assertions” (KrV, A/B). However, Kant blames them for making the mistake of assuming “two eternal and infinite selfsubsisting non-entities (space and time), which exist (yet without there being anything real) only in order to comprehend everything real within themselves” (ibid.). The metaphysicians of nature (i.e., the LeibniziansWolffians), on the other hand, are accused of identifying space and time with relations of appearances abstracted from experience, at the cost of disputing “the apodeictic certainty of a priori mathematical doctrines in regard to real things (e.g. in space)” (ibid.). Caution is in order when reading these passages. Kant’s qualified nod to the mathematical investigators of nature does not amount to any endorsement of Newton’s absolute space, of course. And that Kant was in fact at pain to avoid Newton’s absolute space becomes even more evident if one takes into consideration the second Critique, where we find what is, in my view, one of the most profound and lucid explanations as to why Kant could not endorse Newton’s absolute space: namely, what I called the argument from Spinozism (Massimi b). If God is the cause of the existence of substance, and if we regard ourselves as substances (or things in themselves), then it turns out that God is also the cause, or the determining ground of our human actions. But if so, freedom would be jeopardized and fatalism of action would be rampant. This undesirable conclusion can only be avoided via a two-step maneuver that Kant outlines in a key passage of the second Critique (KpV, :–): first, Kant draws a distinction between the divine existence as the existence of a being in itself and our existence as things in appearance, and, second, he reallocates space and time from “essential determinations of the original being itself” to us and our outer sense. Hence, Kant concludes that the ideality of space and time is the best antidote against the Spinozistic danger lurking in the view that space (and time) are “essential determinations of God.” Interestingly, although no explicit mention is here made of Newton’s absolute space, the notion of God as the cause of existence of substance chimes with Newton’s General Scholium to the Principia where the Lord God Pantokrator “endures always and is present everywhere, and by existing always and everywhere he constitutes duration and space . . . He is omnipresent not only virtually but also substantially; for action requires substance” (Newton , ).
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By reallocating space from an essential determination of God to us and our outer sense, the Spinozistic danger of fatalism of action could be eschewed. Thus, there are three main themes of this short primer on Kant’s lifelong engagement with Newton’s absolute space that bear on the rest of my story in Sections . and .: Kant never endorsed the Newtonian metaphysical view of absolute space as an expression of God’s omnipresence in nature for what I take to be fundamentally moral grounds (i.e., to safeguard human freedom against what he feared was the risk of fatalism of action). . Kant never endorsed either the Newtonian mechanical view of absolute space as a physically empty space, in which physical bodies move or remain at rest. However, his enthusiasm for Newtonian speculative experimentalism (with the ether as a matter of light and fire) informed and shaped his matter theory as early as and – as I argue in Section . – laid the foundations for his mature critical argument for the physical impossibility of a vacuum in the Phenomenology chapter of the MAN. . As early as , Kant did seem to toy with the idea that there must be absolute space as a Grundbegriff to make sense of directions in space, indexicality, and chirality (which the Leibnizians, in his view, were incapable of explaining). A similar line of reasoning – inspired by indexical or, better, broadly perspectival considerations – reappears in the Phenomenology in MAN and (as I point out in Section .) ultimately underpins Kant’s view of absolute space as a necessary idea.
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These three themes can all be found in the Phenomenology chapter of MAN, where Kant gives probably the most lucid and mature argument for absolute space understood this time as a “necessary concept of reason” (MAN, :), and to which I am going to turn next.
. Newton’s Absolute Space in the Phenomenology Chapter of MAN The chapter on Phenomenology that concludes MAN opens with a definition of matter as “the moveable insofar as it, as such a thing, can be an object of experience” (MAN, :). The task of the chapter is to offer an explanation of how matter as the movable – whereby motion is “given only as appearance” like anything else that is represented through the senses – can become an object of experience. In other words, the
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Phenomenology is designed to show how matter (as a material movable thing) can be thought of as “determined with respect to the predicate of motion” (ibid.). The immediate problem with this task is that motion as an appearance is only a “change of relation in space,” as Kant clearly says at the outset. And three types of relations can be envisaged when something (let us call it A) moves with respect to something else (let us call it B): Either of the two correlates (be it A or B) can be said to be moved with respect to the other • Or one of the two correlates (A or B) “must be thought in experience as moved to the exclusion of the other” • Or, “third, both must be necessarily represented through reason as equally moved” (MAN, :). •
Appearances, however – Kant warns us – do not offer us any guidance in determining which of these three types of change of relations in space is at stake whenever we experience matter as the movable in space. Yet for matter as the movable in space to become an object of experience (and not just be an appearance), it “must be determined in one way or another by the predicate of motion” (MAN, :). Thus, the task of the Phenomenology is to clarify the modalities through which we come to experience matter as a material thing (i.e., an object) movable in space, or, in Kantian idiom, how we transform appearances into objects of experience. In the ensuing three main Propositions, Kant sets for himself the task of answering this question by proceeding in a familiar systematic fashion and mapping three main kinds of motion (rectilinear, circular, and “opposite and equal motion”) as, respectively, impossible, actual, and necessary. Unsurprisingly, this has long been read as the chapter that gives a determination of motion with respect to the category of modality (see Friedman , ch. , for a thorough analysis). In what follows, I will not attempt to explicate each and every one of these three main Propositions and my analysis will be very selectively confined to the role of absolute space in the Phenomenology.
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On this influential reading of the Phenomenology, modality does not in fact literally add any “further predicate (or conceptual ‘determination’)” (Friedman , ) over and above those already explicated in the Phoronomy (movable in space), Dynamics (filling of space), and Mechanics (mechanical moving forces). Instead, according to Friedman, modality “describes how the object in question is related to one (and eventually all) of our three intellectual cognitive faculties: understanding, the power of judgment, and reason” (ibid.).
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Absolute space indeed makes its debut right after Proposition , where Kant asserts that rectilinear motion of a matter “with respect to an empirical space, as opposed to the opposite motion of the space, is a merely possible predicate. The same when thought in no relation at all to a matter external to it, that is, as absolute motion, is impossible” (MAN, :). In the following Proof, Kant develops the first aforementioned point by arguing that in experience “there is no difference at all between the motion of the body in relative space, and the body being at rest in absolute space, together with an equal and opposite motion of the relative space” (MAN, :). The argument is somehow reminiscent of Newton’s argument in the Scholium at the outset of the Principia, where Newton discusses the difference between absolute and relative space and absolute and relative motion with reference to the motion of a body inside a sailing ship. For Newton, too, like Kant, saw rest in absolute space (or what he called “true rest”) as the “continuance of a body in the same part of that unmoving space in which the ship itself, along with its interior and all its contents, is moving” (Newton , ). And like Kant, Newton too had already stressed how difficult it is to tell apart relative space from absolute space just on the basis of appearances. But, with a surprising twist, Kant goes on to assert that “absolute space, in contrast with relative (empirical) space, is no object of experience, and in general is nothing” (MAN, :) and that absolute motion qua motion in absolute space is impossible (by contrast with rectilinear motion in relative empirical space that is merely possible). Hence, the first pressing question: (A) Why does Kant assert not just that absolute space is no object of experience, but that absolute space “in general is nothing”? He does not seem to have provided any argument for this conclusion until this point. To complicate matters, in the following Proposition , Kant asserts that “the circular motion of a matter, as distinct from the opposite motion of the space, is an actual predicate of this matter; by contrast, the opposite motion of a relative space, assumed instead of the motion of the body, is
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“For we define all places on the basis of the positions and distances of things from some body that we regard as immovable, and then we reckon all motions with respect to these places, insofar as we conceive of bodies as being changed in position with respect to them. Thus, instead of absolute places and motions we use relative ones, which is not inappropriate in ordinary human affairs, although in philosophy abstraction from the senses is required” (Newton , ).
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no actual motion of the latter, but, if taken to be such, is mere semblance” (MAN, :). In other words, while in the case of rectilinear motion (Proposition ) Kant grants the possibility that either of the two correlates (be it the body A or the relative space B) can be moved with respect to the other, when it comes to circular motion, he denies such possibility and swiftly concludes that a disjunction is here at stake whereby either the body is moved or the relative space is moved, and these two possibilities are at the mutual exclusion of one another. The problem is the following. Rectilinear and circular motions in Newtonian mechanics are governed by a few simple rules: α. Absolute space (and absolute time which I will not discuss here) provide the privileged inertial frame, where Newton’s law of inertia applies (all bodies are either at rest or move with uniform motion with respect to absolute space). β. In addition to absolute space, there might be other inertial reference frames with respect to which bodies may move with uniform (i.e., nonaccelerated) motion – for example, a ship, whereby the rest or motion of, say, a sailor inside can be regarded as relative to the reference frame of the ship itself, or as relative to the reference frame of the sea where the ship is moving, or as relative to absolute space where the ship and the sea are themselves located. γ. Newton’s laws apply to all inertial reference frames, and the relation between inertial frames is governed by Galilean transformations. In other words, it is possible to ask what the relative velocity of, say, a sailor walking inside the ship is either with respect to the motion of the ship itself as an inertial reference frame or with respect to absolute space where the ship and the sea are located.
What is prima facie odd about Kant’s Propositions and is that Kant seems to be defending something akin to tenets β and γ (in Proposition ) without holding α, that is, without assuming absolute space as a privileged reference frame, for he clearly says that absolute space is “nothing.” Even more interestingly in the Remark attached to the Proof of Proposition , Kant refers to Newton’s Scholium to the Definitions as follows: Newton’s Scholium to the Definitions . . . may be consulted on this subject, towards the end, where it becomes clear that the circular motion of two bodies around a common central point (and thus also the axial rotation of the earth) can still be known by experience, and thus without any empirically possible comparison with an external space; so that a motion . . . which is a change in external relations in space, can be empirically given, even
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though this space is not itself empirically given, and is no object of experience. This is a paradox that deserves to be solved. (MAN, :)
Kant seems to be appealing to Newton’s own arguments in the Scholium (where absolute space features prominently) even though Kant repeats once more that absolute space is not empirically given and is no object of experience (no tenet α). Unsurprisingly, Kant seems to be aware that the view he has just put forward is paradoxical by Newtonian lights. Thus, a second pressing question arises: (B) Why does Kant nod to Newton’s Scholium in Proposition where circular motion is regarded as an actual predicate of matter even if absolute space is said not to be empirically given? The paradox becomes glaring once Newton’s Scholium with its famous bucket experiment and two-globe thought experiment is recalled. Newton famously argued that circular motion in absolute space has empirical effects that we can all observe and be familiar with. These concern the observable effects of inertial forces acting on a body that is rotating and accelerating at a constant rate. Consider a bucket full of water and hanging from a long cord that gets twisted and released. The walls of the bucket might act as a reference frame for the motion of the material body (water), and one can easily observe that at the beginning (when the cord is released and the bucket begins to spin) the water and the bucket are at relative rest with respect to one another. But as soon as the bucket continues to spin and accelerates, the water starts receding from the center and rises up the sides of the bucket. This distinctive observable concave shape of the water’s surface is the product of inertial forces acting on it. Newton concluded that “this endeavour showed the true circular motion of the water to be continually increasing and finally becoming greatest when the water was relatively at rest in the vessel” (Newton , ). In other words, the true circular motion of the water (and of “each revolving body”) is not relative to the surrounding bodies (be it the bucket for the water or the “heavens of the fixed stars” for planets). But it is relative to absolute space. Indeed, this is the empirical evidence for absolute space, according to Newton. To buttress this conclusion, at the end of the Scholium, Newton presents a second example, this time a thought experiment concerning yet again inertial forces acting on a cord connecting two globes which are rotating with respect to one another. In this case, too, the tension on the
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cord is the sign of inertial forces acting on the globes and making them recede from the axis of rotation. By extrapolation from familiar experiences with cords, Newton concludes that “both the quantity and the direction of this circular motion could be found in any immense vacuum, where nothing external and sensible existed with which the balls could be compared” (ibid., ). In the nineteenth century these well-known passages of the Scholium became the focus of Ernst Mach’s criticism, which opened the path to Einstein’s special relativity and the eventual demise of Newton’s absolute space as a privileged reference frame. For our purposes here, let me just briefly canvass two possible replies on Kant’s behalf to some of these paradoxical aspects of his view of absolute space in MAN. The answers to questions (A) and (B) can both be found in the final General Remark to Phenomenology. I state them here in brief. In reply to question (B) as to why Kant seems to nod to Newton’s Scholium in Proposition even if absolute space is said not to be empirically given, the answer should be found, in my view, in the way Kant comes to rethink Newton’s absolute space. In Kant’s hands, this is no longer a privileged inertial reference frame, but instead an idea of reason fulfilling an important indexical and perspectival function in our ability to locate and refer to motion of material bodies. There is some continuity here with the relationalism of the pre-critical period, and most importantly with what I’d like to call the ‘perspectival’ analysis of ideas of reason that Kant had already given in the first Critique, Appendix to the Transcendental Dialectic, to which I return in more detail in Section .. In reply to question (A) as to why Kant asserts not just that absolute space is no object of experience, but that absolute space “in general is nothing,” I think the answer lies in Kant’s lifelong dynamical argument against the idea of an empty space (Newton’s vacuum), continuing on the well-established line of argument that began with Physical Monadology as discussed in Section .. Let me substantiate this point in the remainder of this section. In the General Remark at the end of the Phenomenology chapter, Kant clarifies the grounds on which he had formerly asserted that absolute space “in general is nothing.” These grounds are primarily of dynamical, rather than phoronomic or mechanical nature. For Kant distinguishes “various concepts of empty space” (MAN, :). In a phoronomic sense, empty space “which is also called absolute space, should not properly be called an empty space; for it is only the idea of a space, in which I abstract from all particular matter that makes it an object of experience.” In a dynamical sense, by contrast, empty space is “that which is not filled, that is, in which
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no other moveable resists the penetration of a moveable, and thus no repulsive force acts; it can either be empty space within the world (vacuum mundanum), or, if the latter is represented as bounded, empty space outside the world (vacuum extramundanum)” (ibid.). In a mechanical sense, empty space is in turn said to be the “emptiness accumulated within the cosmos to provide the heavenly bodies with free motion” (MAN, :). And it is clear from the rest of the discussion that Kant is primarily concerned with the dynamical sense of empty space when he asserts that absolute space is not an object of experience and “in general is nothing.” The reasons for the claim are first and foremost of dynamical rather than phoronomic or mechanical nature. Newton himself did not clearly distinguish among these different senses of empty space. Despite absolute space acting as a privileged reference frame, Newton characterized it in terms of an “immense vacuum” in the Scholium to the Definitions (, ); and again in the General Scholium at the end of the Principia, after mentioning Boyle’s experiments with the air pump, Newton concluded that “the case is the same for celestial spaces, which are above the atmosphere of the earth. All bodies must move very freely in these spaces, and therefore planets and comets must revolve continually in orbits given in kind and in position, according to the laws set forth above” (Newton , –). But Kant could not countenance the idea of a dynamically empty space for reasons he had already abundantly elucidated in his pre-critical theory of matter, inspired in turn by Newton’s Opticks and speculations about the ether (as we saw in Section .). At the very end of the General Remark to Phenomenology, unsurprisingly, Kant returns precisely to his matter theory and the ether to argue both against the possibility of a vacuum mundanum and a vacuum extramundanum. Against the former, Kant maintains that although it is logically possible to assume empty space in the pores of physical bodies, there might be dynamical reasons why this is physically impossible. Namely, it might be that the cohesion of matter, instead of being caused by an attractive force, is in fact the effect “of a compression by external matter (the aether) distributed everywhere in the universe, which is itself brought to this pressure only through a universal and original attraction, namely gravitation (a view that is supported by several reasons)” (MAN, :). Kant does not list or indicate the several reasons for this dynamical view. He does not need to. For this is a view that – as Section . briefly illustrated – he had already abundantly explored and developed since the time of Physical Monadology with his sui generis explanation of the specific densities of different bodies.
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As for the vacuum extramundanum, Kant again resorts to a dynamical refutation in terms of the attractive force acting on the ether “(which encloses all these bodies, and, driven by that force, conserves them in their density by compression)” (MAN, :). Kant claims in a rather speculative way that if there were an empty space outside the ether (that is assumed to be present among heavenly bodies), as the distance among bodies increases, the attractive force would decrease, and, with it, the density of the all-encompassing ether would also decrease indefinitely “but nowhere leave space completely empty” (ibid.). The ether that in Universal Natural History as a primordial fine substance had offered Kant a tool to explain planet and star formation now provides an argument for preventing heavenly bodies from receding from one another as the attractive force decreases at a distance. But while Kant warns his readers that these refutations of empty space are “entirely hypothetical” and speculative, he seems to be more sanguine in his answer to the aforementioned question (B), namely: How can circular motion be an actual predicate of matter if absolute space is not empirically given? In other words, how can Kant nod to Newton’s bucket and two-globe experiments when he has just denied the very absolute space that these Newtonian thought experiments were meant to provide empirical evidence for? As anticipated, the answer lies in the way Kant continues the journey started in the pre-critical period and the way he reassigned Newton’s absolute space to an idea of reason fulfilling an important indexical and perspectival role, as I explain in the next section.
. What Becomes of Absolute Space? Ideas of Reason as Foci Imaginarii Let us then return one more time to the striking claim that Kant makes right at the outset of the General Remark to Phenomenology, and with which I opened this chapter, namely: Absolute space is therefore necessary, not as a concept of an actual object, but rather as an idea, which is to serve as a rule for considering all motion therein merely as relative; and, all motion and rest must be reduced to absolute space, if the appearance thereof is to be transformed into a determinate concept of experience (which unites all appearances). (MAN, :)
How can Kant argue that absolute space is necessary, that motion and rest must be reduced to it for appearances to become objects of experiences, after he has just repeatedly claimed that absolute space qua empty space
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does not exist and is physically impossible (although logically noncontradictory)? Recall this is the paradox that Kant set for himself to solve. And the solution, I suggest, has to be found in the way Kant reinterprets Newton’s arguments for absolute space. In Section ., I briefly mentioned how difficult it is to make sense of the Directions in Space arguments from indexicality and incongruous counterparts qua intended arguments for absolute space. Eighteen years later, the Critical Kant develops a much more persuasive argument from indexicality and perspectivity that in my view has the power to explain at once why absolute space becomes an idea of reason (and a necessary one too), and, relatedly, why absolute space is said to be crucial in transforming appearances into objects of experience, when in fact very little is left of Newton’s original notion. I proceed as follows. First, I explicate what is at stake in Kant’s claim that “Absolute space is therefore necessary, not as a concept of an actual object, but rather as an idea, which is to serve as a rule for considering all motion therein merely as relative” by looking at the regulative role of ideas of reason in the Appendix to the Transcendental Dialectic of the first Critique. Second, equipped with a perspectivalist reading of ideas of reason, I return one more time to Kant’s Phenomenology and try to shed light on the paradox of absolute space. In the Appendix to the Transcendental Dialectic in KrV, Kant presents the faculty of reason in its hypothetical use as being directed “at the systematic unity of the understanding’s cognitions, which is the touchstone of truth for its rules” (A/B). A little earlier in the Appendix, he had asserted that ideas of reason have an excellent and indispensably necessary regulative use, namely that of directing the understanding to a certain goal respecting which the lines of direction of all its rules converge at one point, which, although it is only an idea (focus imaginarius) – i.e. a point from which the concepts of the understanding do not really proceed, since it lies entirely outside the bounds of possible experience – nonetheless still serves to obtain for these concepts the greatest unity alongside the greatest extension. Now of course, it is from this that there arises the deception, as if these lines of direction were shot out from an object lying outside the field of possible empirical cognition (just as objects are seen behind the surface of a mirror); yet this illusion (which can be prevented from deceiving) is nevertheless
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In what follows, I draw on Massimi (d) and () for a perspectivalist analysis of the Appendix and the role of ideas of reason. For an excellent discussion of the transcendental illusion in the context of the regulative role of ideas of reason, see Spagnesi ().
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indispensably necessary if besides the objects before our eyes we want to see those that lie far in the background, i.e., when in our case, the understanding wants to go beyond every given experience . . . and hence wants to take the measure of its greatest possible and uttermost extension. (KrV, A/B)
Why are ideas of reason said to be illusory? And how can illusory vanishing points nonetheless play an “indispensably necessary regulative use”? Kant’s so-called doctrine of the transcendental illusion continues to remain one of the most puzzling and intriguing aspects of Kant’s theoretical philosophy, in my view. Especially puzzling are some of the examples that Kant gives in the Appendix, where in addition to the three official transcendental ideas of God, soul, and world, he mentions the ideas of “pure earth, pure water, pure air” and of “fundamental power.” A first possible way of thinking about the indispensably necessary role of such foci imaginarii is that they play a role analogous to Plato’s ideas as “archetypes of things themselves” (KrV, B). Yet by contrast with Platonic ideas, Kant’s ideas of reason are illusory in creating the deception of such archetypes (nowhere to be found either within the bounds of possible experience or beyond, given the limits of our knowledge). By distinguishing between appearances and things in themselves, and by relegating the latter outside the boundaries of human knowledge, Kant, under this first reading, would find himself, however, in the difficult position of having to explain why ideas of reason – qua illusory archetypes of things in themselves – are nonetheless “indispensably necessary.” What good are ideas of reason so understood for theoretical knowledge, and in particular for scientific knowledge? Scholars adopting this interpretive stance have emphasized the direct link between the archetypical reading of ideas as illusory objects and the regulative role of reason in seeking after systematic unity as necessary for a correct use of the understanding. Kant’s quest for systematicity has often been read as the quest for ideal unconditioned ‘objects’ that would inevitably take the understanding beyond its proper domain and remit. This interpretive stance enjoins us to think that the illusion of thinking about an ideal ground, substrate, or unconditioned behind the appearances is necessary in motivating the operations of the understanding, albeit in a merely regulative (non-constitutive) way. Attractive and well established as it might be, this interpretive stance faces a problem. The necessity that attaches to the regulative use of
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See Grier () and (, ch. ) as an example of this interpretive stance.
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transcendental ideas remains to be explained. For it is not the kind of modal necessity originating from the understanding. Modal necessity features prominently in the Postulates of Empirical Thinking in General, where it captures the way in which the Postulates operate with respect to the Analogies to make experience of nature possible for us (KrV, A/ B). Thus, we are left with the open and pressing task of explaining how and why exactly – under this first interpretive reading – ideas of reason would in fact be necessary in motivating the operations of the understanding. In what sense is seeking after an ideal ground – we might want to call it an archetype or unconditioned – (non-modally) necessary for a correct empirical use of the understanding? What is clear is that ideas of reason are not necessary in the sense of contributing (in a constitutive way) to the activity of the understanding in delivering objects of possible experience. At best, they would seem desirable (but not really indispensable) in giving legitimacy to otherwise subjective rational maxims, which we might find useful in empirical investigation (e.g., think of nature as if it were systematic; think of chemical reactions as if there were pure air; and so on). On a variation of this first interpretive reading, ideas qua foci imaginarii are necessary because reason – in its hypothetical use – is said to provide us with the universals (e.g., “pure water,” “fundamental power”) under which particulars can be subsumed. These universals are required because without them the understanding could not even deliver true universal generalizations in science. The faculty of understanding, with its a priori categories and principles, can at best establish that for each event there is some cause; but not that causes of type X are followed by effects of type Y. For inductive generalizations of this nature to be possible, reason in its hypothetical/regulative use has to complement the understanding by providing ideas of reason. This alternative interpretive stance goes some way toward explaining the necessity that attaches to the use of regulative ideas of reason. However, it has its own difficulties. For example, it proves difficult to square this interpretation with the three official transcendental ideas (God, soul, and world) in the second part of the Appendix. How are the ideas of God, soul, and world related to our ability to draw universally valid inductive inferences?
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Universals should be such that they are neither given a priori (and hence empirically unrevisable) nor empirically given (otherwise, they would not be able to fulfill their taxonomic task of providing an ideal ground for subsuming particulars). For this reading of the ideas of reason, see Allison (, –) and Buchdahl (a, ).
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Elsewhere (Massimi d, ), I have suggested a different interpretive reading of the transcendental illusion. Core to it is the following move. Do not think of the regulative role of ideas primarily in terms of their acting qua proxies for noumenal objects (the unconditioned). Think of their regulative function instead first and foremost in the etymological Latin sense of regula, that is, as “rules” for a correct empirical use of the very same faculty of understanding. Ideas in their regulative role should not mistakenly be hypostatized into placeholders for an unconditioned, which although epistemically unknowable, act nonetheless as an ideal ground for the unity that reason seeks out. Kant chose a peculiar terminology that in my view speaks against any temptation to reify the regulative role of ideas of reason into their being proxies for noumenal objects. For he enjoins us to posit “an idea only as a unique standpoint from which alone one can extend the unity that is so essential to reason and so salutary to the understanding; in a word, this transcendental thing is merely the schema of that regulative principle through which reason, as far as it can extends systematic unity over all experience” (KrV, A /B ; my emphases). Kant uses the language of “standpoint,” “focus imaginarius” (vanishing point), and “rules” to explicate the regulative role of ideas of reason without surreptitiously assuming that ideas accomplish such a role by standing for noumenal objects. Think instead of ideas as ‘imaginary standpoints,’ as I’d like to call them, whose role is to define the abstract space of reason within which the understanding’s cognitions are located. In perspectival drawing in art, vanishing points are the necessary points where the lines converge so as to give an impression of depth to the represented scene. Mastering perspectival techniques in the art via instruments such as the perspectograph allowed painters such as Piero della Francesca and Albrecht Du¨rer to transform the represented scene into a “window on reality” (Panofsky , –). Analogously, ideas as vanishing points transform an aggregate of cognitions by the understanding into a systematized and unified experience. We can think of the abstract space of reason as a ‘perspectival space of reason’ generated by ideas qua foci imaginarii in analogy with perspectival drawing in art. Out of metaphor, ideas of reason can be thought of as acting qua a “shared conversational scoreboard” (to echo David Lewis’ [] apt expression): they allow individual judgments to reach unanimity and universality, which for Kant are the hallmark of bona fide knowledge against doxastic or bogus knowledge. That is why the indispensably necessary role of ideas of reason cannot be restricted to inductive
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inferences, in my view. For it extends to the very possibility of knowledge in general, that is, the possibility of a correct empirical use of the faculty of understanding (recall Kant’s wording in the Appendix where the hypothetical use of reason is defined as the “touchstone of truth” for the systematic unity of the understanding’s cognitions; see KrV, A/B). Kant seems to be saying that the understanding can only guarantee the validity of the judgments it subjectively produced (hence a lingering threat of transcendental solipsism that some scholars have seen at work in Kant’s critical project). Therefore, a correct empirical use of the understanding requires ideas of reasons. But in what sense does reason offer a ‘touchstone of truth’ for the understanding’s cognitions? I suggest that we read this complex passage of the transcendental illusion along the following lines: reason provides a touchstone of truth because it offers ideas as imaginary standpoints necessary to confer on the understanding’s cognitions the unanimity and universality that they would otherwise lack. Our ability to veridically judge that things are a certain way demands the hypothetical use of reason to supplement with ideas as rules the workings of the faculty of sensibility and understanding. In support of this reading, consider what Kant says in the Canon of Pure Reason in KrV, where he seems to suggest that a condition for true judgments is the ability to communicate them so that inter-conversational agreement can be reached: Truth, however, rests upon agreement with the object, with regard to which, consequently, the judgments of every understanding must agree (consentientia uni tertio, consentiunt inter se). The touchstone of whether taking something to be true is conviction or mere persuasion is therefore, externally, the possibility of communicating it and finding it to be valid for the reason of every human being to take it to be true; for in that case there is at least a presumption that the ground of the agreement of all judgments [der Grund der Einstimmung aller Urtheile], regardless of the difference among the subjects, rests on the common ground, namely the object, with which they therefore all agree and through which the truth of the judgment is proved. (KrV, A–/B–)
Without reason and ideas as rules for inter-conversational agreement, I contend, there cannot be any guarantee that my judging that things are a certain way matches with other people’s judgments that things are indeed that way. To secure inter-conversational agreement with the object, ideas qua imaginary standpoints are required: they offer a ‘shared
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See Moore ().
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inter-conversational scoreboard,’ where it is possible to reach agreement and establish the universal and unanimous validity of true judgments. True judgments are then not effected by the faculty of understanding alone; they are ultimately effected by the faculty of reason in its indispensably necessary regulative role. For it falls within reason’s remit to test how each individual judgment would fare on the inter-conversational scoreboard (consentientia uni tertio, consentiunt inter se). This might sound surprising and baffling. Has not Kant, after all, abundantly explained how we all share a priori forms of intuitions, categories, and principles of the understanding so as to make experience possible? What do ideas of reason add to this story, given that they do not have any constitutive role? To clarify this point, let us return to Kant’s examples of “pure earth, pure water, pure air” in the Appendix. Suppose I have encountered a number of phenomena concerning water: rain from the clouds, filling oceans, dew dropping from a blade of grass, and so on. On each and every occasion, I have encountered these phenomena by applying categories and principles of the understanding to the spatiotemporal manifold. For example, the principle of causality might be at play in me having the phenomenal experience of rain filling up a reservoir over time; and substance might be responsible for my ability to have the phenomenal experience of the ocean being the same ocean (despite regular cycles of evaporation and raining). The faculty of sensibility and the faculty of understanding, in their constitutive function, are all that is needed for the ocean and the rain filling up the reservoir to become objects of experience for me. Yet for them to be objects of experience not just for myself, but also for my friends Martha, Paula, and every other human being – namely, for my individual judgments about these objects of experience to have universality and unanimity – something else is needed: ideas of reason. For if I did not have the idea of “pure water” as an (entirely regulative) tertium, there would be no guarantee that my experience of the ocean as the same ocean would match Martha’s experience and Paula’s experience and any other human being’s experience of the ocean as the same ocean. If we did not share a faculty of reason with ideas (not as proxy for metaphysical objects but as a shared inter-conversational scoreboard), each of us could claim to know the ocean as an object of experience without having in fact any way of comparing that my judgment about this object of experience agrees indeed with Martha’s and Paula’s and everyone’s else too. Sharing a priori forms of intuition, categories, and principles of the understanding – in and of itself – cannot guarantee that our claims of knowledge match those of our fellow human beings. For their role is to
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make possible individual objects of experience, not to judge whether individual objects of experience (afforded by different individuals) do indeed agree with one another and have the universality and unanimity required for us as humankind to make true judgments that things are indeed a certain way. Without ideas of reason, our individual ability to come to know ‘objects of experience’ – via our individual faculties of sensibility and understanding alone – would be akin to Wittgenstein’s famous scenario (, ) where everyone has a box with something in it which we call a ‘beetle,’ and everyone claims to know what a beetle is only by looking at their own beetle and without ever being able to look at anyone’s else beetle. We cannot look into the inner workings of other people’s faculties of sensibility and understanding to come to the conclusion that indeed we do know what a beetle is, that we do know that this is the same ocean, or that planetary motion is some kind of circular motion. For us humankind to reach such universal and unanimous judgments, ideas of reasons are required as imaginary standpoints, external to all of us so that “consentientia uni tertio, consentiunt inter se.” If my reading of the regulative role of ideas of reason as imaginary standpoints is on the right path, we can begin to catch a glimpse as to why in MAN Kant reintroduces absolute space as an idea. More to the point, this reading enables us to reply to question (B) as we left it in Section ., that is, why Kant seems to nod to Newton’s Scholium in Proposition even if absolute space is said not to be empirically given (for all the dynamical reasons Kant had against the vacuum, as I already explained in Section .). For relative motion as an appearance to become experience, that is, for my individual experience of motion in relative space to become knowledge that we can all agree upon and share, there has to be a ‘concept of absolute space’ as a ‘mere idea’ (MAN, :). Absolute space “as an idea, which is to serve as a rule for considering all motion therein as merely relative” (ibid.) is necessary, I suggest, like any other idea of reason is indispensably necessary. Namely, not because it is a proxy for a noumenal entity (i.e., Newton’s absolute space) as some unconditioned ground required for the regulative role of reason. But instead because it acts itself like a focus imaginarius to create a perspectival space so to speak, within which individual judgments of motions of material bodies can all be located. Absolute space is itself only an imaginary standpoint, nonetheless a necessary one to secure the unanimity and universality of our individual judgments about motions of material bodies.
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For when we leave what Friedman aptly calls “our parochial perspective here on earth” (, ) to progress “to the center of gravity of the solar system, from there to the center of gravity of the Milky Way galaxy, from there to the center of gravity of a rotating system of such galaxies, and so on ad infinitum,” we need a way of securing that each individual judgment about these different rotational motions of heavenly bodies (my judgment, Martha’s, Paula’s, and that of anyone else) agrees with a third party (i.e., the Kantian absolute space as a mere idea) so that they can agree with one another (consentientia uni tertio, consentiunt inter se). Thus, Friedman and I agree on the regulative function of absolute space as an idea. But while Friedman sees absolute space as performing the task of some privileged inertial reference frame (even if a purely ideal one for Kant), I see its necessary regulative function as linked to our ability to veridically judge that things are a certain way. I do not see Kant’s absolute space in MAN as a proxy for an ideal reference frame to which all the relative reference frames (e.g., our solar system, the Milky Way, and so on) can ultimately be reduced to. For I do not see ideas of reasons explicating their regulative role by acting as placeholders for noumenal entities in general. Under the perspectivalist reading I have suggested, the idea of absolute space acts instead as a “rule” that guides my individual judgment about, say, planetary motions or the Milky Way’s motion qua circular motions and make it possible for them to agree with Martha’s judgment and Paula’s judgment and everyone else’s judgment about planetary motions or the Milky Way’s motions. That is why, in my view, Kant can rightly nod to Newton’s thought experiments while also denying that absolute space is empirically given to us. Absolute space is never empirically given to us, for it is only an idea of reason for Kant. Yet the idea is at work (and necessarily so) behind our individual cognitions of circular motions and offers a shared scoreboard, against which we can confer on our individual judgments the universality and unanimity that they would otherwise lack. That is why in my view Kant can rightly contend that “motion and rest must be reduced to absolute space, if the appearance thereof is to be transformed into a determinate concept of experience (which unites all appearances)” (MAN, :). For matter as the movable “insofar as it, as such a thing” to become an object of experience – despite motion being only the appearance of a change of relation in space – the idea of absolute space is necessary, again “not as a concept of an actual object” (MAN, :) but as a “necessary concept of reason, and thus nothing more than a mere idea” (:).
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Absolute space allows us to think of each motion and empirical relative space where it occurs as contained in a space of greater extent, and take the latter to be at rest . . . and so on to infinity, without ever arriving by experience at an immoveable (immaterial) space, with respect to which either motion or rest might absolutely be attributed to any matter . . . Rather, one must think a space in which the latter [relative space] can itself be thought as moved, but which depends for its determination on no further empirical space, and thus is not conditioned in turn – that is an absolute space to which all relative motions can be referred . . . so that in it all motion of material things may count as merely relative with respect to one another, as alternatively mutual, but none as absolute motion or rest. (MAN, :)
To conclude, absolute space is a mere idea. Yet it is necessary to secure that individual empirical representations of space (in which material bodies move by changing relations) do not remain mere appearances. Absolute space as a necessary idea can transform appearances into objects of experience by unifying (or uniting) all appearances with respect to a perspectival space of reason where ideas themselves are nothing over and above vanishing points. Although the determination of motion with respect to the categories of quality, quantity, relation, and modality is a constitutive procedure delivered by the faculty of understanding, ultimately such constitutive procedure needs be supplemented by a regulative procedure that can secure the unanimity and universality of our synthetic a priori judgments on the rotational motions of heavenly bodies as we move from the earth, to the center of the solar system, to the Milky Way, and so on. Such regulative procedure requires absolute space, stripped of its Newtonian metaphysical and mechanical attributes and transformed into an idea. Thus, there is for Kant no absolute space as either a determination of God’s presence or as a self-subsisting entity in nature. And yet its concept is needed for directing the understanding toward a vanishing point that although “lies entirely outside the bounds of possible experience – nonetheless still serves to obtain for these concepts the greatest unity alongside the greatest extension” (KrV, A/B).
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I thank the editor, Michael Bennett McNulty, for inviting me to contribute to this volume. Earlier versions of this chapter were presented at Groningen, New York University, and the Kant Congress in Oslo. I thank the audiences for incisive and constructive comments. This essay is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon research and innovation programme (grant agreement European Consolidator Grant H-ERC--CoG Perspectival Realism: Science, Knowledge, and Truth from a Human Vantage Point).
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Proper Natural Science and Its Role in the Critical System Michael Friedman
. Introduction The concept of proper natural science is introduced in the Preface to the Metaphysical Foundations of Natural Science (), and it has attracted increasing interest in recent scholarship on Kant’s philosophy of physical science. I shall argue here, however, that it also plays a central role in the critical system as a whole as it develops from the first edition of the Critique of Pure Reason () to the Critique of the Power of Judgement () and beyond. In particular, there is an ongoing entanglement between the physical science of Kant’s day (including, especially, the science of physical chemistry) that eventually extends to all of the central aspects of his critical system – from the necessity of particular empirical causal laws to the relationship between mechanism and teleology and even to Kant’s most developed treatment of the relationship between theoretical and practical reason at the end of his critical period.
. Proper Natural Science and Causal Necessity in the Metaphysical Foundations It is clear and uncontroversial that the Metaphysical Foundations articulates principles of what Kant calls pure natural science, which directly instantiate, via what Kant calls the empirical concept of matter, the pure (transcendental) categories and principles of the understanding in the Critique of Pure Reason. Among such principles of pure natural science, in particular, are what Kant calls the three Laws of Mechanics: the conservation of the total quantity of matter, inertia, and action equals reaction. These laws are especially important, of course, because they directly instantiate the Analogies of Experience in the first Critique.
Kant states these laws in the third, or Mechanics, chapter as instantiations of the relational categories and principles. It is noteworthy that in the Introduction to the second edition of the Critique Kant
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What Kant calls pure natural science thus consists of synthetic a priori propositions, such as, paradigmatically, his three Laws of Mechanics. In the Preface to the Metaphysical Foundations, however, Kant also develops a distinction between pure and proper natural science, which, in my view, is of fundamental importance: Since the word nature already carries with it the concept of laws, and the latter carries with it the concept of the necessity of all determinations of a thing belonging to its existence, one easily sees why natural science must derive the legitimacy of this title only from its pure part – namely, that which contains the a priori principles of all other natural explanations – and why only in virtue of this pure part is natural science to be proper science. Likewise, [one sees] that, in accordance with demands of reason, every doctrine of nature must finally lead to [proper] natural science and conclude there, because this necessity of laws is inseparably attached to the concept of nature, and therefore makes claim to be thoroughly comprehended [durchaus eingesehen]. Hence, the most complete explanation of given appearances from chemical principles still always leaves behind a certain dissatisfaction, because one can adduce no a priori grounds for such principles, which, as contingent laws, have been learned merely from experience. (MAN, :–)
Pure natural science consists of propositions that are strictly synthetic a priori. Proper natural science, however, is the broader concept, containing pure natural science as a sub-part (as its “pure part”). Nevertheless, proper natural science, as such, still consists of necessary laws – which necessity is “inseparably attached to the concept of nature.” Proper natural science includes particular empirical laws, which, precisely in virtue of the pure part of natural science (consisting of synthetic a priori laws), still count as necessary (“only in virtue of this pure part is natural science to be proper science”). I have argued that, just as Kant’s three Laws of Mechanics are paradigmatic of the synthetic a priori propositions of pure natural science, Newton’s law of universal gravitation is paradigmatic, for Kant, of a particular empirical law of nature that still counts as necessary in virtue of precisely its relationship to the three Laws of Mechanics. In particular,
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presents these same laws as undoubted examples of “pure natural science” (KrV, B–n): “[O]ne need merely consider the various propositions that come forth at the outset of proper (empirical) physics, such as those of the permanence of the same quantity of matter, of inertia, of the equality of action and reaction, etc., and one will be quickly convinced that they constitute a physica pura (or rationalis), which well deserves to be separately established, as a science of its own, in its whole extent, whether narrow or wide.” Kant makes it clear that he takes the law of universal gravitation to be empirical in the second, or Dynamics, chapter of the Metaphysical Foundations: “[N]o law of either attractive or repulsive force may be risked on a priori conjectures. Rather, everything, even universal attraction as the cause of weight [Schwere], must be inferred, together with its laws, from data of experience” (MAN, :).
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the law of universal gravitation is inferred from Kepler’s so far merely inductive “rules” of planetary motion, but, for Kant, it is still determined from these rules by geometry and the three Laws of Mechanics. The inference in question is thus what Newton himself calls a “deduction from the phenomena” (via geometry and the Newtonian Laws of Motion). The law of gravitation, for both Newton and Kant, is not a mere hypothesis for explaining the Keplerian phenomena, and, in this way, it now counts for both as more than merely inductive. For Kant, in particular, it now counts as a necessary and universally valid genuine empirical law of nature. I have also argued that the relevant kind of necessity is precisely that characterized in Kant’s discussion in the Postulates of Empirical Thought in general of the category of necessity (KrV, A/B–). Kant is here describing a three-stage procedure in which we begin with the formal a priori conditions of the possibility of experience in general, obtain perceptions of actual events and processes given in sensation, and then assemble these perceived events and processes together in a unified experience via necessary connections (notwendige Verknu¨pfungen) using the general conditions of the possibility of experience with which we began. In his detailed discussion of the third Postulate (KrV, A–/ B–) Kant makes it clear that he is referring, more specifically, to causal necessity and to particular (empirical) causal laws. He mentions two essentially different types of laws: “general laws of experience” (KrV, A/ B), such as the Analogies of Experience, and “empirical laws of causality” (KrV, A/B) – that is, particular causal laws relating particular kinds of events. Indeed, the very concept of causality with which Kant is operating demands such a (universal) empirical law in each case: “The schema of the cause and of the causality of a thing in general is the real upon which, whenever it is posited, something else always follows” (KrV, A/B; emphasis added). Kant is suggesting, therefore, that the “material” necessity in question is precisely that of the causal connections among diverse events whose (objective) necessity Hume had denied.
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I have made this argument in a number of places. I am here following the discussion in Friedman (), which also contains references to earlier such discussions. This procedure for transforming perceptions into experience is reflected in Kant’s reformulation of the general principle governing all three Analogies of Experience in the second () edition: “Experience is possible only through the representation of a necessary connection of perceptions” (KrV, B) I discuss the relationship between these passages from the first Critique and Kant’s official “answer to Hume” in the Prolegomena () in Friedman ().
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We have seen that, for Kant, the three Laws of Mechanics in the Metaphysical Foundations are instantiations of the three Analogies of Experience – the very principles that are constitutive, for Kant, of (causal) necessity. In order to see that Newton’s law of universal gravitation is paradigmatic, for Kant, of a necessary but still empirical (causal) law of nature, we need briefly to consider the fourth or Phenomenology chapter of the Metaphysical Foundations – which, in turn, corresponds to the Postulates of Empirical Thought in general. The role of the Phenomenology, in this context, is to explain how attributions of motion and rest to bodies can be successively determined under the modal categories of possibility, actuality, and necessity – thereby resulting in a distinction between “true” and merely “apparent” motion. Kant, on my reading, here develops a reconstruction of Newton’s deduction from phenomena of the law of universal gravitation in Book of the Principia. We begin, from the observable (Newtonian) “Phenomena” described by Kepler’s rules: the merely relative motions of the satellites in the solar system with respect to their corresponding primary bodies (the Moon relative to the Earth, the moons of Jupiter and Saturn relative to the planets in question, and the planets relative to the Sun). Since we have not yet introduced a distinction between true and apparent motions, however, the corresponding merely relative motions thus count (so far) as merely possible. At the next stage we use the law of inertia (Kant’s second Law of Mechanics) to derive inverse-square (centripetal) accelerations of its satellites directed toward each (corresponding) primary body in the solar system (the Moon toward the Earth, the moons of Jupiter and Saturn toward their primary bodies, and so on): we now have true (as opposed to merely apparent) orbital rotations in each case, which thus now count as actual. At the third stage, finally, we find both that the accelerations in question are directly proportional to the quantities of matter of the corresponding primary bodies and that such accelerations are also everywhere mutual between any two massive bodies. In accordance with the equality of action and reaction (Kant’s third Law of Mechanics), therefore, we now have what Kant calls necessary equal and opposite motions, where the accelerations of any two gravitationally interacting bodies are oppositely directed and in inverse proportion to their masses. Finally, since each of these mutual accelerations has just been determined as necessary in accordance with the Postulates of Empirical
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I here, once again, follow Friedman (); I develop this reading of the Phenomenology in fullest detail in Friedman ().
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Thought, the law of universal gravitation has itself been determined as (conditionally) necessary in the same sense – relative to (conditional upon) the initial Keplerian Phenomena from which we began. The law of universal gravitation, in other words, is determined in its connection with the actual in accordance with the general conditions of the possibility of experience: the three Analogies of Experience as further specified by Kant’s three Laws of Mechanics. The point is that, whereas Kepler’s rules are (so far) merely inductive generalizations and, as such, are not yet grounded in a priori laws of the understanding, the law of universal gravitation is grounded by applying such a priori laws to the Keplerian Phenomena. And, in precisely this way, the law of universal gravitation thereby acquires a more than inductive material necessity in the specific sense of the Third Postulate. Kant emphasizes in the Preface to the Metaphysical Foundations that all proper natural science is grounded in pure natural science and that the latter consists in a combination of both metaphysics and mathematics (MAN, :). The three Laws of Mechanics are synthetic a priori principles, because they result from instantiating the three Analogies of Experience by what Kant calls the empirical concept of matter (or body) – the concept of the movable in space. Such metaphysical principles belong to what Kant calls special as opposed to general metaphysics, that is, the transcendental philosophy of the first Critique (MAN, :–). And, as such, they necessarily involve mathematical synthetic a priori principles as well (MAN, :). Thus, for example, the law of conservation of the total quantity of matter involves a precise quantitative instantiation of the category of substance; the law of inertia involves precise quantitative instantiations of the category of causality and the predicable (derivative category) of force; and the law of the equality of action and reaction involves a precise quantitative instantiation of the category of community or interaction (Wechselwirkung). It is just this feature of special metaphysics, which, in the present case, enables a fruitful collaboration between the special metaphysics of corporeal nature and what Kant calls the “mathematical doctrine of motion [mathematische Bewegungslehre]” – where the
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The precise sense in which this concept is empirical, for Kant, raises a number of difficult questions, which cannot be pursued here. Friedman () again contains my most detailed discussion of these questions. The important point here, however, is that the dependence of pure natural science on an empirical concept, for Kant, does not compromise its a priori status – so long as it depends on nothing going beyond the content of this concept. By contrast, an empirical law of proper natural science (like the law of universal gravitation) depends on given empirical facts going beyond such conceptual content (in this case the initial Keplerian Phenomena).
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latter is developed, paradigmatically, in Newton’s Mathematical Principles of Natural Philosophy. Kant, in his treatise, is providing the required metaphysical principles. An especially significant case of the relationship between the principles of general and special metaphysics is that between the principle of causality in the first Critique and the law of inertia in the Metaphysical Foundations. Kant states this law in the Mechanics as the third proposition of this chapter and appends the following “proof”: Second Law of Mechanics. Every alteration of matter has an external cause. (Every body persists in its state of rest or motion, in the same direction, and with the same speed, if it is not compelled by an external cause to leave this state.) Proof. (From general metaphysics we take as basis the proposition that every alteration has a cause, and here it is only to be proved of matter that its alteration must always have an external cause.) Matter as mere object of the outer senses, has no other determinations except those of external relations in space, and therefore undergoes no alteration except by motion. With respect to the latter, as alteration of one motion into another, or of a motion into rest, or conversely, a cause must be found (by the principle of [general] metaphysics). But this cannot be internal, for matter has no essentially internal determinations or grounds of determination. Hence every alteration in a matter is based on an external cause (that is, a body persists, etc.). (MAN, :)
Thus it is clear, in the first place, that this law of special metaphysics results from instantiating the more general principle of causality formulated in general metaphysics (the transcendental philosophy of the first Critique) in a way that restricts the possible causes to those acting externally (rather than internally). And, in the second place, the parenthetical insertions at the end of both the statement of the proposition and its proof make it clear that Kant takes this law to be essentially equivalent to the law of inertia formulated as the first of Newton’s three Laws of Motion in the Principia: “Every body perseveres in its state of being at rest or moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed” (Newton , ). Kant’s law of inertia, like
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In the penultimate paragraph of the Preface Kant expresses the hope that “mathematical natural scientists should find it not unimportant to treat the metaphysical part, which they cannot leave out in any case, as a special fundamental part in their general physics, and to bring it into union with the mathematical doctrine of motion” (MAN, :). The following final paragraph then explicitly cites Newton and the Principia.
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Newton’s, presupposes the concept of Newtonian impressed forces – or, in Kant’s more general terminology, externally acting causes. Kant calls such externally acting forces – causal actions exerted by material (lifeless) substances in accordance with the law of inertia – “moving forces [bewegende Kräfte].” And, when discussing the a priori concepts of substance, causality, action, and force in the Second Analogy (in both editions of the Critique), Kant appeals to such forces as the main example of what provides the particular content, rather than the mere universal form, of an alteration of state: “For this acquaintance with actual forces is required, which can only be given empirically, e.g., the moving forces, or, what is the same, certain successive appearances (as motions), which indicate such forces” (KrV, A/B). Moreover, as Kant explains in a footnote to the main paragraph, the law of inertia is therefore necessarily involved: “One should well note that I do not speak of the alteration of certain relations in general, but rather of alteration of state. Therefore, if a body moves uniformly it does not alter its state (of motion) at all, but it certainly does if its motion increases or decreases [i.e., accelerates or decelerates]” (KrV, A/Bn). Mathematically described moving forces, appropriately determined from given phenomena by mathematics and the Newtonian Laws of Motion (or, for Kant, his three Laws of Mechanics), constitute proper natural science. And the law of universal gravitation, in particular, is thus paradigmatic of such science. An important question, however, is the scope of this kind of science and whether, more specifically, it extends beyond the Newtonian mathematical science of motion. In the Preface to the Metaphysical Foundations Kant is famously doubtful about chemistry: So long, therefore, as there is still for chemical actions of matters on one another no concept to be discovered that can be constructed, that is, no law of the approach or withdrawal of the parts of matter can be specified according to which, perhaps in proportion to their density or the like, their motions and all the consequences thereof can be made intuitive and presented a priori in space (a demand that will only with great difficulty ever be fulfilled), then chemistry can be nothing more than a systematic art
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From this it follows that we not only need to rely on experience to provide real possibility for any supposed actual forces but also need to rely on experience to give determinate particular content to the general concept of an alteration: i.e., to know what counts as an alteration of state in any particular case. Nevertheless, the form of an alteration in general, Kant says, can still be known a priori (KrV, A/B). This a priori general form involves succession in time in accordance with a universal and necessary rule (whatever this rule may be), and such succession, moreover, must unfold continuously in time: see the entire discussion surrounding the paragraph under consideration (KrV, A–/B–).
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or experimental doctrine, but never a proper science, because its principles are merely empirical, and allow of no a priori presentation in intuition. Consequently, they do not in the least make the principles of chemical appearances conceivable with respect to their possibility, for they are not receptive to the application of mathematics. (MAN, :)
Kant here finds it difficult to grant the status of proper natural science to anything not very close to the Newtonian mathematical science of motion – and not very close, in particular, to the law of universal gravitation. Nevertheless, in the climactic § of the Transcendental Deduction in the second edition of the Critique Kant illustrates the relationship between mere perception and full-blooded experience (note above) by the examples of applying the (mathematical) categories of quantity to the perception of a house and applying the (dynamical) category of causality to the transition from a state of fluidity to one of solidity in the freezing of water (KrV, B–). The category of causality in the second example is applied to a clearly chemical phenomenon. And, since Kant’s first example applies the categories of quantity to a perception, it seems that the second example involves (or presupposes) a quantitative characterization of this chemical phenomenon as well. It is obvious that the state transition in question involves the freezing point of water (! Celsius). But it is also clear, I believe, that the caloric theory of heat and the states of aggregation of various kinds of substances are also salient here. By this time () Kant had acquired a good understanding of Joseph Black’s conception of latent heat, where, in the example, a quantifiable measure of heat (caloric) is lost in the freezing process with no accompanying change in the temperature of the water. Here, in particular, we have both a quantitatively conserved substance (caloric) and a quantitative treatment of its causal action in an important change of state (solidification). This might have given Kant reason to take the prospects for a proper science of chemistry more
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That Kant takes the law of gravitation to be paradigmatic of proper natural science is clear from the first quotation in this chapter, which asserts: “every doctrine of nature must finally lead to [proper] natural science and conclude there, because this necessity of laws is inseparably attached to the concept of nature, and therefore makes claim to be thoroughly comprehended [durchaus eingesehen]” (MAN, :; compare note above). That chemical “laws” cannot be so thoroughly comprehended is already clear from the final sentence of the paragraph (:–). Note: “Einsicht ” (insight) is simply the noun corresponding to the verb “einsehen,” so that “durchaus eingesehen” means (literally) thoroughly seen-into. Kant chose a new textbook for his lectures on theoretical physics in , which developed the caloric theory of heat in detail, including the concept of latent heat: see Friedman (b, –, together with n. beginning on ).
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seriously than he had before – for this science was now becoming mathematized at the empirical level without needing a Newtonian mathematical theory of motion at the (so far unobservable) micro-level. In the following year (), moreover, Kant describes chemical laws in considerably less pessimistic terms than those he had used in . In the second remark to Proposition II in the first chapter of the Analytic of Pure Practical Reason in the Critique of Practical Reason, Kant emphasizes that all genuinely objective laws – whether theoretical or practical – must be grounded in a priori principles. Considering the theoretical case, in particular, Kant contrasts mechanical and chemical laws as follows: “Even the rules of concordant experiences are only called laws of nature (e.g., mechanical [laws]) if one actually cognizes them a priori, or (as in the case of chemical [laws]) one assumes that they would be cognized a priori from objective grounds if our insight [Einsicht] went deeper” (KpV, :). By contrast, in the two passages touching on chemistry from the Preface to the Metaphysical Foundations (MAN, :– and –), Kant appears to have no room for (justifiable) assumptions concerning possible extensions of our insight. On the contrary, especially in the second passage emphasizing the importance of the application of mathematics in proper natural science, Kant presents us with a binary choice: either one has already “specified” a mathematical law of motions akin to the law of universal gravitation or one has not. This is one of the reasons, in my view, that Kant’s discussion of the freezing of water in the second edition of the Transcendental Deduction is significant for his intellectual development. As already remarked, in particular, Kant had now acquired a good understanding of Joseph Black’s conception of latent heat in the context of
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Recall that proper natural science is possible only on the basis of both metaphysics and mathematics – where, in particular, the latter provides quantitative realizations of the pure categories of metaphysics, especially the dynamical categories: see the paragraph to which note above is appended. Kant is specifically calling chemical laws “laws” in reality, insofar as it is legitimate to assume that they would be grounded in a priori objective grounds if our insight were to go deeper. Indeed, Kant is even suggesting that chemical (causal) laws, in that case, would be mechanical laws – empirical laws of proper natural science. In the second passage Kant is clear that if such a mathematical law of motion has not yet been specified, “then chemistry can be nothing more than a systematic art or experimental doctrine, but never a proper science” (MAN, :; emphasis added). Kant appears to be a bit more open to future developments in the first passage (:), because of its emphasis on “demands of reason” and the suggestion that one is only making a “claim” to thorough comprehension. But Kant appears to squash this openness in the following sentence: “Hence, the most complete explanation of given appearances from chemical principles still always leaves behind a certain dissatisfaction, because one can adduce no a priori ground for such principles, which, as contingent laws, have been learned merely from experience” (MAN, :; emphasis added).
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caloric theory and thus a concrete example of a new kind of application of mathematics to a chemical phenomenon – one that also provides a mathematical realization of the category of substance and its quantitative conservation. There is a tension, therefore, between Kant’s initial strictly Newtonian view and a more relaxed view associated with caloric theory. The year was an especially busy one for Kant. During his work on the second edition of the Critique of Pure Reason he was already seriously working on the second Critique, which was originally intended as an appendix to the B edition of the first. Just a few weeks after completing the manuscript for the second Critique, moreover, Kant made it clear, in a letter to Karl Leonhard Reinhold in December of , that the same topics on which he had lately been working so intensively had also led him to the idea of a third Critique in which the three main branches of critical philosophy – “theoretical philosophy, teleology, and practical philosophy” (Br, :) – would all be united in one system. It is only in this system that the two apparently incompatible realms of nature and freedom, theoretical science and morality, are to be finally successfully integrated with one another.
. Mechanism and Teleology in the Third Critique In the first three sections of the (published) Introduction to the Critique of the Power of Judgement () Kant begins with the distinction between the a priori legislation of the faculty of understanding to nature on behalf of theoretical cognition and that of the faculty of reason to the will (the faculty of desire) on behalf of purely practical cognition. In the following §IV on “the power of judgment as an a priori legislative faculty” (KU, :), Kant introduces a sharp distinction between determining and reflecting judgment. The former takes a general concept or principle as already given and seeks to subsume particular instantiations under it. The latter, by contrast, takes the particular as already given and seeks to find appropriate general concepts and principles under which to subsume this particular. Moreover, the distinction between determining and reflecting judgment is to be applied, first and foremost, to the problem of the relationship between universal transcendental principles of the
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For the circumstances surrounding Kant’s writing of the second Critique, see Allen Wood’s General Introduction to Gregor (Wood , xxv–xvi). For this background to the third Critique, see the Editor’s Introduction to Guyer (, xiii–xiv), which includes a translation of a substantial part of the letter to Reinhold.
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understanding and particular empirical laws. Thus, the same problem Kant addressed in the first Critique by his doctrine of the regulative use of reason in its attempt to find systematic unity among all such empirical laws he now addresses by the idea of reflecting judgment and its new a priori transcendental principle of “the purposiveness [Zweckmäßigkeit] of nature in its multiplicity” (KU, :). The following §V continues the discussion of the principle of purposiveness and begins with the determining power of judgment involved in the pure principles of the understanding governing the concept of a nature in general – which principles are both a priori and necessary. Kant then observes that particular empirical laws of nature, of course, cannot be necessary in this sense: “Thus we must think in nature, with respect to its merely empirical laws, a possibility of infinitely manifold empirical laws, which are nonetheless contingent for our insight [Einsicht] (cannot be cognized a priori), and with respect to which we judge [beurtheilen] the unity of nature in accordance with empirical laws and the possibility of the unity of experience (as a system in accordance with empirical laws) as contingent” (KU, :). Kant concludes, nevertheless, as follows: But since such a unity must still necessarily be presupposed and assumed, for otherwise no thoroughgoing interconnection of empirical cognitions into a whole of experience would take place, because the universal laws of nature yield such an interconnection of things with respect to their genera, as things in nature in general, but not specifically, as such and such particular beings in nature, the power of judgement must thus assume it as an a priori principle for its own use that what is contingent for human insight [Einsicht] in the particular (empirical) laws of nature nevertheless contains a lawful unity – which, to be sure, is not fathomable by us but still thinkable – in the combination of the manifold into one whole of experience [Erfahrungsenthalte] possible in itself. (KU, :–)
Although our insight will never get to the bottom of the “lawful unity” underlying the particular empirical laws of nature, the reflecting power of judgment assumes that it is there to be found – and that we can, and indeed must, therefore seek to find it. Since to call something a “law” is
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This kind of purposiveness, however, is merely subjective, insofar as the power of judgment thereby prescribes a purpose only to itself – in seeking always for systematic unity in the multiplicity of particular empirical laws. In sum, the purposiveness in question has no specific conceptual content: it is a “merely formal purposiveness, i.e., a purposiveness without purpose [Zweckmäßigkeit ohne Zweck]” (KU, :). It thus also has a locus in the aesthetic judgment of taste, which subsumes a particular not under any additional concept but rather under a feeling of pleasure.
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thereby to attribute necessity to it, there must be a necessity to be sought (at least in principle) in the case of any particular genuine empirical law. If we now turn to § of the Critique of the Aesthetic Power of Judgement (the first of the last three sections of this part of the third Critique), we find a striking discussion of what Kant calls “free formations of nature,” exemplified by the solidification of a fluid due to a loss of the caloric that had maintained it in its originally fluid state. Kant describes this process as one of crystallization, which he then exemplifies with the freezing of water: The formation in such a case takes place through precipitation, i.e., through a sudden solidification, not through a gradual transition from the fluid to the solid state, but as it were through a leap, which transition is also called crystallization. The most common example of this sort of formation is freezing water, in which straight raylets of ice form first, which then join together at angles of degrees, while others attach themselves at every point in exactly the same way, until everything has turned to ice, so that during this time, the water between the raylets of ice does not gradually become more viscous, but remains as completely fluid as it would be if it were at a much higher temperature, and still it is fully as cold as ice. The matter that separates itself, which suddenly escapes at the moment of solidification, is a considerable quantum of caloric, the departure of which, because it was required only for maintaining a fluid state, leaves what is now all ice not the least bit colder than was the water that shortly before was still fluid. (KU, §, :)
Kant is now perfectly explicit about the role of caloric in this process (considered as quantifiable) as well as the concept of latent heat. Kant is also describing the process of solidification itself in mathematical (geometrical) terms, so that both the action of caloric (as a cause) and the result of this action in the fluid (as an effect) are now characterized mathematically. The significance of this kind of free formation of nature, in the context of the Critique of the Power of Judgement as a whole, is clarified in the preceding discussion. Kant is here discussing the difference between
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Compare again the Preface to the Metaphysical Foundations at :– (in the paragraph to which note above is appended). See note above, together with the paragraph to which it is appended. In the paragraph in question I was considering Kant’s example of freezing water, and I asserted that (assuming an understanding of both caloric and latent heat) “we now have both a quantitatively conserved substance (caloric) and a quantitative treatment of its causal action in an important change of state (solidification).” What we do not see in , however, is a mathematical treatment of the process of solidification – which, as we have just seen, Kant does state explicitly in .
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aesthetic and genuinely teleological judgments. The purposiveness of the former, as we have seen, is merely formal and subjective, while that of the latter is both material and objective. The reason is that living or organic material systems, for Kant, are characterized by the specific objective processes of nutrition, growth, reproduction, and inheritance. We shall see below why, for Kant, purely mechanical processes cannot, for our particular kind of understanding, fully explain such processes – so that teleological conceptualization and explanation, in these cases, is unavoidably necessary for us. According to the paragraph immediately preceding the one quoted above, however, this is emphatically not the case for the free formations involved in crystallization: [N]ature displays everywhere in its free formations so much mechanical tendency to the generation of forms that seem as if they are made for the aesthetic use of our power of judgement without giving us the slightest ground to suspect that it requires for this anything more than its mechanism, merely as nature, by means of which it can be purposive for our judging even without being based on any idea. (KU, :)
The discussion of crystallization and the freezing of water then follows. It seems clear, therefore, that, while Kant does find a limit to the “mechanical” laws of proper natural science in biological phenomena, he finds no such limit, at this point, in chemical phenomena (at least in what we now call physical chemistry).
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Kant is not quite ready in to suggest that chemistry has now entered the secure path of a natural science (presumably, a proper natural science). But he comes extremely close to asserting precisely this after he has adopted the chemistry of Lavoisier in the later s. In The Metaphysics of Morals (), for example, we find that “the moralist rightly says that there is only one virtue and doctrine of virtue, that is, a single system that connects all duties of virtue by one principle; the chemist, that there is only one chemistry (Lavoisier’s); the teacher of medicine, that there is only one principle for systematically classifying diseases (Brown’s)” (MS, :). And in the Anthropology from a Pragmatic Point of View () Kant asks: “What amount of knowledge, what discovery of new methods would now lie already in store, if an Archimedes, a Newton, or a Lavoisier had, with their same industry and talent, been favored by nature with a lifetime lasting through a century of undiminished vitality?” (Anth, :). Kant takes Lavoisier’s chemistry to be the only possible one in ; and the sequence of Archimedes, Newton, and Lavoisier in suggests that the development beginning with statics, continuing with Newtonian mechanics, and followed by (physical) chemistry represents the order in which each first entered the secure path of a (natural) science. It may have also suggested to Kant that Lavoisier’s application of the conservation of matter to all of the agents in chemical reactions taken together – solid, liquid, and gaseous – introduces an already established conservation principle, governing quantity of (ponderable) matter, into chemistry, while essentially interacting, at the same time, with the specifically chemical conservation principle governing the quantity of (imponderable) caloric. All Kant appears to need, in this connection, is that both conserved quantities (ponderable and imponderable) act as “external” forces in the sense of his Second Law of Mechanics. Evidence for this reading can be
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In order fully to appreciate the very different ways in which Kant in the third Critique treats the two cases of chemistry and biology, we now turn to the Critique of the Power of Teleological Judgement – and, in particular, to the Antinomy of the Power of Judgement with respect to “mechanical” and “teleological” modes of explanation (§§–). This Antinomy, in particular, involves an apparent conflict between two (regulative) maxims of reflective judgment: one according to which “[a]ll generation of material things and their forms must be judged as possible in accordance with merely mechanical laws,” the other according to which “[s]ome products of material nature cannot be judged as possible according to merely mechanical laws (judging them requires an entirely different law of causality, namely that of final causes)” (KU, §, :). It is in the course of this discussion that Kant finally resolves the apparent conflict between mechanism and teleology to his satisfaction, and it also becomes clear, in particular, that the former is essentially framed by the conception of mechanical moving forces developed in the Metaphysical Foundations. Kant’s elaboration on the relationship between mechanism and teleology in his further reflections on the Antinomy culminates in a consideration of “the final purpose [Endzweck] of the existence of a world, i.e., of creation itself” (KU, §, :), leading up to the discussion of Kant’s “moral proof of the existence of God” with which the third Critique – and thus the critical system as a whole – concludes (§§–). As first explained in §, Kant takes properly teleological judgments to apply to a distinctive subset of natural beings, which he calls “organized [organisirten]” beings (KU, :). Teleology, for Kant, is thereby employed in the conceptualization of living or organic material systems such as plants and animals. We take such systems to have a “formative power [bildene Kraft],” which is responsible for the development, maintenance, and reproduction of the special organization of material parts characteristic of the organic system in question. And, at least when judged by the standards of our understanding, this formative power cannot itself be understood as an action of “moving force [bewegende Kraft],” and thus not as a product of the mechanism of nature. Nevertheless, Kant insists
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found in the discussions of quantity of matter and the balance in the earlier parts of the Opus postumum from the years –, but I cannot go into it further here. See § (KU, :): “An organized being is thus not merely a machine – for that has solely moving force; rather [an organized being] possesses a formative power within itself, which it communicates to the matters that do not have [this power] (it organizes them). It therefore possesses a selfreproducing [sich fortpflanzende] formative power, which cannot be explained by the capacity for motion alone (mechanism).”
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at the same time that the concept of a natural purpose is not a constitutive concept of the understanding or reason but rather a purely regulative concept of the faculty of reflecting judgment: The concept of a thing as in itself a natural purpose is thus not a constitutive concept of the understanding or reason, but it can still be a regulative concept of the power of reflecting judgment, in accordance with a distant analogy with our causality according to purposes as such, in order to guide our investigation of objects of this kind and to consider their highest ground. This latter [is undertaken] not on behalf of the knowledge of nature or its primordial ground, but rather precisely [on behalf of] the same practical faculty of reason in us, in analogy with which we consider the cause of this purposiveness. (KU, §, :)
The “distant analogy” in question is that we consider natural ends as analogous to our own technical products, which are formed in accordance with our conscious intentions. We thereby represent these products of nature as if they had been designed by a supersensible intelligence as their “primordial ground.” Nevertheless, since we can have absolutely no theoretical cognition of such an intelligence, the ultimate motivation for even considering such a “highest ground” can only lie in practical reason. The following paragraph goes on to consider the objective (empirical) reality of the concept of an organized being or natural purpose: Organized beings are thus the only ones in nature, which, even when one considers them for themselves and without a relation to other things, must still be thought as possible purposes of nature, and which thus first provide the concept of a purpose that is not a practical one but rather a purpose of nature with objective reality – and thereby [provide] a basis in natural science for a teleology, i.e., a mode of judging of its objects in accordance with a special principle, of such a kind that one would simply not otherwise be justified in introducing into [natural science] (because one can in no way comprehend [einsehen] such a type of causality a priori). (KU, :–)
Since Kant has just said in the preceding paragraph that the concept of a natural purpose is a merely regulative concept of reflecting judgment, the “special principle” in question is introduced in accordance the faculty of reflecting judgment. What Kant is saying in the present paragraph, therefore, is that the concept of a natural purpose does have objective (empirical) reality, insofar as there do happen to exist beings in nature with dispositions for nourishment, growth, reproduction, and self-maintenance. And, although we have no theoretical cognition of the underlying cause of these dispositions by the determining power of judgment, the faculty of reflecting judgment can justify our introduction of a teleological causal
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principle into the study of nature, where, however, it thereby serves only as a regulative principle compensating for the necessary limitations of our properly constitutive theoretical cognition. As we have seen, the Antinomy of the Power of Judgement (first articulated in §) concerns an apparent conflict between two regulative maxims of reflective judgment, one governing our search for purely mechanical explanations of the generation of all material things, the other stating that there will always remain cases of material generation where purely mechanical explanations are not to be found and recommending the use of teleological explanations instead. The first point Kant then makes is that there is no genuine antinomy at all (no contradiction between two incompatible principles) in the case of two such merely regulative maxims of reflecting judgment. On the contrary, the two would be genuinely antinomial only if they were mistakenly formulated as “objective” (constitutive) principles of determining judgment (KU, :). In the case of reflecting judgment, Kant continues, the purely regulative maxim with respect to mechanism has this formulation: “I should always reflect on [all events in material nature] in accordance with the principle of the mere mechanism of nature and thus investigate them as far as I can, because without taking this principle as the basis for investigation no proper cognition of nature is possible” (KU, :). Since all such investigation is potentially infinite, however, we will relatively quickly arrive at a point where some other regulative principle, such as the principle of teleology, is needed to supplement the principle of mechanism for some natural forms (organized bodies). But there is no antinomial conflict here: “Reflection in accordance with the first principle [of mechanism] is not thereby abolished; it is rather requested that it be pursued as far as one can, and it is not thereby also asserted that such forms would not be possible in accordance with the mechanism of nature” (KU, :) Indeed, our human reason is absolutely incapable of determining whether, “in the for us unknown inner ground of nature itself, the physical-mechanical [combination] and the combination in accordance with purposes may not cohere together in the same things in one principle” (ibid.). As Kant reiterates at the end of the following §, therefore, there can be no incompatibility at all between the two regulative maxims (KU, :). Kant proceeds in § by considering the “various systems concerning the purposiveness of nature” (KU, :). The main question, he says, is
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See the paragraph following the one to which note above is appended for Kant’s precise statement of the Antinomy (KU, :).
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whether the principle of teleology is “merely subjectively valid” (as a regulative maxim), or is also “an objective principle of nature, according to which there would pertain to it, in addition to its mechanism (in accordance with mere laws of motion) yet another kind of causality, namely that of final causes, under which the first kind (that of moving forces) would stand only as intermediate causes” (KU, :–). Kant then classifies the various systems in question and argues, in the following § (:), that “[n]one of the above systems accomplishes what it pretends to do.” In particular, what Kant calls the physical system of a realism of purposes is characterized at the end of §: “[This system] bases purposes in nature on the analogue of a faculty acting in accordance with an intention, on the life of matter (in it, or also through an animating inner principle, a world-soul), and is called hylozoism” (KU, :). And the way in which Kant dismisses this alternative in the following § is of particular interest: “[T]he possibility of a living matter (the concept of which contains a contradiction, because lifelessness, inertia, constitutes its essential characteristic), cannot even be conceived” (KU, :). So when Kant speaks of “moving forces” and “laws of motion” here he means moving forces and laws of motion as explained in the Mechanics of the Metaphysical Foundations – where the law of inertia (and thus the Newtonian concept of externally acting impressed forces) is centrally and essentially involved. Therefore, when Kant speaks of “mechanics” and the “mechanism of nature” here he specifically has in mind proper natural science as explained in the Preface to the Metaphysical Foundations. This makes good sense, for Kant, because only what he calls proper natural science can lead to full insight (Einsicht) into the necessity of particular empirical laws. I now turn, against this background, to the pivotal §: “On the peculiarity [Eigenthu¨mlichkeit] of the human understanding, whereby the concept of a natural purpose becomes possible for us” (KU, :).
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It is important to note, however, that there is a fundamental asymmetry between the regulative principles of mechanism and teleology. While any particular mechanical explanation of a phenomenon purports to be a constitutive (and therefore objective) characterization of it, no particular teleological explanation of a phenomenon can be constitutive in this way: compare the passage from § (KU, :) quoted in the paragraph to which note above is appended. The principle of teleology, in this sense, is regulative twice over. The quotation in the paragraph following note above explained how proper natural science can provide maximal insight into the necessity of particular empirical laws, such as, paradigmatically, the law of universal gravitation (note above). In my discussion of §§IV and V of the Introduction to the third Critique I suggested that the possibility of such insight must be assumed (as possible) for the reflective power of judgment (see note above, together with the paragraph to which it is appended).
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Indeed, Kant’s elucidation of the peculiarity of our understanding also involves explaining, in a certain sense, how it makes teleology (and thus the concept of a natural purpose) necessary for us. Kant has already stated in § that, although “we can in no way prove the impossibility of the generation of organized natural products through the mere mechanism of nature” (KU, :), “it is just as indubitably certain, relative to our cognitive faculty, that the mere mechanism of nature can supply no explanatory ground for the generation of organized beings” (KU, :; emphasis added). Moreover, in § he has similarly but more dramatically made the following famous claim: It is completely certain that we cannot even become sufficiently acquainted with organized beings and their internal possibility in accordance with merely mechanical principles, let alone explain [such beings] to ourselves. Indeed, it is so certain that one can boldly say that it is absurd for human beings even to make such an attempt or to hope that some day perhaps a Newton could still arise who will make comprehensible even the generation of a single blade of grass in accordance with laws of nature that have not been ordered by any purpose [Absicht]. Rather, one must completely deny this insight to human beings. (KU, :)
The question then inevitably arises as to the basis for this (negative) certainty, and the task of § is finally to make this clear. It lies, as Kant suggests, not so much in the nature of the objects in question as in the nature of our specifically human understanding. The main idea to which Kant appeals in elucidating this “peculiarity” of our understanding is a contrast between the character of our human understanding, in proceeding from the “analytically universal [AnalytischAllgemeinen] (from concepts) to the particular (to the given empirical intuition),” and the character of another quite different understanding, “which, because it is not discursive like ours but intuitive, proceeds from the synthetically universal [Synthetisch-Allgemeinen] (from the intuition of a whole as such) to the particular, i.e., from the whole to the parts” (KU, :). Two pages earlier, moreover, Kant suggests that the idea here is analogous to one already used in the first Critique in elucidating the special character of our human intuition by contrast with “another possible intuition” – so as then to take “our intuition as a particular kind, namely one for which objects are valid only as appearances” (KU, :). And, two pages after drawing the contrast between the analytically universal procedure of a discursive understanding like ours and the synthetically universal procedures of an intuitive understanding (), Kant illustrates the synthetically universal procedure in question by
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“the unity of space” – which although it is “no real ground of [natural] generations but only their formal condition,” still has “some similarity with [such a real ground], in so far as no part within it can be determined except in relation to the whole (whose representation thus lies at the basis of the possibility of the parts)” (KU, :). It seems clear, moreover, that the property of space at issue here is the one appealed to in the third argument of the Metaphysical Exposition of the Concept of Space in the Transcendental Aesthetic of the first Critique. Space is not a “discursive, or, as one says universal [allgemeinen] concept” but a “pure intuition,” because “one can only represent a single space, and, if one speaks of many spaces, one understands them as only parts of one and the same unique [alleinigen] space[; t]hese parts can also not precede the single allencompassing [allbefassenden] space as its constituents (out of which a composition [Zusammensetzung] would be possible), but [can] only be thought within it” (KrV, A/B). In § of the third Critique, however, what is primarily at issue is Kant’s contrast between our discursive understanding as proceeding in an analytically universal fashion and our synthetically universal pure intuition based on a geometrical part-whole relationship between a region of space and a larger such region. But the latter also involves, in the context of natural teleology, a causal-dynamical relationship between a spatial part of an organized body in nature (e.g., an organ) and the material whole (organism) of which it is a (spatial) part. While our discursive understanding can only proceed from a part to a greater whole, the very concept of an organized (i.e., organic) body requires that there be a special kind of “reciprocal [wechselseitig]” causal relationship among each part, the whole, and all the other parts. And it is only in virtue of this property of organized bodies that they function as “self-organizing” (and “self-reproducing”) beings. Kant therefore contrasts a discursive understanding like ours with an intuitive understanding as follows in §: “In accordance with the constitution of our understanding, by contrast, a real whole of nature is to be regarded only as the effect of the concurrent moving forces of the
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See §: “In such a product of nature each part is thought as existing only through all the others, and also for the sake of the others and on account of the whole, i.e., as an instrument (organ) – which, however, is not sufficient (for it could also be an instrument of art and thus be represented as possible at all only as a purpose). Rather, [it is thought] as an organ that produces the other parts (and thus each of the others reciprocally), such that it cannot be an instrument of art, but only of the nature that supplies all the materials for instruments (even those of art); and only then and for this reason can such a product be called, as an organizing and self-organizing being, a natural purpose” (KU, :–). (For “self-reproducing,” see note above.)
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parts” (KU, :; emphasis added). His point, therefore, involves precisely the concept of “mechanical moving forces” developed in the Metaphysical Foundations. So, once again, the concept of proper natural science plays a central role in Kant’s argument. But “mechanical moving forces,” as we have seen, conform to Kant’s Second Law of Mechanics, and therefore involve only external causal influences – between one body or part of a body and others external to it. There can be no reciprocal such causal influence between a part and a whole of which it is a part. And it is for precisely this reason, for Kant, that mechanical interactions in accordance with proper natural science can never explain the generation and properties of self-organizing and self-reproducing organized beings. So, since our understanding is discursive, the only way that we can even think such a possibility is by invoking another kind of understanding, completely different from ours, that is itself intuitive rather than discursive. Due to the same “peculiarity” of our understanding, however, we – unlike the intuitive understanding itself – can never think or conceive the relevant mode of generation in which the whole is the ground of all of its parts except by thinking of such an intuitive understanding as producing or creating this whole through an idea or representation of its purpose. We must conceive the intuitive understanding as intentionally (and thus teleologically) productive of the organized body on analogy with our own intentional production of artifacts.
. The Bridge between Nature and Freedom The Antinomy of the Power of Judgement occupies §§– of the third Critique. It is followed by a final discussion on the Methodology of the Teleological Power of Judgement occupying §§–. This final discussion culminates in the moral proof of the existence of God with which both the third Critique and the entire critical system conclude (at least as it stood in ). The second section of the Methodology (§) is entitled “On the necessary subordination of the principle of mechanism to the teleological principle in the explanation of a thing as a natural purpose” (KU, :). Kant begins by asserting that, although our authorization to seek for mechanical explanations is unlimited, we are nevertheless bounded by the capacity of our particular understanding for comprehending the organized beings that actually exist. We therefore require, in particular, fundamental teleological principles of original organization. In § of the Methodology, however, Kant is by no means dogmatic about the latter claim. The fifth paragraph of § envisions an
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“archaeologist of nature,” who hypothesizes a “universal mother” from whom all organized creatures have emerged – the “maternal womb of the earth, which has just emerged from a condition of chaos” (KU, :). Kant continues: But he must in the end nevertheless attribute to this universal mother a purposively posited organization for the sake of all of these creatures, for otherwise the purposive forms of the products of the animal and vegetable kingdoms cannot be thought at all in accordance with their possibility.* In that case, however, he has only further postponed the ground of explanation and cannot presume to have made the generation of these kingdoms independent of the condition of final causes. (KU, :–)
Kant explains in the footnote that his negative judgment concerning the possibility of a generation of organic from inorganic matter ultimately rests on nothing more nor less than the fact that this kind of generation, “so far as our empirical knowledge of nature extends, is nowhere to be found” (KU, :n). But the judgment in question is not simply that, as a matter of brute contingent fact, no such generation has yet been discovered. For Kant takes “our” empirical knowledge to be limited in principle by the nature of our human understanding, which is by no means contingent from our point of view. From the point of view of the intuitive or archetypal understanding proceeding in accordance with “synthetic universality,” however, it is contingent: there can be no proof from this point of view that the problematic form of generation is absolutely impossible. In the fourth paragraph of § Kant emphasizes that an “analogy of forms” among many different animal species, in terms of skeletal structure, for example, or other such systematic relationships among other bodily parts (e.g., organs), “allows the mind an at least weak ray of hope that something may be accomplished here with the principle of the mechanism of nature, without which there can be no natural science at all” (KU, :). Kant concludes: This analogy of forms, in so far as in all variety they seem to have been generated in accordance with a common prototype [Urbild], strengthens the suspicion of an actual kinship among them in the generation from a common proto-mother [Urmutter] by means of a gradual approach of one animal species, in which the principle of purposes appears best confirmed, namely human beings, down to polyps, and from them even further to mosses and lichens, and finally to the lowest level of nature noticeable by us, to raw matter: out of the forces of which, in accordance with mechanical laws (like those active in the generation of crystals), the whole technique of nature, which is so inconceivable to us in organized beings that we believe
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ourselves forced to think up another principle for this, seems to derive. (KU, :–)
This striking passage appears to be the closest Kant ever comes to suggesting that organized beings may possibly be generated by mechanical laws. As we have seen, however, Kant cancels this possibility for us. Indeed, in his earlier discussion of crystallization in §, toward the end of the Critique of the Aesthetic Power of Judgement, Kant has made it perfectly clear that this process is quite unproblematically mechanical. And it is especially striking, in particular, that Kant, at the beginning of his climactic discussion of the Methodology of Teleological Judgement, recurs again to his earlier (and more detailed) discussion of crystallization located in an analogous position toward the end of the Critique of the Aesthetic Power of Judgement. Kant’s mature view of the relationship between what we now call physical chemistry and the sciences of life is therefore an important component in his mature view of the latter sciences. The centerpiece of Kant’s teleological Methodology features a juxtaposition of natural and moral teleology. And it is found most explicitly in the transition from §, on the ultimate purpose (letzten Zweck) of nature as a teleological system, to §, on the final purpose (Endzweck) of the existence of a world, that is, of creation itself. The ultimate purpose of nature as a teleological system (§) involves a systematic relationship among all the organized beings (plants and animals) in nature, where, on Kant’s view, human beings are the only such organized beings that can possibly constitute the ultimate natural purpose of the whole system. This, for Kant, is because only human beings are capable of representing purposes to themselves and then acting in accordance with them. But what human end or purpose is capable of organizing the totality of this being’s purposes? Kant argues that the only two possibilities at this point, where we are restricting ourselves to purposes that can be found within rather than beyond nature, are happiness and the cultivation of a human being’s rational capacity to set and pursue arbitrary ends. Since the content of happiness varies widely (and indeed arbitrarily) among different human beings, however, the cultivation of rationality – prudential rationality as such – is the only determinate purpose that can be pursued by all.
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See again: “[N]ature displays everywhere in its free formations so much mechanical tendency to the generation of forms that seem as if they are made for the aesthetic use of our power of judgement, without giving us the slightest ground to suspect that it requires for this anything more than its mechanism, merely as nature” (KU, :). There is no objective (or immanent) purposiveness in the “free formations” of nature, such as, paradigmatically, crystallization. Here I draw on Friedman ().
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In §, by contrast, we are considering the final purpose (Endzweck) of the existence (and creation) of the world. This purpose is final in the sense of being unconditional, and there is absolutely nothing within nature that is not conditional on something else. For Kant, therefore, the only possible final purpose, in this sense, is the Highest Good unconditionally commanded by morality, that is, by pure practical reason itself: Now we have only a single kind of being in the world whose causality is teleological, i.e., directed at ends [Zwecke], and yet at the same time so constituted that the law according to which they have to determine ends for themselves is represented by themselves as unconditioned and independent of natural conditions, but also as in itself necessary. The being of this kind is the human being, but considered as noumenon: the sole being in nature in which we can nonetheless cognize, from the side of its own constitution, a supersensible faculty (freedom) and even the law of its causality together with its object, which it can set for itself as the highest purpose (the Highest Good in the world). (KU, :)
It is in the crucial transition from prudential to pure practical reason, therefore, that we move from nature to freedom – and, accordingly, not only conceive the ultimate purpose of nature as a teleological system, but also become (morally) worthy of occupying the position of ultimate purpose as part of our very highest purpose: the Highest Good in the world. But how do we move from the cultivation of prudential rationality to pure practical reason, from § to §, from nature to freedom? Toward the end of § Kant is emphasizing the problem of “unsocial sociability” discussed earlier in his Idea for a Universal History with a Cosmopolitan Aim (). Here Kant, following Rousseau, is considering the rise of inequality in all forms of human progress, all of which are necessarily communal. The problem is to explain how the fearsome inequalities inevitably resulting from our unsocial sociability can be mitigated. One idea stems from the social contract tradition via an “invisible hand” explanation, which begins with self-interested, purely prudential rationality and arrives at a just society as the only possible solution (for prudential reason) of the problem of ever-worsening inequality. Kant, however, is not, in the end, completely satisfied with this kind of explanation, and he pursues in addition the idea of a gradual social cultivation of our specifically moral capacities – an education of the sentiments also inspired by Rousseau.
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See Wood (), which examines the conception of unsocial sociability in detail and also Kant’s debt, in this context, to Rousseau.
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This idea is developed in the final paragraph of § focusing on “a purposive striving of nature towards an education or cultivation [Ausbildung], which makes us receptive to higher purposes than nature itself can afford” (KU, :). The paragraph ends by invoking the special kind of pleasure that is characteristic, for Kant, of aesthetic experience: Beautiful art and sciences, which, by means of a pleasure that can be universally communicated, and by elegance and refinement for society, make human beings, if not morally better, then at least better mannered for society, reduce very much the tyranny of what is attached to the senses, and thereby prepare human beings for a sovereignty in which reason alone is to have the power; while the evil that afflicts us, partly from nature, partly from the intolerant selfishness of human beings, at the same time calls forth, strengthens, and steels the powers of the soul not to be subjected to the former, and thus allows us to feel an aptitude for higher purposes that lie hidden within us. (KU, :–)
There appears to be no doubt that the “higher purposes” in question are those unconditionally commanded by morality, that is, by pure practical reason, so that now, at the moment of transition from § to §, we stand at the very doorstep of morality. This, of course, is only the beginning of Kant’s Rousseau-inspired conception of the moral education of our natural sentiments, but here is not the place to develop it further. So I shall conclude, rather, by calling attention to an important point of agreement between this argument from the third Critique and Kant’s earlier discussion in the Groundwork of the Metaphysics of Morals () of a Kingdom of Ends – a point that further emphasizes the importance of his bridge between natural and moral teleology. Kant introduces the Kingdom of Ends as a variant of the formula of autonomy, namely: “The concept of each and every rational being, which must consider itself through all the maxims of its will as universally legislating, in order to evaluate itself and its action from this point of view [the principle of autonomy], leads to a very fruitful concept depending on it, namely that of a Kingdom of Ends” (GMS, :). Kant further characterize this last concept, that of a Kingdom of Ends, as follows: By a Kingdom, however, I understand the systematic combination of various rational beings through communal [gemeinschaftliche] laws. Now because laws determine ends in accordance with their universal validity, if one abstracts from the personal differences between rational beings, as likewise from every content of their private ends, a whole of ends – (of rational begins as ends in themselves as well as of their own ends, which
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each may set for himself ) in systematic connection, i.e., a Kingdom of Ends – can be thought, which is possible in accordance with the above principles. (GMS, :)
And he rounds off his introduction of this concept by giving even more prominence to the formula of humanity: For rational beings all stand under the law that every one of them ought to treat itself and all others never merely as means, but always at the same time as end in itself. From this, however, arises a systematic combination of rational beings through communal objective laws, i.e., a kingdom that, because these laws have as their aim the relation of these beings to one another as ends and means, can be called a “Kingdom of Ends” (obviously only an ideal). (ibid.)
Thus Kant here emphasizes that the Kingdom of Ends is not only a procedural device for “abstract[ing] from the personal differences between rational beings” in order to arrive at truly universal moral laws but also an end (in itself ) to be pursued. Kant explicitly links the idea of a Kingdom of Ends with that of a Kingdom of Nature a few pages later in the Groundwork. He is considering a progression of three subsidiary variants, corresponding to the three categories of quantity, of the original universal formula of the categorical imperative reached at the end of the first part of the Groundwork. The third variant (the Kingdom of Ends) corresponds to the category of allness or totality, and it thereby corresponds, Kant says, to a complete determination of all maxims by means of that [variant] formula, namely, that all maxims from one’s own legislation are to harmonize with a possible Kingdom of Ends as a Kingdom of Nature.* A progression takes place here, as through the categories of the unity of the form of the will (its universality), the plurality of the matter (of objects, i.e., of ends), and the allness or totality of the system of these [ends]. (GMS, :).
What is most important in this context is the footnote Kant attaches to the juxtaposition of the Kingdom of Ends with a Kingdom of Nature: * Teleology considers nature as a Kingdom of Ends, morality a possible Kingdom of Ends as a Kingdom of Nature. In the former the Kingdom of Ends is a theoretical idea for explaining that which exists. In the latter, it is a practical idea in order to bring about that which does not exist but can become actual through our deeds and omissions – and, indeed, in accordance with precisely this idea. (GMS, :n)
This explanatory footnote in the Groundwork appears to fit very well with the transition from § to § in the third Critique. Section §
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belongs to natural teleology, and, as such, it considers a teleological organization of the whole of nature under a hierarchical system of purposes. In this sense, the ultimate purpose of nature as a teleological system is a theoretical idea for (teleologically) explaining that which exists under the superordinate purpose (or purposes) of the human species – which, for Kant, do not yet amount to the specifically moral purposes prescribed by pure practical reason. In §, by contrast, Kant is considering the final purpose of the existence of a world, that is, of creation itself. And this, as we have seen, can only be the Highest Good in the world – an ideal of pure practical reason that we can asymptotically approach through our own “deeds and omissions” but never completely attain. Nevertheless, this idea of pure practical reason functions precisely to guide our actions in nature toward the Highest Good as much as it can or, more precisely, as much as we can embrace it. That Kant is here constructing a bridge between nature and freedom means that the resources of both theoretical natural science and practical moral philosophy are implicated in his momentous intellectual pursuit. What I have attempted to show in this chapter is that and how the changing status of physical chemistry in his critical system – a development revealed in the history of this science in the years through and beyond – is inextricably entangled with the parallel development of Kant’s philosophy as a whole, including, in particular, his moral philosophy. Just as he had to deny knowledge in order to make room for (practical) faith, Kant had also to make room for physical chemistry within proper natural science while simultaneously elevating natural teleology to the status of an unavoidable part (the first part) of his long-sought-for philosophical bridge.
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See note for the culmination of Lavoisier’s physical chemistry in the late s – and, for the moral philosophy, see the analysis, in the last two paragraphs, of Kant’s explanatory footnote in the Groundwork (GMS, :n).
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Warda, Arthur. . Immanuel Kants Bu¨cher. Berlin: Breslauer. Warren, Daniel. . Reality and Impenetrability in Kant’s Philosophy of Nature. New York: Routledge. . “Kant on Attractive and Repulsive Force: The Balancing Argument.” In Discourse on a New Method: Reinvigorating the Marriage of History and Philosophy of Science, edited by Mary Domski and Michael Dickson, –. Chicago: Open Court. . “Kant on Mathematical Force Laws.” In Kant and the Laws of Nature, edited by Michela Massimi and Angela Breitenbach, –. Cambridge: Cambridge University Press. Washburn, Michael. . “The Second Edition of the Critique: Toward an Understanding of Its Nature and Genesis.” Kant-Studien : –. Watkins, Eric. . “The Laws of Motion from Newton to Kant.” Perspectives on Science : –. a. “The Argumentative Structure of Kant’s Metaphysical Foundations of Natural Science.” Journal of the History of Philosophy (): –. b.“Kant’s Justification of the Laws of Mechanics.” Studies in the History and Philosophy of Science : –. a. “Introduction.” In Kant and the Sciences, edited by Eric Watkins, –. Oxford: Oxford University Press. ed. b. Kant and the Sciences. Oxford: Oxford University Press. . Kant and the Metaphysics of Causality. Cambridge: Cambridge University Press. . “On the Necessity and Nature of Simples: Leibniz, Wolff, Baumgarten, and the Pre-Critical Kant.” Oxford Studies in Early Modern Philosophy : –. . Kant on Laws. Cambridge: Cambridge University Press. Watkins, Eric, and Ina Goy, eds. . Kant’s Theory of Biology. Berlin: De Gruyter. Watkins, Eric, and Marius Stan. . “Kant’s Philosophy of Science.” In Stanford Encyclopedia of Philosophy. Edited by Edward Zalta. https://plato .stanford.edu/entries/kant-science/. Westphal, Kenneth. a. “Does Kant’s Metaphysical Foundations of Natural Science Fill a Gap in the Critique of Pure Reason?” Synthese (): –. b. “Kant’s Dynamic Constructions.” Journal of Philosophical Research : –. Willaschek, Marcus. . Kant on the Claims of Metaphysics: The Dialectic of Pure Reason. Cambridge: Cambridge University Press. Willaschek, Markus, and Eric Watkins. . “Kant on Cognition and Knowledge.” Synthese : –. Wilson, Curtis. . “The Great Inequality of Jupiter and Saturn, from Kepler to Laplace.” Archive for History of Exact Sciences (–): –. Wittgenstein, Ludwig. . Philosophical Investigations. th edition. Oxford: Wiley Blackwell. Originally published .
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Wolff, Christian. . Vernu¨nfftigen Gedancken von Gott, der Welt und der Seele des Menschen, auch allen Dingen u¨berhaupt. Abt. , Bd. of Wolff’s Gesammelte Werke. Hildesheim: G. Olms Verlag. Originally published . Wood, Allen. . “Unsocial Sociability: The Anthropological Basis of Kantian Ethics.” Philosophical Topics (): –. . “General Introduction.” In Practical Philosophy, edited by Mary Gregor, xiii–xxxiii. Cambridge: Cambridge University Press. Zach, Richard. . “Hilbert’s Program.” In The Stanford Encyclopedia of Philosophy. Edited by Edward Zalta. https://plato.stanford.edu/archives/ fall/entries/hilbert-program/. Zammito, John. . “‘This Inscrutable Principle of an Original Organization’: Epigenesis and ‘Looseness of Fit’ in Kant’s Philosophy of Science.” Studies in History and Philosophy of Science (): –. . “‘Proper Science’ and Empirical Laws: Kant’s Sense of Science in the Critical Philosophy.” In The Palgrave Kant Handbook, edited by Matthew C. Altman, –. Basingstoke: Palgrave Macmillan.
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Index
acceleration, , , , , , Adams, Robert, Adelung, Johann Christoph, Adickes, Erich, , , , Allison, Henry, –, , –, , Anthropology from a Pragmatic Point of View, , appearance, –, , , , –, – Archimedes, Bartoloni-Meli, Domenico, Baum, Manfred, Baumgarten, Alexander, , , Beck, Dominicus, Beck, Johann Sigismund, Beck, Lewis White, – Beiser, Frederick, Bernoulli, Daniel, biology, , , , –, – Biwald, Leopold, Black, Joseph, – Blomme, Henny, body, –, –, , , , , . See also matter doctrine of, , , , , –, , , , , essence of, organized. See biology; teleology rigid, – Boerhaave, Herman, Breitenbach, Angela, –, , , , – Brewer, Kimberly, Brittan, Gordon, , , , , , , , , , , Buchdahl, Gerd, –, –, , , , , bucket experiment, – Buffon, Georges-Louis Leclerc, Comte de, Butts, Robert, ,
caloric, –, – Carrier, Martin, , , Cassirer, Ernst, categories, –, –, , , , , –, , – of modality, , , – predicables, , of quality, , , , of quantity, , , of relation, , , ; see also causality causality, –, –, –, –, , –, –, – principle of, –, , certainty, –, , –, –, chemistry, , , , –, , –, –, –, –, Clagett, Marshall, Cohen, Alix, cohesion, , , –, Concerning the Ultimate Ground of the Differentiation of Directions in Space, , , , continuity, , , , , , , Correspondence, , , , , cosmology, , , , Cramer, Konrad, Critique of Practical Reason, , , Critique of Pure Reason, –, –, , , , , , –, –, –, –, , –, , –, , –, , –, , –, –, , –, , Analogies of Experience, , , , –, , , , , , Anticipations of Perception, –, Antinomy of Pure Reason, Appendix to the Transcendental Dialectic, , –, –, – Architectonic of Pure Reason, , , –,
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Index
Critique of Pure Reason (cont.) Axioms of Intuition, , , , Canon of Pure Reason, – Discipline of Pure Reason, , , –, Postulates of Empirical Thought in General, , – Transcendental Deduction, , – Critique of the Power of Judgment, , , , , , –, – crystallization, –, – d’Alembert, Jean le Rond, , –, De Bianchi, Silvia, , , , , , – de Risi, Vincenzo, density, –, , – Descartes, René, , , , Detlefsen, Michael, du Châtelet, Emilie, Dunlop, Katherine, , Dyck, Corey, , – Eberhard, Johan Peter, , Edwards, Jeffrey, , elasticity, –, – electromagnetism, Ellington, James, Emundts, Dina, , , ends. See also purposiveness final end, , Kingdom of Ends, – of science, –, –, –, ultimate end of nature, –, Engelhard, Kristina, – Erxleben, Johann Christian Polycarp, –, , essence, –, , –, , ether, –, –, – Euler, Johann Albrecht, Euler, Leonard, , , –, –, , , , experience, , , –, , , –, –, –, , –, , –, –, , –, –, –, , , , Falkenburg, Brigitte, , The False Subtlety of the Four Syllogistic Figures, filling space. See matter, as filling space finitism, – force, , –, –, – action of, –, –, – immediacy, –
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attractive, , , , –, –, , –, , –, –, –, , , –, centrifugal, derivative, –, –, fundamental, , –, –, , –, gravitational, , – impressed, moving, , , , , , – reduction of, repulsive, , , , , –, –, –, , –, –, –, , , Förster, Eckart, , , , freedom, , –, , –, , , Friedman, Michael, –, –, –, , , , , , , , , , , –, , , , –, , –, , , –, , , , , –, , –, , , –, –, , , –, , Frierson, Patrick, Galilei, Galileo, , Garber, Daniel, , Gaukroger, Stephen, , , Gava, Gabriele, Ginsborg, Hannah, Glezer, Tal, , , , –, Gloy, Karen, , , God, , , –, , –, , Gouaux, Charles, , Goy, Ina, Grier, Michelle, grounding, –, –, , Groundwork of the Metaphysics of Morals, , , – Guyer, Paul, , , Hacking, Ian, Harman, Peter, Hatfield, Gary, Hauser, Berthold, heat, , , –, – Heath, Peter, Hebbeler, James, , Heidemann, Dietmar, Heilbron, John Lewis, Heis, Jeremy, , , Hellwag, Christoph Friedrich, – Herschel, William, highest good,
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Index Hoefer, Carl, Höfler, Alois, Hogan, Desmond, , , Holden, Thomas, Hoppe, Hansgeorg, , Horváth, Johann Baptist, Howard, Stephen, , Hume, David, Huygens, Christiaan, Hyder, David, , , Idea for a Universal History with a Cosmopolitan Aim, Inaugural Dissertation, , , , incongruent counterparts, , – inertia, –, , , , , , , – inertial frame, , , , , , intuition, –, –, empirical, , , –, – pure, –, , –, , , , –, –, –, Jäsche Logic, , , , – judgment, alternative, determining, –, – disjunctive, – distributive, reflecting, – Kahn, Samuel, , Kannisto, Toni, Kant, Immanuel. See titles of individual works Keill, John, , –, – kinematics, – Kitcher, Patricia, Kitcher, Phillip, , , , , knowledge, , –, –, –, –, , , , –, , , –, , , , –, , Körner, Stephen, – Kraus, Katharina, , Kreines, James, –, , Kru¨ger, Lorenz, Kuehn, Manfred, Lagrange, Joseph Louis, , – Lambert, Johann Heinrich, , , –, – Laudan, Larry, Lavoisier, Antoine, – laws, law of universal gravitation, , , – derivation, –
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of dynamics, –, of diffusion of force, –, universal law of dynamics, –, , –, – of mechanics, –, , , –, , , derivation, , –, –, law of action and reaction, , , –, , –, –, , , , –, , , – law of inertia, , , , , , , , , –, of motion, , , , –, of nature, , , , , , –, – apriority of, , –, –, –, –, , , best systems account, , –, derivation account, dispositional essentialism, necessitation account, , , Laywine, Alison, , , , –, Le Sage, Georges-Louis, Lectures on Logic, , Lectures on Metaphysics, , , , Lectures on Physics, , – Lefèvre, Wolfgang, Leibniz, Gottfried Wilhelm, , , –, , –, , , , Lequan, Mai, Lewis, David, , Lichtenberg, Georg Christoph, light, , –, Lind, Gunter, Linnaeus, Carolus, logical derivation, –, – Longuenesse, Béatrice, Lu-Adler, Huaping, Lyre, Holger, –, , – Mach, Ernst, magnitudes extensive, , –, , – intensive, , –, , , , – Makkreel, Rudolf, Maronne, Sébastien, , Massimi, Michela, –, , , , , , , –, –, , mathematical-mechanical mode of explanation, –, , –, , –, , – mathematics, , application of, –, –, –, , , , –, –, , –, –, –, , –, –,
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Index
mathematics (cont.) arithmetic, , , construction, , –, –, , –, of communication of motion, , – of intensive magnitudes, – of matter, –, , –, –, –, –, – of motion, –, –, , , of space filling, –, –, –, , –, –, , – geometry, –, –, , , , – matter, , analysis of, –, –, –, divisibility of, –, , –, exposition of, –, , as filling space, –, –, , , , –, –, , , –, as having moving force, , , , , impenetrability, –, –, –, , , ; see also matter, as filling space; force, repulsive as movable, , , , , , , , as object of experience, –, , –, – quantity of, –, , , , , , specific variety of, , –, –, , state changes, –, – Maupertuis, Pierre Louis, – McLaughlin, Peter, McLear, Colin, , McNulty, Michael Bennett, –, , , –, –, –, , , , , , , , , , , , McRobert, Jennifer, – measurement, , –, – mechanical philosophy. See mathematicalmechanical mode of explanation mechanism, –, , –. See also mathematical-mechanical mode of explanation Meier, Georg Friedrich, Mensch, Jennifer, Messina, James, , , , , , metaphysical-dynamical mode of explanation, –, , , –, – metaphysics, –, metaphysical construction, –, of nature, , general, –, , –, – special, –, , –, –, –, , , – completeness of, –, –
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relation to mathematics, , –, –, –, –, , –, –, , – relation to science, –, –, Metaphysics of Morals, , –, , Miller, David Marshall, Mischel, Thomas, , Mittelstaedt, Peter, –, monadology. See substance, monads Moore, Adrien, moral philosophy, , – Morrison, Margaret, motion, , –, –, –, –, –, – absolute, , – acceleration, , , , , , circular, –, –, communication of, , –, , composition of, –, –, , – curvilinear, quantity of, , , , , , rectilinear, , –, – relative, –, , –, –, –, representation of, –, – rest, , speed, – Nassar, Dalia, nature, , . See also metaphysics, of nature; science, natural doctrine of. See physics formal, , –, material, , – Nayak, Abhaya, , Negative Magnitudes, – New Doctrine of Motion and Rest, , New Elucidation, –, Newton, Isaac, , –, , , –, –, , – Opticks, , Philosophiæ naturalis principia mathematica, , , –, , , –, , –, , , –, , – Of the Different Races of Man, Okruhlik, Kathleen, On Fire, On the Use of Teleological Principles in Philosophy, The Only Possible Argument in Support of a Demonstration of the Existence of God, Onnasch, Ernst-Otto,
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Index Opus postumum, , , , , transition project, Oresme, Nicole, –
Rush, Fred, Rusnock, Paul,
Palter, Robert, Panofsky, Erwin, Panza, Marco, , parallelogram rule, , – apriority of, – Parsons, Charles, , , Patton, Lydia, , , , , , –, , perspectivalism, – Physical Monadology, , , , –, –, , , , , , – physics, , Einsteinian, –, , empirical, – physica generalis, , – rational, –, , special, Plaass, Peter, , , –, , –, , , –, , Pollok, Konstantin, , , , , , , , , , –, , , , , , , , , , principle of succession, –, , principle of sufficient reason, , Prize Essay, –, Prolegomena to Any Future Metaphysics, –, –, , –, , , , , , , purposiveness, , –, , – quantum mechanics, Quarfood, Marcel, reason, –, , , , , –, , –, ideas of, –, , , –, , –, – regulative use of, , , –, practical, – systematicity, –, –, , –, –, , –, Reflections, , , , , Reichenbach, Hans, Reinhold, Karl Leonhard, resistance. See matter, as filling space; matter, impenetrability Richards, Robert, Röd, Wolfgang, , Rohault, Jacques, Rousseau, Jean-Jacques,
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’s Gravesande, Willem, Sarmiento, Gustavo, Schäfer, Lothar, Schönfeld, Martin, , , , Schopenhauer, Arthur, Schröter, Johann Hieronymus, Schulting, Dennis, Schu¨tz, Christian Gottfried, science, –, –, –, –, –, , definition of, –, – natural, , –, –, –, –, , , , –, classification of, demarcation of, proper, , –, –, , –, –, pure, –, – rational, , Shabel, Lisa, , , Sieg, Wilfred, , Smit, Houston, , Smith, Sheldon, , solidity. See matter, impenetrability Sotnak, Eric, , space, –, , , , , –, –, , , –, absolute, , –, –, , , , , , , as idea of reason, –, , –, , – efficient and equitable character, , –, – empty, –, –, , , – filled. See force, repulsive; matter, as filling space as ground of laws of nature, , –, –, –, – as infinite magnitude, relationalism, , –, relative, , –, , , Spagnesi, Lorenzo, speed, – Spinozism, – Stan, Marius, , , , , , , , , , , , , , , Stang, Nicholas, , , , –, , Stevens, Stanley Smith, Storrie, Stefan, , Stratmann, Joseph, Strawson, Peter,
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Index
Sturm, Thomas, –, , , , , , , –, , – substance, , , monads, –, , –, , , , – phenomenal, , , – Suisky, Dieter, – Sutherland, Daniel, , , , –, Swineshead, Roger, Sylla, Edith Dudley,
Tait, William, , , teleology, , , – natural and moral, – Thöle, Bernhard, , , Thorndike, Oliver, , Thoughts on the True Estimation of Living Forces, , time, –, , , , –, , –, absolute, , as infinite magnitude, temporal order, –, , , Tolley, Clinton, transcendental idealism, , , , – transcendental illusion, – transcendental philosophy, , , relation to natural science, –, –, –, , –, –, – transcendental realism, , Tuschling, Burkhard, ,
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understanding, , , , –, , – Universal Natural History and Theory of the Heavens, , , , , unsocial sociability, van den Berg, Hein, , , , , , –, Walsh, William Henry, Warda, Arthur, Warren, Daniel, , , , , , , , , , , , –, , , , , , Washburn, Michael, Watkins, Eric, –, , –, –, , , –, , , , , , Westphal, Kenneth, , , What Real Progress Has Metaphysics Made in Germany since the Time of Leibniz and Wolff ?, , Willaschek, Marcus, , Wilson, Curtis, Wittgenstein, Ludwig, Wolff, Christian, , , , , Wood, Allen, , Wunderlich, Falk, Zach, Richard, Zammito, John, , ,
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Titles published in this series (continued): Kant’s Lectures on Ethics Kierkegaard’s Fear and Trembling Kant’s Lectures on Anthropology Kant’s Religion within the Boundaries of Mere Reason Descartes’ Meditations Augustine’s City of God Kant’s Observations and Remarks Nietzsche’s On the Genealogy of Morality Aristotle’s Nicomachean Ethics Kant’s Metaphysics of Morals edited by Spinoza’s Theological-Political Treatise edited by . . Plato’s Laws edited by Plato’s Republic edited by . Kierkegaard’s Concluding Unscientific Postscript edited by Wittgenstein’s Philosophical Investigations edited by Kant’s Critique of Practical Reason edited by Kant’s Groundwork of the Metaphysics of Morals edited by Kant’s Idea for a Universal History with a Cosmopolitan Aim edited by é Mill’s On Liberty edited by . . Hegel’s Phenomenology of Spirit edited by
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