Diagrams, Visual Imagination, and Continuity in Peirce's Philosophy of Mathematics 3031232445, 9783031232442

Table of contents :
Preface
Contents
Chapter 1: Introduction
Two Approaches to Diagrams
Peirce’s Vision
What This Book Is About
Chapter 2: Meritocratism, Errors, and The Community of Inquiry
Meritocratism and Science
Peirce’s Maxim
Chapter 3: Logic and Mathematics
Self-Interpretation, Conventionality, and The Language of Thought
Two Kinds of Minds
Fast, Pedestrian, or Both?
Chapter 4: Peirce’s Transcendental Deduction and Beyond: Categories, Community, and the Self
The Kantian Legacy
Peirce’s Deduction
Some Consequences of Peirce’s Deduction: The Community of Inquiry and the Self
Fixation of Belief
Chapter 5: Sign Relation
Kinds of Relations
Degenerate Relations
Categories and Math
Chapter 6: One, Two, Three
Abstractions, Things, and Signs
Modes of Being
Chapter 7: Iconicity, Novelty, and Necessity
Novelty
Necessity
Inference as Observation
Chapter 8: The General and the Particular
Mathematical Diagrams as Icons
The General and the Particular Fused Together
Chapter 9: Diagrams Between Images and Schemata
The Role of Mathematical Cognition in Kant’s 1st Critique
Diagrams and Schemata
Chapter 10: Existential Graphs
Visualized Inferences and Inferential Visuality
The Graphs Explained
The Graphs Contextualized
Chapter 11: Iconicity, Similarity, and Habitual Action
Three Formulations of the Maxim
What Is Likeness?
Chapter 12: Mapping Philosophy: Peirce’s Quincuncial Projection
Language and Maps
Quincuncial Map
The Virtual
Chapter 13: L’image-Mouvement, Mathematically Sublime, and the Perception of Totality
Synthetic Unity vs. Dynamic Totality
The Mathematically Sublime
The Architectonic Role of Mathematics
Chapter 14: The Metaphysics of Continuity
Reality, Generality, and Continuity
The Analytic Approach to Continuity: Cantor and Dedekind
Continuity and Infinitesimals
Chapter 15: Conclusion
References

Citation preview

Mathematics in Mind

Vitaly Kiryushchenko

Diagrams, Visual Imagination, and Continuity in Peirce’s Philosophy of Mathematics

Mathematics in Mind Series Editor Marcel Danesi, University of Toronto, Canada

Editorial Board Louis H. Kauffman, University of Illinois at Chicago, USA Dragana Martinovic, University of Windsor, Canada Yair Neuman, Ben-Gurion University of the Negev, Israel Rafael Núñez, University of California, San Diego, USA Anna Sfard, University of Haifa, Israel David Tall, University of Warwick, United Kingdom Kumiko Tanaka-Ishii, University of Tokyo, Japan Shlomo Vinner, The Hebrew University of Jerusalem, Israel

The monographs and occasional textbooks published in this series tap directly into the kinds of themes, research findings, and general professional activities of the Fields Cognitive Science Network, which brings together mathematicians, philosophers, and cognitive scientists to explore the question of the nature of mathematics and how it is learned from various interdisciplinary angles. Themes and concepts to be explored include connections between mathematical modeling and artificial intelligence research, the historical context of any topic involving the emergence of mathematical thinking, interrelationships between mathematical discovery and cultural processes, and the connection between math cognition and symbolism, annotation, and other semiotic processes. All works are peer-reviewed to meet the highest standards of scientific literature.

Vitaly Kiryushchenko

Diagrams, Visual Imagination, and Continuity in Peirce’s Philosophy of Mathematics

Vitaly Kiryushchenko Department of Philosophy York University Toronto, ON, Canada

ISSN 2522-5405     ISSN 2522-5413 (electronic) Mathematics in Mind ISBN 978-3-031-23244-2    ISBN 978-3-031-23245-9 (eBook) https://doi.org/10.1007/978-3-031-23245-9 © Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

I think there could be a mathematical explanation of how bad your tie is Russell Crowe as John Nash in Ron Howard’s “A Beautiful Mind” Just as we say that a body is in motion, and not that motion is in a body, we ought to say that we are in thought and not that thoughts are in us Charles S. Peirce

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Preface

This book is about the relationship between necessary reasoning and visual experience in Charles S. Peirce’s mathematical philosophy. As we know from Kant, vision (as a part of human sensibility) and responsiveness to reasons (as supported by our overall conceptual capacities) are related with one another through the imagination. Mathematics is an expression of this relation based on our most fundamental intuitions about space and time. Peirce went a long way to develop Kant’s take on the nature of mathematics, and central to his interpretation of it was the idea of diagrammatic reasoning. According to Peirce, in practicing this kind of reasoning, one treats diagrams not simply as external auxiliary tools, but rather as immediate visualizations of the very process of the reasoning itself. As thinking, in this case, is actually performed by means of manipulating images, seeing and understanding become one. Defining diagrammatic reasoning as a fusion of vision and thought helped Peirce find some intriguing answers to questions concerning the nature of mathematical knowledge, many of which could not even be as much as formulated by Kant. What is the role of observation in mathematics? How can we explain the fact that mathematical reasoning is deductive and, at the same time, capable of the discovery of new truths? How is mathematical necessity reconciled with the essential incompleteness and indeterminacy of our ordinary visual experience? What exactly is the relationship between the particularity of a mathematical diagram and the generality of the meaning it conveys—and what is the difference (if any) between mathematics and natural languages in this respect? and so on. Peirce’s life-long, if unsystematic, work on the issues that are associated with the questions above created an intricate conceptual puzzle. The driving motivation of the research this book represents is to show that tackling this puzzle requires something more than sifting through the wealth of available historical and philosophical material. While the histories of science and philosophy do provide separate bits of the puzzle, Peirce’s theoretical interests, by his own admission, appear to be closely intertwined with certain facts of his personal history. In light of this, without considering relevant biographical data, in Peirce’s case, there is no way to understand how the pieces of the puzzle actually fit together. Due to the plurality of data impelled by the task, this book is vii

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Preface

addressed both to those specializing in philosophy, mathematics, and intellectual history, and to a wider audience that might be interested in what all those areas have in common in Peirce’s case. Last but not least, this book could not have been written without the support of family, friends, and colleagues. I am especially grateful to Eric Bredo, Marcel Danesi, Paul Forster, Nathan Houser, Henry Jackman, Steven Levine, Mark Migotti, and James O’Shea. Of great importance for the book were my conversations with Kathleen Hull and Thomas L. Short. As to the biographical part of the study, I am indebted to Joseph Brent, the author of the most comprehensive biography of Charles Peirce to date. Finally, I owe much more than I can tell to my constant companions and interlocutors, Zina Uzdenskaya and Gleb Kiryushchenko. Toronto, ON, Canada

Vitaly Kiryushchenko

Contents

1

Introduction����������������������������������������������������������������������������������������������    1 Two Approaches to Diagrams������������������������������������������������������������������     1 Peirce’s Vision������������������������������������������������������������������������������������������     4 What This Book Is About������������������������������������������������������������������������     6

2

 Meritocratism, Errors, and The Community of Inquiry���������������������   11 Meritocratism and Science����������������������������������������������������������������������    11 Peirce’s Maxim����������������������������������������������������������������������������������������    15

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Logic and Mathematics ��������������������������������������������������������������������������   21 Self-Interpretation, Conventionality, and The Language of Thought������    21 Two Kinds of Minds��������������������������������������������������������������������������������    25 Fast, Pedestrian, or Both?������������������������������������������������������������������������    29

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Peirce’s Transcendental Deduction and Beyond: Categories, Community, and the Self ������������������������������������������������������������������������   31 The Kantian Legacy ��������������������������������������������������������������������������������    31 Peirce’s Deduction ����������������������������������������������������������������������������������    35 Some Consequences of Peirce’s Deduction: The Community of Inquiry and the Self ����������������������������������������������������������������������������    39 Fixation of Belief ������������������������������������������������������������������������������������    43

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Sign Relation��������������������������������������������������������������������������������������������   49 Kinds of Relations������������������������������������������������������������������������������������    49 Degenerate Relations ������������������������������������������������������������������������������    58 Categories and Math��������������������������������������������������������������������������������    60

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One, Two, Three ��������������������������������������������������������������������������������������   63 Abstractions, Things, and Signs��������������������������������������������������������������    63 Modes of Being����������������������������������������������������������������������������������������    68

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Contents

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 Iconicity, Novelty, and Necessity������������������������������������������������������������   75 Novelty����������������������������������������������������������������������������������������������������    75 Necessity��������������������������������������������������������������������������������������������������    77 Inference as Observation��������������������������������������������������������������������������    81

8

 The General and the Particular��������������������������������������������������������������   85 Mathematical Diagrams as Icons ������������������������������������������������������������    85 The General and the Particular Fused Together��������������������������������������    88

9

 Diagrams Between Images and Schemata ��������������������������������������������   93 The Role of Mathematical Cognition in Kant’s 1st Critique ������������������    93 Diagrams and Schemata��������������������������������������������������������������������������    98

10 Existential Graphs ����������������������������������������������������������������������������������  105 Visualized Inferences and Inferential Visuality ��������������������������������������   105 The Graphs Explained������������������������������������������������������������������������������   107 The Graphs Contextualized����������������������������������������������������������������������   112 11 Iconicity,  Similarity, and Habitual Action ��������������������������������������������  117 Three Formulations of the Maxim ����������������������������������������������������������   117 What Is Likeness?������������������������������������������������������������������������������������   120 12 Mapping  Philosophy: Peirce’s Quincuncial Projection������������������������  125 Language and Maps ��������������������������������������������������������������������������������   125 Quincuncial Map��������������������������������������������������������������������������������������   128 The Virtual ����������������������������������������������������������������������������������������������   138 13 L  ’image-Mouvement, Mathematically Sublime, and the Perception of Totality ����������������������������������������������������������������  141 Synthetic Unity vs. Dynamic Totality������������������������������������������������������   141 The Mathematically Sublime������������������������������������������������������������������   144 The Architectonic Role of Mathematics��������������������������������������������������   149 14 The  Metaphysics of Continuity��������������������������������������������������������������  153 Reality, Generality, and Continuity����������������������������������������������������������   153 The Analytic Approach to Continuity: Cantor and Dedekind������������������   156 Continuity and Infinitesimals������������������������������������������������������������������   159 15 Conclusion������������������������������������������������������������������������������������������������  163 References ��������������������������������������������������������������������������������������������������������  167

Chapter 1

Introduction

Two Approaches to Diagrams Mathematicians apply visual diagrams1 in their work all the time, whether they want to make special use of Euclid’s fifth postulate, to prove Fermat’s principle, or to extract an algorithm that defines the seemingly chaotic movement of pigeons picking bread crumbs from the ground. Using diagrams helps mathematicians identify patterns that solve particular mathematical problems by making the force of necessary reasoning visually given. An important question, though, is: What is it that a mathematician actually manages to see when using visual diagrammatic representations? A mathematical diagram, a paradigmatic use of which is exemplified in Euclid’s Elements, is an individual image that instantiates necessary relations. A mathematical diagram, therefore, is a relational image of some universal mathematical truth. As such, it has a dual nature. On the one hand, as an observable entity, it allows a mathematician to experiment upon it and to visually demonstrate the necessity of a given conclusion. On the other hand, it represents an abstract mathematical entity that cannot be reduced to a sum total of its particular representations. It follows, then, that a diagram partakes the characteristics of both an individual image observed at this particular instant and an object of a general nature that makes a thing what it is at any particular moment of its existence. In combining the characteristics of individual images and abstractions, diagrams show the essential in a thing at the expense of the features that are less prominent or less relevant to the case. A diagram may thus be considered a visible form (είδος) that, as the Platonic geometers once believed, represents an immediate union of knowing and seeing. But again: Seeing what exactly? There are two ways to define what using mathematical diagrams actually amounts to, depending on what kind of questions we would like to ask. For example, our interest might be in the practicalities of using diagrams. In this case, we normally do well simply by establishing that, when using those diagrams with the aim to

 Peirce’s idea of diagrammatic likeness (or parallelism in relations rather than in appearance taken simpliciter) indeed goes beyond visual perception. However, the current research concentrates on the latter exclusively, so throughout this book a “diagram” will always refer to vision. 1

© Springer Nature Switzerland AG 2023 V. Kiryushchenko, Diagrams, Visual Imagination, and Continuity in Peirce’s Philosophy of Mathematics, Mathematics in Mind, https://doi.org/10.1007/978-3-031-23245-9_1

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visualize something that is, by its nature, general, we apply those diagrams as auxiliary tools. In doing so, we facilitate our reasoning by some external means and then translate the results into a formal symbolic calculus in order to make inferences. From this perspective, although the diagrams are constructed as elaborately staged observations that make certain steps of a mathematical proof visually available, those diagrams do not constitute an entirely independent mathematical language and are but partial models designed for the purposes of informal demonstration only. Accordingly, on this view, mathematicians do not actually build proofs on visual imagery directly, but rather use the latter to enhance the symbolic formalizations of the former, where the symbolic formalizations are considered as the mathematical language proper. This view does have significant practical merit as it proves suggestive of a variety of particular modes of use associated with diagrams in mathematics. The above, however, is not the only possible approach. Instead of simply accepting the fact that formal proofs are sometimes not enough, we might be interested in figuring out why exactly that is so. In this case, instead of inquiring into certain particular modes of using diagrams in mathematical proofs, we would rather be asking a question about the conditions of the possibility of diagrammatic experience. Our interest would be metaphysical rather than purely practical, and the straightforward approach just outlined would not be of much help. After all, explaining, in general terms, the advantages of using something by saying that it is good for the purpose is like explaining the effects of opium, as the doctor in Moliere’s “Imaginary Invalid” famously puts it, by virtus dormitiva, or its “capacity” to do what it does. Moreover, in this case, simply saying that a diagram is an individual object that represents some abstract entity (as we just established above) would not suffice either. Such is every sign, written and spoken words included. In order to achieve progress on the matter, we would need to make some further assumptions. As diagrams are visual signs that help us observe necessary relations, there has to be something about the very nature of our ordinary visual experience that makes such observation possible. In other words, there has to be a direct link between our most basic mathematical intuitions and the cognitive mechanisms that, in appealing to spatial relations, enable our visual integration. When considering diagrams, we should presuppose a necessary connection between the way deductive inferences generally work and the basic schematisms behind the ordinary perceptions that make those inferences a matter of vision. This, in turn, means that we have to assume further that some mathematical notions are at the core of our ordinary perception. There is, in short, a mathematical a priori of human sensibility, the very possibility of which naturally brings forth the further assumption that mathematical diagrams, properly constructed, might represent an independent deductive language—which is to say that they might themselves be not just auxiliary tools, but immediate visualizations of the mathematical deductive process as such. From this point of view, the necessary character of deductive arguments would not be simply illustrated by diagrams; it would be internal to the diagrams that mathematicians create. There would have to be something in the very way the diagrams are constructed that conditions their deductive force as it is directly given in visual

Two Approaches to Diagrams

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perception. In fact, as will become clear later in the book, in inquiring about the role of visual experience in mathematical cognition, we should assume that there is a reciprocal relationship between our ordinary vision, on the one hand, and the mathematical demonstrations that supposedly constitute an independent mathematical language, on the other hand. The problem though is that (as will be discussed in due course) thanks to its reciprocity, this very relationship also needs to be shown rather than simply described. The two views summarized above constitute the framework within which most of the research on the use of diagrams in mathematics relevant to the current study is situated, with subjects ranging from the general and historical analysis (Brown 1999; Cheng et  al. 2001; Giardino 2017; Mumma and Panza 2012; Otte 1997; Pietarinen 2011, 2016) and the overall logical status of diagrams (Shin 1994, 2002; Sloman 2002) to the role diagrams play in mathematical proofs (Barker-Plummer 1997; Dörfler 2005; Hanna 2007; Kulpa 2009; Moktefi and Shin 2013; Mumma 2010; Sherry 2009), the role of analogical reasoning in scientific concept-formation (Abrahamsen and Bechtel 2015; Nersessian 1992), the use of diagrammatic tools in teaching mathematics (Bakker and Hoffmann, 2005; Boaler 2016; Danesi 2016b; Hegarty and Kozhevnikov 1999; Kucian et  al. 2011; Legg 2017; Murata 2008; Novak 1998; Prusak 2012), and the application of mathematics to the making of geographic maps (Kiryushchenko 2015; Tversky 2000). Whatever the differences between the two views are, there is an underlying assumption shared by the majority of the researchers on both sides of the aisle. The assumption is that, because of the aforementioned dual nature of diagrams (as connecting the particular and the general, images and relations), studying them may be seen as a way of bridging the gap between the purely platonic view of mathematics as a domain of abstract, unchanging forms, on the one hand, and theories that aspire to uncover the experiential basis of mathematical truths, on the other hand (Danesi 2016a, pp. 15–18). This assumption is epitomized in the claim that the cognitive structure of mathematics presupposes a strong connection between mathematical abstractions and metaphorical cognition, where “metaphorical” is understood as providing a link between our conceptual capacities and our bodily experiences. In this view, our ideas of quantity and number systems are linked to our perception through what is known as “conceptual blending” (Lakoff 1999; Lakoff and Núñez 2000). To unpack the metaphors that instantiate such blending is to reveal the cognitive schematism deeply ingrained in mathematical cognition. Revealing this cognitive schematism, in turn, would require showing that both figurative language and complex math are “implanted in a form of cognition that involves associative connection between experience and abstraction” (Danesi 2016b, p. 4). Mathematical diagrams are well-­ suited to be used in order to demonstrate the truth of this assumption. They differ from discrete symbolizations in that they involve both observation and generalization and are capable of effectively combining pictorial and algebraic characteristics. This being the case, a study of diagrams (whether with an eye on particular modes of their application or concentrating on their general metaphysical properties) constitutes a productive way to approach the associative connection in question.

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It is often claimed that all representations, algebraic formulas and linguistic expressions included, always already contain some sort of a diagram (at least in the sense that any information can be diagrammatized, i.e., represented as some sort of a visual schema). Therefore, it might seem that, in presenting diagrams as being in opposition to linguistic expressions and algebraic formalizations, we are trying to describe an opposition between a part and its whole. Yet once we understand clearly what exactly is being opposed to what in this case, the problem disappears. The true opposition here is between a visual representation of a piece of information, where the whole of this information is grasped immediately as such, and a linear string of arbitrary linguistic signs, where grasping the whole is the end result of reading the signs consecutively.

Peirce’s Vision One of the thinkers who pioneered the research of the role diagrams play in logical and mathematical reasoning was the philosopher and mathematician Charles Sanders Peirce (1839–1914). Peirce was one of the earliest authors on visual perception whose account included not just immediately given particulars, or sense data, but also kinds, relations, modal qualities, and other general objects. He is famous for claiming that anything general is, by definition, relational; that a relation is something that can always be represented visually by means of a diagram; and that the capacity to recognize relational similarities is a necessary condition not only for vision but for human perception in toto (CP12: 314; CP6: 190, 595; CP7: 499). As a logician and a mathematician, Peirce also believed that the capacity for the recognition of relational sameness is fundamental for human rationality, and that, if expressed graphically, a proper understanding of this capacity can provide the basis for a deductive mathematical language. This language would amount to a complete system of diagrammatic expression independent from formal symbolic proofs. In using such language, one would be able to accomplish all necessary derivations by applying a set of basic, intuitively grasped graphic conventions rather than an essentially arbitrary system of natural language-based signs. Creating such an independent language and applying it in mathematical proofs, Peirce believed, could show that diagrams might be construed not just as external aids to mathematical reasoning, but rather as immediate visual embodiments of the very process of such reasoning. This, in turn, could address the metaphysical concern about using diagrams in mathematics as described above and help us clarify the reciprocal relationship between mathematical demonstrations and visual experience. It is important for the current study that Peirce’s theoretical convictions were in sync with some of his personal proclivities. Peirce was a deeply visual thinker. So

 Peirce, C. S. (1931–1958). Collected Papers of Charles Sanders Peirce. Vols. 1–8. C. Hartshorne, P. Weiss, & A. Burks (Eds.). Cambridge, MA: Harvard University Press. 2

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much so that he often complained about his personal inability to cope with the limitations of written language and considered his predisposition to use graphical representation and visual images as evidence of his “intellectual left-handedness” (Hardwick 1977, pp. 95–96). As an educator, Peirce went as far as to claim that it would be a good idea if some sort of diagrammatic logic were taught to children in schools before the grammar of a natural language (CP4: 619). As Peirce’s numerous autobiographical sketches also suggest, he was convinced that his aptitude for visual representation had everything to do with his mathematical mindset and that both were responsible for certain pronounced traits of his own personality. As will transpire, being a mathematician, according to Peirce, is an intellectual advantage that comes at a price. Peirce’s aptitude for visual representations, of course, went beyond psychological and educational effect. He firmly believed that the link between this aptitude and his strong mathematical capacity was a necessary one and that this link pointed at something larger than a set of personal intellectual idiosyncrasies. In fact, Peirce was convinced that the link is valid for every mathematician and that there is no mathematical reasoning proper that is not diagrammatic (CP1: 54; CP2: 216; CP5: 148). According to Peirce, although we certainly can express our thoughts without using diagrams, the deductive force of our inferences is best expressed diagrammatically. There is little doubt that this belief was one of the principal driving forces behind Peirce’s work on his system of diagrammatic logic (his Existential Graphs, henceforth EG), which he began to develop in the mid-1880s and continued to improve incessantly until his death in 1914. The system is so designed as to combine some minimal algebraic symbolizations with a set of basic graphical conventions and to permit uninterrupted experimental manipulation with the resultant graph. Peirce devised a full-blown graphical grammar aimed at regulating this manipulation. The design of the grammar is such that the logical form of any graph constructed using this grammar, as well as any derivation the graph may represent, are immediately available to the eye. Deriving and carefully observing, according to Peirce, should be construed as one and the same process. Accordingly, every Peircean graph can do what no inference written in a formal symbolic language can: It can convey information and, at the same time, provide an interpreter with a key to how to decode it. Seeing how a given graph develops through a series of transformations into a meaningful structure, and understanding how this development works is one and the same act the mathematician performs. Conceived in this way, the fusion of seeing and understanding, which Peirce’s graphs represent, provides us with something more than unnecessary visual help to mathematical reasoning. The iconicity of the graphs can tell us something important about the homological relationship between the grammar of the visual language Peirce had in mind and the very machinery of thought, or, as Peirce himself puts it, thinking in actu (NEM43: 239; CP4: 6).

 Peirce, C. S. (1976). The New Elements of Mathematics. C. Eisele (Ed.). Vol. 4. The Hague, The Netherlands: Mouton. 3

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In Peirce’s case, the idea of visual language proves to be crucial not only for the proper understanding of the mode of reasoning specific to mathematics. It is also important for the proper understanding of his pragmatism and his semiotics, the two parts of Peirce’s philosophy which he struggled to reconcile throughout his life and which are permeated with mathematical thought. Seen from this perspective, visualized mathematical reasoning appears to be the very heartbeat that pumps blood through the veins of Peirce’s entire philosophical system and serves as a background for his philosophical realism (sf. Peirce 2010, pp. xvi–xviii; Fisch 1967). Moreover, according to Peirce, not only his own philosophical thought, but philosophy in general stands in dire need of mathematical concepts and can be properly understood as a worthy intellectual enterprise only if considered a consequence of visualized mathematical reasoning (cf. K.L. Ketner’s and H. Putnam’s introduction to Peirce’s Cambridge Conference Lectures in Peirce 1992, pp. 1–104).

What This Book Is About Now that the overall context of Peirce’s interest in visual thinking is established, it is time to explain what this book is about. Peirce’s theoretical intuitions pertaining to diagrammatic thinking represent an intricate knot of relations between written language, ordinary visual experience, necessary mathematical reasoning, and imaginative experimentation with graphically expressed information. The current study is preoccupied with this knot but pursues no ambition to solve it completely. Notably, it does not in any way intend to approach the matter against the background of a comprehensive reconstruction of Peirce’s mathematical philosophy. Neither will it go into great detail about Peirce’s EG, especially as there is ample research on the matter, both classical and cutting edge (see Pietarinen 2011; Roberts 1973; Shin 2002; Zalamea 2003; Zeman 1964 among a plethora of other sources). In considering the relationship between mathematical reasoning, diagrams, and everyday visual experience, we will not pay attention to either the direct cognitive causes of this relationship or the variety of ways in which mathematicians actually use pictures and diagrams in their work (for these topics, see the accounts of how the links between numerical and spatial representations are rooted in the same patterns of brain activity in Gracia and Noël 2008; Hubbard et al. 2005, and the discussions of particular ways in which conceptual material and images are combined in mathematical reasoning in Loeb 2012; Lowrie and Kay 2001; Martinec and Salway 2005; Pinto and Tall 2002). What this study offers instead is an approach that will allow tracing the roots of Peirce’s conception of a diagram in certain patterns of interrelation between his semiotics, his pragmaticist philosophy, his mathematical ideas, bits of his biography, his personal intellectual predispositions, and his scientific practice as an applied mathematician. Every one of those areas is thoroughly researched, yet they have

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never been considered together in one study. Meanwhile, even though challenging (especially stylistically), this approach might prove quite helpful. Discovering the patterns just described can give us an idea of what the mathematical eye actually sees with the help of diagrams and, at the same time, can provide us with a general conceptual imprint of the mathematical mind that possesses the power of such sight. Based on all these considerations, our strategy will be to develop five themes that intertwine—which is the reason why these themes are not discussed in the book one by one in consecutive chapters, but rather create a labyrinth of ideas in which the reader will have to find their own path. First, from the Peircean semiotic perspective, to learn what mathematical diagrams are is, to a large extent, to define what those diagrams are qua signs, which characteristics they share with other kinds of signs, and which characteristics are unique to them. As will transpire, there are three important facts about diagrams that define their similarity to other kinds of signs: their being capable of generating new meanings, their pragmaticist connection to habitual action, and their ability to fuse the general with the particular. At the same time, what distinguishes diagrams as signs is that they make it possible for us to observe a given argument in its development rather than to read it in a linear fashion. This distinctive feature of diagrammatic signs will help us better understand Peirce’s more general ideas of relation, interpretation, and likeness. It will also provide a useful context for discussing the Kantian roots of Peirce’s theory of signs. Diagrams considered as signs will be the primary focus in Chapters 5–8 and 10–11 of the book. Second, Peirce’s views on the role of diagrams in mathematical reasoning owe a great deal to Kant. Kant’s critical philosophy was an object of steady interest for Peirce the mathematician, and Kant’s influence on Peirce was mediated and greatly diversified by lessons in logic and mathematics from his father Benjamin (MS 310, 823; Colapietro 2006, pp. 173–174, 196). This mediation enriched Peirce’s perception of Kant’s critical approach—to such an extent that in his application for a Carnegie grant in 1902 Peirce avouched that “Kant’s criticism was, so to say, my mother’s milk in philosophy” (MS L4 75, pp. 26–27). It is this beverage, according to some, that made Peirce immune to the deficiencies of the classical empiricist approach in philosophy and mathematics (Friedman 1995; Short 2007, pp. 66–67, 81–83). Peirce also praised Kant for the fact that he based metaphysics on logic and for the decisive emphasis Kant laid on the idea of architectonic, or a systematic vision of reason based on the unity of its most fundamental theoretical, practical, and aesthetic ends (Nordmann 2006; Parker 1998, pp. 2–59). Importantly, Kant’s transcendental schematism of the imagination and his idea of the mathematically sublime provide a link between Peirce’s approach to the mathematical notion of continuity and his idea of dynamic totality. This link adds an interesting twist to

 Charles S. Peirce Papers. The Houghton Library, Harvard University, Cambridge, Massachusetts (henceforth, MS or L followed by the manuscript or letter number and, wherever needed, page number). 4

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1 Introduction

Peirce’s philosophy of mathematics, as in Kant’s Critique of Pure Reason, Totalität is one of the three mathematical categories of quantity responsible for the representation of the mathematical idea of number. The Kantian themes relevant to Peirce’s conception of diagrammatic thinking are discussed in Chapters 4, 9, and 13, with Chapter 9 presenting the material central for understanding the current project as a whole. Third, according to Peirce, diagrammatic representation has everything to do with the thinking process forming an uninterrupted continuum. From this perspective, to learn what diagrams are is to understand how mathematicians of Peirce’s day viewed the idea of continuity. Peirce was dissatisfied with what Cantor and Dedekind had to say about the continuity of the real number line and struggled to formulate his own solution to the problem. The novelty of Peirce’s approach is that he moves away from the algebraic understanding of continuity through the idea a sequence of real numbers and links it to the ideas of generality and triadic sign relation. This helps Peirce create a still wider semiotic context for his mathematical studies. The idea of continuity will be the matter of our attention in Chapter 14. Fourth, Peirce’s mathematical views on the nature of diagrammatic representation were deeply impacted by his work as a geodesist at the US Coast and Geodetic Survey. While working at the Survey and at the Harvard Observatory, Peirce made some original contributions as a mathematician in the fields of geodesy, spectroscopy, and astronomy. The Survey published some of Peirce’s ideas on the economy of research and the statistical approach to observational errors (CP3: 140–160; 7: 139–157). But Peirce’s measurements of gravity and his pendulum research apart, the most philosophically interesting and important outcome of Peirce’s work at the Survey was his quincuncial map projection―a version of the conformal stereographic projection created with the application of the theory of functions of a complex variable. The principal feature of Peirce’s map is that it allows continuous tessellation ad infinitum, in which all attributes of each new copy of the map exactly match those of all its immediate neighbors. Thanks to this feature, the two-­ dimensional map reproduces an essential characteristic of the surface of the three-­ dimensional spherical object the map represents: its continuity. As will be demonstrated in Chapter 12, this allows interpreting Peirce’s map as a virtual object and a mathematical diagram of the Earth’s surface. The aim of this chapter is to demonstrate that some of Peirce’s late semiotic ideas about diagrams find support in his early practice as an applied mathematician, thus providing an interesting example of the intersection of scientific practice and philosophical speculation. Finally, as we point out in Chapters 2 and 3, in Peirce’s case, some biographical data and a brief account of certain traits of Peirce’s character are necessary for the proper understanding of how various parts of his theory of iconicity are related to each other. Taken together, these details can shed more light on why exactly visual experience was crucial for Peirce the mathematician and how the idea of it is related to statistical reasoning, Peirce’s social habits, his conception of continuity, and the important distinction Peirce made between mathematics and logic. Our contention is that certain conceptual aspects of Peirce’s theory about diagrams should be

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considered as interrelated with certain existential aspects of his life. Accordingly, we believe that introducing a few bits of Peirce’s intellectual biography in this context is a must. It will prove conducive to the overall aim of the current project if only because a close connection between thinking and living is strongly suggested by Peirce’s own pragmatist methodology.

Chapter 2

Meritocratism, Errors, and The Community of Inquiry

Meritocratism and Science Charles Sanders Peirce was brought up in a family with two other mathematicians: his older brother James Mills and his father Benjamin. Benjamin Peirce (1809–1880), a disciple of a famous American mathematician and cartographer Nathaniel Bowditch (1773–1838), is best known for his book on linear associative algebra (Peirce 1882), his original calculations of Neptune’s orbit, and the so-called “Peirce Criterion,” which is still used today in mathematical statistics for the elimination of suspect experimental data (Kent 2011; Ross 2003). Like his son Charles, Benjamin Peirce was a highly unusual individual. The incomprehensibility and hermetic character of his lectures at Harvard were the subject of many legends and anecdotes. As one of his colleagues once wrote: his intuition of the whole ground was so keen and comprehensive that he could not take cognizance of the slow and tentative process of mind by which an ordinary learner was compelled to make his step-by-step progress. In his explanations he would take giant strides; and his frequent ‘you see’ indicated what he saw clearly, but that of which his pupils could get hardly a glimpse (Cajori 1890, pp. 139).

In an apt phrase of another Harvard colleague, Professor Peirce “was such remote a planet as to be visible only to a few telescopic minds” (Peabody 1927, p. 26). It should be noted though that, in spite of his arcane way to conduct lectures, Peirce’s father had some prominent disciples, among which were three presidents of Harvard, one of whom, Abbott Lawrence Lowell, wrote his thesis on quaternions, the hottest mathematical topic of the time. Benjamin Peirce’s overall attitude toward mathematics was in the general vein of the Puritan views that dominated New England before the publication of Darwin’s works in America. He understood mathematics as a divine language or a universal grammar that alone could provide the right clue to the riddle of Creation (Peirce 1881, pp. 9–38). One of his favorite illustrations of the pervasiveness of the laws of

© Springer Nature Switzerland AG 2023 V. Kiryushchenko, Diagrams, Visual Imagination, and Continuity in Peirce’s Philosophy of Mathematics, Mathematics in Mind, https://doi.org/10.1007/978-3-031-23245-9_2

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mathematics was the fact that ancient Greeks discovered conic sections as a matter of pure geometry, only to have Kepler centuries later rediscover the exact same sections as shapes of planetary orbits (Peterson 1955, p. 106). What made Benjamin Peirce’s lectures controversial was that science in general and mathematics in particular were presented in them as an esoteric art accessible only to a chosen few. According to one of his students: sometimes, even while stating his propositions, he would be seized with some mathematical inspiration, would forget pupils, notes, everything, and would rapidly dash off equation after equation, following them out with smaller and smaller chalk-marks into the remote corners of the blackboard, forsaking his delightful task only when there was literally no more space to be covered, and coming back with a sigh to his actual students. There was a great fascination about these interruptions; we were present, as it seemed, at mathematics in the making; it was like peeping into a necromancer’s cell, and seeing him at work; or as if our teacher was one of the old Arabian algebraists recalled to life. The less we knew of what was going on, the more attractive was the enthusiasm of the man (as quoted in Emerson 1987, p. 96)

This esotericism, a source for both high admiration and numerous complaints of his students, was only partly due to the lack of elocutionary talent that Peirce père himself had always been all too hasty to admit. Aside from the huge gulf that existed at the time between the low standards of Harvard school theory and the mastery of real scientific practice, the more substantial reason was the extreme meritocratism of Benjamin Peirce. As Menand (2001) puts it, “[i]t was said at Harvard that you never realized how truly incapable you were of understanding a scientific matter until Professor Peirce had elucidated it for you” (p. 153). Peirce senior was “a confirmed intellectual elitist, a pure meritocrat, with no democracy about him” (ibid.). He regarded understanding not as a desirable ultimate result of a slow, pedestrian learning process but as a necessary initial requirement. Genius, for him, was the absolute human criterion and sufficient reason for any sort of domination, including political, over the mediocre. As a man of character and a powerful mind who looked down upon society, Benjamin Peirce had always been explicit in his belief that the glory of a university should depend on a small group of gifted individuals. His earnest conviction was that a worthy educational institution had to nourish its scientific elite and that the latter, in return, was to take the entire responsibility for science both as a field of practical experiments and as an institutional enterprise. On this view, an administrator with no serious training in science would do nothing but harm to the university that hired him. Moreover, science, Peirce believed, was not supposed to be under any immediate social obligation, in view of the very character of its activity (Lenzen 1968; Struik 1968, pp. 413–419). He would not go so far as to deny that the social meaning of scientific research as something useful in the short run might occasionally coincide with its long-term goals. Such, for instance, was the case during the Civil War, when the results of the fundamental pendulum experiments performed by the US Coast and Geodetic Survey, where Benjamin Peirce served as Superintendent in the 1870s, proved to be conducive to creating detailed maps of the East Coast, which played a prominent part in the success of the Union forces (Menand 2001, pp. 177–200). Yet, as numerous letters from Peirce family correspondence suggest,

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the practical meaning of science proper, according to Benjamin Peirce, ultimately depended on one and only crucial condition: Science should have nothing to do with any kind of immediate practical application of its results. Peirce the elder treated mathematical (and, more broadly, scientific) inquiry in an Aristotelian vein, as an activity that is worthy of pursuit for its own sake. Aristotle’s practical syllogism tells us that a goal stated in its major premise should differ from a course of action appropriate to it, as brought about by the conclusion of the syllogism. In De Motu Animalium, Aristotle sums it up in an example: “I need a covering; and a cloak is a covering. I need a cloak. What I need, I have to make. … I have to make a cloak” (De Motu, 701a17–19, as in Nussbaum 1978, p. 40). According to Aristotle, in attaining a set goal, as a practical agent, I should be inclined to avoid any significant divergence from the action chosen in accordance with the goal. Thus, if I set out to save money, it would be a blatant contradiction on my part not to put some limit on my expenses. Now science, according to Benjamin Peirce, due to the very nature of its activities, presupposes an entirely different motivational structure. A meritocratic scientist, for whom the pursuit of truth is an activity that has an absolute value in itself, may temporarily diverge from his short-term goal by committing errors, without thereby violating any norms of the logic of his research. Thus, for instance, we think of Ptolemy as a true mathematician in spite of the facts that his Almagest is premised on the geocentric model, that the ecliptic longitudes included in his star catalogue are far from being accurate, and that Hipparchus’s assessments of the precession of the equinoxes used by Ptolemy are off their actual value. Part of the reason a scientist is never swayed by the anomalies and failures of predicted results is that these anomalies and failures are publicly recognized, subject to criticism, and, as such, are the principal building blocks of the scientific practice itself. Errors thus appear to be the necessary elements of scientific research as a self-corrective, self-organized, dynamic process, where those involved in it are capable of choosing their own subjects of interest, irrespective of whether the larger society approves of those subjects. They are also intolerant of any sort of external planning. As Michael Polanyi aptly put it eighty years after the publication of Benjamin Peirce’s Ideality in the Physical Sciences, “[y]ou can kill or mutilate the advance of science, you cannot shape it. For it can advance only by essentially unpredictable steps, pursuing problems of its own, and the practical benefits of these advances will be incidental and hence doubly unpredictable” (Polanyi 1962, p. 64). Following this logic, because the scientific pursuit of truth is self-corrective and performed for its own sake, only science can be considered a domain of human effort where there’s no essential contradiction between belief and action. A researcher is anyone but a struggling Kantian rational agent torn between his personal desires and the universal rule of conduct he knows he must apply. For this agent, acting on a rule does not necessarily follow from his awareness of this rule. Contrary to this, what a scientist believes to be worth pursuing, and what he does in order for his belief to have a practical meaning, amounts to the same sort of considerations: so far as he follows a set of methods and procedures approved by his research community, he always acts in the way he ought to. In this case, the question

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central to deontological ethics (“How do our beliefs motivate our actions?”) simply does not arise. Charles Peirce later paraphrases this conclusion in his “Fixation of Belief” (1877), stating that: to avoid looking into the support of any belief from a fear that it may turn out rotten is quite as immoral as it is disadvantageous. The person who confesses that there is such a thing as truth, which is distinguished from falsehood simply by this, that if acted on it should, on full consideration, carry us to the point we aim at and not astray, and then, though convinced of this, dares not know the truth and seeks to avoid it, is in a sorry state of mind indeed (W31: 257).

As we will see shortly, what serves as a clip that holds together the overall Aristotelian meritocratic stance the Peirces shared and the argument about the nature of scientific inquiry just outlined is not a metaphysically described struggle between one’s personal inclinations and a universal rule of conduct, as a Kantian would suggest, but a statistical interpretation of error. Effectively, within the century between the publication of Kant’s Universal Natural History and Theory of the Heavens (1755), Malthus’s On the Principle of Population (1798), Laplace’s five-volume Celestial Mechanics (1799–1825), and Darwin’s On the Origin of Species (1859), science had been imbued with mathematically grounded statistical methods, for which a concept of error was one of the essentials. By applying probability theory to discrepancies of observation, the nineteenth-­century scientist not only knew that his calculations contained a certain number of errors; he actually needed those errors in order to build his explanatory models. He knew that studying the distributions of the errors meant using those errors as tools for arriving at true beliefs. As Menand (2001) aptly puts it, in showing that every belief, with no exception, is falsifiable, statistical methods “conquered uncertainty by embracing it” (p. 182). Prima facie, it might seem that all “statistical” refers to in the description above is a set of problem-solving techniques demonstrating how self-corrective behavior emerges from patterns of interrelation between our goals and the errors committed on the way to achieve those goals (for example, when various kinds of reinforcement learning techniques or adaptive behavior are applied). This would not be a total misunderstanding of how the Peirces actually treated the idea of the progress of scientific knowledge and would be especially true in the case of Peirce Jr. who, although he had his reservations concerning Darwin and refused to view scientific theories simply as instruments of organic adaptation, endorsed the statistical part of Darwin’s theory and used the notion of biological adaptation in explaining certain purely intellectual phenomena (see, e.g., Skagestad 1979). Be it as it may, there is more to this story, given what has just been said about Benjamin Peirce’s views on scientific teleology.

 Peirce, C.  S. (1982). Writings of Charles S.  Peirce. A Chronological Edition. Vols. 1–6. M.  H. Fisch, E.  Moore, C.  Kloesel, & Peirce Edition Project (Eds.). Bloomington, IN: Indiana University Press (henceforth, W followed by volume number and then page number). 1

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Peirce’s Maxim Recall that, in the case of science, errors become an important public asset. Keeping this in mind, together with Benjamin’s meritocratic perspective on the responsibilities of science before the wider social world as described above, it might be claimed that the Peirces viewed scientific (and, ultimately, mathematical) community as representing not just a set of professional institutions freely pursuing their specific goals, but an ideal pattern of social life sensu lato. Whereas the majority of other social contexts presuppose the need to conceal errors, when statistically approached, errors appear to be the necessary elements of science. Scientific truth appears to be nothing other than a function of those errors. It is important that, according to both Peirces, accepting the idea of mathematical community thus defined, necessarily involves the defiance towards any kind of pre-established unifying forms and conventions that cannot be tested and falsified. In other words, the overall scientific logic, which drives mathematical research, tends to deny any possibility of non-­ reflective actions dictated by some beliefs fixed once and for all. To put it simply, doing math opens the possibility to think outside the box, and both Charles and Benjamin viewed this possibility not as a hindrance to social cooperation (as it might well seem to be to most of us), but as the ultimate source of such cooperation. Like his father, Charles Peirce laid stress on the capacity to err as an essential human capacity and linked it to the evolutionary idea of fortuitous variation. While inanimate objects have no capacity to err, and in lower animals, such capacity is very limited, the tendency to commit errors appears to be essential for being human. This tendency, “when you put it under the microscope of reflection,” shows up as a result of fortuitous micro-variations in our responses to the environment. And as far as such variations allow us humans to learn and self-correct, they are responsible for the plasticity and high complexity of our habits and, therefore, are formative for human intelligence (CP6: 86). The idea of fortuitous variation, applied to the development of science understood as the human endeavor par excellence, serves as a background of Charles Peirce’s definition of the “community of inquiry.” He describes this community in his “Some Consequences of Four Incapacities” (1868) as “a community without definite limits, and capable of indefinite increase of knowledge,” where “[the] two series of cognitions―the real and the unreal―consist of those which, at a time sufficiently future, the community will always continue to reaffirm; and of those which, under the same conditions, will ever after be denied” (W2: 239). Again, what underlies the idea of the end of inquiry is statistical reasoning. As Peirce explains, “judging of the statistical composition of a whole lot from a sample is judging by a method which will be right on the average in the long run” (CP 1:93). On this view, no matter where different members of a community may begin, as long as they follow a certain method, the results of their research should eventually converge toward the same outcome. The method is formulated in Peirce’s maxim of pragmatism: “Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our

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conception of the object” (W3: 266). This formulation suggests that our idea of an object is ultimately our idea of the expected experiential effects of this object. It also suggests that meanings of our ideas depend on our capacity to predict practical outcomes of our experiments with the objects of those ideas, whether those objects are abstract mathematical concepts, chemical substances, or physical bodies. The idea of practical bearings, as applied in the maxim, finds its expression in a set of conditional statements about what would happen, given that such-and-such experimental conditions are in place. Consequently, accepting the truth of a proposition amounts to acquiring a habit of using a variety of conditionals the expression entails. On this view, the meaning of a proposition “spells out how acceptance of the proposition would affect conduct, and indicates what circumstances are relevant to evaluating an assertion of the proposition” (Hookway 1985: 240). To recapitulate an earlier point of Benjamin Peirce, as our ideas about reality are exhausted by conceived conditional resolutions about our possible behavior, in following the maxim as scientists, we cannot but act on our beliefs. And, given that we are persistent enough in following the maxim as a method of reasoning and acting, our results will always be a distribution of statistical errors, which, as our experimentation goes on, always converge to an approximation (CP1: 400–409). In other words, in referring to the full account of meaning as the sum total of practical effects, the maxim represents a method about which we know that its application will yield true beliefs on average in the long run (although we never know in advance what variations will be especially conducive to our efforts). What the method does, statistically speaking, is it enhances our communal cooperation and gradually removes the indeterminism by sorting our conditional expectations into patterns, each of which represents a selection from a larger statistically admissible set. According to Peirce, if followed long enough, the method is supposed to ultimately give us a full conception of the objects of our interest. According to the statistical construal of the maxim, it is rational to act only if the single case we are dealing with presently is considered a member of the infinite series of similar or comparable cases. The series necessarily has to be infinite in order to display the tendency to converge to a result narrow enough to establish a new norm―which we begin to question immediately afterward (CP1: 400–409). Neither you nor I, or any finite number of the members of our research community can actually confront the series as a whole, as it is not just longer than any individual life, but extends beyond the life of any historical community. Therefore, neither you nor I, or any finite number of people who represent such a historical community are being rational unless here and now, in every decision we make, we identify ourselves with a communal “We” understood as an infinite community of future decision makers (W2: 239–242). According to Peirce, it is this community, not any limited historical one, or a particular member of it, that is the proper subject of knowledge. A community every member of which follows the maxim, by definition exceeds any particular set of members due to its continuous reliance on future opportunities to correct beliefs currently held. To sum up, as far as Peirce’s pragmatic maxim tells us that the meaning of any proposition consists in a set of counterfactual expectations, every concept we use by

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necessity addresses itself to possible future interpretation. In order to secure the self-corrective character of inquiry, the reference to the possible future (the reference that, as will be shown in the chapters below, is foundational for understanding of the mathematical grounds of Peirce’s overall philosophical project) presupposes the regulative idea of an ultimate agreement of the future communal “We.” This agreement is a guarantor of the truth of the conceptions that will survive through all statistical distributions of our errors and all our applications of the method. As we move towards this ultimate state of affairs, we communicate our ideas and correct our mistakes. Peirce insists that the very way we go about our experience suggests that the idea of an ultimate agreement is indispensable. Without having it in mind, we simply cannot go on with our inquiry. The agreement is thus defined as “an ideal, regulative, normative notion, providing a reason―an irreducibly normative reason―for continuing inquiry” (Pihlström 2012, p. 243). The notion of the ultimate agreement is regulative in that it both motivates our current inquiry and guarantees its result in the long run. On the level of local social and scientific practices, it cannot secure anything except statistical approximation. The approximation, however, helps us sort out our ongoing disagreements about the meaning of the practices we share. Peirce takes it to be an important advantage of his maxim that, whenever it is applied, the reality the resulting concepts describe is no longer determined by individual will, a priori rules, or even (as Benjamin’s meritocratism would suggest) any existent social contract. If it is determined by anything at all, it is rather by the more basic, pre-contractual fact that every human being is a truth-seeker by nature. Mathematics in its pragmatist understanding is nothing other than an extension of this natural (yet again, as the meritocrat would add, in most cases dormant) human disposition, or a more sophisticated expression of it. What it adds to this disposition is the idea of a method that provides sophisticated statistical tools to enhance our natural inclination toward making the right decisions. In following the method, we are not simply being “rational”—whatever this word might mean in abstracto. Mathematics (and science in general) has little to do with rationality as a fact or a capacity, and it has everything to do with it as a process. Without some sort of an account of how it came to be, this rationality could receive only an a priori definition—and, according to Peirce, a rather shallow definition at that. Peirce lays stress on two important details. First, he emphasizes that our rational capacities have a history, the understanding of which is crucial for the understanding of what the capacities actually are. Second, he claims that, in the case of science, this history is attached to not just any, but to a very particular set of social institutions, which have so far made the best possible use of these capacities (W3: 253–256). It might seem logical to suppose that, if the reference to possible future is indispensable (i.e., if we have to go on in every particular case no matter what), then our inquiry simply cannot terminate in an ideal state of complete information. On the face of it, then, the notion of a future community refers to something that is positively impossible to achieve. Peirce is aware of this difficulty, but cannot help adding some specifically mathematical optimism to the picture:

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2  Meritocratism, Errors, and The Community of Inquiry In many mathematical treatises, the limit is defined as a point that can ‘never’ be reached. This is a violation not merely of formal rhetoric but of formal grammar. True, in the world of real experience, ‘never’ has at least an approximate meaning. But in the Platonic world of pure forms with which mathematics is always dealing, ‘never’ can only mean ‘not consistently with___.’ To say that a point can never be reached is to say that it cannot be reached consistently with___, and has no meaning until the blank is filled up. And thereupon, the mathematical and balanced conception must be that the point is instantly passed through. The metaphysicians have in this instance been clearer than the mathematicians— and that upon a point of mathematics; for they have always declared that a limit was inconceivable without a region beyond it (CP4: 118; emphasis added).

To counterbalance the pessimistic view of the idea of infinite series of interpretation above, Peirce also formulates a more general maxim, which supports the mathematical understanding of limit and which Peirce calls “the principle of an intellectual hope”: [L]ogic forbids us to assume in regard to any given fact of that sort that it is of its own nature absolutely inexplicable. This is what Kant calls a regulative principle, that is to say, an intellectual hope. The sole immediate purpose of thinking is to render things intelligible; and to think and yet in that very act to think a thing unintelligible is a self-stultification. It is as though a man furnished with a pistol to defend himself against an enemy were, on finding that enemy very redoubtable, to use his pistol to blow his own brains out to escape being killed by his enemy. Despair is insanity. True, there may be facts that will never get explained; but that any given fact is of the number, is what experience can never give us reason to think (CP 1.405; emphasis added).

According to this principle, although any particular achievement of science is fallible, and an inquiry can undermine any one of them in the course of time, it cannot undermine all of them at once. No finite set of principles or newly discovered facts can disprove all the experiences we have at any moment in the course of our inquiry. The application of the method of science, as Peirce defines it in his maxim of pragmatism, presupposes converging to a limit (and, therefore, presupposes the idea of such limit), and yet at any point in time in the process of actual inquiry we always find ourselves only in a provisional stage of knowledge, unable to ascertain how far off we actually are from the limit. The future community, associated with the limit, is a regulative ideal that motivates us to go on because it provides us with a warrant that the true beliefs we are after will be established in the long run. To generalize this conclusion, the method-driven statistical approximation that is at work, in this case, establishes the goals of our inquiry as ultimate ideals, or ends. For this reason, everything there is should be defined by the way it conforms to the idea of what it is going to ultimately become. Our efforts to settle with an opinion that we take to be correct can succeed only insofar as we do our best in trying to bring what we currently believe in accord with what we imagine to be the end of our inquiry. Cognizability of reality, understood as a set of opinions which a community will arrive at as a result of an investigation carried sufficiently far, is thus a condition of the possibility of human knowledge, whereas the fallibility and incompleteness of this knowledge is the price human reason pays for its ambition to have it.

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There is one more aspect to the free and open inquiry that Peirce’s meritocratic idea of a future community represents. It will become important in the next chapter, where we will discuss Peirce’s distinction between the logical and the mathematical mindsets. For Peirce, to ensure the correctness and the proper quality of a scientific inquiry, one has to contextualize the facts and conditions that initiated the inquiry. Performing this task amounts to collecting a large volume of historical and technical data, the obtaining of which goes in line with the progress of the inquiry, but the bulk of which may end up being only remotely connected to its ultimate results. For instance, if I do a research on Petrus Hispanus’s logic, I cannot limit my studies to simply reading his Summulae Logicales. Obviously, I have to learn everything I possibly can not only about Aristotle’s philosophy and scholastic accounts of the syllogism but also about theologia naturalis and the contradictions between Aristotle’s logic and the doctrine of the Church. I will also likely need to learn about quite a few more things. For instance, the life of Pope John XXI, with whom Petrus is identified, the thirteenth-century history of Spain, Latin translations of Arabic texts on mathematics at the end of Reconquista, and the host of other facts and theories, all of which I will never need (all they represent is a large set of statistically admissible data), but some of which will prove conducive to my inquiry in the long run (although again, I will not know in advance which ones in particular). The process of creating such a vast context is not bound (at least not directly) by any formal requirements or immediate practical goals. Nobody can tell me as a researcher how to do my job. Yet, although this contextualization is not constrained by anything in particular, without performing it, the long-term goals of my inquiry will inevitably become problematic. It does not simply inform my inquiry but reshapes this inquiry as it progresses by helping me better connect with a wider research community. Ultimately, both the contextualization that accompanies my hypothesis-driven research and the research itself should converge at some limit, which is characterized, as Peirce puts it, by “the ideal state of complete information” (W2: 241). It is probably this theoretical understanding of an inquiry “carried sufficiently far” that Josiah Royce, one of Peirce’s closest friends, had in mind when he pointed at the tight connection between the following two personal facts about Peirce: Peirce was fond of saying that he grew up in a laboratory. Later, he did some good work in the observatory. Still later, he was busy with the conduct of a good many statistical researches in connection with the Coast Survey. He was early and long familiar with exact measurement, and with the theory and practice of the determination of the errors of measurement in the measuring sciences. So, when he spoke of being a scientific philosopher, he was not without a really close knowledge of what scientific method in philosophy ought to mean. That in addition he did not fail to appreciate some at least of the great historical thinkers was due to his wide, manifold, and in some respects, very thorough erudition—an erudition that remained, like many other of his personal possessions, somewhat capricious, despite its frequent thoroughness (Royce and Kernan, 1916, pp. 701–702; emphasis added).

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What Royce is pointing at here is that being an applied mathematician (a geodesist and an astronomer in this particular example) was for Peirce inseparable from being a historian of science, whose “capricious” but “thorough” erudition would always make any measurement or research report into a slow-paced, meticulous, and method-driven process, with the results incessantly refined and quite often ultimately left unpublished.2

 See esp. Peirce’s apologetic introduction to his application for a grant from the Carnegie Institution for Science in 1902 (MS L75). 2

Chapter 3

Logic and Mathematics

 elf-Interpretation, Conventionality, and The Language S of Thought The statistical theory of errors, which Peirce’s father applied in mathematics and astronomy, deeply influenced all principal concepts that cemented the architectonic of Peirce’s pragmatism. Apart from this, during Peirce’s lectureship at Johns Hopkins University (1879–1883), statistical methods instigated his interest in different theories of the natural conditions of criminality and unconventional behavior in general1, and his research into the psychology of great men in particular (W5: 26–106). Benjamin Peirce also nurtured in his son severe intellectual discipline. Combined with Peirce’s logical and mathematical genius, this discipline formed in him a steady habit of self-interpretation―the habit that, with time, developed to the extent that, in one of his letters to his lifelong friend William James, Peirce confessed: “I have been forced to study myself until I have become a devoted seautonologist” (MHFC2 Peirce to James, 07/16/1907). The habit of self-analysis was greatly reinforced by some peculiarities of Peirce’s own personality, by which he himself was deeply puzzled. Among these peculiarities, Peirce laid stress on the troubles he had with written language, his early-­ discovered disposition toward diagrammatical thinking, his poor social skills, and his left-handedness3. Peirce’s diaries, as well as his correspondence with friends and family, strongly suggest that the statistics-based research interests he developed, as  Peirce had a particularly acute personal interest in Galton’s eugenics and Lombroso’s anthropological criminology, with his principal source for the former being Lombroso’s 1883 book called Inquiries into Human Faculty and Its Development. 2  Max H. Fisch Catalogue at Peirce Edition Project, IUPUI. 3  As Joseph Brent notes in his Peirce biography, as a matter of fact, Peirce was able to use both of his hands in writing simultaneously, and from time to time entertained his students by writing on 1

© Springer Nature Switzerland AG 2023 V. Kiryushchenko, Diagrams, Visual Imagination, and Continuity in Peirce’s Philosophy of Mathematics, Mathematics in Mind, https://doi.org/10.1007/978-3-031-23245-9_3

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well as his psychological predispositions and personality traits, were intertwined in a Gordian knot which Peirce the seautonologist craved to untangle throughout his life (Lieb 1953, pp. 33–34; MHFC, CSP-HPE 01.03.06, pp. 1–6). This chapter does not intend to solve this knot completely, but it does intend to use some of the known steps of Peirce’s self-analysis in order to provide an explanation of the relationship between logic and mathematics, the clarification of which Peirce took to be a very important task. It will become clear shortly that in what follows in this chapter, Peirce refers to the relationship not so much between two disciplines, but rather between two different mindsets, or “kinds of minds.” His comparisons of the two, interpreted through the prism of the aforementioned psychological predispositions and personality traits, will equip us with a useful context for the more technical theoretical discussion of the role of diagrams in mathematics later in the book, beginning with Chapter 4. As Peirce confessed on many occasions, he had a strong habit of thinking by means of pictures and mappings rather than linguistic units (MS 619; MS L387; MHFC, Essays toward the Interpretation of our Thoughts, 06.04.09; CP2: 169). Thus, an oft-quoted entry in Peirce’s late diary reads: “I do not think I ever reflect in words: I employ visual diagrams, firstly, because this way of thinking is my natural language of self-communion, and secondly, because I am convinced that it is the best system for the purpose” (MS 619: 8, 1909). He was strongly inclined to attribute his capacity of visual thinking to his mathematical mindset: he thought mathematical reasoning, at bottom, to be an interplay of schemes, maps, and images, rather than a matter of arbitrary symbolic descriptions. Visual, iconic experience, Peirce believed, also was at the core of ordinary linguistic competence in general— to the extent that, as mentioned in the previous chapter, he was convinced that diagrammatic logic should be taught to children before the grammar of any natural language (see also CP2: 778, 3: 418). In his family correspondence over the years, Peirce confessed repeatedly that English, to him, was as foreign as any other tongue, and in his diaries sometimes linked his incapacity of linguistic expression to his left-­handedness (Brent 1998, pp. 39–60). He also insisted that being left-handed is indispensable and necessary for being a capable logician. On top of that, Peirce persistently complained about his inability for ethical self-control and his general disposition against social conventionality. In an early draft of “A Neglected Argument for the Reality of God” (1908), he confessed that, as a logician, he was “accustomed to think of Reason and Authority as opposite ways of determining opinions, and to approve of the former alone” (MS 842, pp. 180–181). One of Peirce’s late autobiographical sketches informs us that, although he received extensive training in aesthetics from his father, he did not inherit Benjamin’s noble character and struggled, with no apparent success, “to acquire a sovereignty” over himself (MS 619). According to one of Peirce’s letters to Victoria Welby, this failure could be partly explained by the fact that he was “brought up with far too lose a rein,” except that he “was forced to think hard and continuously” (Peirce

the blackboard, ambidextrously and simultaneously, a mathematical problem and its solution (Brent 1998, p. 15).

Self-Interpretation, Conventionality, and The Language of Thought

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1958, p.  417). In addition to all that, one of Peirce’s attempted autobiographies contains the claim that his grandfather “was, on account of his shocking opinions, ‘read out’ of that same Quaker society, for belonging to which Southwicks4 were so signally disgraced” (MHFC 01.23.11). This family aptitude for unconventional behavior Peirce, again, blamed on his left-handedness. Thus, one of his often-cited letters to a mathematician Cassius L. Keyser says: But I am left-handed; and I often think that means that I do not use my brain in the way that the mass of men do, and that peculiarity betrays itself also in my ways of thinking. Hence, I have always labored under the misfortune of being thought ‘original.’ Upon a set subject, I am likely to write worse than any man of equal practice (as quoted in Brent 1998, p. 43).

Another Peirce’s letter to Victoria Welby contains a somewhat more extensive and detailed explanation of the link between his left-handedness, his troubles with written language, his disdain for conventionality, and the meticulousness of his thinking habits—in relation to his passion for logic: [A]s a boy I invented a language in which almost every letter of every word made a definite contribution to its signification. It involved a classification of all possible ideas; and I need not say that it was never completed. … The grammar of my Language was, I need hardly say, modelled in a general way after the Latin Grammar …, and it never dawned upon me that they could be other than they are in Latin. Since then I have bought Testaments in such languages as Zulu, Dakota, Hawaiian, Jagalu, Magyar (Basque I have dipped into otherwise; and I learned a little Arabic from Edward Palmer whom I knew in Constantinople and later in Cambridge). These studies have done much to broaden my ideas of language in general; but they have never made me a good writer, because my habits of thinking are so different from those of the generality of people. Besides I am left-handed … which implies a cerebral development and connections of parts of the brain so different from those of right-handed people that the sinister is almost sure to be misunderstood and live a stranger to his kind, if not a misanthrope. This has, I doubt not, had a good deal to do with my devotion to the science of logic. Yet probably my intellectual left-handedness has been serviceable to my studies in that science. It has caused me to be thorough in penetrating the thoughts of my predecessors, not merely their ideas as they understood them, but the potencies that were in them (Hardwick 1977, pp. 95–96; emphasis added).

Taken together, all these letters and notes confirm that, in Peirce’s case, visual thinking, personal difficulties in dealing with written language, the idea of a universal language, left-handedness, and the tendency to disregard social conventions happen to be intimately connected with each other. Peirce not only was fully aware of this interconnectedness but persevered in trying to find some sort of theoretical justification of it. It is our hypothesis that a version of such justification might be found in Peirce’s numerous comparisons of the overall nature and goals of logical and mathematical knowledge—with the idea of visual thinking as a key stone that holds all the pieces together. First of all, reading through the quoted and other materials, one cannot help noticing a contradiction. On the one hand, although Peirce never pretended to be a  What Peirce refers here is the story of Lawrence and Cassandra Southwick, Peirce’s remote paternal relatives who lived in Salem, MA, where they were sentenced to death by the Court of Massachusetts for their Quaker beliefs, but escaped the hanging and later on settled on Shelter Island. 4

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professional linguist, he conducted an experimental statistical research into the grammars of numerous natural languages with just one aim, to find out empirically if there is a perfect universal language whose structure correctly reflects the very fabric of human thought. On the other hand, he was reluctant to accept the very possibility of pre-established unifying forms, which he took to be one of the consequences of his left-handedness and unconventionality. According to Peirce, the two personal traits (one marked by the desire to create an ideal, rule-governed communicative structure, another—by the reluctance to accept the laws that cannot ever be questioned) together result in a contradictory character of “the sinister” who, Peirce claims, “is almost sure to be misunderstood” (Ibid, p.  95)5. The inner conflict between the two selves (a left-handed intellectual who defies the norms of social contract, and a researcher who is captured by the idea of a perfect language for an ideal community), according to Peirce, has everything to do with the art of logic. With this last statement in mind, the urge “to be thorough” mentioned above requires some further attention. Namely, it might be used to explain the specific literalism that Peirce often insisted upon in his business correspondence, his permanent striving to clarify the detail, to precondition every would-be situation. The finest example of this counterfactual literalism (which, as discussed in the previous chapter, is directly suggested by the maxim of pragmatism) is found in Peirce’s letters to Daniel Coit Gilman, the first president of Johns Hopkins University, soon after Peirce’s scandalous resignation from this institution6. The letters, every single one of which is excessively long and detailed, show quite clearly that Peirce’s continuous and pertinacious plea for extensive explanations and clarifications, in which he insists in his correspondence with Gilman in spite of the obviously irremediable social failure, was a plea for the logical consistency and necessary continuity of communication. (MHFC, Peirce-Gilman 1883). Gilman’s response was predictable: Peirce was let go. Obviously, Peirce’s insistence on the total rationalization of every possible practical context seems to stand in stark contradiction with common sense, as well as with the net of usual hesitations, anxieties, and ambitions that configure our everyday social communication. From the point of view of the social conventions that regulated human interactions in Victorian America, the circumstances of Peirce’s forced resignation did not require any further clarification, and Peirce’s insistence simply did not seem to be rational, as it blatantly denied any practical significance to some well-established beliefs. Moreover, in his desire “to be thorough,” Peirce insisted on following a certain method that entailed the rules for achieving a concrete result—the truth—and emphatically not some particular norms of ethically sound conduct. In this case, the method (Peirce’s maxim of pragmatism)

 In calling himself “sinister” in the passage above, Peirce puns on the meaning of ‘left hand’ in Latin in an attempt to provide his friend Victoria Welby with an explanation for his thwarted careers in academia and in the US Coast and Geodetic Survey. 6  As is well known, Peirce’s professorship at JHU was abruptly terminated in 1883, presumably due to his extramarital affair with Juliette Froissy, who later, after Peirce divorced Melusina Fay, became his second wife. 5

Two Kinds of Minds

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is considered as a general formula of conduct, from which the norms are to follow unambiguously. Effectively, this stance brings any socially acceptable relations between theory and practice into disarray insofar as it turns a blind eye to any short-term interests of other people. What is required by the pursuit of truth, at which a theory is aimed, might neither fit within the practical limitations of a particular social contract nor comfort the majority of the members of the social group that supports it. As the pursuit of truth by a scientist is an activity performed for its own sake, the fact that the truth might be uncomfortable is of little concern for the scientist. Consequently, on the one hand, within the framework of existent social relations, the rationalizing effort, which Peirce had in mind in his correspondence with Gilman, was inevitably perceived by outsiders as immature and absurdly egotistic. On the other hand, Peirce often confessed that his chief trouble with Uncle Sam was that he could not make any sense of actions the whole meaning of which was simply to confirm relations already formalized. Given that the pursuit of truth is strongly associated with a specific method-­ driven collective endeavor, Peirce saw the resultant situation not as a conflict between individual rationality and socially grounded authority, but rather as the coexistence of two different social orders, one based on the short-term, backward-­ looking, survival-based rationality of a historical social group, and another representing an open-ended, future-oriented community of researchers. Peirce would, no doubt, readily acknowledge the Durkheimian triviality that the social has a nature and goals of its own, different from those of an individual. But as a logician, he was convinced that those goals were largely characterized by inconsistency and the lack of creativity, and were simply used as an excuse for one’s personal weakness that one needs to hide by using the weaknesses of others. In Peirce’s view, therefore, for one to be logically thorough and honest in one’s pursuits is, in a way, to be unconventional.

Two Kinds of Minds With the contradiction between the extremes of the desire to build a system of perfect communication, on the one hand, and the requirement to question established norms, on the other hand, sublated within the logical mindset, what about mathematics? Peirce saw both logic and mathematics as equally proper applications of his intellectual powers. He claimed that visuality was equally important for both. Yet he introduced a few important distinctions between the two. Overall, while he thought of mathematics as the primary tool for doing science, logic, for him, was first and foremost a formal enhancement of the natural disposition of human beings as truth-­ seekers. In “The Simplest Mathematics” (Chapter 3 of the unfinished Minute Logic, 1902), Peirce claims that mathematics does not in any way depend on logic. While reasoning is an essential part of both, logic might not be able to correct the errors a mathematician makes on the spot. According to Peirce, quite on the contrary, logic

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heavily depends on mathematics, as it is actually incapable of solving any problems of its own without the use of mathematics. “Indeed,” he adds, “all formal logic is merely mathematics applied to logic” (CP4: 228). Where a mathematician has made a mistake, an appeal to logic might not turn out to be all that helpful. But why? Because mathematical reasoning, Peirce says, is “so much more evident than it is possible to render any doctrine of logic proper—without just such reasoning—that an appeal in mathematics to logic could only embroil a situation” (CP4: 243). This is what Peirce wrote circa 1906 in an unfinished draft clarifying the difference between the two by using a very convincing metaphor: The distinction between the two conflicting aims [of logic and mathematics] results from this, that the mathematical demonstrator seeks nothing but the solution of his problem; and, of course, desires to reach that goal in the smallest possible number of steps; while what the logician wishes to ascertain is what are the distinctly different elementary steps into which every necessary reasoning can be broken up. … In short, the mathematician wants a pair of seven-league boots, so as to get over the ground as expeditiously as possible. The logician has no purpose of getting over the ground: he regards an offered demonstration as a bridge over a canyon, and himself as the inspector who must narrowly examine every element of the truss because the whole is in danger unless every tie and every strut is not only correct in theory, but also flawless in execution (MHFC, Fragment on logician and mathematician, c. 1906).

Two years later, in “Some Amazing Mazes” (The Monist, 1908), Peirce makes some notes on the same topic. He claims that logic and mathematics as “the two habits of mind are directly the reverse of each other,” that “a mathematician does not care to go to the trouble (which would often be very considerable) of ascertaining whether the theoric [sic!] step he proposes to himself to take is absolutely indispensable or not, so long as he clearly perceives that it will be exceedingly convenient.” Accordingly, considerations of convenience make a mathematician “introduce theoric steps which relieve the mind and obviate confusing complications without being logically necessary” (CP4: 614). Based on these characterizations, it seems reasonable to suppose that, according to Peirce, a person who has both the logical and the mathematical mindset is intellectually and psychologically apt to experience a significant tension between the two extremes; the tension that should be resolved in achieving some sort of balance between mathematical insight and logical meticulousness. In his Minute Logic (1902), Peirce further deepens the distinction between the two. The logician, he says, is interested in defining the very nature of reasoning rather than in figuring out the consequences of a particular hypothesis. The mathematician, on the contrary, is after the most efficient methods of reasoning, especially if their application can be extended to some anticipated new problems. He is not willing to spend time dissecting the parts of his reasoning whose correctness he currently sees as a matter of course. In dealing with the algebra of logic, “the mathematician asks what value this algebra has as a calculus. Can it be applied to unravelling a complicated question? Will it, at one stroke, produce a remote consequence?” CP4: 240). The logician wants nothing of the sort. Quite on the contrary, “the greater number of distinct logical steps, into which the algebra breaks up an inference, will for him constitute a

Two Kinds of Minds

27

superiority of it over another which moves more swiftly to its conclusions. He demands that the algebra shall analyze a reasoning into its last elementary steps” (Ibid.). What is thus a definite merit for one, is a detriment and a deficiency for another. The difference in question is far more than the one between two disciplines or two points of view. For Peirce, as mentioned earlier, it is a difference between two mindsets. One is at its best studying different ways in which conclusions are drawn, another feels at home actually drawing the conclusions. But by the same token, the two are also closely interrelated in the sense of representing two different aspects of one and the same language of rationality. While logic is about how, given the available means, the chosen end is to be pursued, mathematics is the very practice of such pursuit. Logic is a static grammar that supplies a set of rules for the contemplation of the forms of thought, abstractly defined. Mathematics is a habit-driven activity, a creative practice that puts the rules to work. Accordingly, if mathematics is a science of reasoning, then “just as it is not necessary, in order to talk, to understand the theory of the formation of vowel sounds, so it is not necessary, in order to reason, to be in possession of the theory of reasoning” (CP4: 242). Mathematics, as it were, is always there as an actual reasoning capacity ready to be actualized, unlike logic that provides a theory that gives details on what reasoning really is. Another important distinction between the two mindsets, according to Peirce, has something to do with a difference between hypotheses and facts. Mathematical necessity is something we simply recognize. This recognition requires neither any immediate appeal to logic nor any special warrant from it. Mathematics as a reasoning practice deals with hypotheticals and is not concerned about whether the hypotheticals are existentially true. It only traces out the consequences of hypotheses. Visualized in a diagram, a mathematical hypothesis does not have to relate to any matter of fact. Prima facie, it might seem that a disagreement between two mathematicians will introduce a problem, as, in this case, “the mathematicians find themselves suddenly abutting against brute fact” (CP1: 247). And, according to Peirce, “it is when we pass out of the realm of pure hypothesis into that of hard fact that logic is called for” (CP2: 191). Logic is a science of truth and, by extension, a science of fact, as whether or not there be any such thing as truth is a question of fact. However, as any supposition in a mathematical dispute is of our own making, we end up as competent as we might be to address any problem. And as the dispute relates merely to the consequences of a hypothetical state of affairs, a careful study of the diagrams representing the relevant hypothesis should ultimately provide all the means necessary to resolve it. And such study is a matter of pure mathematics (Ibid.). This argument might well be the end of it. However, Peirce reminds us that mathematical inferences are not only hypothetical but also apodictic. Although, as he notes, “we may reason imperfectly and jump at a conclusion; still, the conclusion so guessed at is, after all, that in a certain supposed state of things something would necessarily be true” (CP4: 233; emphasis added). This feature of mathematical reasoning, together with two other characteristics of the mathematical mindset just discussed (the mathematician’s reluctance to simply follow a path that is

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conventionally approved in advance, and his reasoning being relieved from any responsibility to facts, leaving it to logic), is closely related to what Peirce called “mathematical generalization.” This generalization is essentially different from the logical one understood as a transition from a premise about a number of samples to a conclusion about a class that comprises those samples. The mathematician, unlike the logician, is intensely interested in finding efficient methods of reasoning that can fast-track him to valid results. And he is involved in this quest always with a view to a possible extension of those methods to new problems. Peirce notes that the way the mathematician envisions and creates the possibility for such extension is by means of the generalization that, in the sense Peirce attaches to the term, the logician would be utterly reluctant to perform. In generalizing, the mathematician acts very much like a chess player who decides to play a gambit—an opening in which the player accepts a degree of risk by making a small sacrifice for the sake of a compensating positional advantage that the sacrifice creates. Peirce explains his point by the following examples of the way the mathematician thinks: [R]ather than suppose that parallel lines, unlike all other pairs of straight lines in a plane, never meet, he supposes that they intersect at infinity. Rather than suppose that some equations have roots while others have not, he supplements real quantity by the infinitely greater realm of imaginary quantity. He tells us with ease how many inflexions a plane curve of any description has; but if we ask how many of these are real, and how many merely fictional, he is unable to say. He is perplexed by three-dimensional space, because not all pairs of straight lines intersect, and finds it to his advantage to use quaternions which represent a sort of four-fold continuum, in order to avoid the exception (CP4: 236).

It makes no sense to ask a mathematician or a quantum physicist about whether imaginary numbers exist—in any possible sense of “exist.” Asking such a question is missing the point entirely. In introducing the idea of imaginary number, or of eleventh dimension, according to Peirce, a mathematician aims to exchange a smaller problem involving the uncomfortable exceptions that hamper his advance, for the bigger one that is free of such exceptions. “Generalizing,” in this case, means not simply blurring the mind’s eye focus with more abstraction. It means intervening, adding something new so that “all the relations with which he chooses to deal are of the nature of correspondences” (ibid.). Although both logic and mathematics, as two aspects of one and the same language of thought, ultimately rely on diagrammatic thinking, mathematics also has some specifics to it in this respect. Mathematics, Peirce says: requires a certain vigor of thought, the power of concentration of attention, so as to hold before the mind a highly complex image, and keep it steady enough to be observed; and though training can do wonders in a short time in enhancing this vigor, still it will not make a powerful thinker out of a naturally feeble mind, or one that has been utterly debilitated by intellectual sloth (CP2: 81).

A mathematician visualizes his reasoning process as a complex image that he has to be able to keep “steady enough to be observed.” The complexity of the image is due to the fact that it has to be observed in its totality (the idea of which will be the focus of discussion in Chapter 13).

Fast, Pedestrian, or Both?

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Geometrical graphs represent figures whose transformations are largely based on intuitively grasped spatial relations. Algebra, in turn, represents arrays of characters, with rules of permissible transformations attached to those characters, which are largely based on certain habits of association, so that “the mathematician just as directly perceives that another array is permissibly scriptable, as he perceives that a person talking in a certain tone is angry, or using certain words in such and such a sense” (CP4: 246). Mathematics thus is a direct exercise of a certain habit of mind— in the same sense of “direct” in which, Peirce claims, we directly know when a person is angry. This habit applies equally well to number theory and to topical geometry, and the results of its application are algebraic or geometrical depending on a mathematician’s current goals. What Peirce insists on here is that the representation of this habit is, at bottom, essentially diagrammatic in both cases, i.e., presupposes the capacity to perform experiments with the same visual schematisms. We do not read this representation off a set of consecutive examples, but rather witness it by keeping in mind a “complex image” of all possible transformations it presupposes (observing it in its totality). Mathematical practice of studying hypotheticals, Peirce suggests, also has something to do specifically with our aesthetic experience. On the face of it, deducing the consequences of hypotheses is the primary business of mathematics. This might seem to be all the mathematician does. But, Peirce notes, it should take some very special intellectual capacity to come up with such ideas as those of imaginary quantity and Riemann’s surface, to introduce non-Euclidian measurement, or to imagine ideal numbers and the perfect liquid. What is this special capacity? As the quote below suggests, Peirce hesitates to give his final verdict on the matter. Yet he claims that, while thinking in these terms means applying mathematics to a question that is not mathematical, the imagination of this form and magnitude reminds the one that is at work in poietic genius. He also adds that “if mathematics is the study of purely imaginary states of things, poets must be great mathematicians, especially that class of poets who write novels of intricate and enigmatical plots” (CP4: 238). Mathematics deals with hypotheticals and combines the power of deductive reasoning and creative intuition. Logic deals with facts and combines openness to criticism and the meticulousness of the analysis of different modes of argumentation. Mathematics achieves in practice what logic describes in theory. According to Peirce, the non-conventionality of a logician is, as it were, an imposition of theory onto practice: Full rational transparency and publicity of reasons is a logical requirement for a decent social life. Mathematics, in turn, is the enactment of this requirement by powerful insight.

Fast, Pedestrian, or Both? The distinction between the logical and the mathematical mindsets might also have something to do with two more of Peirce’s “seautonologist” characteristics. Both in his early diaries and in family correspondence, Peirce often used the words “fast”

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and “pedestrian” as self-descriptions, to which he ascribed significant value. Although the meanings of the words are never fully explained, the manner in which the words are used by Peirce suggests that, taken together, they describe a synthetic, logico-mathematical way of thinking; such that allows the pedestrian logical theory of thinking and its fast mathematical practice to coexist consistently. A thought is mathematically fast owing to the fact that there is something genuinely non-­ hermeneutic about it. Going back to what we achieved in the previous chapter, Peirce follows his father Benjamin in claiming that mathematical understanding is not so much a desirable final outcome of a gradually evolving procedure as it is a necessary precondition for a dialogue. As to the logical pedestrianism, given the close attention Peirce the logician paid to terminological continuity, there is a definite possibility this pedestrianism refers to the Aristotelian περιπατητικόσ. Like Aristotle, who walked with Teophrastus and Strato in the gardens of the Lyceum, the pedestrian logician travels along the pathways of thought, tracing every turn, counting the steps made, mapping every dead end, and making sure the pathways are well-trodden. While doing mathematics means using seven-league boots in order to get from point A to point B as soon as possible, logical pedestrianism is a way to keep the speed of mathematical thinking in check. Widening the analogy, the case at hand, then, is a fast intellect making use of the pedestrian leisure as defined by Aristotle throughout Nicomachean Ethics, i.e., making use of freedom and open-­ mindedness as prerequisite for pursuing activities choice-worthy for their own sake. Consequently, knowledge based on this logico-mathematical ideal can be conceived neither as an abstract accumulation of theories nor as an immediate access to practice. It is rather a synthesis of both. Obtaining such knowledge may be conceived as a way to live a life that gives expression to the strife for excellence, or ἀρετή (cf. Ransdell 1977, pp. 167–170). The theory here is endowed both with architectonic (or logical) and operational (or mathematical) meanings unambiguously interpretable one into another. In Peirce’s case, the search for the continuity between these two extremes ultimately results in mathematically grounded diagrammatic logic (to be discussed in Chapter 10).

Chapter 4

Peirce’s Transcendental Deduction and Beyond: Categories, Community, and the Self

The Kantian Legacy Now that we have situated visuality and diagrammatic expression within the Peircean mathematical and logical mindsets and learned what intellectual habits, according to Peirce, condition the aptitude for visual thinking, our next goal is to build a link between Peirce’s idea of a visual diagram and his conception of the universal categories. This task opens a more technical part of the current study. Accomplishing it first requires explaining what the categories are and how Peirce’s conception of them was formed within the framework of his early theory. On the one hand, clarifying what exactly the universality of the categories amounts to will pave the way for introducing Peirce’s mathematical idea of the reducibility of n-adic relations and for the basic diagrammatic representation of this idea in Chapters 5 and 6. On the other hand, this will help us better understand how Peirce’s views on the nature of generality and community of inquiry are related to his idea of continuity (Chapter 14). “On a New List of Categories” (henceforth NL) is an early paper presented by Peirce to the American Academy of Arts and Sciences in 1867. The same year NL was published, Peirce also presented two papers on formal logic with arithmetic relations to the American Academy of Arts and Sciences: “On an Improvement in Boole’s Calculus of Logic” (W2: 12–23); and “Upon the Logic of Mathematics” (W2: 59–69). In spite of its small size, NL represents a significant achievement for Peirce, who claimed proudly that the text was his “one contribution to philosophy” (CP8: 213). At the same time, Peirce confessed that NL was “perhaps the least unsatisfactory, from a logical point of view, that I ever succeeded in producing; and for a long time most of the modifications I attempted of it only led me further wrong” (CP2: 340). Murphey (1961) claims that “certainly of all Peirce’s published papers there is none which is so cryptic in its statement of essentials, so ambiguous in its definition of terms, so obscure in its formulation of the central doctrine, or so © Springer Nature Switzerland AG 2023 V. Kiryushchenko, Diagrams, Visual Imagination, and Continuity in Peirce’s Philosophy of Mathematics, Mathematics in Mind, https://doi.org/10.1007/978-3-031-23245-9_4

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important in its content.” (Murphey 1961, p. 66). According to Short (2007), Peirce’s main goal in the paper was to demonstrate the necessity of his three categories a priori—quality, relation, and representation—the project that was later abandoned by Peirce and that, as Short further suggests, “has, indeed, been a stumbling block to those who have tried to understand his later thought in its terms” (p. 32). NL is indeed a very complex, extremely dense, and highly technical paper, in which Peirce, who was twenty-eight years old at the moment of its publication, directly engages Kant’s philosophy for the first time. This paper is also Peirce’s first published attempt to apply his mathematically grounded ideas about the nature of relations to language in general and to the logical structure of predication in particular—an attempt that paved the way for some of the most important of Peirce’s later achievements. In particular, by Peirce’s own admission, it instigated his research in the logic and mathematics of George Boole and Augustus De Morgan. This study, in turn, helped him fully develop what he later called “the logic of relatives,” or the study of symbolically represented relative terms (Hawkins 1995; Kremer-Marietti 1994; Merrill 1978; CP3: 1–19, 27–98, 104–157, 359–403). It also laid foundation for Peirce’s later version of his doctrine of the three categories―firstness, secondness, and thirdness—which helped Peirce reinvent Kant’s transcendental philosophy, especially Kant’s Critique of Pure Reason (henceforth KRV), in the context of his pragmatism (Gava 2014: 154). Kantian themes in NL are especially important for bringing forth two striking analogies. First, both Kant and Peirce lay stress on a tight link between our capacity to operate concepts and our capacity to use language. As we will also see later in Chapter 9, both Kant and Peirce consider mathematics as the primary justification of this link. In Book I of “Transcendental Analytic” (KRV: A67/B92-B169), Kant describes the categories as pure concepts of the understanding that define the appearance of any object in general. The categories (which Kant divides into two groups, “mathematical” and “dynamical”) are not empirical concepts, but the most basic features of the fabric of the human mind. At the same time, Kant believed that our ability to understand is closely related to our ability to form judgments. Accordingly, every one of his twelve categories has a corresponding form of judgment as its counterpart (for instance, the category of necessity is at work in apodictic judgments, the category of reality is at work in affirmative judgments, etc.). Likewise, NL cashes out Peirce’s own list of the categories that he characterizes as the simplest necessary elements of all possible experience and, at the same time, as the conceptual steps required for constructing a simple proposition of the form “S is P.” The second analogy consists in an essential overlap between what Kant says about “schemata” (mediating concepts that are necessary for the application of the categories—especially mathematical categories—to experience) and the way Peirce characterizes diagrams and their role in mathematical reasoning (to be discussed in Chapters 9 and 12 below). In this context, again, NL is important as the text that opens Peirce’s dialogue with Kant and gives a critical response to the issues Peirce the mathematician scrutinized in Kant’s Critique of Pure Reason. In “Analytic of

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Concepts,” Kant seeks to deduce a definitive set of necessary basic concepts. Peirce is convinced that Kant’s set of categories is not nearly as certain and necessary as Kant suggests and that Kant’s deductive derivation of his categories is flawed and incomplete. To repeat, Peirce’s goal in NL is to show us how the simplest possible ideas (categories, or, in Peirce’s own parlance, “forms of representation”) cluster together to form the most basic propositional structure. His question is how we can conceptually describe the mechanism that makes what simply is into an intelligible representation. What cognitive primitives are at work when we know (and being the knowers that we are, can communicate) that “lobsters are fierce,” that “Huygens was born in 1629” or that “most diamonds  have the hardness of 10 on the Mohs scale”? To achieve this goal, Peirce uses a Kantian conceptual framework. On this early Peirce’s view of Kant, to deduce the categories is to describe how the “manifold of sense” takes up a propositional form, or in other words—to describe how the inflow of perceptual data is organized by our understanding into something we can communicate. A notable difference between Peirce and Kant is that, according to Peirce, the categories are not static a priori forms but consecutive stages of the passage from the manifold of sense (i.e., the manifold of unorganized, raw sensible intuitions) to the unity of representation (i.e., a judgment). This is why Peirce calls his collection of categories a list and not (as Kant does) a table. In Peirce’s case, the categories are derived in a certain order, which can be reverted and read backwards, but cannot be otherwise changed (Ponzio 2019, p. 165–166). It is partly for this reason that Peirce later called his categories “cenopythagorean.” The reference to Pythagoras, according to Peirce, suggests that the categories follow one after another in the way numbers do, which, in turn, prompts the notion of an operator generating a recursive process (for more details, see Esposito 1979: 58–59). In order to set the right context for a discussion of Peirce’s arguments in NL, let us begin with describing how Kant motivates the introduction of his own table of the categories. According to Kant, all we have in appealing directly to our senses is what he calls a “manifold of impressions,” an unorganized flux of impressions presented to the senses, but, Kant says, not yet experienced. Experience, according to Kant, can result only from the activity of the mind that organizes and structures the manifold by means of certain concepts. Without these concepts, by itself, the manifold is based on the receptivity of consciousness that is utterly passive. According to Kant, in order to transform the manifold into an immediately perceived whole, we need another active capacity of the understanding. As Kant himself technically puts it, “the combination of a manifold in general can never come to us through the senses, and cannot, therefore, be already contained in the pure form of sensible intuition” (KRV: B 130). The combination is only possible due to a set of universal concepts of the understanding that have to be applied to the sensible manifold in order to make it into a unity. For instance, I perceive an apple as a concrete unity of different general properties without necessarily being consciously aware, or thinking of it as an object. I do treat it as such, but that, however, only becomes possible as a result of a complex cognitive integration of an indiscriminate flux of sensory

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input, which involves processes that locate the apple’s edges, present it as “round” and having a “core,” identify it as a stable thing, frame my expectations about how it would taste if I took a bite, justify the fact that it can be presented to me simultaneously with other objects, etc. According to Kant, on the part of sensibility, the forms of such integration are time and space, and on the part of the understanding, these forms are the categories. Kant says that a category is not only a universal concept but also a form of judgment. For instance, a judgment “All humans must die” in its form is: (1) affirmative, (2) universal, and (3) apodictic,

because it asserts, that in: (1a) reality, we affirm there exists (2a) a unity of objects (all humans), to which a certain predicate (being dead eventually) is ascribed with (3a) necessity.

This means that, if there are rules according to which we apply concepts in order to have an orderly experience, there should be corresponding rules according to which we form our judgments about the objects of our experience. From this, it follows that, at the most fundamental level, the categories of the understanding should be correlated with the logical functions of judgment and that, ultimately, both a language and what the language is about should have the same rational structure (with mathematics as the perfect representation of this symmetry, as we shall see in Chapter 9). Accordingly, “Analytic of Concepts” does in fact represent not just one table, but two, one containing twelve fundamental concepts, or “forms of appearance,” another containing twelve forms of judgment. Peirce follows Kant in supporting this analogy, when in §4 of NL he claims that: [t]he unity to which the understanding reduces impressions is the unity of a proposition. This unity consists in the connection of the predicate with the subject; and, therefore, that which is implied in the copula, or the conception of being, is that which completes the work of conceptions of reducing the manifold to unity (W2: 49).

Peirce agrees with Kant that the universal concepts, or categories, have to be deduced, which means that Peirce, just like Kant, wants to show that the categories are necessary and that, without the application of the categories, there can be neither experience proper nor the understanding of it or judging about it. Although the effort Peirce makes in NL is based on the acceptance of Kant’s method of deduction, he is not convinced by Kant’s results. Kant’s own table consists of four triads: Quantity, Quality, Relation, and Modality. As Buzzelli (1972) notes, “Peirce discovered that each of these triads was intrinsically related to some other part of the table. In other words, the four classes of categories were not independent of one another but rather depended on some more fundamental structure that Kant had not observed” (p. 64). Peirce, therefore, sets to discover a still more fundamental conceptual structure than that of Kant by deducing a new list of concepts that would be both elementary and universal, in the sense of being inevitably involved in any thought, judgment, and experience, whether actual or possible.

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Peirce’s Deduction With all these differences in mind, in NL, Peirce follows Kant rather closely. Just as, in Kant’s case, the universal concepts are used in order to reduce the manifold of sense impressions to an intelligible unity expressible in a judgment, in Peirce’s case, they are used in order to bring what he, following Aristotle, calls substance (what is, in the most general sense, present) to the unity of being (or the unity of the copula in any proposition of the form “S is P”). Peirce’s list thus represents a logical continuity between substance (Kantian manifold of impressions) and being (Kantian conceptual unity that underlies our capacity to claim that “this is that”). By the end of NL, three elementary conceptions are deduced—quality, relation, and representation—in addition to being and substance. The results of the deduction are ordered in NL in the following manner: BEING (What is) Quality Relation Representation SUBSTANCE (it)

Being and substance mark the beginning and the end of the deduction of the categories and in Peirce’s later works both are dropped altogether. In §3 of NL, Peirce introduces the concept of substance: That universal conception which is nearest to sense is that of the present, in general. This is a conception, because it is universal. But as the act of attention has no connotation at all, but is the pure denotative power of the mind, that is to say, the power which directs the mind to an object, in contradistinction to the power of thinking any predicate of that object,—so the conception of what is present in general, which is nothing but the general recognition of what is contained in attention, has no connotation, and therefore no proper unity (W2: 49).

Substance is the universal conception of that which is “nearest to sense.” It refers to “the present, in general,” to that which is merely recognized as some indeterminate contents of an act of attention, as “it” or “this.” Peirce’s substance is, no doubt, reminiscent of Hegel’s sense-certainty, the chapter on which ends up with the unity of consciousness and its object being reduced to an inexpressible manifold of “thises.” Substance exists as such, without reference to anything else: Before any comparison or discrimination can be made between what is present, what is present must have been recognized as such, as it, and subsequently the metaphysical parts which are recognized by abstraction are attributed to this it, but the it cannot itself be made a predicate. This it is thus neither predicated of a subject, nor in a subject, and accordingly is identical with the conception of substance (Ibid).

In saying that “the general recognition of what is contained in attention … has no connotation,” Peirce simply means that, in being directed at that which is immediately given, the mind cannot define that at which it is directed. The immediacy presupposes no mediating relation and, therefore, no differentiation between the mind itself and that at which it is directed. The mind, therefore, identifies with what is, in the most basic sense, present. In still simpler terms, in an act of attention that is

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directed at the present (say, an apple), the mind cannot obtain an idea of something being a perceiver as distinctly different from the apple that is being perceived. It can neither know itself nor be aware of the apple it supposedly has in front of it. On this, initial stage, I am as far as I can possibly be from being able to say, “This is an apple.” The deduction has just begun, and all I can do is simply point at the thing to express the immediacy of my relation to it: this! §4 of NL introduces the concept of being: The unity to which the understanding reduces impressions is the unity of a proposition. This unity consists in the connection of the predicate with the subject; and, therefore, that which is implied in the copula, or the conception of being, is that which completes the work of conceptions of reducing the manifold to unity (ibid.).

What this technical description means is that the simple togetherness of being and a subject is meaningless. A subject itself does not represent a conception that can unify anything: “S is” is not a proposition. But with the concept of being as the copula “is” obtained through a set of mediating steps, which NL carefully describes, the deduction of the sensuous manifold (substance) to conceptual unity (being) gets completed, with the proposition of the form “S is P” in place. It is important to have a clear understanding of what exactly this Peirce’s analysis is about. In describing the passage from the manifold of perception to the conceptual unity of representation, we do not describe the transition from, say, our vaguely sensing “something” as we enter a room, and concluding that (oh!) “it is a table.” What Peirce presents is a conceptual analysis of the most basic epistemological situation, not an empirically informed research of a cognitive scientist. He is definitely not interested in the latter in any way. Once the notions of the beginning and the end of the deduction are established, Peirce’s line of argument in NL is as follows. According to Peirce, the first conception that mediates between being and substance is that of quality: “A proposition always has, besides a term to express the substance, another to express the quality of that substance; and the function of the conception of being is to unite the quality and the substance. Quality, therefore, in its widest sense, is the first conception in order in passing from being to substance” (W2: 52). Peirce takes it slow from here, carefully stipulating every step of the analysis and admitting only what is truly necessary. In his terminology, in the proposition “This stove is black,” “this stove” is a more immediate concept (the one that is closest to the simple, purely denotative ‘this’ of substance), and “black” is a more mediate one (that of a quality). Since the proposition asserts the applicability of the latter to the former, i.e., claims that S is indeed P, “the more mediate conception is clearly regarded independently of this circumstance, for otherwise the two conceptions would not be distinguished, but one would be thought through the other, without this latter being an object of thought, at all” (W2: 52). Accordingly, in addition to a purely denotative ground, we have an idea of a quality. Peirce notes that “quality seems at first sight to be given in the impression” (ibid.). This, however, is wrong. We need to distinguish, Peirce says, between substance and quality before we can apply the latter to the former. Once this is done, we see clearly that a qualitative

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aspect (“blackness”) should therefore first be considered, not as applied to an object (or be perceived as “given in the impression” of an object—the stove), but as a simple monadic unity, as it is in itself. But, Peirce says, considered in this way, it can be nothing but a general notion, a universal: while there are black stoves, blackness qua blackness is something only possible, or potential. As such, it is something we can think about only in abstracto. While I can think of a variety of black objects, blackness itself cannot be made into a mental representation; it cannot be seen. In and of itself, it is only a possibility of cognition, not something actually cognized. It is what it is: an abstract notion. Since as such, unembodied, quality is not immediately available in perception (what exactly is ‘blackness’ as such?), to learn about a quality, we need some other quality. We can predicate a quality only by means of contrasting it with (or recognizing its analogy to) something this particular quality itself is not. Therefore, the reference to it alone cannot do all the work necessary to bring the manifold of sense-impressions to conceptual unity. Blackness can be actualized in an object only in some respect—if only in the sense that black things are always more or less so. It can become an object of perception or imagination only as having some particular form (as this black object). Reference to a correlate (or relation of some one thing to some other) is in place, Peirce says, on some particular occasion of reference to a quality. We can predicate a quality only by means of relating it to something else, its correlate. By actually claiming that “this stove is black,” we recognize the fact that the quality is actually conceptually integrated into the constitution of an object. There is something else to what we have now than the quality of blackness itself; something that is correlated to it. Accordingly, by this logic, whenever we comprehend some one thing in concreto, we comprehend its relation to some other thing. Therefore, in the very act of our comprehension of what “this is,” we have already introduced a duality. To take a simple example, whenever we think about a murderer, we think of him in relation to a murdered person, and whenever we think of an equivalent of “man” in French, we find “homme” appropriate—with all further inferential implications involving likenesses and contrasts presupposed by those correlates. Whereas quality is something potential and abstract, relation is always about some matter of fact. Relation, which represents the second step in Peirce’s deduction, is a conception that accounts for externality. Whereas quality refers only to itself, relation represents a reference to something other, i.e., something external to a quality taken as such. Of course, it might be claimed that a relation need not be existential but can be purely hypothetical, and the “is” may well categorize in the abstract rather than make an existential claim. But that is irrelevant to the case. Due to the very character of the task—a deduction of the set of the most basic concepts—the load every new move in the deduction brings with itself is minimal. All we need at this point is to note that, as Peirce himself puts it, “we can know a quality only by means of its contrast with or similarity to another” (W2: 53). The whole point of the deduction is to add nothing beyond the bare minimum that every step necessarily requires. Obtaining the distinction provided by the category of relation, according to Peirce, is not the end of the story. In addition to quality and relation, on every

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occasion of reference to a correlate, we also conceive a third element that mediates between the other two. Looking for the word “homme” in a French dictionary, for instance, will lead us to the word “man” “which, so placed, represents homme as representing the same two-legged creature which man itself represents” (W2: 53). Peirce calls this mediating representation an interpretant, “because it fulfills the office of an interpreter, who says that a foreigner says the same thing which he himself says” (W2: 54). Just as an interpreter is a middleman who says something (for instance, “this stove is black”) and, in doing so, also says that he says the same thing as does the foreigner, Peirce’s interpretant is defined as “a mediating representation which represents the relate to be a representation of the same correlate which this mediating representation itself represents” (W2: 53). The passage just quoted marks a magic moment in the paper. What happens when an interpreter does the two things mentioned above simultaneously (saying something and, in doing so, saying that he is saying the same thing that the foreigner does) is he addresses his thought to a future interpretation. Selfconscious understanding, in which I realize that I think, and communicating with the other are accomplished in one and the same act. Self-reflection, then, is by definition a dialogue between two different phases of a reflecting ego. A comparison between the following well-known paragraphs from Kant’s “Transcendental Analytic” and Peirce’s NL will help us better understand what this means. Both are extremely important for our purpose here and are worthy of quoting in full. (Kant): For the empirical consciousness which accompanies different representations is in itself diverse and without relation to the identity of the subject. That relation comes about, not simply through my accompanying each representation with consciousness, but only in so far as I conjoin one representation with another, and am conscious of the synthesis of them. Only in so far, therefore, as I can unite a manifold of given representations in one consciousness, is it possible for me to represent to myself the identity of the consciousness in these representations. In other words, the analytic unity of apperception is possible only under the presupposition of a certain synthetic unity. The thought that the representations given in intuition one and all belong to me, is therefore equivalent to the thought that I unite them in one self-consciousness, or can at least so unite them; and although this thought is not itself the consciousness of the synthesis of the representations, it presupposes the possibility of that synthesis. In other words, only in so far as I can grasp the manifold of the representations in one consciousness, do I call them one and all MINE.  For otherwise I should have as many-colored and diverse a self as I have representations of which I am conscious to myself (KRV: B133). (Peirce): If we had but one impression, it would not require to be reduced to unity, and would therefore not need to be thought of as referred to an interpretant, and the conception of reference to an interpretant would not arise. But since there is a manifold of impressions, we have a feeling of complication or confusion, which leads us to differentiate this impression from that, and then, having been differentiated, they require to be brought to unity. Now they are not brought to unity until we conceive them together as being OURS, that is, until we refer them to a conception as their interpretant. Thus, the reference to an interpretant arises upon the holding together of diverse impressions, and therefore it does not join a conception to the substance, as the other two references do, but unites directly the manifold of the substance itself. It is, therefore, the last conception in order in passing from being to substance (W2: 54).

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As the first passage from Kant’s “Analytic of Concepts” indicates, for Kant, self-­ consciousness implies that the notion “I think” must accompany all other notions for them to be comprehensible. What I need for my acts of understanding to work, according to Kant, is a thought of a special kind. This thought, although it neither amounts to the consciousness of the conceptual synthesis itself nor has anything to do with my empirical self (for instance, my capacity to self-identify when looking in the mirror or my knowledge of any external fact in general, for that matter), should still be there for the synthesis to be possible. Accordingly, what the “I think” does is it formally secures the numerical identity of myself in all of my various experiences—the identity that precedes all data of the intuitions and confirms the unity of all the various experiences that I have (KRV: A 107, A 113). In other words, as Kant says, I need to be able to grasp the manifold of sense-impressions as mine. Without this conceptual addition, I cannot make external reality the proper object of my understanding. The second passage describes Peirce’s interpretant, which, unlike the Kantian self-consciousness, preserves “as many-colored and diverse a self” as there are representations. For Peirce, self-consciousness is a useless concept. It represents neither the inner feeling of “being oneself” nor a special kind of thought representing the basic “principle” of cognition. As Peirce indicates in the closing paragraph of NL, his list of conceptual primitives, or categories, which constitute his account of the most basic structure of experience and meaning, leads directly to his first classification of signs, or, as he puts it in NL, “modes of representation.” Thus, Peirce’s reinterpretation of Kant’s idea of the synthesis of impressions is, at the same time, an introduction to a rudimentary idea of sign. The classification of signs in NL is much less refined than those later developed by Peirce, and comprises what he terms likenesses, indices, and symbols—three elements, each of which receives in NL only a brief sentence-long definition. Likenesses are defined as “those whose relation to their objects is a mere community in some quality,” indices—as “those whose relation to their objects consists in a correspondence in fact,” and symbols—as “those the ground of whose relation to their objects is an imputed character” (W2: 56). In other words, the three basic categories essential to experience and thought that Peirce derives from his analysis of Kant’s table of twelve categories of the understanding relate to signs of quality (like a vague feeling on the surface of one’s skin), signs of factual connection (like an observed causal correlation between a weathervane and the direction of the wind), and signs of symbolically articulated similarity (like a noun that refers to an appropriate set of objects).

 ome Consequences of Peirce’s Deduction: The Community S of Inquiry and the Self In “Some Consequences of Four Incapacities,” a much less technical paper published a year after NL in The Journal of Speculative Philosophy, Peirce goes as far as to compare a man with a word. We are accustomed to think that a man has a

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conscious life, while a word is not. But the distinction is deemed to remain vague until “consciousness” is clearly defined. As Peirce notes, consciousness “may mean that emotion which accompanies the reflection that we have animal life. This is a consciousness which is dimmed when animal life is at its ebb in old age, or sleep, … which is the more lively the better animal a man is, but which is not so, the better man he is” (W2: 240). While we distinguish between the meaning of a word and its embodiment as written or vocalized, we do not attribute consciousness to words because having it seems to require the possession of an animal body. Again, following Kant, we might identify consciousness with the “I Think,” or the unity of thought. The basic idea of Kant’s transcendental apperception, as has just been discussed, is that any time I have an experience, I also know that it is I that have that experience, and knowing that is equivalent to knowing that this experience belongs to the same self that has all other experiences; the self that is numerically identical throughout all my experiences. This unity, according to Kant, is purely formal and cannot be reduced to any sort of inner feeling of self. Peirce is dissatisfied with Kant’s solution; according to him, all this formal unity amounts to is the recognition of the consistency of thought. And, as far as thought is discursive, its consistency is a matter not of introspection, but of interpreting one word into another. Peirce concludes: Man makes the word, and the word means nothing which the man has not made it mean. … But since man can think only by means of words or other external symbols, these might turn round and say: “You mean nothing which we have not taught you, and then only so far as you address some word as the interpretant of your thought.” … [T]here is no element whatever of man’s consciousness which has not something corresponding to it in the word; and the reason is obvious. It is that the word or sign which man uses is the man himself. For, as the fact that every thought is a sign, taken in conjunction with the fact that life is a train of thought, proves that man is a sign; so, that every thought is an external sign, proves that man is an external sign. That is to say, the man and the external sign are identical, in the same sense in which the words homo and man are identical. Thus my language is the sum total of myself; for the man is the thought (Ibid: 241).

To repeat, what one might take to be an act of introspection is, according to Peirce, either a glimpse into one’s animal nature (some sort of basic psychological response to a generalized feeling of one’s physical body), or an abstract principle of cognition. And while the former, as Peirce claims, on closer consideration has nothing to do with anything specifically human, the latter, in signifying the unity of our thought, in fact signifies only its capacity to go on in a logically consistent way. The interpretive relation between a man and a word is structurally the same as the one between the English word “man” and the Latin word “homo.” Twenty-six years after the publication of NL, in his “Questions on William James’s Principles of Psychology” (Ms. 1099, 1894), Peirce is even more explicit: “Everybody will admit a personal self exists in the same sense in which a snark exists; that is, there is a phenomenon to which that name is given. It is an illusory phenomenon; but still it is a phenomenon. It is not quite purely illusory, but only mainly so. It is true, for instance, that men are selfish, that is, that they are really deluded into supposing themselves to have some isolated existence; and in so far, they have it” (CP8: 82;

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emphasis added). In describing self-consciousness as an idea immediately available to us on reflection, Peirce calls it the “Snark,” the hunting of which, as Lewis Carroll tells us in the end of The Hunting of the Snark, brings no sensible result. An interpretant unites the manifold of impressions not in a self-consciousness that allows me to grasp the manifold of impressions as mine, but by correlating the impressions to each other in such a way that their unity is comprehended only through its further correlation to something else. It is, Peirce says, in so correlating various parts of the manifold to each other that the impressions are grasped as ours. By allowing us to so grasp the impressions, the interpretant replaces the Kantian synthesis of apperception in a single self-consciousness (my numerically identical self) with the idea of an intersubjective synthesis of meaning addressed to future interpretation (a community). The idea of a unity, according to Peirce, is logically inseparable not from the numerical identity of a self, but from the idea of mediating representation. As Peirce puts it elsewhere, while the idea of one yields no relation, and the idea characteristic of two simply brings forth otherness, the idea of three is the one of a true medium, which, Peirce claims, is just a narrower version of uniter (CP1: 476). The principal takeaway here is that understanding and being able to use language is not about the synthetic activity of an individual mind, but rather about an interpretive and self-corrective communal effort addressed to future interpretation. And while this effort is embodied in languages we use in our everyday life, there is an ideal model of it represented in the community of inquiry. Something is comprehensible for me only if it is interpretable by others. As Apel (1980) aptly puts it, Peirce re-describes the conditions of the possibility of knowledge by shifting attention from the domain of the Kantian pure consciousness to the domain of the consciousness that is linguistically and intersubjectively constituted (Ch. 3). Peirce thus tries to get rid of any kind of subjectivist approach to understanding and interpretation. In the process of transformation that takes place in-between being and substance, a mediating (and thus uniting) interpretant gives a representation its ultimate form and functions as a dynamic principle of coordination between the other two elementary parts of representation (the other two universal categories). Rather than being a static form imposed on representations by the Kantian synthesis of the understanding, Peirce’s interpretant is an interpretive force, or a form of a relation that unifies the sensible manifold in a representation by throwing it forward, towards its further interpretation by some future self. There are two points that are important to keep in mind for the sake of further discussion. First, NL describes the unity of impressions not as an “art concealed in the depths of the human soul” (KRV: B 180) but as an act of correlation. It is not that a unity is achieved through a continuous act of correlation; it is that a unity is such an act. The point is not that, in offering you an interpretation that is mine, I thereby also make it a part of some impersonal Platonic storage of ideas, but that my idea becomes intelligible (acquires an intelligible content) only as tested and approved by innumerable others. According to Peirce’s analysis in NL, the unity taken as a process acquires cognitive value through a set of modes of reference—quality, relation, and representation—which constitute three consecutive steps to cover what might be called “a logical distance” between the manifold of impressions and their

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conceptual synthesis in a proposition. If we conceive the three categories as the three most primitive kinds of relation, then our basic cognitive capacities are represented by a synthesis between unary, binary, and triadic relations (or between a quality, a fact, and a proposition). The second point of importance is the fact that, according to Peirce, we should consider understanding not as the capacity to grasp the manifold of impressions in one consciousness but as a result of the correlation to something else, which entails that understanding is a function of the reference to the future, the performance of which cannot be reduced to an introspective process. An act of understanding, whether it amounts to finding a solution to a complex mathematical problem or simply to the realization of the truth of a single fact), presupposes manipulation of outward signs. What is important, again, is not that I grasp the impressions as mine, but that the representations resulting from the synthesis of those impressions are available for interpretation by others. As the process of interpretation, conceived in this manner, is necessarily an open-ended one, the “We” (or the community as it was discussed in Chapter 2), which is the proper subject of the synthesis, cannot be limited to any finite group of people existing somewhere at some particular moment of time. It is always a future community. The way Peirce defines such an interpretation-­ related future community in “Some Consequences” is closely related to the idea of reality. As Peirce suggests, the very first, uncritical idea of reality as something external comes from the discovery of the fact that some of our beliefs are erroneous or have objects that are illusory, unreal. The discovery is uncomfortable as it destroys our idiosyncratic self-assurance. But errors made by individuals are corrected by a communal inquiry which, Peirce says, is destined to result, sooner or later, in the state of complete information, the latter being the final goal towards which every inquiry should be ultimately directed. He concludes: The real, then, is that which, sooner or later, information and reasoning would finally result in, and which is therefore independent of the vagaries of me and you. Thus, the very origin of the conception of reality shows that this conception essentially involves the notion of a community, without definite limits, and capable of a definite increase of knowledge. And so those two series of cognition – the real and the unreal – consist of those which, at a time sufficiently future, the community will always continue to re-affirm; and of those which, under the same conditions, will ever after be denied (W2: 239).

In correcting individual errors, the inquiry results in two sets of beliefs: ones that it will reaffirm and ones that it will deny until sometime sufficiently far into the future. The notion of an extended community guarantees that we will have knowledge about things as they really are in the long run and that it does not prevent us from having such knowledge on numerous occasions at any given moment of time. Although the reference to the future community cannot make us absolutely certain about actually having objective knowledge on any particular occasion, given that we follow the maxim of pragmatism—that is to say, if we approach the meaning of every concept we use as a set of conceivable outcomes of our possible interactions with the objects to which the concept applies—it makes us justified in holding our beliefs and treating them as true.

Fixation of Belief

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Fixation of Belief NL is an early Peirce’s paper that treats the categories as basic concepts that help us make a complex and varying sensible manifold into the unity of a proposition. In other words, Peirce the meritocrat first approaches his new list from a rather narrow logical point of view. However, an overwhelming consensus among Peirce scholars is that there is an unbroken continuity in the development of Peirce’s thought1 and that its connection with Kant at any given point of its development should not be questioned. Even in his late recollections, Peirce noted that, in the years following NL, his “Kantism got whittled down to small dimensions. It was little more than a wire, – an iron wire, however” (MS 317). This self-characterization will prove especially important in Chapters 9 and 12 below where Peirce’s conception of diagrams will be discussed in the context of Kant’s philosophy of mathematics. With all this in mind, NL is usually analyzed either on its own merits or as Peirce’s early semiotic work, which provides grounds for the later developments in his theory of signs. Meanwhile, there is a way to look at it in a different light, which might help us see beyond Peirce’s semiotics and into his pragmatism. In 1877–1878, Appleton’s Popular Science Monthly published a series of Peirce’s articles known as Illustrations of the Logic of Science. The first two of them, “The Fixation of Belief” and “How to Make Our Ideas Clear” (henceforth FB and HMIC) are commonly taken to be based upon a nameless paper read by Peirce before the Metaphysical Club in November 1872, just about the time Peirce’s Kantism “got whittled down to small dimensions.” FB, apart from the fact that it includes preliminary notes to what in the next article of the series would appear as the maxim of pragmatism, is also remarkable on account of its composition. Namely, it may be read as presenting the argument that gives a consistent pragmatist reinterpretation of the semiotic categories described by Peirce one year earlier in NL. It is important that this reinterpretation also lays out a social theory that may help us relate Peirce’s semiotics to his mathematically grounded attitude towards social conventions and thus can serve as a further development of the themes discussed in Chapters 2 and 3 above. Recall that Peirce’s NL rests on the assumption that there’s a conceptual space between being and substance covered by a set of consecutive media—quality, relation, and representation―that are needed to get the process of interpretation started. An interpretant, the last general term in the sequence, makes us conceive our impressions together as being ours and thereby allows us to grasp the manifold of them in a propositional form. Furthermore, in allowing us to so grasp the manifold, an interpretant addresses any given expression to its further interpretation. These early semiotic intuitions, which were inspired by Peirce’s studies of Kant’s table of categories, proved to be of crucial importance for Peirce’s later theoretical  Lane (2004) provides an elaborate account showing that Peirce’s thought reveals some strong ontological intuitions as early as 1868, the year “Some Consequences of Four Incapacities” was published in the Journal of Speculative Philosophy. But also see Goudge (1950). 1

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developments. In particular, the conceptual amalgam brought forth by the notion of an interpretant already contained the germ of Peirce’s idea of a community of inquirers, the regulative future us that any interpretation guided by an appropriate method is inevitably aimed at. As will be shown in the remainder of this chapter, there is an important analogy between NL and FB. The latter may be understood as resting on the assumption that there is a conceptual space between doubt, which Peirce defines as “an uneasy and dissatisfied state from which we struggle to free ourselves” and belief he described as “a calm and satisfactory state which we do not wish to avoid, or to change to a belief in anything else” (W3: 247). A doubt irritates us and initiates a struggle to cease it, which ends up in establishing a belief. A belief or a set of beliefs that “guide our desires and shape our actions” is what Peirce calls inquiry (ibid.). Doubt is an immediate cause of inquiry and its starting point, whereas settling doubt and fixing belief is its sole purpose. Just as the passage from being to substance is accomplished by means of a logical synthesis that brings the manifold of impressions to unity, the passage from doubt to belief might be understood as accomplished by a practical synthesis that results in a certain mode of action. The conceptual space between doubt and belief can only be passed by fixing our beliefs in one way or another, and the role of consecutive media here is played by four different methods of fixing beliefs. The methods are those of tenacity, authority, a priori, and science, which may be organized in three consecutive steps. That said, to justify the hypothesis about the structural concordance between NL and FB, a brief summary of the four methods is now in order. The method of tenacity is applied when a man holds a self-satisfied opinion so that “the pleasure he derives from his calm faith overbalances any inconveniencies resulting from its deceptive character” (W3: 249). In terms of NL, at this first step, any belief is nothing more than a quality in itself. Like a self-satisfied and self-­ contained opinion held by an individual, a quality, which Peirce compared with the Kantian manifold in intuition (or, with reservations, with Hegelian sense-­ certainty) is: …an instance of that kind of consciousness which involves no analysis, comparison or any process whatsoever, nor consists in whole or in part of any act by which one stretch of consciousness is distinguished from another, which has its own positive quality which consists in nothing else, and which is of itself all that it is, however it may have been brought about; so that if this feeling is present during a lapse of time, it is wholly and equally present at every moment of that time (CP 1.306).

Tenacity, intellectual inelasticity is a feature of any individual mind as far as it equals itself, is considered as it is in itself regardless of anything else and presupposes no distinction or comparison. In this case, an individual mind is incapable of any information exchange and is confined to an immediate experience of a self-­ satisfactory feeling of assurance, a kind of Hegelian sinnliche Gewißheit, a naïve and immediate unity of subject and object, an abstraction of “here and now” (for the reference to Hegel in this context, see also Stern 2005: 67). Just as blackness as such cannot be made into a mental representation (we need something to ascribe it to), a

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tenacious intellect represents only a possibility of cognition, not any actual knowledge, which always requires an act of correlation. The second method represents an opinion enforced by an authority, be it an individual or some sort of political or religious institution. Here, a belief is in the form of action-reaction or, in Hegelian terms, “negative unity,” and is therefore of relational character: This conception, that another man’s thought or sentiment may be equivalent to one’s own, is a distinctly new step, and a highly important one. It arises from an impulse too strong in man to be suppressed, without danger of destroying the human species. Unless we make ourselves hermits, we shall necessarily influence each other’s opinions (W3: 250).

A collision between the institutionalized social reality and the blind tenacity in following one’s belief gives rise to the first objectified form of rationality: the idea of the existence of the other (or, in terms of NL, the idea of relation). The other here is reduced to a general expression for a set of opinions held by a certain historical social group, be it a state, a religion, or any other set of social institutions. In this case, an opinion held by an individual cannot be considered simply in itself anymore, as it turns out to be confronted by another opinion with which it enters into a recognitive relationship. The blind immediacy of tenacity is replaced by a direct experience of the other. Sooner or later in any given society, Peirce says, “some individuals will be found who … possess a wider sort of social feeling” (W3: 252; emphasis added). This wider social feeling allows those individuals to see most of the laws of a certain cultural, political, or religious tradition (which are taken at their face value at the previous stage) as mere historical accidents and products of public opinion manipulation by an authority. The individuals then propose the new a priori method of fixing beliefs which excludes the possibility for a belief to depend on the idiosyncratic whim of either an individual or a law-like power of society. This method predetermines the choice of opinion by bringing it (as philosophers of the a priori believe) into harmony with natural causes. Peirce claims that “this method is far more intellectual and respectable from the point of view of reason than either of the others which we have noticed” but that there is a principal downside to it. It is that: …it makes of inquiry something similar to the development of taste; but taste, unfortunately, is always more or less a matter of fashion, and accordingly metaphysicians have never come to any fixed agreement, but the pendulum has swung backward and forward between a more material and a more spiritual philosophy, from the earliest times to the latest (W3: 253).

Every person, says Peirce, is a truth-seeker by nature. All we need is to acquire a method that would put this natural disposition to work in the most effective way possible. And it is science, according to Peirce, that is the ultimate source of the next and final method of fixing beliefs, because science, in its pragmatist understanding, is nothing but a more logically complex and sophisticated expression of our natural disposition to seek the truth. Only the scientific method is capable of fixing our beliefs in such a way that they are “determined by nothing human, but by some external permanency – by something upon which our thinking has no effect” (W3:

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253; emphasis added). Later in the text of FB, Peirce adds that “our external permanency would not be external, in our sense, if it was restricted in its influence to one individual. It must be something which affects, or might affect, every man” (W3: 253–254). Still, later Peirce also refers to “real things whose characters are entirely independent of our opinions about them” (W3: 254), but the text of FB does not provide any clear arguments as to whether those things shall be considered as anything different from the Kantian things-in-themselves (Short 2007, 46–48). Peirce’s notion of external permanency as “something upon which our thinking has no effect” may be read as a terminological allusion to the concept of “something permanent” (etwas Beharrliches), which occurs at the end of Kant’s “Analytic of Principles.” Kant needs it for the refutation of Descartes’ problematic idealism according to which the reality of things outside me is ultimately indemonstrable, the only ultimately irrevocable claim being that “I am.” Kant uses this concept to show that, as far as there’s nothing permanent in self-perception (I always think of myself as a subject, while I can experience myself only as an immanent object), the very continuity of experience, including my inner experience of “myself,” are necessarily bound up with the existence of something external (KRV: B 275–276). Be it as it may, the meaning of the term “external” as applied to reality is partly clarified only in HMIC, the next paper in the series: Different minds may set out with the most antagonistic views, but the progress of investigation carries them by a force outside of themselves to one and the same conclusion. This activity of thought by which we are carried, not where we wish, but to a foreordained goal, is like the operation of destiny. […] The opinion which is fated to be ultimately agreed to by all who investigate, is what we mean by the truth, and the object represented in this opinion is the real (W3: 273; emphasis added).

If the object of the final opinion is the real, the truth of a belief as it is described in HMIC, just like the unity of substance as it is described in NL, is defined as the ultimate agreement of the community of “all who investigate.” And it is this ultimate agreement that, according to Peirce, marks the external permanency upon which individual thinking has no effect. Referring to the method of science, Peirce supplements the notion of external permanency by the metaphor of some force operating outside individual minds. Right after the quoted passage, he undertakes to explain how the idea of such force is to be brought into compliance with his earlier definition of reality as independent from what is thought of about it: … reality is independent, not necessarily of thought in general, but only of what you or I or any finite number of men may think about it; and […] though the object of the final opinion depends on what that opinion is, yet what that opinion is does not depend on what you or I or any man thinks. […] Our perversity and that of others may indefinitely postpone the settlement of opinion. […] Yet even that would not change the nature of the belief, which alone could be the result of investigation carried sufficiently far (W3: 273).

Reality, towards which individual opinions are forced or, as Peirce puts it, destined to converge, is independent of what any finite number of individuals might believe about it. But it is not independent of “thought in general.”

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The challenge here is to understand what “thought in general” is, given that Peirce most certainly was not a realist of the Platonic stripe and held that beliefs fixed by an appropriate method are always shared by a finite number of individuals. A related puzzling fact is that there’s no direct indication of the distinction between having a belief and having a true belief in the text of either FB or HMIC. On the contrary, in the very beginning of FB, before setting about to describe the four methods, Peirce stipulates that fixing beliefs does not presuppose any epistemologically legitimate difference between the two: With the doubt, therefore, the struggle begins, and with the cessation of doubt it ends. Hence, the sole object of inquiry is the settlement of opinion. We may fancy that this is not enough for us, and that we seek, not merely an opinion, but a true opinion. But put this fancy to the test, and it proves groundless; for as soon as a firm belief is reached we are entirely satisfied, whether the belief be true or false. […] The most that can be maintained is, that we seek for a belief that we shall think to be true. But we think each one of our beliefs to be true, and, indeed, it is mere tautology to say so (W3: 247–248).

As far as there are different sorts of social practices that yield appropriate methods of reaching firm beliefs, one may choose the method which would bring him satisfaction. But even though, as far as a belief is firm, we are satisfied no matter what method of fixing beliefs we follow, it is the scientific method that Peirce characterizes by the end of FB as “the only one of the four methods which presents a distinction of a right and a wrong way” (W3: 254; emphasis added). There must be, then, a solid criterion which would help us understand why only scientific method is capable of fixing our beliefs in such a way that they are properly determined by external reality. Naturally, anybody may accept false premises and come to false conclusions. Science, as Peirce sees it, does not offer a way to get rid of this problem in any given “here and now,” but offers a mode of action, a method which ascribes practical meaning to our natural inclination towards making right decisions in the long run. As discussed in Chapter 2, in HMIC Peirce gives the following formulation of the method of science as the fourth and final method of fixing beliefs: “Consider what effects that might conceivably have practical bearings you conceive the object of your conception to have. Then, your conception of those effects is the whole of your conception of the object” (W3: 266). According to this formula, what makes a conception contentful is neither a mental proxy of its object nor an idea of the object’s qualities directly perceived, but a set of conceivable effects that our experiments with the object may bring about. To put it differently, conceptions mean something to us humans only relative to purposeful human behavior. In FB, the method of science is seen by Peirce as an extension and a correction of the a priori method by applying it to experience, so the two methods together may be taken to form the third and last step in the practical synthesis, as analogous to the logical one described in NL. In NL, the three categories are necessary steps in the process of reducing the manifold of impressions to unity, which is accomplished by the reference to an interpretant. Likewise, in FB, each method of fixing beliefs is the necessary step in appreciating the advantages of scientific inquiry. It is only the reference to an interpretant that brings impressions to unity by making them

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available for further interpretation by the unlimited future “we” of inquiry. It is only the reference to the scientific method that shows the way to the formation of general opinions we are fated to obtain in the long run, provided our investigation, according to this method, is carried sufficiently far. An interpretant unites my diverse impressions not by joining any other concept to the diversity, but by appealing to its further interpretation by another interpretant. Likewise, in Peirce’s maxim certain conceivable practical results of actions and perceptions are united into a concept, where “the test of whether I am truly following the method is not an immediate appeal to my feelings and purposes, but, on the contrary, itself involves the application of the method” (W3: 255). In applying an interpretant, we appeal to something that transcends both the abstraction of a single quality and the mere fact of existence of something external to us. Similarly, in the use of the scientific method, reality is no longer determined either by individual will or by social contract. On this view, while an individual will is captured by egotistic impulses, a social order, accepted uncritically, is little more than an excuse for logical inconsistency and lack of creativity. Thus, FB may be seen as a rewrite of Peirce’s earlier logical deduction of the categories in terms of practical philosophy. It represents the categories as the ways to conduct inquiry (or methods of fixing beliefs). In FB and HMIC, Peirce further develops his semiotic conception of interpretation in the context of his pragmatist approach to meaning and his idea of the universal language of science. A scientific community is not just about following a set of pre-determined conventions in order to establish an objective relation between mind and world, but rather about a set of forward-looking adaptational techniques. These techniques, Peirce believed, could play a dual role. They can help us organize our scientific inquiries and, at the same time, work out principles for a broader communal life.

Chapter 5

Sign Relation

Kinds of Relations In NL, Peirce does three things: He (1) defines his three universal categories (quality, relation, representation), (2) gives his initial classification of signs into likenesses, indices, and symbols, and (3) shows that his explanation of how the universal concepts unify experience yields the idea of continuous mediation which replaces the Kantian idea of the unity of experience in self-consciousness. Mediating representation leads us from unorganized sense data to general concepts that unify the manifold of the data. With this in mind, we are now set to approach the idea of a diagram in the context of Peirce’s definitions of sign and sign relation. Peirce defines a sign as something that “involves a plural relation, for it may be defined as something in which an element of cognition is so embodied as to convey that cognition from the thought of the deliverer of the sign, in which that cognition was embodied, to the thought of the interpreter of the sign, in which that cognition is to be embodied” (On the Logic of Quantity, MS 16, c. 1895). This definition is important as it tells us that a sign has a relational structure, that this structure embodies a cognition, and that the embodiment is such that it allows to convey the cognition from one mind to another. Note that the only constant here is “cognition.” Peirce is careful not to talk about a particular thought, a content or a mind as staying the same throughout the entire process of “conveyance.” Although Peirce refers to signs here as consisting of plural relations, he claims that only genuinely triadic relations are indispensable and necessary for signification. Accordingly, a sign is also defined by Peirce with reference to a triadic relation of something standing for something else, called its object, to a third, called its interpretant. Revoking his definition of interpretant in NL, Peirce adds that “this triadic relation [is] such that the representamen determines its interpretant to stand in the same triadic relation to the same object for some interpretant” (CP1: 541). So what stays the same throughout interpretation is the triadic structure and that only in its numerical value and an © Springer Nature Switzerland AG 2023 V. Kiryushchenko, Diagrams, Visual Imagination, and Continuity in Peirce’s Philosophy of Mathematics, Mathematics in Mind, https://doi.org/10.1007/978-3-031-23245-9_5

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appropriate quality. The sameness here is the one of relations, not substances or things. Elsewhere, Peirce gives us more details on the matter: A sign, or representamen, is something which stands to somebody for something in some respect or capacity. It addresses somebody, that is, creates in the mind of that person an equivalent sign, or perhaps a more developed sign. That sign which it creates I call the interpretant of the first sign. The sign stands for something, its object. It stands for that object, not in all respects, but in reference to a sort of idea, which I have sometimes called the ground of the representamen. ‘Idea’ is here to be understood in a sort of Platonic sense, very familiar in everyday talk; I mean in that sense in which we say that one man catches another man’s idea, in which we say that when a man recalls what he was thinking of at some previous time, he recalls the same idea, and in which when a man continues to think anything, say for a tenth of a second, in so far as the thought continues to agree with itself during that time, that is to have a like content, it is the same idea, and is not at each instant of the interval a new idea (CP2: 228).

Although Peirce here refers to an idea “understood in a sort of Platonic sense,” he confines his reference to the everyday use of the term, and the “likeness” it presupposes is only the thought agreeing with itself as it lasts. Put in somewhat less technical terms, by a sign Peirce broadly means anything (a thought, an emotion, a name, a mathematical function, an argument of a mathematical function, an existent physical object, a natural kind, a tool, the word “tool,” a proposition in which the word “tool” is used, a musical concerto, an action, a picture, a graph, a move in a chess game, a law of nature—anything at all) capable of standing for something else, to someone who can give it a new interpretation based on its relation to the same object (cf. CP2: 228). Thus, “Santiago” in “Charles Santiago Sanders Peirce” stands for the name of a Biblical character to someone who knows that it represents a Spanish equivalent of “St. James,” and who can interpret it further as an expression of Peirce’s gratitude to his friend William James. “Qf4” stands for Bobby Fischer’s move in game six of the 1972 World Chess Series, to Boris Spassky who, while pondering over his unfortunate situation, was preparing to resign. “Black Square” stands for a visual expression of the mathematical regularities that represent the basic relations between form and color in suprematism, to an art critic who is writing a book about Kazimir Malevich. “10” stands for tetractys, an equilateral triangle consisting of ten points arranged in four rows of 1, 2, 3, and 4, to a Pythagorean mathematician Eudoxus, who further interprets it as a mathematical expression of the universal cosmic harmony. The same number also stands for “impossible to be scratched by a piece of corundum” to a mineralogist as a description of diamond’s hardness according to Mohs’ scale, etc. (cf. EP2: 3261). The triadic relation of a sign, A, standing for its object, B, to some further interpretant, C, presupposes a more abstract generic idea of triadicity as a part of Peirce’s logic of relatives. In this logic, monads represent a one-valence structure that results from the association of a single object with some quality (as in “___ is algebraic”). Dyads represent a two-valence structure that results from the existential connection between two objects (as in “___ attracts ___”). Triads represent a three-valence

 Peirce, C.  S. (1992–1998). The Essential Peirce. Selected Philosophical Writings. Vols. 1–2. N. Houser & C. Kloesel (Eds.). Bloomington, IN: Indiana University Press. Henceforth, EP followed by volume number, then page number. 1

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structure that results from three objects standing to each other in relation of interpretation (as in “___ gives ___ to ___”). What is notable about this approach is that it extends the notion of a proposition beyond the binary form and adopts a mathematical formalism in which a function can have any number of arguments. According to Peirce, the necessary character of triadic relations follows from the fact that all other relations are either reducible to them or, in some special sense, are contained in them. The rules are as follows. A relation is contained in a triad if its arity is less than three, and the reducibility case holds true for any relation whose arity is greater than three. In other words, tetrads, pentads, etc., can be completely analyzed in terms of triads only, and triads only, in turn, can produce relations with arity greater than three. Triads (three-valence structures) contain both dyads (two-­ valence structures) and monads (one-valence structures) in the sense that a lower-­ arity relation forms an organic part of a higher-arity one:

An existential relation between two objects presupposes some qualities associated with each of those objects but as such cannot be constructed as a sum of these associations. This relation involves something more due to its specific character as a dyad. Similarly, a triad (or, in terms of NL, an “interpretation”) presupposes some existential relation, but as such cannot be constructed by a simple addition of a monad to a dyad. It involves something more due to its specific character as a triad. Put differently, that the lower-arity relations thus form an organic part of the higher ones means that neither can a triad be broken down into dyads and monads nor can a dyad be broken down into monads alone, while keeping its original meaning (Burch 1992; Hereth and Pöschel 2011; Kleinert 2007). One relation being an organic part of another is also illustrated by the idea of prescision. In NL, Peirce gives the following example: “I can prescind red from blue, and space from color (as is manifest from the fact that I actually believe there is an uncolored space between my face and the wall); but I cannot prescind color from space, nor red from color” (W2: 51). Prescision, thus, is not a reciprocal process. By analogy with Peirce’s example, dyads cannot be prescinded from monads (yet monads can be prescinded from dyads), and triads cannot be prescinded from dyads (yet, again, the reverse is possible). From a stock limited to monadic relations (a), we can construct, by joining, a set of medads (b), i.e., relations whose arity is zero2: 2  Peirce’s term medad comes from the Greek μη for “not.” Peirce, whose first degree was in chemistry from Lawrence Scientific School at Harvard, had a tendency to draw broad parallels between his logic of relatives and the idea of chemical valence. In particular, he compared logical medads

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(a)

(b)

Here, (a) shows two independent unary relations, such as the monadic predicates “__ is a number” and “__ is odd.” Their combination results in a complete proposition, or a closed relation containing no blanks, where, for whatever X we choose, if X is as a number, X is odd (as in the first part of b), a particular case of which would be a proposition presupposing identity, x = x (as in the second part of b). Peirce’s graphical notation changed and evolved over time. For instance, in his Monist article on the logic of relatives (1897; CP3: 456–552), in which Peirce applies the concepts of Boolean algebra to relations, he uses the following graph for medads: h

d

Here, h and d are monads (say, “__ is a number” and “__ is odd,” respectively). If we agree that the enclosure around h separates the antecedent from the consequent, then h reads “anything whatsoever, if it is a number,” and the graph as a whole expresses the same proposition as above. Examples of Peirce’s notation from the Monist article just quoted above are useful, because they reveal one important feature of Peirce’s analysis. In most of the cases, Peirce carefully leads his reader through the process of decoding of a given diagram, however simple, leaving no detail unattended. Quoting just one such explanation in full is justified here as, following Peirce’s meticulous instructions, one can almost visualize his pointer moving along the paper and directing the reader’s eye exactly where it needs to be going:

l

v

w [In this] figure, l is the dyad ‘___ loves ___,’ and it is important to remark that the bond to the left is the lover and that to the right is the loved. Monads are the only relatives for which

to saturated chemical compounds—such that may result, for instance, from joining two bonds of a bivalent radicle (cf. CP3: 421).

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we need not be attentive to the positions of attachment of the bonds. In this figure, w is the monad ‘___ is wise,’ and v is the monad ‘___ is virtuous.’ The l and v are enclosed in a large common circle. Had this not been done, the medad could not be read…, because it would not consist of antecedent and consequent. As it is, we begin the reading of the medad at the bond connecting antecedent and consequent. Every bond of a logical graph denotes a hecceity; and every unencircled bond (as this one is) stands for any hecceity the reader may choose from the universe. This medad evidently refers to the universe of men. Hence the interpretation begins: ‘Let M be any man you please.’ We proceed along this bond in the direction of the antecedent, and on entering the circle of the antecedent we say: ‘If M be.’ We then enter the inner circle. Now, entering a circle means a relation to every. Accordingly we add ‘whatever.’ Traversing l from left to right, we say ‘lover.’ (Had it been from right to left we should have read it ‘loved.’) Leaving the circle is the mark of a relation ‘only to,’ which words we add. Coming to v we say ‘what is virtuous.’ Thus our antecedent reads: ‘Let M be any man you please. If M be whatever it may that is lover only to the virtuous.’ We now return to the consequent and read, ‘M is wise.’ Thus the whole means, ‘Whoever loves only the virtuous is wise’ (CP3: 475).

But let us continue with Peirce’s arithmetic of valences. It will be shown shortly that monads and dyads together give us only those relations whose arity is 0, 1, or 2, while a combination of triads can result in a relation whose arity may be represented by any number greater than 3. Dyads result from two objects being bound by some factual relation (as in “___ loves ___”). But the factual relation is not reducible to a total sum of qualities involved in it and cannot produce a surplus of valences. Thus, according to the first of the three graphs below, (a), joining, for instance, the two-place predicate “___ loves ___” (as in “Helena Jans loves Rene Descartes”) and the two-place predicate “___ loves ___” (as in “Descartes loves analytic geometry”) still yields the two-place proposition “___ loves somebody that loves ___.” According to Peirce, any number of dyads, no matter how great, will yield a dyad, as, for instance, in “__loves somebody that loves somebody that serves somebody that loves__” (CP3: 421). Along the same vein, according to the second graph below, (b), the two-place predicate “Xanthippe loves Socrates” and the one-place predicate “Socrates is mortal” result in a one-place predicate “__ loves something that is mortal.” Finally, the two blanks of a dyad may also be joined to make a complete proposition, or a closed relation; in this case, “___ loves ___” might mean, for instance, that something loves itself (c):

(a)

(b)

(c)

Peirce often uses an analogy between predicate structures, n-adic relations, and the chemical idea of an unsaturated bond. Thus, using his concept of rhema (a propositional symbol with some of its parts erased), Peirce gives the following important chemical justification for some of the cases just described:

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5  Sign Relation A rhema is somewhat closely analogous to a chemical atom or radicle with unsaturated bonds. A non-relative rhema is like a univalent radicle; it has but one unsaturated bond. A relative rhema is like a multivalent radicle. The blanks of a rhema can only be filled by terms, or, what is the same thing, by ‘something which’ (or the like) followed by a rhema; or, two can be filled together by means of ‘itself’ or the like. So, in chemistry, unsaturated bonds can only be saturated by joining two of them, which will usually, though not necessarily, belong to different radicles. If two univalent radicles are united, the result is a saturated compound. So, two non-relative rhemas being joined give a complete proposition. Thus, to join ‘__ is mortal’ and ‘__ is a man,’ we have ‘X is mortal and X is a man’ or some man is mortal. So likewise, a saturated compound may result from joining two bonds of a bivalent radicle; and, in the same way, the two blanks of a dual rhema may be joined to make a complete proposition. Thus, ‘__ loves __,’ ‘X loves X,’ or something loves itself. A univalent radicle united to a bivalent radicle gives a univalent radicle (as H—O—); and, in like manner, a non-relative rhema, joined to a dual rhema, gives a non-relative rhema. Thus, ‘__ is mortal’ joined to ‘__ loves __’ gives ‘__ loves something that is mortal,’ which is a non-relative rhema, since it has only one blank. Two, or any number of bivalent radicles united, give a bivalent radicle (as —O—O—S—O—O—), and so two or more dual rhemata give a dual rhema; as ‘__ loves somebody that loves somebody that serves somebody that loves __.’ Non-relative and dual rhemata only produce rhemata of the same kind, so long as the junctions are by twos … (CP3: 421).

Similar regularities are observed once we join triadic relations and any other relations whose arity is less than three. Thus, the predicate “___ orders ___ to do ___,” joined with an additional characterization (either monadic or dyadic) of any one of the three places, will still yield a predicate with the number of places three or less:

In his “Logic of Relatives,” Peirce also considers cases, in which, using his own notation, two triads result in a dyad, x

y

one triad is turned into a monad, x

and an even number of triads becomes a medad.

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x

y

x

w

In “The Basis of Pragmatism in Phaneroscopy” (1906; EP2: 364), Peirce gives a somewhat simpler and more diverse illustration of the combinations of triads resulting in relations whose arity is other than three:

Monad

Triad

Medad

Dyad Tetrad Composed of 2 Triads

Composed of 1 Triad

Monad

Triad Pentad Composed of 3 Triads

Medad

Dyad Tetrad Composed of 4 Triads

Hexad

Peirce attached special significance to the triadic relations he considered genuine (truly necessary or irreducible) as opposed to a variety of degenerate ones. Although the graphs above may make the irreducibility of triads a matter of direct visual intuition, the visual form by itself cannot explain what the irreducibility actually consists in. Take, for instance, the triadic relation “A lies between B and C,” and then consider a particular case of this relation, which specifically describes the face of a properly running clock. In this case, the relation amounts to a subset of the set A × B × C that contains such triples as {11, 4, 8} and {9, 2, 6} but does not contain such triples as {3, 2, 7} or {1, 10, 2}. This relation can be represented as a table, as a picture, as a mapping pattern, or as a graph. And it is, according to Peirce, not genuine. The question is: Why?

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To begin answering this question, let us take one of Peirce’s own examples. Compare the triads “New York is between Boston and Washington” and “A gives B to C.” The former is easily analyzable into the conjunction of the dyads “New York is south of Boston” and “New York is north of Washington.” In a similar manner, “12 is between 8 and 4 on the face of a running clock” is analyzable into the conjunction of the dyads “12 is before 4” and “12 is after 8.” “A gives B to C,” on the contrary, cannot be analyzed into, say, “A threw B” and “C picked up B,” because something that makes a triad a triad proper will be lost. Namely, this analysis would disregard both A’s intention that C own B and B’s willingness to accept C as a gift. In his “A Guess at the Riddle” (1888), Peirce offers the following clarification of the case: The fact that A presents B with a gift C, is a triple relation, and as such cannot possibly be resolved into any combination of dual relations. Indeed, the very idea of a combination involves that of thirdness, for a combination is something which is what it is owing to the parts which it brings into mutual relationship. But we may waive that consideration, and still we cannot build up the fact that A presents C to B by any aggregate of dual relations between A and B, B and C, and C and A. A may enrich B, B may receive C, and A may part with C, and yet A need not necessarily give C to B. For that, it would be necessary that these three dual relations should not only coexist, but be welded into one fact. Thus we see that a triad cannot be analyzed into dyads (EP1: 252).

In making a promise, I do not have an exclusive right over my own decision to act in a certain manner. I also cannot simply take it for granted that my having a good reason to do something means that you would also have a good reason to do the same thing were you in circumstances similar to mine. Actions that involve giving or promising—anything that presupposes the idea of a contract or a deal—result in a normative bond and the formation of the single common will that creates an overarching unity between the parties involved. Peirce also makes an important point that, in this case, “there need be no motion of the thing given. Giving is a transfer of the right of property” (CP1: 345). However, although giving requires “no motion of the thing given,” the transfer A and C are involved in is not an abstract idea external to what A and C actually do. Peirce is not a mathematical platonist. According to him, interpretations that realize a given set of relations are practices; they are not abstract platonic entities entirely independent of intelligent agents and the language those agents use. There is, then, a natural link between the triadic character of the genuine thought and the openness of the thought in addressing itself to future interpretations leading to the final resolve in action. Of course, a promise can be broken and a contract deemed null and void, but that does not make the established practices, or the very institution of promising or contracting any less powerful. Those are owned by a community, whose openness, as we have already learned from Peirce, makes any error, violation of a norm, or misuse of terms a public fact. Genuine triads bind their variables not by association (like monads) or by fact (like dyads), but by an established, institutionally embodied habit of mind sealing the bonds between their elements by the force of necessity. As mentioned above, according to Peirce, genuine triads alone are capable of producing relations whose arity is greater than three. If we construct a relation using triads alone, the arity of this relation will increase (CP1: 347, 363). As Peirce explains,

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Take the quadruple fact that A sells C to B for the price D. This is a compound of two facts: first, that A makes with C a certain transaction, which we may name E; and second, that this transaction E is a sale of B for the price D. Each of these two facts is a triple fact, and their combination makes up as genuine a quadruple fact as can be found. The explanation of this striking difference is not far to seek. A dual relative term, such as “lover” or “servant,” is a sort of blank form, where there are two places left blank. … But a triple relative term such as “giver” has two correlates, and is thus a blank form with three places left blank. Consequently, we can take two of these triple relatives and fill up one blank place in each with the same letter, X, which has only the force of a pronoun or identifying index, and then the two taken together will form a whole having four blank places; and from that we can go on in a similar way to any higher number. … Thus any number, however large, can be built out of triads; and consequently no idea can be involved in such a number, radically different from the idea of three. (EP1: 252).

This explanation has the following diagrammatic equivalent:

x

Thus, for instance, “___ gives ___ to ___” combined with “___ takes ___ from ___,” yields a four-place predicate “___ gives ___ to someone who takes ___ from ___.” And the result, when combined with another tetrad “___ sells ___ to ___ for ___” adds two more blanks, connecting the two into the sextuple relation “___ gives ___ to someone who takes ___ from someone who sells ___ to ___ for ___” (CP3: 421).3 The overall result can be represented diagrammatically like this:

 The portions of the text that signify the blanks of the two tetrads merge are marked in italics.

3

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Peirce explains that, whenever we join two dyads, we will end up with no more than just two blank spaces in the resultant combination, just like, whenever we build a road with three forkings, it will be able to have any number of termini bigger than three, an effect no number of straight pieces of a road put end on end will ever have (EP1: 252). Here is, again, Peirce’s own detailed illustration of this same effect. As Peirce’s diagram of forking paths above shows us immediately, nowhere do we need an intersection of more than three ways, and yet, using such intersections only, we can obtain any number of termini we like:

A road with a fork in it is the analogue of a triple fact, because it brings three termini into relation with one another. A dual fact is like a road without a fork; it only connects two termini. Now, no combination of roads without forks can have more than two termini; but any number of termini can be connected by roads which nowhere have a knot of more than three ways. See the figure, where I have drawn the termini as self-returning roads, in order to introduce nothing beyond the road itself. Thus, the three essential elements of a network of roads are road about a terminus, roadway-connection, and branching; and in like manner, the three fundamental categories of fact are, fact about an object, fact about two objects (relation), fact about several objects (synthetic fact) (W5: 244; MS 546).

Degenerate Relations According to Peirce, relations come in grades. As has already been mentioned above, there are triads and dyads that are genuine, and there are also those that are not genuine, or degenerate. A lower, degenerate grade of a relation originates in this relation borrowing some characteristics of a relation with smaller arity. The same notion of degeneracy also applies to Peirce’s broader definition of sign as something, A, that stands for something else, B, to an interpreting unit C. A triadic sign relation is degenerate if it can be analyzed into a simple aggregate of monadic and dyadic relations between its elements. As Peirce deliberately uses the mathematical notion of degeneracy, he naturally finds analogies to the idea of degrees of the triadicity in geometry. In particular, he mentions conic sections that can be either genuine (when they form a parabola, a hyperbola, or an ellipse) or degenerate (when they form either two intersecting lines or—when the plane intersects the cone only in the vertex—a single point) (EP1: 255). In a draft of the second Lowell lecture (1903), Peirce also gives the following comment on the matter:

Degenerate Relations

59

I borrow the designations genuine and degenerate from the geometry of plane curves. A curve is said to be of the second order if every ray cuts it in two points, real, coincident, or imaginary. Accordingly, the ellipse, parabola, and hyperbola, which only differ metrically,—that is to say, in respect to a matter of detail,—are curves of the second order, conics. But a pair [of] rays in the plain is cut in two points by every ray. It is therefore a curve of the second order; but we call it a degenerate curve of the second order, because it is really nothing more than two curves of the first order; while a conic which cannot be so resolved, is termed a genuine curve of the second order (MS 304).

Accordingly, there are no grades of degeneracy to monads, one grade to dyadic relations, and two grades to triadic ones. The difference between degenerate and genuine dyads is that the former combine their objects into one fact by virtue of resemblance only, while the latter create a stronger causal bond between the objects. Here is Peirce’s own description of the difference: Rumford and Franklin resembled each other by virtue of being both Americans; but either would have been just as much an American if the other had never lived. On the other hand, the fact that Cain killed Abel cannot be stated as a mere aggregate of two facts, one concerning Cain and the other concerning Abel (EP1: 254).

Peirce also gives an interesting example of a degenerate dyad as an expression of identity: Identity is the relation that everything bears to itself: Lucullus dines with Lucullus. Again, we speak of allurements and motives in the language of forces, as though a man suffered compulsion from within. So with the voice of conscience: and we observe our own feelings by a reflective sense. An echo is my own voice coming back to answer itself. So also, we speak of the abstract quality of a thing as if it were some second thing that the first thing possesses (CP1: 365).

Following similar logic, degenerate triads come in two degrees. To the first degree belong those that Peirce calls “accidental thirds.” They act as dyads that have the arity of 3 only in the sense in which a mixture has three ingredients which it brings together by nothing more than simply containing each. To the second degree belong those that Peirce calls “thirds by comparison”: “In natural history, intermediate types serve to bring out the resemblance between forms whose similarity might otherwise escape attention, or not be duly appreciated. In science, a diagram or analogue of the observed fact leads on to a further analogy” (EP1: 255). Genuine triadicity, unlike a degenerate one, has something to do with a reciprocal attunement between the mathematical mind and nature: If two forces are combined according to the parallelogram of forces, their resultant is a real third. Yet any force may, by the parallelogram of forces, be mathematically resolved into the sum of two others, in an infinity of different ways. Such components, however, are mere creations of the mind. What is the difference? As far as one isolated event goes, there is none; the real forces are no more present in the resultant than any components that the mathematician may imagine. But what makes the real forces really there is the general law of nature which calls for them, and not for any other components of the resultant (CP1: 366).

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Categories and Math Peirce took clarifying the idea of the three basic kinds of relations and analyzing the nature of each to be essentially a mathematical task because “[the reason] why there should be these three categories and no others … will be found to coincide with the most fundamental characteristic of the most universal of the mathematical hypotheses, I mean that of number” (CP1: 421). Another explanation of the mathematical character of the inquiry into the nature of the categories that Peirce has to offer consists in the reference to the link between the relational character of the categories, visual experience and mathematical reasoning. According to Peirce, deductive logic is best understood only as paired with the study of the logic of relations, which “corrects innumerable serious errors into which not merely logicians, but people who never opened a logic-book, fall from confining their attention to non-relative logic” (CP3: 641). Both the elements and the process of deductive reasoning must be observable, i.e., must contain iconic representations of relations connected by analogy, in such a way that “we cannot fail to remark that it is by observation of diagrams that the reasoning proceeds in such cases” (Ibid.). The analogical connections allow us to come to a conclusion by simplifying the relations. Moreover, a logic based on manipulating observable relational structures can show very effectively that not just one, but numerous lines of inference may be taken, independent from one another. Peirce also notes that the same effect is evident without the use of the logic of relatives when we consider how many theorems can be deduced from just a few simple premises of number theory. In the above, as well as in several other places, Peirce keeps bringing up the concept of number with reference to the relationship between the diagrammatic representation of n-adic relations and the way mathematicians demonstrate the deductive necessity of their conclusions. There is a hint here at the link between (1) the basic cognitive schematisms that are responsible for our capacity to match pieces of visual information together, (2) the idea of continuous observation understood as the process of inference, and (3) the idea of number. Peirce never goes into detail as to why the latter is important in this context, but a conjecture about his reasoning on the matter will be presented in Chapter 9 below, where the relationship between Peirce’s diagrams and Kant’s schemata will be discussed. By mid-1880s, Peirce became convinced that the categories, once the process of thinking is represented diagrammatically, could be directly extracted out of the representation. It is worthwhile mentioning that around the time Peirce’s “A Guess at the Riddle” was written, in 1886, Philosophical Transactions of the Royal Society of London published “A Memoir on the Theory of Mathematical Form” by Alfred Bray Kempe, a British mathematician best known for his proof of the four-color theorem (later shown incorrect). The opening paragraph of the memoir reads: My object in this memoir is to separate the necessary matter of exact or mathematical thought from the accidental clothing—geometrical, algebraical, logical, &c.—in which it is usually presented for consideration; and to indicate wherein consists the infinite variety which that necessary matter exhibits. … The memoir is confined to the exposition of fundamental principles, to their elementary developments, to their application to such a variety of

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cases as will vindicate their value, and to a description of some simple and uniform modes of putting the necessary matter in evidence (Kempe 1886: 2).

The pure form of the mathematical thought, separated from the accidental geometrical, algebraical, and logical clothing, was presented by Kempe as a system of diagrams that consisted of spots connected by different types of lines. Kempe’s diagrams were supposed to express the universal form common to algebraic and geometrical representations that would reveal a “deep grammar” of mathematical thinking. Peirce read the Memoir and later, in 1892, commented on it, noting that “… it is to be remarked that Mr. Kempe’s conception depends upon considering the diagram purely in its self-contained relations, the idea of its representing anything being altogether left out of view; while my doctrine depends upon considering how the diagram is to be connected with nature” (CP3: 423). It is partly in view of this criticism that four years later, in 1896, Peirce redefined his list of categories in his paper “The Logic of Mathematics: An Attempt to Develop my Categories from Within” (CP1: 417–519; Esposito 1979, pp.  57–58). Beginning with this paper, Peirce’s theory of relations makes a decisive step beyond purely mathematical considerations. The three kinds of relations—monadic, dyadic, and triadic—now represent three elementary ideas endowed with both linguistic and ontological significance. As will be shown, they give form to our judgments and formulas and, at the same time, refer to the basic features of whatever our judgments or formulas are about. While a one-place predicate ascribes a quality to an object, a genuine monad is an embodiment of some quality. While a two-place predicate connects two objects into a single fact or event, a genuine dyad fleshes out causal relation. While a three-place predicate relates two objects by virtue of a mediating third, a genuine triad expresses a regularity or a natural law.

Chapter 6

One, Two, Three

Abstractions, Things, and Signs Throughout his career, Peirce moved gradually from his early understanding of the categories as the universal characteristics of cognition and judgment to a broader understanding of them as the most basic mathematical elements inherent in nature and thought. From this perspective, Peirce’s early trichotomy of categories (quality, relation, and representation) may be restated as comprising abstractions, experiences, and signs proper, or, as Peirce preferred to call his categories from the mid-­1880s on, firstness, secondness, and thirdness. An abstraction, or a universal, is a quality that is a one predicable of many (there are many things that are characterized as “black”). An abstraction always equals itself, refers to nothing but itself, and is never present in experience as such. To use Peirce’s example from NL again, in saying “this stove is black,” we do not mean to say that blackness as such is in the stove. Qualitative abstraction (or, as Peirce also calls it, suchness) is expressed numerically by one. From this new perspective, it is not just one of the three necessary conceptual elements of predication alone, but, more generally, the first elementary idea of relatedness, which contains no reference to any kind of comparison, juxtaposition, causality, or resistance. It is not yet an idea of a thing or an object. It is an idea of likeness—the idea that is functionally related to sameness or identity. Something, as being the same, can be reproduced or repeated. It might be said that a qualitative abstraction is an internal moment of a repetition of one and the same before the very fact of the repetition itself became an object of reflection. The repetition itself is in place, but the duration, the time during which it happens is not apprehended yet. It is a pure duration, not yet a stretch between our memories of the past and our expectations about the future. It is a note repeated without an underlying harmony, or a syllable repeated without creating

© Springer Nature Switzerland AG 2023 V. Kiryushchenko, Diagrams, Visual Imagination, and Continuity in Peirce’s Philosophy of Mathematics, Mathematics in Mind, https://doi.org/10.1007/978-3-031-23245-9_6

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either alliteration or a rhyme. It is a situation when the repetition is not a part of something else and, therefore, itself remains unapprehended as such. According to Peirce’s own description, there are certain qualities of feeling, such as the color of magenta, the odor of attar, the sound of a railway whistle, the taste of quinine, the quality of the emotion upon contemplating a fine mathematical demonstration, the quality of feeling of love, etc. I do not mean the sense of actually experiencing these feelings, whether primarily or in any memory or imagination. That is something that involves these qualities as an element of it. But I mean the qualities themselves which, in themselves, are mere may-bes, not necessarily realized (CP1: 304).

In identifying the first category with abstraction, Peirce makes a reference similar to that of John Austin who notes in his “Are There A Priori Concepts?” (1939) that, speaking carefully, we can sense “red” with the same level success as we can sense “resemblance” or “quality.” We do sense something of which we might, on occasion, say “this is red,” just as we do sense something of which we might report “this resembles that” or “this ‘red’ is similar to that ‘red’” (Austin 1961, pp. 17–20). For this very reason, taken as a category, quality, according to Peirce, is not an empirical, but a hypothetical concept. For instance, although we recognize the feeling of pleasure that we experience in different circumstances and in relation to different things, the quality of this feeling as such (in the categorial sense of the term) has nothing to do with the dynamic conditions of our sensibility and can be abstracted from the latter. Once it is abstracted, what the quality is as such becomes unavailable, but it can be pointed at and named in particular circumstances we find ourselves (“this feels good” or “that looks red”). The resulting abstraction represents an uncontextualized or pure repetition in the sense just described. As such, it can only refer to a unique series of events and, therefore, does not constitute a true generality, a generality of a law that connects events through cycles of comparisons, juxtapositions, and causal relations discernible in different contexts as parts of some behavioral patterns or habits. Peirce comments both on the temporal and on the spatial aspects of the immediacy of quality (and of the underlying idea of sameness). In the third of his Lowell lectures (1903), he says that “[t]he immediate present, could we seize it, would have no character but its Firstness. Not that I mean to say that immediate consciousness (a pure fiction, by the way), would be Firstness, but that the quality of what we are immediately conscious of, which is no fiction, is Firstness” (CP1: 343; emphasis added). Firstness is one of the universal elements framing our experience. In “Meaning” (1910), Peirce makes the following observation: “On a map of an island laid down upon the soil of that island there must, under all ordinary circumstances, be some position, some point, marked or not, that represents qua place on the map, the very same point qua place on the island” (CP 2:230). There is going to be one point on the surface of the map that is also a point on the surface of the mapped area (CP 2:230). As it is a geometric point, one can label it with a pin, but one can neither actually define its spatial limits, nor mark a particular moment when it is passed as a part of a route laid through it on the map. This example is as close as we can get if we want to attach a visual image to the idea of a quality. Quality

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signifies the immediacy that is commonly—yet quite mistakenly—ascribed to perception. This immediacy is localized as something present “here and now.” But the “here” and the “now” change. Now is day, but time goes by and what is now becomes night. Similarly, here is a tree, but I turn my head and what is here becomes a house. In fact, both “here” and “now,” as the forms of this localization, turn out to be incapable of containing or presenting any sensible particular, and so the qualitative immediacy they localize turns out to be the most abstract and poorest notion of all. Unlike a quality, or abstraction, the second category—a thing or a matter of fact―is something the existence of which follows from experience. Unlike a quality, it never equals itself, because whatever comes with experience cannot be simply given. The existence of a thing or a fact is always known only through one’s acquaintance with some other thing or fact, with which the former is causally connected. Thus, a weathervane is causally connected with the direction of the wind, the mercury stem in a thermometer is causally connected with the temperature, and one’s index finger is causally connected with the direction one is pointing at with it. For some one thing to come into existence, there must be another thing in place: An essence of some one thing (in the categorial sense of the term) thus consists in how a second thing is (CP1: 24). Existence is binary. It is a dyad, and its idea is expressed numerically by two. In a dyad, two objects are existentially differentiated from one another and, by virtue of this very differentiation, connected. That by which they are connected does not constitute a third object of the relation, but rather represents a monadic aspect of it, a monad within a dyad. Thus, according to one of Peirce’s examples quoted above, the fact that Cain killed Abel is not a mere aggregate of two facts, one concerning Cain and the other concerning Abel; there is a third fact of murder involved. Yet, the fact of murder does not constitute the third element of the relation but only something that Cain and Abel have in common—not in themselves (as each of two different qualities would), but through a causal connection (EP1: 254). What Cain did and what Abel suffered from are connected causally. The juxtaposition of one quality with another in a dyad amounts to a relation between ego and non-ego, which yields the feeling of externality and constraint. [There is a category] which the rough and tumble of life renders most familiarly prominent. We are continually bumping up against hard fact. […] You get this kind of consciousness in some approach to purity when you put your shoulder against a door and try to force it open. You have a sense of resistance and at the same time a sense of effort. […] It is a double consciousness. We become aware of our self in becoming aware of the not-self. … (CP1: 324–325).

The idea of second reveals itself through causation and static forces; a car hitting a wall, a key hanging on a keyholder, a string attached to a bow—in all these examples, the forces involved occur between pairs. According to Peirce, our senses (touching or smelling something), our will (a set of internal events leading up to an act), and our memories (pieces of our past directly affecting our perceptual capacities) are also examples of dyads or reactions of secondness between the ego and the non-ego. In these reactions, we are either agents (as in the case of an at-fault car accident) or patients (as in the case of memories and perceptions). Our perceptions

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result from nerve reactions and muscular efforts, one of our memories suddenly interrupts another, in all those cases we experience a kind of forcefulness, the characteristic of the double consciousness Peirce calls “secondness.” In describing how constraint is an indispensable characteristic of secondness, Peirce also implies two important things. First, he says that dyads, unlike monads, have a relation to time. Time is not considered as just a pure duration marked by an uncontextualized repetition (i.e., not as an unmeasured duration between two similar events). Time is the way the past acts on us through memory. In his Monist paper “Issues of Pragmaticism” (1905), Peirce gives us more details: That Time is a particular variety of objective Modality is too obvious for argumentation. The Past consists of the sum of faits accomplis, and this Accomplishment is the Existential Mode of Time. For the Past really acts upon us, and that it does, not at all in the way in which a Law or Principle influences us, but precisely as an Existent object acts (EP2: 357; CP5: 459).

Just as perception tells us something about that object over there, as distinct from the one over here, memory tells us something about an event as taken place then rather than now (sf. CP1: 24). Secondness, as an indispensable element of our experience, transfers the immediacy of the universal “here and now” into the direct consciousness of something being “there and then.” An abstraction, which was lacking the embodiment, enters into existence. As perception, this direct consciousness, therefore, is of an object distanced from us by one (however small) unit of space; and as memory, it is of an object distanced from us by one (however small) unit of time. Those units are not “seconds” and “millimeters,” but certain minimal samples of duration and extension required for us to distinguish “here” from “there” and “now” from “then.” What we have in this case is not an abstract quality anymore, but an infinitesimal quantity (the concept which is of utter importance for Peirce in relation to the idea of continuity, to be discussed in Chapter 14). Just as the link, which holds together two facts in a dyadic relation, does not constitute a third fact proper, space and time, as the forms of secondness, do not yet represent the true continuity. The former is a mere togetherness of objects (this picture on the wall, above that table) and their properties (this salt is white and also tart, and also cubical in shape, etc.). The latter is a mere succession of events, one following another. Secondness organizes qualities in mathematical sets that can be mapped one onto another. Importantly though, as the direct consciousness of “there and then,” it does what no diagram or any other iconic sign can do: The where and the when of the particular experience, or the occasion or other identifying circumstance of the particular fiction to which the diagram is to be applied, are things not capable of being diagrammatically exhibited. Describe and describe and describe, and you never can describe a date, a position, or any homaloidal quantity (CP3: 419).

Even a gesture intended to direct our attention towards an object, which requires some discursive context in order to be properly interpreted, and a map, which is a diagram that actually indicates localities and orients us in space, can be used only once we understood the law of projection according to which this map is created.

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The pure denotative force of secondness is non-discursive. Peirce goes on to claim that “If a diagram cannot do it, algebra cannot: for algebra is but a sort of diagram; and if algebra cannot do it, language cannot: for language is but a kind of algebra. It would, certainly, in one sense be extravagant to say that we can never tell what we are talking about; yet, in another sense, it is quite true” (Ibid.). We do have a distinction between places and times in language, and yet no description can actually locate objects as being “there” and events as happening “then.” For instance, when we learn from Chapter 1 of James Joyce’s Ulysses that “Stately, plump Buck Mulligan came from the stairhead, bearing a bowl of lather on which a mirror and a razor lay crossed,” (Joyce 1993, p. 3) we need indefinitely more knowledge about where the stairhead was, what exactly the bowl looked like, and other details that might help our imagination visualize the scene. And whatever information we have, will still not be enough. Although secondness can get objects and events properly located, it cannot coordinate them in a way their relation to one another could be interpreted. While firstness does not have the forcefulness of secondness, and so cannot by itself originate a higher arity relation, secondness lacks the mindedness of thirdness, and so cannot originate a higher-arity relation either. As Peirce the pragmatist would suggest, secondness cannot subject the way its objects are related to a certain habit of mind (cannot organize them into a relation of meaningful repetition). Now a sign proper differs both from abstractions and from things. It is something that connects the former with the latter and—which is important—shares the characteristics of both. Like an abstraction (and unlike a thing), a sign is a one predicable of many (thus, for instance, there is more than one way to determine temperature or the direction of the wind). And like a thing (and unlike an abstraction), it never equals itself because, as a sign, it always stands for something that it itself is not (thus, for instance, a word that names an object is not an object itself). A sign’s identity, in other words, is in its reference to something other than itself: whenever there is a sign, there is a reference to some other quality, thing, or fact. However, unlike something that is merely a thing (or a dyad), a sign refers to its object not through a causal connection (although as a sign it can be about such a connection), but through a connection by means of habits, where a habit means a general disposition to behave in a certain way, an internalized rule. Thus, an English phrase, a facial expression of sentiment, a theatre ticket, or a mathematical equation are all examples of a sign. Each is a concrete occurrence characterized by some quality (a loud phrase, a joyful expression, a paper ticket, an elegant equation) and refers to something it itself is not (a norm prescribing how to understand the phrase, to react to the expression, to use the ticket, and to apply the equation) by means of expected behavior. Insofar as greeting someone, going to a theatre, solving an equation, and other patterns of behavior become routine, they become habits or generalized tendencies to act in certain ways. It is in virtue of this reference to practices, to behavioral patterns, and not in virtue of reference to some private episode of one’s inner mental life, that a concrete occurrence of a sign provides some ground for its future interpretation: I use “How are you doing?” to greet a friend, a ticket ensures my access to a theatre performance, I express joy when I win a lottery, and I apply eiπ = 1 to

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express Euler’s identity. In these examples, a sign brings together (or, in Kant’s parlance, “performs a synthesis of”) the concept of a thing, the concept of a quality that grounds the relation of this thing to something else, and the concept of expected behavioral outcomes that the use of this sign implies for an interpreter. A sign proper is thus a triadic relational structure, which mediates between things, thoughts about those things, and habits of action entailed by those thoughts. While an abstraction, through pure repetition, relates me to the empty present, and an existent thing makes me aware of the causally conditioned immediate past, a sign always relates me to the possible future. And, in doing so, it induces me towards and makes me a part of a certain communal effort. Because it involves a relation to the possible future, exchanging signs organizes our perception of time. Time as a relation between abstractions, things, and signs (or between firstness, secondness, and thirdness) is thus a relation between empty present, experienced past that fills the void of the present, and expected future that tells us what we need to do for the present to make sense.

Modes of Being In his later writings, Peirce often laid stress on the dependence of logic on phenomenology (CP1: 191; 2: 120; 5: 39; 8: 297), which “ascertains and studies the kinds of elements universally present in the phenomenon,” where the phenomenon is defined as “whatever is present at any time to the mind in any way” (CP 2.186). Just as in the case of his earlier versions of the list of his three categories, Peirce made it explicit that the task of defining such universal elements has everything to do with the foundations of mathematical knowledge: “The real aim is to find an indisputable theory of reasoning by the aid of mathematics. The first step in the order of logic towards this end … is to formulate with mathematical precision, definiteness, and simplicity, the general facts of experience” (CP3: 618). Peirce’s firstness, secondness, and thirdness, as the three fundamental elements that are present to one’s mind at all times, represent a phenomenological version of the earlier triad that underlies his idea of the logic of relations. Given that any deduction, according to Peirce, involves observation, the phenomenological categories are, by extension, indispensable for mathematical deductive reasoning. Besides phenomenological interpretation, Peirce also suggests an ontological one, which, in turn, is closely related to the modal understanding of the three categories as possibility, actuality, and necessity (Short 2007: 60–90). If, phenomenologically speaking, one, two, and three comprise what is observed by the mind, from the ontological standpoint, Peirce says, they represent certain “modes of being.” Thus, firstness is more than simply a conceptual element in a theory of predication and more than a context-free pure repetition:

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Firstness is the mode of being which consists in its subject’s being positively such as it is regardless of aught else. That can only be a possibility. For as long as things do not act upon one another, there is no sense or meaning in saying that they have any being, unless it be that they are such in themselves that they may perhaps come into relation with others. The mode of being a redness, before anything in the universe was yet red, was nevertheless a positive qualitative possibility (CP1: 25; emphasis added).

Whatever is in itself but may, perhaps, enter into relation with others, is like a geometrical point that, if connected to another point on a plane, may turn into a line, or like an instant that, if connected with another instant, may turn into a duration. Secondness is the actuality of a thing; it is “a mode of being independent of any qualities or determinations, or, as we may say, having brute fighting force, or self-­ assertion” (CP1: 434). It is a “mode of being over and above what its mere inward suchness involves … gained by its opposition to another, that suchness does not avail to confer” (CP1: 456). Actuality, however, lies not simply in its being a dyad, but in its more global relation to the universe of all other existent things, facts, and events (CP1: 24). It is, Peirce says, “the force of existence” (CP4: 172) or “the mode of being consisting in its forcing itself into a place in the world” (CP1: 492). Just as, in Peirce’s logic of relations, dyads are not something extraneous to monads, so secondness is not something extraneous to firstness, but incorporates it organically. A mathematical relation that is demonstrable graphically in the logic of relations also constitutes a part of the external world. A possibility is, in some sense, an element of actuality, and actuality is a possibility actualized. Peirce further makes a clear distinction between actuality as a characteristic of secondness, which pertains to the universe of individual things and facts, and reality as a characteristic of thirdness. The latter, he says, is “a mode of being which consists … in the fact that future facts of Secondness will take on a determinate general character” CP1: 26). Reality, then, according to Peirce, is of the nature of laws and regularities. It also pertains to what Peirce calls “would-bes.” When I claim that something would happen were the circumstances such-and-such, I establish a connection between two conceivable situations and then define what the possible future outcomes of me believing such connection to be true might be. Thus, for instance, the probability of a die thrown from a dice box to turn up a number divisible by three is one-third. This statement means that the die has a certain “would-be”; that is to say, relative to the conditions established, it has a property analogous to a habit that a man might have (CP2: 664). The result of every particular throw is always probabilistic, and in making my predictions, I can never exhaust the whole series of events of an appropriate kind. If I were to throw a die, a number on it, divisible by three, would turn up one-third of the time provided that I kept throwing the die long enough. While this result is not fully guaranteed at any particular moment of time, it will, at some point, reveal a tendency to converge to the probability ratio (one-­third) based on a set condition (“divisible by three”). Every time my resolution to do what I intend to do actually brings about the outcomes I foresee, I should take my conception of the connection between the circumstances and the outcomes (or what a sign that stands for this connection to me tells me) as true. And if, on the contrary, my resolution to do

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what I intend to do falls short of the outcomes I foresee, I should do whatever it takes to redefine the set condition, re-examine the overall situation, and repeat the procedure. Importantly, what it all means is that my resolution to act in the present (e.g., my resolution to throw a die) always appears to be affected by the possible future (for instance, by my betting on a certain outcome when playing craps). This, according to Peirce, is exactly what defines my connection to reality. But why exactly the would-bes Peirce mentions in this context are real? There is nothing either mysterious or excessively metaphysical about this kind of realism. As has just been mentioned, taking the possible future outcomes into consideration (or predicting possible outcomes of actions we are prepared to perform) obviously affects what we do here and now. The future that our statistically framed experiments expose as likely lays constraint on what we can think and do in the present (Hausman 1993, pp.  167–168). The constraint affects our behavior not simply causally (in the way actual objects affect each other, i.e., one billiard ball hitting another), but in terms of correcting our habits—not just individual actions, but modes of conduct. A substantial part of the progress of our inquiry consists in figuring out what exactly went wrong, and what type of action we need to choose to correct our errors. Thus, the possible future, in reshaping our current habits, introduces an element of self-control both to our thought and to our conduct. For instance, if our study of the relationship between the temperature and pressure of an ideal gas heated in a closed container showed that the equation pV = nRT does not hold, we would have to look for what might have introduced an error. In a similar manner, if we claim a liquid to be ethanol, and yet its fermentation by acetic acid bacteria does not give us vinegar, we have to figure out what exactly made our claim false. In this case, simply referring to our rationality would not make any difference whatsoever. We would have to make actual changes to the initial setting, check the equipment, revaluate some of our basic assumptions about residual properties, repeat the experiment, introduce a new hypothesis, and, if nothing else works, take a blind guess. In short, we actually need to do something in order to clarify what we mean. Actual objects, facts, and their relations to each other are characterized by brute causal force. Objects and facts are existent in their reactive effects, which by themselves do not constitute any real constraint in the above sense. Would-bes, on the contrary, are real as far as they are characterized by a prognosticated law-like behavior. A stone, when released, would fall in a predictable manner because it has a general property of being so disposed. The law-like behavior of the stone is due to a set of features that define its real nature, and not merely its existence as a member of some actual causal chain (Margolis 1993: 295–300). Laws (e.g., the law of gravitation) do not exist as such, but they are operative in existent things and events (e.g., objects falling down) which are their concrete realizations (CP8: 12). A law is real because it cannot as such be exhausted by all interactions that did, or ever will, exemplify it, and so it always goes beyond any particular collection of facts: “No collection of facts can constitute a law; for the law goes beyond any accomplished facts and determines how facts that may be, but all of which never can have

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happened, shall be characterized” (CP1: 420)1. In his 1910 reflections on the doctrine of chances, Peirce writes: It is very true, mind you, that no collection whatever of single acts, though it were ever so many grades greater than a simple endless series, can constitute a would-be, nor can the knowledge of single acts, whatever their multitude, tell us for sure of a would-be. But there are two remarks to be made; first, that in the case under consideration a person is supposed to be in a condition to assert what surely would be the behavior of the subject throughout the endless series of occasions—a knowledge which cannot have been derived from reasoning from its behavior on the single occasions; and second, that that which in our case renders it true, as stated, that the person supposed “ipso facto knows a would-be of that subject,” is not the occurrence of the single acts, but the fact that the person supposed was in condition to assert what would surely be the behavior of the subject throughout an endless series of occasions (CP2: 667).

Reality, which characterizes general objects, can be exhausted by actuality with the same level of success as the Greek language, when we talk or think about it, can be reduced to such Greek words as we happen to use at the time we do so (CP5: 504). If reality could be exhausted by actuality, no normativity would be even remotely possible: we would be able to talk only about what there is and would not be able to tell anything at all about what there ought to be. According to Peirce, to say that a law goes beyond any collection of facts is to say that both the generality of a law and its predictive power amount to its capacity to relate certain relevant known facts to their possible future interpretations. Now, to this realist claim a nominalist might object that the relation in question is of purely intellectual nature, and that, while facts are particulars, or something that actually happens, laws are mere abstract forms, or the ways we describe particulars. The third of eight lectures Peirce gave at the Lowell Institute in Boston in 1903, under the general title “Some Topics of Logic bearing on Questions now Vexed,” contains the following response to the nominalist: To say that a prediction has a decided tendency to be fulfilled, is to say that the future events are in a measure really governed by a law. If a pair of dice turns up sixes five times running, that is a mere uniformity. The dice might happen fortuitously to turn up sixes a thousand times running. But that would not afford the slightest security for a prediction that they would turn up sixes the next time. If the prediction has a tendency to be fulfilled, it must be that future events have a tendency to conform to a general rule. “Oh,” but say the nominalists, “this general rule is nothing but a mere word or couple of words!” I reply, “Nobody ever dreamed of denying that what is general is of the nature of a general sign; but the question is whether future events will conform to it or not. If they will, your adjective ‘mere’ seems to be ill-placed” (CP1: 26).

Let us say that you, like Guildenstern from Tom Stoppard’s Rosencrantz and Guildenstern are Dead, consistently, time after time, bet tails on the flip of a coin, and ninety-two coins flipped come down heads ninety-two consecutive times. On the one hand, if you believe exclusively in the existence of particulars, you might explain your gambling misfortune away by saying that, because each individual  Elsewhere, Peirce also makes the following statement: “No agglomeration of actual happenings can ever completely fill the meaning of a ‘would-be’” (CP 5.467). 1

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coin, flipped individually, is as likely to land heads as tails, it should cause no surprise that each individual time it actually does land either one or the other way. On the other hand, you might know that a set of ninety-two flips of a coin gives you 292 possible ways the entire sequence of your flips has developed so far, and that, as you keep flipping, the probability of each outcome approaches the bell curve as a limit, with the continuation of the series of heads becoming progressively less and less likely. Some logicians (quite wrongly) call this the “Gambler’s fallacy.” From the Peircean perspective, this explanation should suffice to dissuade you from setting up to re-examine the fundamentals of the law of probability right away and begin more modestly, by simply acknowledging that, given how it all has been playing out so far, it probably just isn’t your day. The difference between the two explanations is obvious. While the second explanation will ultimately help you succeed in predicting future outcomes of your actions based on the rule it formulates, the first one (whatever some logicians might say) seems to refer to a mere regularity that has no predictive power whatsoever. It can be observed that Peirce’s late understanding of the nature of laws is closely connected to what he had to say in his earlier works (NL and “Some Consequences” in particular) about the ideas of an interpretant and a community of inquiry. An interpretant, as it is first introduced in NL, is a triadic mediating representation that brings the manifold of impressions to unity (or subsumes them under a general concept) by addressing them, in this very act of unification, to possible new interpretations. Peirce’s deduction in NL thus demonstrates that generality is not a characteristic of an abstract content in one’s individual mind, but is, essentially, a mediating capacity. And, just like the meaning of a law cannot be exhausted by a finite set of facts, the community of inquiry Peirce has in mind cannot be limited by any particular set of members. What makes it a community is the maxim-driven desire of any one of its members to make whatever is currently known available for further interpretations, and to use those interpretations to correct their current beliefs. Peirce insists on a strong connection between laws exceeding any finite collection of facts, triadic mediating relations, and the human capacity of making predictions based on maxim that organizes our experience with reference to a regulative ideal. From this, ontological perspective, categories are no more seen as just Kantian basic a priori concepts which, applied to the forms of intuition, allow us to form judgments about the phenomenal world. They are essential features of the world itself. To be, therefore, according to Peirce, is to partake in the nature of the categories and, ultimately, to be is to be a sign. As Fisch (1986) notes, at this point “Peirce’s general theory of signs [becomes] so general as to entail that, whatever else anything may be, it is also a sign” (p. 357). Now, as it might seem, given that everything is a sign, and that every sign is addressed to its future interpretation, any future interpretation cannot result in anything but the creation of a new sign, which inevitably leads to the creation of another sign, etc., ad infinitum. But this is not quite so. On the one hand, any and every interpretation has a long-term goal defined by its reference to the idea of extended community. As has been discussed in Chapter 2, no matter where different members of a community may begin, as long as they

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follow the method formulated in the maxim of pragmatism, the results of their research should eventually converge toward the same outcome. On the other hand, any and every interpretation has a short-term goal. Believing a proposition is true amounts to being prepared to act habitually on this belief when the occasion presents itself. For instance, believing in the proposition “Gödel’s Second Incompleteness Theorem shows that Peano arithmetic cannot prove its own consistency” amounts to being prepared to employ Cantor’s diagonal argument, or to clarify the idea of self-­ reference in Russell’s paradox. Besides, the maxim of pragmatism tells us that the meaning of whatever sign we use consists in practical effects of such habitual actions in the form of outcomes to be expected (in terms of the example above, this might mean a proof that uncountable sets exist, or the fact that the construction of a formal theory of truth faces a significant barrier, etc.). Now given the fact that believing something consists in being prepared to act on the belief according to a habit, together with the fact that meaning something by using a sign amounts to the sum total of practical effects of conduct based on the habit, we may, after Peirce, conclude that the short-term goal of every new sign is the formation of a habit of action. Signs bring habits about and, at the same time, they are also catalysts that cause those habits to be reinforced or changed. This presupposes that, although semiosis is—theoretically—unlimited, it is, according to Peirce, short-circuited by our practices of habitually using signs.

Chapter 7

Iconicity, Novelty, and Necessity

Novelty It is a well-known fact that Peirce treated visual perceptions as results of unconscious inferences (Hull 2017, p.  150). On his view, any percept is essentially a product of a long history of gradually habitualized, piecemeal adjustments and readjustments to the ever-changing environment. Every visual experience, considered as such readjustment, Peirce says, is “constructed at the suggestion of previous sensations,” all of which are “quite inadequate to forming an image or representation absolutely determinate” (W2: 235). Peirce’s overall conclusion here is that “when we see, we are put in a condition in which we are able to get a very large and perhaps indefinitely great amount of knowledge of the visible qualities of objects” (W2: 236; emphasis added). All this knowledge is constitutive for our vision, and yet what it represents cannot become an object of perception in its entirety as a set of fully determinate particulars. This means that, in order to see and make perceptual judgments, we need to be able to lay stress on some features and drop some others. This inferential selectiveness of vision results from a process that is beyond our conscious control. During the construction of mathematical diagrams, the same selective process takes place, but in this case, we carefully and attentively skeletonize whatever is available within our visual field. Diagrammatization, which, according to Peirce, plays a prominent role in mathematics, is, therefore, inevitably (albeit in a somewhat different vein) performed in every act of vision. The selectiveness of perception also implies that, as Peirce has it, we cannot but admit that “either we perceive some indeterminate properties or we perceive nothing at all” (Wilson 2017, p. 16). Whatever is dropped out from our direct awareness, still remains within our perceptual field and plays a constitutive part in our visual integration. Just as there is nothing that is absolutely determinate in perception at any given moment, there is also nothing in it that is absolutely raw and singular, as anything that is absolutely raw and singular in perception at any given moment is only a brute © Springer Nature Switzerland AG 2023 V. Kiryushchenko, Diagrams, Visual Imagination, and Continuity in Peirce’s Philosophy of Mathematics, Mathematics in Mind, https://doi.org/10.1007/978-3-031-23245-9_7

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existence, a purely denotative “that.” The very immediacy of the presence of such raw data would not allow us to say anything about it and, therefore, to treat it as a sign proper. Once the idea of such immediacy is disproved, all we are left with is what can be used to form perceptual judgments, which are but reports on perceptions, however imperfect. Although those reports do not provide us with immediate access to any perceptual material, they allow us to infer errors and inconsistencies in the former reports, with which they are discursively consistent. And beyond the record those judgments leave us, says Peirce, we cannot go (CP5: 151–157). But if there is nothing that is absolutely raw and singular in perception, then there must be something general in it. And whatever is general, according to Peirce, is not such as it is because it possesses some general “nature,” but just as long as it addresses itself to possible further interpretations. Any perception, in referring to its object, points at some general way to respond to it every time certain conditions hold, even if those responses represent adjustments so miniscule as to be barely detectable (cf. Peirce and Jastrow 1884). With this in mind, to be able to refer to an object in this way is, according to Peirce, to be a sign. Now, as has been discussed in previous chapters, any sign, as far as it is interpretable, is esse in futuro. It is what it is going to become as interpreted in the future, and anything of value in it lies in conditional expectations implied by its possible future interpretations. Just as every thought, every visual perception contains something expected, which, in fact, is inseparable from what this visual perception, allegedly, simply is (CP2: 146). From this, it follows that novelty is a common characteristic of visual perceptions as far as they are signs. Accepting the considerations above means conceding to the fact that vagueness and generality, in the form of a potential or conditional future, are given to us (in some sense of “given”) both in thought and in visual experience. Mathematicians are in possession of specific diagrammatic instruments that allow them to use this fact deliberately instead of simply unconsciously accepting it. These instruments enable them to discern universal truths in particular diagrams due to their capacity to make informative guesses about the novel results obtained from possible transformations of those diagrams. To sum up, what our capacity of perceiving indeterminate properties prima facie means is that every act of visual experience—although what it delivers cannot be changed at will—presupposes interpretation and leaves space for errors, interpretive hypotheses, and imaginative musings about its object (Paavola 2011, p. 305; Vargas 2017). This fact has two further implications. First, if all perception, due to its indeterminacy, involves interpretation, and if iconicity and diagrammatic selectiveness is an integral part of all reasoning, then diagrammatic aspects of reasoning should be considered responsible for the creativity not only of mathematical cognition but also of human cognition in general (Paavola 2011, p. 298). Christopher Hookway takes this argument even further by claiming that the very resemblance between objects as we perceive it is determined by interpretive novelty and that the former cannot be present without the latter: The key of iconicity is not perceived resemblance between the sign and what it signifies but rather the possibility of making new discoveries about the object of a sign through observing features of the sign itself. Thus, a mathematical model of a physical system is an iconic

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representation because its use provides new information about the physical system. This is the distinctive feature and value of iconic representation: a sign resembles its object if, and only if, study of the sign can yield new information about the object. (Hookway 2000, p. 102)

The second implication is that diagrammatic iconicity must be an important characteristic of any natural language. Thus, Peirce, for instance, claims that “in the syntax of every language, there are logical icons of the kind that are aided by conventional rules” (CP2: 280). The suggestion here is that a language is capable of storing old and conveying new information not only because it symbolically encodes this information and refers to appropriate external objects but also due to the fact that its syntax iconically frames our perception. In a sense, the way we write and read prompts the way we think. On this view, the very order of meaning to some extent depends on the visual schematisms set up by the general syntactic arrangement of a given language, and our capacity to produce new meanings has something to do with this very arrangement. Given everything that has been said about the relationship between our perception and our capacity to produce new meanings, mathematical diagrams represent an interesting case. The indeterminacy they involve is not due to our perception being overwhelmed by  visible qualities, as is the case with ordinary vision, but rather owes to the fact that there is always an array of possible transformations of our diagrammatic experiences constrained by the basic spatial relations that mathematics (and geometry in particular) deliberately uses in order to solve problems. Although (as will be discussed in detail later in the book) these transformations are implied by the very way a given diagram is constructed, not all of those transformations will ever be enacted in the reasoning patterns of which the diagram will form a part. Just as we have no control over the selectiveness of our ordinary vision, we cannot predict in advance what particular transformations out of the array of the possible ones will be selected and acted upon (e.g., what particular route a given proof of a geometric theorem will take), and what the ultimate result of those transformations will be (Stjernfelt 2007, pp. 81–83). Mathematical diagrams show some relations that are constitutive of their objects and, at the same time, hide some others that may be discovered later. A diagram connects us to the inexhaustible possibilities of further interpretation, but the way mathematicians construct their diagrams, together with the rules of transformation implied by the construction, make it the case that where the construction ultimately leads us is beyond our personal idiosyncrasies and the whims of our personal imagination.

Necessity There is a problem, however. On the one hand, the construction of mathematical diagrams presupposes creativity. On the other hand, mathematical reasoning is deductive in its character. Diagrammatization thus may be understood as “a sort of self-controlled management of one’s own thoughts” (Hoffmann 2004, p.  133), where our imaginative musings are tamed by visually given constraints. True, our

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capacity to predict possible transformations suggested by a diagram stimulates the introduction of new truths (CP4: 233). What we see when manipulating diagrams in order to achieve a desired proof, is a sort of a vague generalized image of all possible moves and transformations—an image that is not unlike what chess players sometimes describe as a dynamic picture of possible developments in a game, which does not allow seeing every particular move, but rather gives a general idea of where the game is currently going. In this case, I know what I need to do based on a set of generalized expectations of a certain result, but I am also prepared for something unexpected and leave the “why” to the logical analysis that I can use later, after the game is over. But at the same time, again, mathematical diagrams are visualized deductions whose primary goal is to represent patterns of necessary thinking. Peirce believed that this duality of novelty and necessity is, at least in part, reconciled by the fact that mathematical reasoning always involves observation. Observation is essential for mathematical reasoning (and for any deductive reasoning, for that matter) because it enables us to manipulate a diagram and to discover certain relations among its parts that previously went unnoticed. In Peirce’s own words, a deduction “always consists in constructing an icon or diagram the relations of whose parts shall present a complete analogy with those of the parts of the object of reasoning, of experimenting upon this image in the imagination, and of observing the result so as to discover unnoticed and hidden relations among the parts” (CP3: 363; cf. an excerpt from Peirce’s entry on logic for Mark Baldwin’s Dictionary of Philosophy and Psychology in CP2: 216). One of the principal distinguishing properties of mathematical icons is that careful direct observation of them allows one to discover new facts about their objects and, at the same time, forms a part of the deductive process. What we first need is to construct a schematic image, or a diagram (in geometry, a set of figures composed of lines with linguistic symbols attached; in algebra, an array of letters and numbers arranged in a pattern, etc.). The diagram, Peirce says, “is constructed so as to conform to a hypothesis set forth in general terms in the thesis of the theorem. Pains are taken so to construct it that there would be something closely similar in every possible state of things to which the hypothetical description in the thesis would be applicable, and furthermore to construct it so that it shall have no other characters which could influence the reasoning” (CP 4:233). The completion of the diagram does not make whatever is asserted in the theorem evident, “nor will any amount of hard thinking of the philosophers’ corollarial kind ever render it evident” (Ibid.). In lecture VI of his Harvard Lectures on Pragmatism (1903), Peirce explains that, upon the completion of the diagram, we proceed to observe it: This observation leads us to suspect that something is true, which we may or may not be able to formulate with precision, and we proceed to inquire whether it is true or not. For this purpose, it is necessary to form a plan of investigation and this is the most difficult part of the whole operation. We not only have to select the features of the diagram which it will be pertinent to pay attention to, but it is also of great importance to return again and again to certain features. Otherwise, although our conclusions may be correct, they will not be the particular conclusions at which we are aiming. (CP 5.162; emphasis added)

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Based on these, as well as other, more complicated examples, Peirce further shows that it is never the case that, in solving a mathematical problem, simply working out a solution in thought is enough. “It is necessary,” he says, “that something should be done. In geometry, subsidiary lines are drawn. In algebra, permissible transformations are made. Thereupon, the faculty of observation is called into play. Some relation between the parts of the schema is remarked” (CP 4:233; cf. Hull 2017, p. 149; Joswick 1988, p. 113). Summing up, mathematics essentially is an activity, an orderly, habit-driven, and yet creative practice rather than a static deductive grammar that supplies rules for the contemplation of abstract mathematical forms (Campos 2009; Hull 2017). Within mathematical reasoning as a practice, in which a crucial part is played by observation, creative imagination that brings new meanings into existence has a threefold role to play. First, a mathematician forms a skeletonized iconic representation or a diagram (whether geometrical or algebraic) of the facts she is interested in considering. The principal purpose of the initial skeletonization of the problem for a mathematician, Peirce says, “is to strip the significant relations of all disguise” so that “only one kind of concrete clothing is permitted—namely, such as, whether from habit or from the constitution of the mind, has become so familiar that it decidedly aids in tracing the consequences of the hypothesis” (CP3: 559). Second, a mathematician observes this diagrammatic picture until, at some point, “a hypothesis suggests itself that there is a certain relation between some of its parts.” Third, he experiments upon the diagram in order to test his hypothesis and then attempts to make it the case that, given this newly discovered relation, “it is seen that the conclusion is compelled to be true by the conditions of the construction of the diagram” (CP2: 278; cf. CP3: 560; Joswick 1988, pp. 108–109). Thus, observation provides us with a new material and, at the same time, guarantees us that, if the initial construction of a diagram is correct, the conclusion is necessarily true. This, though, is not the end of the story. According to Peirce, there is also an analogy between the cogency of mathematical reasoning and the compulsiveness of perception. Here is how Peirce describes it. Our knowledge is supposed to conform to hard facts, where the “hardness” is essentially a result of an insistency of percepts. Our sensibility is passive, and we have no rational say in the way our perceptual capacity works. In our perceptual judgments, we accept this insistency as a matter of course without any assignable reason. This, according to Peirce, is the way secondness operates in the world. However, Peirce then makes an unusual claim: This indefensible compulsiveness of the perceptual judgment is precisely what constitutes the cogency of mathematical demonstration. One may be surprised that I should pigeon-­ hole mathematical demonstration with things unreasonably compulsory. But it is the truth that the nodus of any mathematical proof consists precisely in a judgment in every respect similar to the perceptual judgment except only that instead of referring to a percept forced upon our perception, it refers to an imagination of our creation. There is no more why or wherefore about it than about the perceptual judgment, ‘This which is before my eyes looks yellow’. (CP7: 659)

Given that mathematics, as Peirce insists, is preoccupied with hypotheticals rather than existential facts, this analogy might seem surprising. On a closer look,

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however, in using this analogy, Peirce simply follows Kant. What he refers to in this passage is the relationship between perception and imagination in Kant’s philosophy of mathematics (which will become our primary topic in Chapter 9). Kant tells us that the receptivity of our senses is passive, while the imagination is our active capacity. Accordingly, what Peirce seems to be saying here is that perceptual contents forced on the passive receptivity of perception and imprints produced by my own active power of imagination share the same phenomenological quality. In both cases, “there is no more why or wherefore” about that “which is before my eyes.” This leaves us with a few questions. What is the relationship between the essential vagueness of perception and its compulsiveness? How is the element of ingenuity and productive creativity of mathematical thinking reconciled with the compulsiveness of perception? How, in view of this particular analogy, we are to explain the fact that mathematical diagrams express necessity and, at the same time, have the capacity to introduce new truths? Although, as Eisele puts it in The New Elements of Mathematics (1979), Peirce was “hot in pursuit” of answers to all these questions, the overall result of the pursuit turns out to be rather frustrating: But all the consequences not readily seen, are only to be discovered or proved after making changes in the image, requiring more or less ingenuity. How the mathematician can guess in advance what changes to make is a mystery that deserves a life-time study. Books have been written about it; but they are mere collections of empirical recipes for accomplishing what has been substantially done before, and are no help to any great step, because they do not go into the rationale of the matter. The truth is that mathematicians have not at present any knowledge of how to plan an attack upon a problem altogether novel. (NEM4: 215)

Peirce illustrates this unnerving conclusion by the example of the four-color map theorem. The proof of this theorem (or of the fact that no more than four colors are required to mark the regions of a given map in such a way that no two adjacent regions have the same color) does require some pre-theoretical spatial intuition that goes beyond an appropriate formalization. One has to perform some rearrangements of regions in the map before any formalization can become possible. But how is this possible? Peirce does not go into any significant detail either about this particular matter or about mathematical creativity in general, but I suggest that one might try and concisely unpack what he says as follows. Based on what has been discussed so far in this chapter, according to Peirce, in diagrammatical thinking, imagination provides us with a relational picture, or a diagram, whose parts are analogous to those of the object of our reasoning. We then experiment upon this diagram and observe new relations between its parts that have not been previously noticed. For the experimentation to become possible, perception and imagination should necessarily interact. The interplay between the two means that, whatever the creativity of imagination throws with an irresistible thrust at us, perception seconds with the “indefensible compulsiveness of the perceptual judgment” (CP7: 659). As mathematics deals with hypotheticals, perception then returns the results of the perceptual judgment back to imagination—not as a mere unnecessary aid that simply facilitates the reasoning process, but as a visual material which, sifted through the compulsive receptivity of our senses and diagrammatized, turns the imagined into a proper object of

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experimentation. Put simply, what can be imagined, can be presented in perception by being schematized and depicted; and what can be schematized and depicted, can be changed and experimented upon. Experimentation based on observation thus provides a link between the inner musings of imagination (which, due to its essential vagueness, allows new interpretations) and outer perceptual changes (which impose themselves compulsively on our senses). It is probably with this in mind that Peirce admits that the kind of experimentation a mathematician performs with diagrams is analogous to the kind of experimentation that is implemented in physical sciences: [R]easoning of much power has, as a historical fact, never been performed by means of words, or other sounds, nor even to any great extent by means of pure retinal sensations, but by means of muscular sensations and visual images which have in the imagination been put in motion, so that a sort of imaginary experiment is made; and the result has been observed inwardly, as that of a physical experiment is outwardly. (NEM, p. 378; cf. CP 4.530)

Inference as Observation Peirce claims that, in any particular instance of mathematical reasoning (not only in the case of geometry but also in the case of syllogistic and algebraic equations), “there must be something amounting to a diagram before the mind’s eye,” and that “the act of inference consists in observing a relation between parts of that diagram that had not entered into the design of its construction” (NEM 4:353; cf. CP 2:279). Inferring, then, is observing attentively what an experiment brings about. For example, a particular case of Aristotle’s “Barbara,” written down correctly, represents a simple diagram that clearly shows the relationship between the three terms involved and actually exhibits the fact that the middle term occurs in both premises and is dropped out from the conclusion. Likewise, an algebraic equation is a rule that maps one relation between variables onto another in such a way that further manipulation could lead inferentially to the discovery of a series of new facts. Thus, the very way algebraic formulae visually structure mathematical argumentation predefines certain inferential moves (CP2: 279). Formulae can be manipulated, the results of the manipulation carefully observed, and new properties found that were not previously discerned. Thus, [t]he letters of applied algebra are usually tokens, but the x, y, z, etc., of a general formula, such as (x + y)z = xz + yz, are blanks to be filled up with tokens, they are indices of tokens. Such a formula might, it is true, be replaced by an abstractly stated rule (say that multiplication is distributive); but no application could be made of such an abstract statement without translating it into a sensible image. (CP3: 363)

The variables here are slots filled with symbols that allow inferences to be made essentially in the diagrammatic way Peirce suggests. In terms of classical geometry, any of Euclid’s theorems is first formulated in abstract symbolic terms. In the Elements, such abstract statement, from which only some trivial truths may be deduced, is followed by the construction of a particular geometrical figure, and then

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the initial statement is reformulated in new terms—this time, with reference to the figure constructed. This, in turn, is followed by modifying the figure (by moving certain parts of it, or adding new lines, or both), and ascertaining whether the modifications hold good relative to the second formulation. Once this is done, the words “which had to be demonstrated” follow without any further restatement of the result in abstract terms. We then finally stop at what is visually presented. As Peirce further notes, [i]n like manner when we have finished a process of thinking, and come to the logical criticism of it, the first question we ask ourselves is ‘What did I conclude?’ To that, we answer with some form of words, probably. Yet we had probably not been thinking in any such form—certainly not, if our thought amounted to anything. … What the process of thinking may have been has nothing to do with this question. (CP2: 55; emphasis added)

There is, in the end, a moment at which the result is shown by the speediest way possible and after which thought can only idle in constructing trivial corollaries. Likewise, there are numerous proofs of Pythagoras’ theorem, but all of them require that, in order to explain the relation among the three sides of a right triangle, a geometer should make a certain rearrangement that actually represents the consecutive inferential steps necessary to arrive at the conclusion. In the initial, Pythagoras’s own version of the proof, it is the rearrangement of the four identical right triangles whose hypotenuses form a square. All we need after that is to stop and stare at the result, as the conclusion requires neither further moves nor any comments. Again, according to Peirce, geometry represents only one of many possible cases in which this ultimate point is revealed. In any logical process whatsoever, our thinking about a premise amounts to a mental perception of the link between its truth and the truth of the conclusion—the link, Peirce insists, that, once all the observational/inferential steps are made, gives us direct certainty of the latter. “Now this no symbol can show; for a symbol is an indirect sign depending on the association of ideas. Hence, a sign directly exhibiting the mode of relation is required” (CP4: 75). The point of a representation in this case is to help regulate an action aimed at an object so that the results can be predicted. This representation is not to “mirror” the object—simply because there is no way of knowing in advance what exactly is being “mirrored” and whether a “mirror” is a good one, other than acting on the object and seeing how the action works out. And this is just another way to express what Peirce has in mind in the last paragraphs of NL. To say that an interpretant brings the manifold of diverse impressions under the unity of a general concept by addressing them to possible new interpretations is to say that an interpretant resembles its object through demonstrating how it would react to a certain change, by reproducing the general way it reacts. Ambrosio (2014) comes up with a similar thesis: The very process of constructing an icon matters for Peirce, as it reveals the very respects in which a particular sign stands for its object. What seems to emerge from Peirce’s account is that the very relation of representation is itself the result of a process of discovery: ‘constructing’ an icon amounts to discovering, and selecting, relevant respects in which a representation captures salient features of the object it stands for. (p. 257)

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Once the construction of a diagram is initiated, the whole array of the inferential elements we might need is already there, in the relations between its parts. Because those relations are already there, they are inferential threads that can potentially be visualized. Meanwhile, we began this chapter by claiming that any perception is itself essentially a product of a long history of gradually habitualized, piecemeal adjustments and readjustments to the ever-changing environment. Those adjustments themselves form inferential ties with each other. Simply put, in order to respond correctly with the word “red” in the presence of a red object, one should have both perceptual intake and an appropriate conceptual network at hand, i.e., should already know, for instance, that red lollipops taste like cherry, that red traffic light means “stop,” and that “red” is a color. This being so, perception itself appears to be based on our inferential capacities. At the same time, in the case of mathematical diagrams, we have the situation vice versa: not visual perceptions made possible via inferential ties, but inferences visualized. Our hypothesis (to be tested in detail in Chapter 10) is that it is the exchange between the former and the latter that makes the observation performed by a mathematician productive.

Chapter 8

The General and the Particular

Mathematical Diagrams as Icons Although a diagram is constructed according to an abstractly stated precept (CP2: 216), not all possible relations between the parts of the diagram are initially predefined in the precept (for instance, a diagram of a square may not show diagonal lines connecting its vertices, a given number sequence may not reveal certain new patterns unless it undergoes certain change, etc.). In this respect, mathematical diagrams are like any other language in that the array of possible interpretations provided by their initial construction always exceeds the array of new interpretations suggested by our current goals and points of view. In using diagrams, what we have is, as it were, a system of keyholes, through which we see something only because we do not see all the rest. We can focus on a detail only if the overall picture is vague and, as a whole, available only through the active processes of construction, manipulation, and anticipation. Even though the possibilities are limitless, the mathematician is capable of anticipating changes between the parts of a given diagram that are specifically characterized by necessity. Peirce admits that the nature of the relationship between novelty and necessity presents an unresolved problem. In spite of all his musings, he confesses that “how the mathematician can guess in advance what changes to make is a mystery” (NEM4: 215). But, based on the analysis in the previous section, we can conclude that this capacity has something to do with the interplay between two pairs of concepts: the deductive force of mathematical reasoning vs. the compulsive force of perception, and the active power of the imagination vs. the passive receptivity of perception. In the former case, just as a percept is forced upon our perceptive capacity, a mathematical truth is forced upon our imagination; there is no “why or wherefore” about either of the two. In the latter case, there is a parallelism between the imaginative experimentation with diagrams (the capacity to predict the dynamic pattern of future changes) and visual perception based on the capacity to adapt to © Springer Nature Switzerland AG 2023 V. Kiryushchenko, Diagrams, Visual Imagination, and Continuity in Peirce’s Philosophy of Mathematics, Mathematics in Mind, https://doi.org/10.1007/978-3-031-23245-9_8

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the ever-changing perceptual environment. But before this analogy between the diagrammatic mathematical visuality and our ordinary, everyday visual experience can be clarified, we need first to address a more general issue of how diagrams frame the relationship between the general and the particular, or between concepts and percepts. It is important to demonstrate conclusively that spatial imagination and abstract reasoning are involved in the process of manipulating diagrams not as two distinct mental faculties, but as two aspects of the same activity put to work together. The point is aptly summarized in Hull (2017): “Peirce’s conception of a diagram is fundamentally and inseparably both conceptual and spatial insofar as reasoning by diagrams engages the continuum of spatial extension in the reasoning process” (p. 147). Mathematics, then, is a practice that makes use of a set of specific cognitive mechanisms in order to creatively schematize together the general and the particular, abstractions and images. A mathematician is capable of both conceptualizing and directly observing the world as an ordered variety of different forms of relations, because, in mathematical thought, visual integration and conceptual synthesis are as mutually interdependent as the two sides of a sheet of paper. Now, diagrams are iconic signs. While Peirce mentions a number of important respects in which visual diagrams in general, and geometrical diagrams in particular, stand out among icons as a special class of them, what is essential for visual diagrams qua iconic signs is the immediacy of diagrammatic representation. While an icon is a sign that refers to its object by resembling it in a certain respect, a geometrical diagram substitutes its object so completely that the distinction between the two becomes problematic (What is the object of a geometrical diagram as a part of a demonstration of Pythagoras’ theorem apart from what this diagram itself directly shows?). A diagram, though, just as any other iconic sign, contains some general elements and, therefore, cannot be considered a pure icon, i.e., a sign that is indistinguishable from its object, or, in Peirce’s own words, a sign that “represents whatever it may represent, and whatever it is like, it in so far as it is an affair of suchness only” (CP5: 74). A pure icon, as Peirce puts it elsewhere, is a sign that “can convey no positive or factual information; for it affords no assurance that there is any such thing in nature. But it is of the utmost value for enabling its interpreter to study what would be the character of such an object in case any such did exist” (CP4: 447). Recall that Peirce believed in the advantage of visual representation compared to written language and was prone to reflecting in diagrams rather than words. The quote above gives us some detail about what the advantage is. The less generality is incorporated into an icon, the closer it is to representing its objects through “suchness only,” and the harder it becomes to distinguish between the icon and its object. The more hypothetical the object of an iconic sign (such as a diagram) is, the less time we need to spend concentrating on distinguishing between the sign and its object, and, accordingly, the more experimental we can get with the sign itself. It appears, then, that in the case of iconic signs, we have a special relation between sameness and novelty: The closer the diagram we are using is to being the mere suchness of the object it represents, the more refined our capacity to experiment

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with it and to discover new truths about its object is. The closer similar objects are to being the same, the more we can achieve as mathematicians. Mathematicians do usually talk about mathematical objects without raising any ontological concerns (in the sense that they say, “42 is an even number” or “the tenth dimension in superstring theory is a single point” in exactly the same way others say, “masala dosa is a food”). At the same time, this does not prevent the fact that there are limits to what mathematicians can change in conducting their diagrammatic experiments. Thus, they cannot simply decide, for the sake of convenience, that π equals exactly 3, just as others cannot simply decide, for the sake of convenience, that glass is a mineral. Yet, as discussed in Chapter 3 above, they can perform what Peirce called “mathematical generalization.” They can trade a smaller problem that involves exceptions they are not willing to accommodate for a larger problem that has no such exceptions. As a result, in dealing with pure hypotheticals, they can introduce imaginary numbers and new dimensions, use vector calculus, etc., without the threat of contradicting any of the existent mathematical conventions or making the signs they use untrue to their objects. As Peirce insists in NL, every sign, even if it is not actually interpreted, in order to be a sign, should be capable of being interpreted. The meaning of any and every sign thus consists not in grasping some unseen internal object residing somewhere in the recesses of an individual mind, but in the sign’s interpret-ability. From this, it follows that a sign’s identity always lies in its reference to some other thing. Given that a sign can be understood only as addressed to its possible future interpretations, an interpretation can result in nothing but the creation (or discovery) of a new sign. In Peirce’s semiotics novelty, therefore, appears to be a necessary characteristic inherent in every act of interpretation. On the one hand, then, the productivity and the novelty of interpretation directly depend on signs’ being capable of resembling their objects as closely as possible (in other words, on those signs being as close as possible to pure icons—which is precisely the case with mathematical objects in relation to their representations). On the other hand, in NL, Peirce tells us that new meanings arise due to the very way every interpretant concludes the transition from the empty abstraction of a quality to a general concept, addressing it to future interpretations. What is exactly the mechanism which connects generality and iconicity and which makes novelty an essential characteristic of interpretation (mathematical interpretation in particular)? So far in this chapter, we have learned two things. First, we have learned that, because mathematical demonstrations are as close as anything can be to what they are demonstrations of, creativity and novelty are essential and unalienable traits of mathematical reasoning. Second, we have learned that novelty, as an essential and unalienable trait of reasoning in general has something to do with genuine (non-degenerate) signs embodying a triadic relation (or thirdness) which is not reducible to either dyadic relation (or secondness) or a monadic relation (or firstness). And just as Peirce’s logic of relations demonstrates the role of novelty visually by means of a variety of graphs representing the three-way forking, Peirce’s NL demonstrates it by means of the conceptual analysis of the way being and substance are joined together in a basic propositional form “S is P.” In other

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words, what Peirce’s conceptual analysis describes in NL as applied to language, his logic of relations shows to be the case as applied to vision. Simple graphical conventions expressing three basic kinds of relations thus diagrammatize the conceptual story—the story which, according to Peirce’s NL, underlies our discursive capacities. What it all means is that we have to have some fundamental iconic sign structures in order to have language, and these same iconic structures also happen to represent the most basic mathematical intuitions. A diagram, according to Peirce, is a vehicle that mediates between language and vision as it combines the visuality of a particular image and the generality of a concept. In producing new meanings, it partakes both triadic conceptual synthesis (i.e., it acts as a language) and the basic visual schematisms that condition our ordinary perception (i.e., it acts as a picture). It provides a link between our visual schematisms and arbitrary symbolisms of a natural language. It is this feature of diagrams, as will be shown in Chapter 9 below, that, among other things, helps Peirce demystify and radically reformulate one of the most difficult sections of the Critique of Pure Reason, where Kant describes the schematism of the imagination.

The General and the Particular Fused Together What any language prima facie does is bring individual objects under general concepts. In NL, Pierce tells us just that. A visual language is characterized by its own way to connect the general and the particular, which differs from the conceptual synthesis performed by a natural language, such as English. A visual diagram represents relations as connections between certain elements in immediate perception in the way neither a word nor a sound ever can do. It might be objected that, if not a sound or a word as such, then a sound pattern or a written sentence do exactly that. However, as we claimed at the outset of the current study, whether this is true or not, depends on what we agree to include in what we call “a diagram.” There is some diagrammatic element in a sound pattern, meaning that we do perceive connections between different parts of the pattern over time. The repetition that supports the connections in this case is being fixed as our experience goes on. It is directly perceived by us in the limited sense that it frames our attention during the entire time we perceive the pattern. In a similar manner, there is some diagrammatic element in a written sentence, meaning that the syntactic structure, which holds the sentence together and defines its possible transformations, is an iconic sign. It is directly given to us in perception in the limited sense that it frames our attention during the entire time we read the sentence. It is a vertical paradigmatic structure that contextualizes our perceptions and keeps them together during our act of reading. We do, in some sense, perceive a musical harmony while listening to a melody, just as we do, in some sense, have some perception of a paradigm while following a linear string of syntagms. But in neither of these two cases, the very structure of the message is given to us diagrammatically as

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such at any given moment of observation, in the way it is given to us in a visual diagram. In mathematical observation, the structure of the message (the way the message is constructed) is given through the necessary character of relations involved, and this necessity is available for immediate observation. Expressed in diagrams, the “deductive must” of the conclusion of a mathematical proof becomes visually evident. And the evidence of such conclusion, according to Peirce, “consists in the fact that the truth of the conclusion is perceived, in all its generality, and in the generality the how and the why of the truth is perceived” (NEM4: 317; emphasis added). Thus, for instance, the slave boy in Plato’s Meno, when asked what the increase in the length of the sides of a square should be if we want to double its area, at first erroneously concludes that doubling the sides should do the trick. But then Socrates shows him (or makes him infer) that it is not the square with doubled sides, but the one on the diagonal that is twice as big as the original square (Figure 8.1). What is thus given in a visual diagram is not a simple sum of sensible particulars (dots, lines, and shapes), but a general truth proven by means of a particular image. Again, it might seem that this is precisely what the words of any natural language, such as English, do. However, in the case of mathematics, this image always happens to be in full compliance with the general truth it refers to, because, being a particular image that it is, it directly visually translates certain general relations that are invariable—which an image of no other kind can do. This characteristic of mathematical diagrams will be discussed in detail in the next chapter. For now, it is important to establish that what is given in a visual diagram is not simply a flow of experience. And in the case of mathematical diagrams, any sequence is perceived as such primarily because it is organized as a visualized proof. So the general truth is made available through the process of visual transformations that lead to a particular image of a special kind, about which, again, “there is no more why or wherefore” (CP7: 659). Figure 8.1  The geometry experiment in Plato’s Meno

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Every instance of the visual diagram is a representation of a skeleton idea of relations between certain things either in the mind or “out there” in the world (CP7: 426). On the one hand, a visually represented relation may carry with it the full history of particular responses that made it possible (for instance, a seashell that shows the history of its own development in its very form). On the other hand, the kind of connection it visually represents is an object of a general nature (for instance, a particular graph that shows an annual stock price change in accordance with a Fibonacci series). Consequently, it might be said that, in some sense, generality is immediately perceived in a visual diagram, in the way it is not perceived in a sound pattern as it is being listened to, or in a written sentence as it is being read. Peirce is aware that, for an empiricist of a certain stripe, this claim might seem not just improbable, but utterly ridiculous. As he puts it, “Bishop Berkeley and a great many clear thinkers laugh at the idea of our being able to imagine a triangle that is neither equilateral, isosceles, nor scalene. They seem to think the object of imagination must be precisely determinate in every respect” (CP5: 371). But Peirce still insists that, if we consider perception carefully enough, contrary to what an empiricist has to say, some form of realism about generality is indispensable. Here is one way to unpack the reasoning behind this belief. In any kind of observation, including mathematical, there is a uniformity in things we directly observe. The uniformity is directly visually given in acts of repetition―not of singular facts and events (or sets of facts and events), but of their modus operandi. A reiterated particular is just this: a reiterated particular; it does not form in us any sort of expectation or generalized image of its possible consequences. It would be fair to say that, according to Peirce, in this case, we do not really see anything. A repetition that reproduces a mode of action (or a habit) is a repetition of a pattern establishing strong connections between particular events, attitudes, or actions. The regularities, which bring forth the uniformity and which underlie our capacity to see something, are, in some sense, visually exposed, and grasping those regularities is a part of our visual experience. Mathematics simply exaggerates this feature of perception, putting our shared basic spatio-temporal intuitions to a particular use. It is noteworthy that the construction of a mathematical diagram shows how the situation it depicts can take place. “Knowing how,” in this case, amounts to knowing about what is possible, as opposed to the customary philosophical “knowing why,” which involves some sort of ontological claim, often unjustified or misconstrued by philosophers (CP4: 176). Mathematics, Peirce says, studies only what is true of hypothetical states of things. To know why, we have to have some sort of strong ontological commitment. To know how, we simply need to construct a schema that would correspond to the hypothesis we have, reveal something similar in every possible situation to which the hypothesis might be applicable, and contain no elements irrelevant to the hypothesis (CP 4.233). Unlike badger-legged philosophers, a mathematician is fast. All he wants is “a pair of seven-league boots, so as to get over the ground as expeditiously as possible” (MHFC, Fragment on logician and mathematician, c. 1906). Naturally, as has already been noted above, the capacity to express general ideas by means of arranging groups of individual objects is a feature possessed by most

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of the languages. Strings of words, hieroglyphs, or mathematical symbols, interconnected by means of certain syntactic, anaphoric, inferential, and other relations, are capable of conveying general ideas. Yet in the case of mathematics, diagrams represent a unique compromise between the universal and the particular—the compromise that, according to Peirce, results in a peculiar relationship between factual certainty and conceptual generality: Among the minor, yet striking characteristics of mathematics, may be mentioned the fleshless and skeletal build of its propositions; the peculiar difficulty, complication, and stress of its reasonings; the perfect exactitude of its results; their broad universality; their practical infallibility. It is easy to speak with precision upon a general theme. Only, one must commonly surrender all ambition to be certain. It is equally easy to be certain. One has only to be sufficiently vague. It is not so difficult to be pretty precise and fairly certain at once about a very narrow subject. But to reunite, like mathematics, perfect exactitude and practical infallibility with unrestricted universality, is remarkable. (CP4: 237)

In mathematical reasoning, certainty and generality not just become compatible, but get reconciled in their extreme forms. Each example of this reasoning shows how exactly perceptive selectiveness and conceptual generalization work together, and how, in doing so, it provides the most telling evidence that there is no sharp discrimination of the intuitive and the discursive processes of the mind. As an observable individual image that possesses some relational properties, a mathematical diagram represents a seamless merger of perceptual intuition and ratiocination. On the one hand, it does not seem right to reduce mathematical thought to perception as a mere psychological process. On the other hand, although there are objective, mind-independent relations out there that represent necessity (e.g., those “recollected” by Meno’s slave boy thanks to Socrates’s majevtic questions), mathematical thought is always embodied in particular token-signs. Just as it is impossible to get at the heart of an onion by peeling off all its skins, it is impossible to get at the heart of a necessary relation by stripping it off whatever particular symbols that happen to signify it (CP4: 6, 87). Peirce’s overall point here is that mathematical necessity cannot be reduced to a visual representation that is simply attached to an idea; it should be considered as a real characteristic of mathematical signs qua signs—a characteristic that is available not only in an act of eidetic contemplation but also in an act of observation. According to Peirce, what distinguishes diagrams from other kinds of signs is that they achieve the fusion of the general and the particular most effectively—by making this fusion immediately visually available based on our natural visual intuitions. And, in the case of mathematical reasoning, this fusion results in direct perception of necessary relations. As Catherine Legg puts it, in the case of diagrams, “[n]ecessary reasoning is in essence just a [visual] recognition that a certain structure has the particular structure that it in fact has” (2012, p. 1). In his discussions of diagrammatic reasoning, Peirce often compares his approach to Kant’s. He praises Kant for his attempt to mediate between empiricism and rationalism as two philosophies laying stress on perceptual intake (particulars) and innate conceptual endowment of the human mind (generalities), respectively (CP1: 374–375; CP2: 62; CP3: 422). He also finds quite remarkable the systematicity of

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this attempt (CP1: 177, 522). And he notes Kant’s acknowledgment of the importance of trichotomic distinctions between groups of universal categories that this attempt implied (CP1: 369; CP2: 381). Nevertheless, he criticizes Kant’s approach incessantly, claiming that: [Kant] drew too hard a line between the operations of observation and of ratiocination. He allows himself to fall into the habit of thinking that the latter only begins after the former is complete; and wholly fails to see that even the simplest syllogistic conclusion can only be drawn by observing the relations of the terms in the premisses and conclusion. His doctrine of the schemata can only have been an afterthought, an addition to his system after it was substantially complete. For if the schemata had been considered early enough, they would have overgrown his whole work. (CP1: 35)

Apart from this, Peirce leveled criticisms against Kant’s nominalism (CP1: 19) and Kant’s being “poorly equipped” to draw up a classification of the functions of judgment based on his table of categories (CP2: 346). As Kant’s idea of transcendental schemata provides an extremely important context for understanding Peirce’s perspective on the role of diagrams in reasoning, Peirce’s criticism of Kant needs now to be unpacked.

Chapter 9

Diagrams Between Images and Schemata

The Role of Mathematical Cognition in Kant’s 1st Critique In order to reform the German idealist tradition without radically undermining it, Kant undertook to reconcile early modern empiricist sense-data theories and classical rationalism. On the one hand, empiricists believed in sense-contents as the pre-­ theoretical empirical basis of knowledge and in theory-independent data immediately accessible to us in either outer or introspective sensation. According to this view, there are judgements (like “this is red”) that form the basis of empirical knowledge but which themselves are beyond defeat. I simply cannot go wrong when I attentively restrict my judgements to a report of what, as I take it, “appears” to me. On the other hand, rationalists relied on the pre-empirical innate knowledge that, due to its infallibility, could never betray the knower. A piece of wax has certain visible shape, size, color, palpable texture, and smell—all of which change as I move the piece closer to the fire. The only characteristics that remain unchanged are the extension and changeability of the piece, neither of which is perceived directly through the senses or the imagination. On this view, what helps me hold on to the concept of wax and understand what it is, cannot be reduced to senses but can be ultimately achieved by my mind alone. According to Kant, neither of the two views is capable of providing the basis for knowledge separately from the other. Rationalists, in claiming that there are ways in which our concepts are gained independently of sense experience, lose the world as we know it altogether. Empiricists, in claiming that sense experience is the ultimate source of all knowledge, fail to explain how we proceed from perceiving a host of disconnected qualities to having a general idea of an object that possesses those qualities. The former lack experience, while the latter cannot find an explanation for conceptual unity. In an attempt to avoid the two extremes, Kant chooses not to treat knowledge either exclusively as an outcome of sensory intake or as an immediate and infallible translation of relations between things in the world into self-evident states of one’s

© Springer Nature Switzerland AG 2023 V. Kiryushchenko, Diagrams, Visual Imagination, and Continuity in Peirce’s Philosophy of Mathematics, Mathematics in Mind, https://doi.org/10.1007/978-3-031-23245-9_9

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mind. But given this choice, in order to demonstrate how knowledge is possible, Kant needs to explain how exactly our capacity of understanding subsumes particular bits of empirical data under general concepts. For example, we cannot apply the concept “gold” to a lump of metal simply because we find it to be heavy, yellow, malleable, etc. A simple collection of particular sensible qualities can neither constitute a general concept nor explain how it is that we are able to attach this concept to something, “it,” of which we, unarmed with any prior conceptual knowledge, do not even know in advance whether it is an “object.” According to Kant, we need something in addition to those qualities that would synthesize them into a general concept capable of referring to an appropriate particular as possessing those qualities. Because sensory data does not come all on its own, somehow mysteriously already partitioned into rationally related bits and pieces by the world itself, we have to assume our already possessing the concepts that do the job. As Kant famously summarizes the dilemma, “thoughts without content are empty, intuitions without concepts are blind” (KRV: A51/B76; emphasis added). In order to respond correctly with the word designating a certain color in the presence of an appropriately colored object, one should have both perceptual intake and a conceptual network at hand, i.e., one should already know a number of other ways to apply the word. According to Kant, however, it is not nearly enough to practically master the inferential relations of the concept “red” to a variety of other descriptive concepts and thus locate it correctly in the space of implications that these other concepts constitute. In Kant’s view, in order to count as knowledge, one’s learning experience must also comply with certain a priori conditions of knowledge. Given that both conceptual capacities and perceptual intake are equally necessary components of experience, we need some sort of a “bootstrapping mechanism,” which would help us build a connection between the two and explain how experience-­based conceptual knowledge is possible. To this end, Kant postulates that both thought and perception presuppose unifying forms. Accordingly, in “Transcendental Aesthetic” and “Transcendental Analytic,” he introduces two forms of representation: time and space as two pure forms of sensible intuition and the categories of the understanding as a set of a priori concepts which, applied in a judgment, represent intuitions as intelligible wholes. It is precisely these representations that Peirce describes as a “unity.” The unity, according to Peirce, consists in the connection of the predicate with the subject of a proposition and is achieved in NL through the introduction of the cenopythagorean categories of quality, relation, and representation. But again, even though, according to Kant, we thus correlated concepts and percepts by assigning an appropriate form to each, unless we explain why it is not the case that just one of these forms is enough to provide a solid basis for knowledge, we would still have to yield to either the empiricist or the rationalist epistemology. Neither the pure intuitions as the basis for sensibility nor pure concepts as the basis for the understanding are sufficient on their own. To explain how intuitions are organized by concepts into a judgment, we need two different capacities that interact. This interaction, according to Kant, has everything to do with mathematics. Let us see first what significance mathematical concepts, according to Kant, play in our sensibility.

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Space and time, Kant argues, are a priori forms that determine what is provided to us through the sensibility (or, in Kant’s preferred terminology, “in intuitions”). Neither space nor time is an empirical concept; we do not make sense of space and time by first observing certain objects and then, based on our experience, inferring certain features that those objects have in common. As Kant puts it briefly, “space is not an empirical concept that has been drawn from outer experiences,” because “in order for certain sensations to be related to something outside me … the representation of space must already be their ground” (KRV: A23/B 38). By the same token, Kant claims, time is the necessary ground for the representations of temporal objects, not an idea drawn from our experiences of the duration of different experienced intervals. In short, Kant’s overall claim is that unless I already possess some a priori notions of space and time, I can never experience objects as spatially or temporally ordered. Moreover, while the sensory intake itself is the external matter (ὕλη) of intuitions, time and space, as the forms (μορφές) of these intuitions, lie in us. They are the ways in which this intake is organized, and they are our ways. Kant also notes that we always represent space and time as singular, and so we must have a pure intuition of each, since singular representations are intuitions. We represent particular spaces and times only as regions of a larger, surrounding space or time (KRV: A25/B 39, A31–32/B 47). Part of the reason for this is that we conceive of particular spaces and times not as instances of the general concepts “space” and “time,” but rather as parts of the larger individuals, space and time. This, according to Kant, is enough to establish the fascinating fact that not only space and time are a priori concepts, but they are also themselves available as pure intuitions. Accordingly, Kant takes mathematics to be an exploration of space and time understood as such pure intuitions, with geometry inquiring into the most basic properties of space, and arithmetic being a research of the most basic characteristics of time. Because of the very nature of space and time, as both forms of sensibility and pure intuitions, Kant took mathematical concepts to be unique. On the one hand, what they represent involves no empirical content whatsoever. The idea of number does not require images of particular things, and a geometrical figure does not necessarily need such representation in order to be understood as such. On the other hand, in proofs of theorems, a mathematical idea is always paired with a singular representation (say, of a particular two-dimensional, three-sided object). Mathematical ideas, then, are pure but also sensible. Thus, for instance, the concept of a triangle, just like the purely empirical concept of a dog, is sensible since it is always constructed by a mathematician in concreto, as having features that are common to several essentially similar objects (there are several kinds of triangles, just as there are several kinds of dogs). Yet unlike the concept of a dog, it is also pure and strictly a priori in origin; as such, it is based not on visual images of concrete triangles, but on a rule that exists only in thought. It is a rule that allows the imagination of a mathematician to mentally construct a pure, general geometrical form that is related to nothing but certain general properties of space. In this particular case, the properties that condition the invariable equivalence of π radians and the sum of measures of the three angles of any triangle in a Euclidean space. These general properties are always there no matter what kind of triangle one has in mind.

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Similarly, algebraically speaking, we can understand that the change in internal energy of a system equals the sum of all heat transfer into and out of the system minus the net work done by the system (ΔU = Q−W ) without actually observing a boiling kettle. The distinctive feature of mathematical representations mentioned above owes, in part, to the fact that mathematical proof is based on a particular construction—in a specific, Kantian sense of the term. The overall meaning of Kant’s “constructivism” about mathematical concepts may be summarized as follows. In natural sciences, we begin with sensuous intuitions of particular objects that are subsequently conceptualized. The intuitions, in this case, are “richer” than their concepts. While concepts predict what happens in general, intuitions beget the nuances of particular existences, all of which the concepts are unable to convey. Thus, to use a familiar Gregory Bateson’s example, I can predict that a stone, if thrown at a window, under appropriate circumstances would produce a crack in the glass in a star-shaped pattern. But, as Bateson puts it, “within the conditions which produce the star-shaped break, it will be impossible to predict or control the pathways and the positions of the arms of the star” (Bateson 1979, p. 41). And conversely, in the case of metaphysics, a priori concepts are “richer” than individual tokens of their empirical usage because they tell us something important about the very conditions of all possible experience. In neither of the two cases, then, is there a full compliance between concepts and the objects to which they refer. Mathematics constitutes a special case in that it mediates between science and metaphysics. This is where the idea of “construction” plays its part: Take, for instance, the concepts of mathematics, considering them first of all in their pure intuitions. Space has three dimensions; between two points there can be only one straight line, etc. Although all these principles, and the representation of the object with which this science occupies itself, are generated in the mind completely a priori, they would mean nothing, were we not always able to present their meaning in appearances, that is, in empirical objects. We therefore demand that a bare concept be made sensible, that is, that an object corresponding to it be presented in intuition. Otherwise the concept would, as we say, be without sense, that is, without meaning. The mathematician meets this demand by the construction of a figure, which, although produced a priori, is an appearance present to the senses. In the same science the concept of magnitude seeks its support and sensible meaning in number, and this in turn in the fingers, in the beads of the abacus, or in strokes and points which can be placed before the eyes. The concept itself is always a priori in origin, and so likewise are the synthetic principles or formulas derived from such concepts; but their employment and their relation to their professed objects can in the end be sought nowhere but in experience, of whose possibility they contain the formal conditions. (KRV: A163/B204)

We construct a mathematical concept based on an a priori intuition which is introduced by a definition (say, of a triangle as isosceles, or of “two” as a simple even number), where our construction directly presents to our senses a particular illustration of certain basic spatio-temporal properties (the equality of lengths, or simplicity and parity). An individual triangle, as constructed based on a definition (whether in imagination or on paper), is an empirical object. At the same time, being the particular object that it is, it serves to express a concept, to which the definition

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refers, without doing any damage to the universality of the concept. It is constructed in concreto, and yet it is impersonal: we prove a theorem not specifically for this particular triangle, but for any object of an appropriate type, as constructed. It is important that the impersonality of a mathematical object differs both from the generality of empirical concepts and from eidetic intuition. The former is obtained by selecting certain characteristics of particular things for the purpose of their classification into genera and species; the latter results in an immediate, non-empirical grasp of είδος, or a universal form. A pure spatio-temporal intuition, which corresponds to a certain mathematical concept, always must be a single object (this triangle). At the same time, this single object, represented in its concreteness as a part of a mathematical proof, expresses universal validity for all other pure intuitions that fall under the same concept. A mathematical object, constructed based on an a priori intuition, unlike the objects of science and metaphysics, is thus in full compliance with its concept. Thanks to this characteristic of its objects, mathematical knowledge, Kant says, always sees the universal in the particular, the necessary in this single instance, “though always a priori and by means of reason” (KRV: A714/B742). Following Kant, then, mathematics is exactly where the two rival schools (empiricists and rationalists) are reconciled. In compliance with the rationalist argument, it should be acknowledged that we could never have experience that contradicts mathematical proofs. Mathematical deductions make use of the most basic properties of space and time, which, according to Kant, are a priori forms that condition any human experience. Any a priori notion is, therefore, at the core is mathematical. Accordingly, no experience can possibly contradict the very forms in which it is given and basic intuitions that constitute it as such. Mathematics thus provides the necessary ground for the relation of all our spatio-temporal representations to the objects of those representations. At the same time, according to Kant, an empiricist should be happy to admit that, although mathematical truths are grounded in pure intuitions and do not require any experience, the definitions, based on which we construct mathematical concepts, are always paired with an immediately evident, readily empirically available representation of a particular mathematical object. The construction a mathematician performs is indifferent as to whether what is used in a proof is delivered by pure intuition only or, in accordance therewith, also by empirical intuition (say, as drawn on a piece of paper). In both cases, it is completely a priori, without having borrowed anything in the pattern of the proof it produces from any experience. Consequently, whereas Kant’s overall epistemological constructivism, according to which the subject knows only what it itself can construct, may sound radical, his mathematical constructivism comes as a part of common sense. It simply makes an uncontroversial statement that our mathematical knowledge has its origins in our own capacities—with an important proviso that, in expressing our basic spatio-temporal intuitions, mathematical notions condition the conceptual integration of our experience in general. But how does the application of concepts to percepts work? On the one hand, Kant’s categories of the understanding are a priori and, therefore, are not simply abstractions from sense perceptions but owe their origin to the nature of the mind

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itself. Each category serves as an attribute of any possible object (an object in general) and, therefore, has merely logical content. Kant’s category of substance, for instance, is simply a characterization of something as the subject of a predication, the category of negation is simply a logical denial of the quality of “being something,” etc. On the other hand, our experience does not immediately present itself in logical terms. As such, it is messy; it is an experience of objects that are near to or far from us, now present and now gone, short-lived or long-lasting, etc. If the categories that the understanding applies to individual phenomena, as something general, are thus not homogenous to those phenomena, the question is: “How do we pass from sensible particulars to general concepts?” For Kant, as for Peirce, the possibility of bringing the forms of sensible intuition and the categories together amounts to adding another, third dimension to the dyadic relationship between experience and the unifying principles of the understanding. For Kant, just as for Peirce, it requires that there is a mediator, something connecting categories, as a priori general concepts, with the manifold of experience. In other words, the unity of the manifold of experience in a concept, according to both Kant and Peirce, is achieved by appealing to some third, intermediate structure that reconciles sensibility and understanding as two interdependent stems of knowledge. Kant calls this intermediate structure the “transcendental schema.” The schema is neither a percept, nor a concept, but something that, as Peirce would put it, interprets one into the other. It is homogeneous with categories of the understanding on the one hand and with corresponding appearances on the other hand. And it is this homogeneity that makes the application of the categories of the understanding to appearances possible (KRV: A 138/B 177).

Diagrams and Schemata Recall that, as discussed in Chapter 4, according to Peirce, an interpretant “arises upon the holding together of diverse impressions, and therefore it does not join a conception to the substance, … but unites directly the manifold of the substance itself” (W2: 54). Likewise, a schema, Kant says, is always a product of imagination, and yet as such “aims at no special intuition, but only at unity in the determination of sensibility” (KRV: A 140/B 179). For this reason, a schema, according to Kant, has to be strictly distinguished from an image. To use Kant’s own mathematical example, five points placed in line alongside one another ( . . . . . ) give me an image of the number five. But in thinking about a number in general, whether it be five, a hundred, or one thousand, I refer not to an image, but to a method according to which this or that multiplicity is brought to conformity with a concept. No image proper would be adequate to the number 1,000,000, just as no image could represent a triangle in general, because “it would never attain that universality of the concept which renders it valid of all triangles, whether right-angled, obtuse-angled, or acute angled; it would always be limited to a part only of this sphere” (KRV: B 179–180).

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For this reason, we always need a schema formulating such a method in the form of a rule. To contextualize the difference between images and schemata, later on in the transcendental deduction, Kant introduces a further distinction between two kinds of imagination: productive and reproductive. The reproductive imagination enables one to imagine a phenomenon when it is absent (for instance, I can form an image of an apple on a plate in front of me if I close my eyes). What this imagination produces never reaches the generality of a concept; it is always limited to a vague image, as a re-production of this particular object. The productive imagination, through applying schemata, which mediate between our conceptual capacities and the manifold of sensible data, enables us to form an image that exemplifies a corresponding concept: [T]he image is a product of the empirical faculty of reproductive imagination; the schema of sensible concepts, such as of figures in space, is a product and, as it were, a monogram, of pure a priori imagination, through which, and in accordance with which, images themselves first become possible. These images can be connected with the concept only by means of the schema to which they belong. In themselves they are never completely at one with the concept. On the other hand, the schema of a pure concept of understanding can never be reduced to any image whatsoever. It is simply the pure synthesis, determined by a rule of that unity, in accordance with concepts, to which the category gives expression. (KRV: A 142/B 181)

Because schemata are determinations of objects in general, and not of specific, individual objects, they simply cannot as such be represented in particular images. A schema prescribes the way to relate a pure concept to an object according to a rule. It is noteworthy that Kant begins expounding schemata with mathematical concepts. As has been briefly discussed above, this choice is not accidental: basic mathematical concepts, according to Kant, are uniquely apt for the job because they are pure but also sensible. On the one hand, a particular triangle is always in full compliance with the concept it represents; on the other hand, it is also pure and relates to certain general properties of space only. As Kant’s idea of schemata is extremely important for understanding the roots of Peirce’s conception of diagrammatic thinking, we will need to follow Kant’s description of schemata for each of his categories in some detail and to show how basic mathematical notions are revealed at the core of every possible experience. The first of the four triads of Kant’s categories of the understanding is the triad of magnitude, or quantity (quantitas). The triad of quantity includes the categories of unity, plurality, and totality. According to Kant, the function of the schema of quantity is played by number or a representation that comprises the successive addition of homogeneous units. Kant does not assign a separate schema for each of the three of the categories comprising the first triad. Quantity, according to Kant, is associated with the idea of successive numerical time series. As Monck (2007) puts it, “in apprehending an object I always successively add part to part, and thus generate a series of determinate magnitude” (p. 43). In a series, unity is expressed as one moment in time series, or the initial unit of a series, plurality as several moments, or an addition of unit to unit (or one part of an object to another) without a definite

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limit, and totality as all units of the series (or an object as a sum of its parts). Thus understood, the idea of number, Kant says, is related to every possible experience, as what it demonstrates is “simply the unity of the synthesis of the manifold of a homogeneous intuition in general” (KRV: A 143/B 182), where “totality” is responsible for our overall capacities of special integration and object recognition. Kant explains this claim later in the text of the “Analytic of Principles,” when he discusses principles of pure understanding: As regards magnitude …, that is, as regards the answer to be given to the question, ‘What is the magnitude of a thing?’ there are no axioms in the strict meaning of the term. … The propositions, that if equals be added to equals the wholes are equal, and if equals be taken from equals the remainders are equal, are analytic propositions; for I am immediately conscious of the identity of the production of the one magnitude with the production of the other. (KRV: A 164)

In this case, the schema simply connects our analytic knowledge of arithmetic operations with equal units (objects or numbers) to or capacity to reproduce the results of those operations in various ways in terms of images. Quantity, of which number is a schema, thus amounts to nothing other but the immediate consciousness of the synthesis of the manifold in intuition. The second mathematical triad of the categories—reality, negation, and limitation—represents being (or non-being) of any object (such as a dog or a table) that remains the same throughout time or of a quantity insofar as it fills time. The schema, in this case, “is just this continuous and uniform production of that [quantity] in time as we successively descend from a sensation which has a certain degree to its vanishing point” (KRV: B 183). The third and the fourth triads of the categories are dynamical (non-­mathematical). The third one comprises substance, cause and community. Kant defines the schema of substance as the permanence of whatever is real in time, or as something that abides while all else changes (we recognize an object as a dog even if we have never seen this particular dog before). The schema of a cause is defined as “the real upon which, whenever posited, something else follows”; and the schema of community as coexistence of substances in time in the reciprocal causality of their accidents, or the objects acting on each other simultaneously—say, several dogs playing in a yard (KRV A 144/B 184). To the last triad (that of modality) correspond the schema of possibility as “the determination of the representation of a thing at any time whatsoever,” the schema of actuality as “existence [of a thing] in some determinate time,” and the schema of necessity as “existence [of a thing] at all times” (ibid.). Upon presenting the schemata for all 12 categories, Kant notes immediately that, in the case of every category, a schema determines the manifold of impressions by the a priori conceptual material in relation to time. He then concludes that “[t]he schemata are thus nothing but a priori determinations of time in accordance with rules. These rules relate in the order of the categories to the time series, the time-­ content, the time-order, and lastly to the scope of time in respect of all possible objects” (KRV: B 185). This is crucial. Time, according to Kant’s analysis, has the most fundamental relation to any experience. For this reason, Kant defines it as the purest possible schema of all concepts. As a schema, time is homogenous with

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appearances in that it is itself a pure intuition (whatever happens, happens in time, so time belongs—in some sense of “belongs”—to all objects of experience). It is, Kant says, “contained in every empirical representation” (KRV A 139/B 178). Meanwhile, time is also homogenous with the categories in that it is a universal concept that rests on an a priori rule. Thus, Kant concludes, “an application of the category to appearances becomes possible by means of the transcendental determination of time, which, as the schema of the concepts of the understanding, mediates the subsumption of appearances under the category” (ibid.). The important result of introducing the idea of schemata is thus a description of the relationship between percepts and concepts that does not require a resemblance between the two—the resemblance which empiricists would unconditionally require and always let pass unquestioned, and which rationalists would simply dismiss as useless in this case. According to Hume, resemblance is one of the three principles of association our imagination applies in using simple ideas to build ones that are more complex (Treatise I, 1 [Hume 1975, pp.  1–4]). From a rationalist point of view, on the contrary, the resemblance thesis, according to which qualities in objects resemble our sensations of them, is an unjustified prejudice and a grave cognitive mistake (Principles I, 66–72 [Descartes 1983, pp. 26–34]). According to Kant, perceptual intuitions and a priori categories possess no property in common whatsoever and yet can work together as mediated by the schematism of the imagination. Again, schemas are neither images nor concepts. They are rules that allow our understanding to subsume the latter under the former. Peirce, although his idea of mediation owes a great deal to Kant, is obviously dissatisfied with this solution. The first problem he identifies, as already quoted, is that Kant “drew too hard a line between the operations of observation and of ratiocination. He allows himself to fall into the habit of thinking that the latter only begins after the former is complete” (CP1: 35).1 The second problem is that Kant simply postulates the synthesis of impressions and does not explain how it is accomplished, i.e., what are the consecutive steps of this accomplishment. Kant himself readily admits this when he says that “[t]his schematism of our understanding, in its application to appearances and their mere form, is an art concealed in the depths of the human soul, whose real modes of activity nature is hardly likely ever to allow us to discover, and to have open to our gaze” (KRV: B 180). The fact of the logical unity between percepts and concepts, therefore, ultimately remains a mystery and thus leaves within the Critique of Pure Reason a gap which Kant fills with the notion of the transcendental scheme. One way to demystify the schematism would be to open up the logical structure of it; to describe it, as it were, from within. But how is this to be done? And what exactly does Peirce’s dissatisfaction with Kant’s schematism consist in? Admittedly, many of Peirce’s comments on this matter are confusing. On the one hand, he claims that Kant “fails to see that even the simplest syllogistic conclusion can only be

 Recall that, as shown in Chapter 7 above, according to Peirce, mathematical deductive reasoning always involves observation. 1

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drawn by observing the relations of the terms in the premisses and conclusion” (CP1: 35). The gist of Peirce’s criticism here is that, according to Kant, there is a rather sharp distinction between images, which are available for observation and experimentation, and schemata, which, although they are homogenous with appearances, act as algorithms for image production and themselves allow neither observation nor experimentation of any kind. On the other hand, Peirce at times draws direct analogies between his idea of a diagram and Kant’s schemata. For instance, in his entry on modality for Baldwin’s Dictionary of Philosophy and Psychology, Peirce claims: “Kant holds that all the general metaphysical conceptions applicable to experience are capable of being represented as in a diagram, by means of the image of time. Such diagrams he calls ‘schemata’” (CP2: 385; emphasis added). In the entry on the nature of proof, also written for Baldwin, Peirce says: It is either mathematical demonstration; a probable deduction of so high probability that no real doubt remains; or an inductive, i.e., experimental, proof. No presumption can amount to proof. Upon the nature of proof see Lange, Logische Studien, who maintains that deductive proof must be mathematical; that is, must depend upon observation of diagrammatic images or schemata. (CP2: 782)

In an unpublished survey of the problems of pure mathematics, Peirce also defines schemata as either geometrical or algebraical: “Geometrical schemata are linear figures with letters attached. … The algebraical schemata are arrays of characters, sometimes in series, sometimes in blocks, with which are associated certain rules of permissible transformation” (CP4: 246). Now whatever Kant means by a schema, the above does not seem to fit any of Kant’s own definitions of it. Besides, whereas the idea of diagrammatic thinking forms an organic part of Peirce’s own architectonic, he claims that “[Kant’s] doctrine of the schemata can only have been an afterthought, an addition to his system after it was substantially complete” (CP1: 35). As if this confusion was not enough already, another comparison of diagrams to schemata can be found in MS 293 (1906): It is a very extraordinary feature of Diagrams that they show,—as literally show as a Percept shows the Perceptual Judgement to be true,—that a consequence does follow, and more marvelous yet, that it would follow under all varieties of circumstances accompanying the premises. … And so the Iconic Diagram and its Initial Symbolic Interpretant taken together constitute what we shall not too much wrench Kant’s term in calling a Schema, which is on the one side an object capable of being observed while on the other side it is General. (NEM 4:314–318)

Whether Peirce wrenches Kant’s term in in the fragment quoted above is an open question. He does establish an analogy between a Kantian schema and a combination of a diagram plus a symbol that interprets it, but he does not give either any further clarification or any examples of the aforementioned combination. It is, therefore, not easy to see what the analogy actually amounts to. According to Kant, schemata are procedural rules, or formulas, within the framework of which the unity of appearances becomes thinkable. The schema of a category is not a relational copy of this category (which a diagram definitely is), but the key to its employment. When enacting a schema, we “set it alongside the category, as its restricting condition, and as being what may be called its formula” (KRV: B 224; emphasis added).

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A diagram is different. It definitely can be a formula, but it by no means can be only that. It should always be remembered that a Peircean diagram is a sensible representation of the same logical relations that are found in its object. It makes those relations observable and open for further experimentation, where the diagram that results from the experimentation is an image of the original diagram, and both are images of the initial object (Paolucci 2017, p. 79). A schema creates a condition under which an a priori concept is applied. A diagram itself always initially has an object, an image of which it shows. I suggest that what Peirce might mean by his comparisons is this. A diagram, as Peirce understood the term, presupposes a set of rules or principles that define its possible transformations. For instance, the very way an algebraic formula is constructed presupposes that there are things we can, and things we cannot do in order to move from the mathematical problem it represents to a solution of this problem. Likewise, Kantian schematisms, as they are applied when a geometrical figure is constructed, presuppose certain a priori intuitions, which make it visually evident at all times that, for instance, two straight lines cannot enclose a space, or that it is impossible to construct a square with the same area as a given circle, etc. In both cases, there are limitations creating an array of possible transformational patterns, which a mathematician can schematically see in advance, and which guide him through his reasoning process. In this respect, diagrams and schemata are alike in the sense that they both select, or organize the manifold of impressions by creating conditions, under which we see some configurations of the manifold only because we do not see all the rest. This being said, Peirce’s idea of a diagram seems to be an improvement on the Kantian schematism of imagination. It is that the logical form of a diagram is not a product of some internal structure or deep grammar hidden from the eye, but an integral part of the observable surface structure of the diagram itself. As a result, learning how the statement a diagram conveys is constructed does not presuppose any metalanguage that would require us to raise one level up relative to learning what is stated. Moreover, one has to make changes to a diagram in order to understand where the argument it demonstrates ultimately leads him. The understanding of what a diagram says is thus achieved by making changes to how it is constructed. This open and highly productive exchange between the “how” and the “what” of diagrammatic communication creates a link between thought and perception: in the case of diagrams, seeing something is, in fact, inseparable from understanding how it all works. This (I speculate that Peirce would agree) means that we can replace the idea of schematism as an “art concealed in the depths of the human soul” (KRV: B 180), with the idea of diagrammatic reasoning as a direct and seamless fusion of percept and concept.

Chapter 10

Existential Graphs

Visualized Inferences and Inferential Visuality In a Peircean rethinking of the Kantian visual schematisms, as presented in the previous chapter, the visual and the conceptual are closely intertwined. Peirce accepts Kant’s overall treatment of mathematical knowledge, which implies that what is available in ordinary visual experience, already contains pure mathematical intuitions. According to Kant, this is one of the reasons why mathematics is a deductive science. Even if, as Kant says, I construct a geometrical object “in empirical intuition” (on paper), I still do it a priori, without having borrowed the patterns obtained in my construction from any experience (KRV: A714/B742). As a result, we have necessary deductive inferences encoded in our visual perceptions. Peirce uses this result to show that some basic intuitions of ordinary perception can be used in order to visualize mathematical reasoning. Meanwhile, as we discussed in Chapter 7, according to Peirce, visual experience itself involves unconscious inferences. There is nothing immediate about it, and its very production presupposes a long history of piecemeal adjustments to the environment, where every new encounter is characterized by vagueness and builds on previous sensations (W2: 235). From this perspective, we have images that are conditioned, qua images, by the inferential ties that hold together our linguistic competence. This allows us to arrive at a rather interesting conclusion. As Peirce seems to imply, there is an exchange between the immanent, diagrammatic imagery of mathematical thought (inferences visualized) and the external, inferentially informed imagery of ordinary perception (visualizations that, in order to be acknowledged as such, require us to engage our inferential capacities). The sense of sight we require in order to observe inferences is intertwined with the inferential capacities we require in order to have the sense of sight. While an ordinary person is content with the passive exercise of (mathematically conditioned) external perception, a mathematician makes good use of the exchange between the external visuality of objects © Springer Nature Switzerland AG 2023 V. Kiryushchenko, Diagrams, Visual Imagination, and Continuity in Peirce’s Philosophy of Mathematics, Mathematics in Mind, https://doi.org/10.1007/978-3-031-23245-9_10

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that surround us and the immanent visuality of inferences we make about those objects. Moreover, a mathematician does it in such a way that geometrical figures and algebraic expressions, as used in his mathematical demonstrations, are always paired with particular diagrammatic images fully compliant with the concepts they represent. This, together with Kant’s claim that mathematical objects are constructed based on our pure intuitions of space and time, means that, in what for a non-mathematician is just an ordinary visual experience of things out there in the world, a mathematician sees the essential—a skeletonized image, or a diagram that reveals a necessary pattern.1 In this context, the very idea of reasoning undergoes a dramatic change. As Peirce puts it in a letter to a colleague, “reasoning … is a kind of experimentation” which “is not done by the unaided brain, but needs the cooperation of eyes and hands,” and in which abstract thought plays only a subsidiary part (MHFC, CSP-JMH, 29.03.87; emphasis mine). Diagrams provide a priori mathematical concepts with perceptual form. At the same time, as Peirce suggests, perceptual judgments are reports of the percepts that already presuppose some conceptual input. What does this reciprocity suggest, and why do mathematicians need it so much? Following Peirce’s logic, to answer this question is to unpack the Kantian schematism of the imagination. Kant defines what the schematism is and why it is conceptually necessary as a part of his theory. But he does not tell us what it consists in, or what the inner workings of it are. To repeat, the schematism of the imagination, according to Kant, is “an art concealed in the depths of the human soul, whose real modes of activity nature is hardly likely ever to allow us to discover, and to have open to our gaze” (KRV: B 180). All we know is that schemata are not images and that they are “nothing but a priori determinations of time in accordance with rules” (KRV: B 185). They are conditions for appearances as subsumed under concepts in relation to time. Does Peirce have anything interesting to add to these definitions? I believe he does. According to Peirce, we can explain the correlation between the internal visuality of inferences and the external visuality of ordinary experience if we can discover the visuality that pertains to the very act of schematization, in which thinking is performed by means of manipulating images. In this way, we will be able to understand what cooperation between reasons, eyes, and hands actually consists in and what the inner workings of this cooperation are. In Peirce’s terms, it means that we are required to discover the fundamental relational structure that underlies all thought and experience and that is available to direct observation. A language of thought itself.

 Thus, for instance, in Ron Howard’s A Beautiful Mind (2001), Russell Crowe, playing a troubled genius mathematician J. F. Nash, sees mathematics everywhere. Throughout the movie, he keeps trying to capture the mathematics of real-life scenarios, such as feeding patterns of pigeons, going on a date, chasing a thief running away with a stolen purse, etc. In one of the scenes, Nash, looking through his dorm window at his peers playing football in the yard, begins to cover the windowpanes with formulas describing the game as it goes on. The formulas, in this case, are directly mapped onto the moving image they purport to describe. There is, thus, a direct correlation between ordinary perception and mathematical inferences. 1

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The Graphs Explained The idea of such relational structure was introduced by Peirce in full force by the end of the 1890s in the form of a diagrammatic logical system known as “Existential Graphs.” The system of EG consisted of graphic conventions supplemented by a set of transformational rules that were meant to replace linear symbolic modes of expression with relational pictures of propositions and arguments in a state of continuous change. Pierce’s diagrams essentially are basic spatial intuitions used to visually embody propositional and argumentative structures as they are developed in reasoning through time. The principal idea behind EG is to replace formalized linear successions of syllogistic structures characterized by the arbitrary connection to the thoughts they express with a set of diagrammatic pictures conceived as schematic visual expressions of relations inherent in the act of thinking itself. The proper object of EG is thus the very machinery of ratiocination: It is requisite that the reader should fully understand the relation of thought in itself to thinking, on the one hand, and to graphs, on the other hand. Those relations being once magisterially grasped, it will be seen that the graphs break to pieces all the really serious barriers, not only to the logical analysis of thought, but also to the digestion of a different lesson, by rendering literally visible before one’s very eyes the operation of thinking in actu. (CP 4.6)

EG are so designed as to show an immediate logical continuity of thinking in the form of a dialogue between two imaginary parties: the “Graphist” and the “Interpreter.” As Peirce puts it, “thinking always proceeds in the form of a dialogue—a dialogue between different phases of the ego” (ibid.). To this end, EG are meant to replace literal signification of functions, logical connectives, and quantifiers with shapes mapped onto each other and composed of a variety of graphical conventions. The overall system of EG consists of three parts: alpha (corresponding to sentential logic), beta (corresponding to first-order predicate calculus), and gamma (roughly corresponding to modal logic). For all three parts, the first convention is “the sheet of assertion,” a blank sheet, on which all graphs are being created and which signifies the initial status quo, i.e., assumes that the truth of whatever is to be stated on it is a matter of agreement between the Graphist and the Interpreter. The rudimentary conventions of alpha graphs represent a set of basic intuitive ideas that support our spatio-temporal ability. The list of the four most important ones is as follows: (1) Objects placed in close proximity to each other tell us that we should consider them in conjunction (A & B). A

B

(2) A linear separation, which cuts whatever objects it encloses off from the sheet of assertion (a “cut”), introduces negation (¬A).

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A

(3) A double cut (i.e., “two cuts one of which has the enclosure of the other on its area and has nothing else there”—CP 4:414) represents the negation of the negation (A).

A

(4) A cut that encloses an area of the sheet of assertions that is empty (i.e., contains no statement), is a pseudo-graph that represents a statement that negates its own truth.

Let us “A” mean “2 is the smallest prime number,” and let us “B” mean “2 is an even number.” In this case, (1) above reads “The smallest prime number is even,” (5) below reads “The smallest prime number is not even,” (6) reads “2 is neither the smallest prime nor an even number.” (7) reads “If 2 is the smallest prime number, then it is even.” If we read the diagram from outside in, it shows that it is not the case that 2 is the smallest prime number, and yet it is not even. Finally, (8) reads “2 is either the smallest prime or an even number” (because, as the diagram shows, it is not the case that it is neither of the two). A

B A & ¬B (5)

A

¬A & ¬B (6)

B

¬(A & ¬B) (7)

B

A

A→ B

A

B

¬(¬A & ¬B) (8)

A B

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Peirce insists that “a diagram ought to be as iconic as possible; that is, it should represent relations by visible relations analogous to them” (CP 4:433; emphasis mine). In compliance with this requirement, as shown in (1–8), Alpha Graphs offer a fine example of the economy of notation. They introduce all four essential logical connectives (negation, conjunction, disjunction, and implication) by using no symbolic notation at all as applied to connectives themselves. All we have here is different combinations of conjunction, which is signified by the intuitively grasped spatial idea of the proximity of two objects to one another, and negation, which is signified by the act of cutting relevant objects out of the sheet of assertion. We do not need two different formulas to express each of the equivalences in (7) and (8). Each new part of EG introduces new diagrammatic devices. Thus, for instance, Beta Graphs use “lines of identity,” or lines connecting objects placed at their extremities and, in virtue of this connection, asserting an existential relation between those objects. Among other things, this allows Peirce (again, without using any formal symbolic notation) to deal with predicates and quantified statements. For instance, to use a familiar example from Peirce’s “Logical Tracts, No.2” (c. 1903) and his 1906 “Prolegomena to an Apology for Pragmaticism” (cf. Roberts 1973, pp. 50–52), (9) below represents the existential claim “There exists something that is not ugly,” and (10) introduces the universal claim “Everything is ugly” by denying that there exists something that is not.

is ugly is ugly

¬(∃x) ¬F(x) ⇔(∀x)F(x)

(∃x) ¬F(x) (9)

(10)

Making use of the lines of identity, (11) introduces two individuals. The extremity of the line referring to a catholic is enclosed once only, thereby negating their existence, while the other extremity is enclosed twice. Accordingly, reading the graph from outside inward, we have “It is not the case that there is a catholic who does not adore some woman.” is a catholic

Is a catholicis a

ador

woman

(11)

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As EG present information in a non-linear manner, most of the graphs allow multiple readings, since we perceive each of them differently, depending on where we begin and how we carve it up. Thus, (11) may also be read as making the universal claim that every catholic does adore some woman, or it also may be taken to express a conditional saying that if there is a catholic, then he adores some woman. We do not need to introduce new graphs to allow all three readings (yet we do need a new expression and new derivation rules if we want to use formal symbolic language instead of EG).2 As shown in (12), the extremity of the line referring to a woman is moved outside of the scope of the bigger cut. This results in the existential claim that there is a woman every catholic adores. is a catholic

Is a catholic

ador

is a woman

(12)

Finally, as shown in (13), removing the outer cut results in the existential claim “Some catholic does not adore some woman.” is a

catholic is a woman

ador (13)

The graphs also introduce a set of rules of transformation, which define permissible ways of building new graphs and serve as means of visualizing successive steps of proofs. In EG, a proof is presented graphically as an exchange between the Graphist and the Interpreter, which results in a succession of changes in a given graph reflecting the flow of the argument it embodies. Thus, in “Prolegomena,” Peirce opens his exposition by a graphical analysis of syllogism Barbara, which shows how, by inserting, erasing, and iterating graphs, cuts and lines of identity, one

 Although overall, beyond the current simple exposition, the rules of Peirce’s Graphs might seem complicated and difficult to use, the “multiple readings” feature contributes to the naturalness of expression and makes the cost of reading somewhat lower (Shin 2002, pp. 1–10). 2

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can visualize, in seven steps, the move from “All men are animals” and “All animals are mortal” to “All men are mortal,”3 or from is an

is a man is an animal

and

is mortal woman

is a man

to

is mortal animal

The point here is not to demonstrate the full arsenal of the rules Peirce uses in developing his system of graphs but simply to illustrate the fact that, in manipulating the graphs, one can actually observe a given argument visualized by an array of transformational rules as a number of continuously changing pictures and thus experience the meaning of the argument visually as a set of transitional states.4 It is for this reason that Peirce sometimes called his graphs the “moving pictures of thought,” (CP 4.8), or “a portraiture of thought” (CP 4.11). An important feature of the Graphs is that the conventions and transformational rules, which constitute the grammar of the graphs, are devised as a surface structure that is not separated from what the graphs actually convey. In other words, the logical form of every graph appears to be an integral part of its message. Every graph thus conveys information and simultaneously provides a key to how this information is to be decoded. Because every graph (including the pseudo-graph and the sheet of assertions itself) offers a direct visualization of the way its messages are encoded, using EG, or the moving pictures of thought, blurs the distinctions between the internal and the external, ratiocination and observation, code and message, and form and content. It is important that EG do not simply show transitions from one thought to another so that we could further translate them into a formal language. They are not used as external aids to the analysis they embody. They represent an identity between the very action of thought and the continuity of movement in a given graph as it changes in space (Paolucci 2017, pp. 84–85). As discussed in the previous chapter, according to Kant, geometrical proofs use particular images of geometrical figures in order to not just illustrate but to directly embody the universal rules that reveal certain general characteristics of space in this particular image and in the transformations it undergoes. Similarly, according to Peirce, EG use particular diagrams in order to not just illustrate but to directly embody the universal rules that reveal certain general characteristics of thought in this particular diagram. In both cases, there is a compliance between an image and the concept it represents. Just as no particulars (like the magnitude of the sides of the figures used, etc.) matter for a geometric proof, neither the size nor particular

 CP 4: 571. For the overall account of alpha and beta rules of transformation, see Roberts 1973, pp. 40–44, 56–63. 4  For further sources on ways to read and the rules of inference in EG, see Bellucci and Pietarinen 2016; Ma and Pietarinen 2017. 3

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shape of cuts and lines of a diagram matters for the argument it represents. What matters is the kind of relations the diagram puts to work. Apart from all the specifics discussed above with regards to space, Peirce’s graphical analysis of the process of thinking also shows the importance of the time-­ related ideas of continuity and change. It shows that there is a dynamic element to our perception, which serves as a bedrock for visual integration, both in the case of the internal visuality of inferences and in the case of the external visuality of ordinary experience. We know that motion in many cases can tell us more than where an object is going, whether this object is a stock price or a point on a map. It can also tell us what the object is. A pencil bouncing off a table, a butterfly in flight, or a closing door support the recognition of those objects as such. An object and its stereotypical, habit-driven motion in some important sense make the object what it really is. What this means in the case of EG is that some facts that are true of the architecture of ordinary visual recognition are also true of mathematics, or even of thought itself. The dynamic element, along with all the aforementioned features of visual perception (its essential vagueness, its interconnectedness with the inferential structure of language, etc.) can also be used in order to grasp the continuity of thinking itself. Moving pictures are needed in order to turn thought into a proper object. Accordingly, if there is a system of graphic conventions, there has to be a corresponding system of moves. Only then thinking can be caught in action. In what it does. What is at stake here is a logical demonstration of the continuity of thought; diagrams make visible the very workings of thinking through the process of continuous translation of one diagrammatically expressed transitional mental state into another. Instead of displaying a linear succession of formally encoded patterns of ratiocination, they visualize the very machinery of it so that we could read it off a particular set of graphs in the same manner we usually read an emotion off a person’s face.

The Graphs Contextualized If this interpretation is correct, the idea of EG serves as a linchpin that holds together several themes that are important both for understanding Peirce’s overall theoretical edifice and for putting this edifice into a wider cultural and historical context. Peirce himself admitted that EG may be understood as “a guide to Pragmaticism” that “holds up thought to our contemplation with the wrong side out, as it were; showing its construction in the barest and plainest manner” (CP 4:7; emphasis added). There are five ways to show how this construction can be used to provide the context in question. First, EG provides a link between Peirce’s broad idea of diagrammatic thinking and his earlier logical work on the new list of categories. Recall that Peirce’s main purpose in NL is to represent Kant’s transcendental synthesis of imagination not as a unity of impressions in a single consciousness, but as a continuous act of correlation addressed to the possible future. The synthesis, Peirce claims, acquires its value through the application of three modes of

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reference—quality, relation, and representation—to experience in order to achieve the conceptual unity of it in the propositional form. The reference that completes the unity is embodied in an interpretant. What is achieved by an interpretant is not grasping the manifold of impressions as mine, but correlating a current state of affairs to a set of expected ones. On this view, understanding consists not in getting immediate access to some hidden recesses of one’s own mind, but rather in making whatever beliefs one has available for future interpretations by other members of their community. Also recall that, according to Peirce, the reference to the possible future turns interpretation of one sign by another into a continuous, uninterrupted relationship between triadic patterns. Construed properly, triadicity is an expression of continuity.5 Given that, according to Kant, the continuity of time is the purest possible schema of all priori concepts, both in NL and in EG, Peirce tries to tackle one and the same problem. Namely, the problem of continuity. The second important role that Peirce’s visual logic plays in his overall system of thought is serving as an illustration of the relationship between logic and chemistry, which was of a high personal significance for Peirce. Peirce had a degree in chemistry from Lawrence Scientific School at Harvard and drew broad analogies between his logic of relations and chemical valences. The analogies, some of which are mentioned briefly above, are well known and thoroughly studied (see Parker 1998; Roberts 1973; Samway 1995). However, one relevant historical aspect behind them is rarely mentioned. Namely, it is the metamorphosis, which had taken place in chemistry in the mid-1840s, and which was triggered by the formulation of the chemical type theory. The new idea that the type theory and, later, the theory of valences brought about was that chemical compounds could be studied not as mixtures of actual substances but as relational pictures, or visual, diagrammatic schemes. Chemists discovered that the relational structure of a molecule and transformations of chemical compounds could be depicted in a certain way, with the use of rather simple graphic conventions. Thus, it is the idea of valences that actually gave birth to the first fully developed language that provided the outward visual projection of the inner life of nature. Just as in Peirce’s logic of relatives (as well as in EG), in the language of chemical valences, how a compound is constructed forms a part of the surface structure of a diagram that depicts it. By analogy with EG, every message formed in this chemical diagrammatic language says what the case is and, simultaneously, shows how it is to be read. It is also noteworthy that, in using the term “graph,” Peirce refers to William Clifford and James Joseph Sylvester who were the first mathematicians to note the importance of the graphical representation in chemistry: “By a graph … I, for my part, following my friends Clifford and Sylvester, the introducers of the term, understand in general a diagram composed principally of spots and of lines connecting certain of the spots” (CP 4: 535, 3.418,  We have already learned that it is also an expression of generality. One way to appreciate the link between the two notions, according to Peirce, is to think about the fact that, while particulars are, by definition, discrete (“I need this ruler,” “this number is even,” etc.) general notions are emphatically not so, as none of those notions can be properly described by enumerating a set of objects it denotes/refers to. 5

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468; see also Clifford’s “Remarks on the Chemico-Algebraic Theory,” Clifford 1882). A.B. Kempe, whose “Memoir on the Theory of Mathematical Form” was an object of inspiration for Peirce (as discussed in Chapter 5), in introducing his version of graphical notation, had in mind the same transition from the substantialism to the visual representation of molecular structure in chemistry. The third important role of EG is to finally provide a clear explanation of how Peirce saw the relationship between the logical and the mathematical mindsets. As discussed in Chapter 3, the distinction between the two reflects a tension between two Peirce’s personality traits: his “pedestrianism,” combined with his great attention for detail (representing the “logical side” of Peirce’s character) and his being a fast thinker, combined with the unconventional élan of his social views (representing the mathematical side of Peirce’s character). “The sinister” misunderstood by others is thus a result of the unresolved tension between the two. One might speculate that because EG represent a logical device that makes some use of the diagrammatic mathematical form (in the described above Kantian sense of “mathematical”), for Peirce, his system of the Graphs was a shortcut aimed to resolve both the central problem of his overall theory and his inner personal conflict. The idea of the graphical expression of logical relations also has something to do with the way Peirce treated classificatory principles as those are applied in natural sciences—the principles that Peirce, by his own admission, learned from a famous Swiss-born American biologist Louis Agassiz when he was a student at Harvard in the late 1850s (CP1: 205 fn.). Agassiz’s Essay on Classification was published the same year as Darwin’s On the Origin of Species by Means of Natural Selection. Agassiz’s lessons allowed Peirce to see clearly what logical mechanisms are at work when a naturalist applies the idea of continuous change in taxonomic classification of plants and animals: … [T]he conception of continuous quantity has a great office to fulfill, independently of any attempt at precision. Far from tending to the exaggeration of differences, it is the direct instrument of the finest generalizations. When a naturalist wishes to study a species, he ­collects a considerable number of specimens more or less similar. In contemplating them, he observes certain ones which are more or less alike in some particular respect. They all have, for instance, a certain S-shaped marking. He observes that they are not precisely alike, in this respect; the S has not precisely the same shape, but the differences are such as to lead him to believe that forms could be found intermediate between any two of those he possesses. He, now, finds other forms apparently quite dissimilar—say a marking in the form of a C―and the question is, whether he can find intermediate ones which will connect these latter with the others. This he often succeeds in doing in cases where it would at first be thought impossible; whereas, he sometimes finds those which differ, at first glance, much less, to be separated in Nature by the non-occurrence of intermediaries. In this way, he builds up from the study of Nature a new general conception of the character in question. He obtains, for example, an idea of a leaf which includes every part of the flower, and an idea of a vertebra which includes the skull. I surely need not say much to show what a logical engine is here. It is the essence of the method of the naturalist. (СР 2.646; emphasis added)

A naturalist is a professional in creating new concepts that he puts to work in order to explain the continuity of a species change. A natural classification essentially is a search for a mediating form (an interpretant) aimed to show where our

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interpretation of a particular character should go on and where it should be terminated, with a resultant new general classificatory unit in place. For instance, evolutionary biology defines homology as a similarity in structure or function of bodily parts of different origins based on their descent from a common ancestor. Thus, we can establish a homological relation between the forelimbs of humans and bats based on the fact that the functional formation and the number of bones in each are almost identical. This mediating relation allows to interpret the succession of changes in the two cases as a series of adaptive modifications of the forelimb structure of their shared ancestor. This being the case, Peirce’s early studies as a naturalist laid foundations for his later mathematical interest in the problem of continuity. Both rules of transformation in EG and natural classification thus understood presuppose the application of the idea of continuous quantity in search for mediating forms of thought. The last theme to be discussed in relation to EG in this section will help us put Peirce’s idea of graphical logic into some wider cultural context. As already mentioned, Peirce characterized EG as “moving pictures of thought” (CP4: 8, 11). The theoretical reasons for this characterization are explained above. Meanwhile, at the end of the nineteenth century, another technological incarnation of the idea of the moving image—cinematography—captured the imagination of people on both sides of the Atlantic. In the United States, the first cinema shows were organized in 1896 by Thomas Edison, who introduced a film projector called vitascope, the patent for which he purchased a year before from an engineer named Thomas Armat (for a brief account see also Pietarinen 2006, pp.  108–109). Apart from a loose chronological coincidence (Peirce began writing on graphical logic in the mid-­1880s and definitely visited some of Edison’s shows in New York and Philadelphia in the late 1890s), this historical detail provides an important point of intersection between philosophy and technology. An existential graph, considered at a random moment of its development before the argument it demonstrates is completed, cannot be considered a photograph of a thought. Likewise, a randomly selected film frame cannot be considered a photograph of an event separated from the sequence of frames represented by the film show as a whole.6 Given that our choice of a film frame is random, the overall entourage, the positions of the objects within the scene that the frame represents, and the poses of the actors it involves do not represent any complete, meaningful scene, as they are not structured by the selective eye of a photographer. Plainly speaking, the frame, taken in separation from the sequence of which it is a part, represents nothing. There is no particular plot related to this particular frame, if selected randomly. The frame is just a moment when the film show suddenly comes to a full stop. As it stands alone, the frame has no content whatsoever. In a similar vein, a single graph can be considered only as a “Dedekind cut”  It is true that, in both cases, the persistence of our vision blends a succession of still pictures together, thus producing an image that we perceive as “moving.” However, that the movement itself is a construction (i.e., something of our own making) does not constitute a problem—as far as we acknowledge the two facts as they are stated above in Chapter 6: (1) we can construct something only if we use signs, and (2) for anything all, to be what it is to be a sign. 6

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within a real continuum. What a series of graphs show is not a sequence of separate images following one another, but the continuity of a thinking process represented visually. If a representation involves no continuity, it has no meaning. In fact, Peirce finds some important analogies between meaning and physical motion as early as 1868. Thus, in a passage from “Some Consequences of Four Incapacities,” partially quoted above, he writes: [N]o present actual thought … has any meaning, any intellectual value; for this lies not in what is actually thought, but in what this thought may be connected with in representation by subsequent thoughts; so that the meaning of a thought is altogether something virtual. It may be objected, that if no thought has any meaning, all thought is without meaning. But this is a fallacy similar to saying, that, if in no one of the successive spaces which a body fills there is room for motion, there is no room for motion throughout the whole. At no one instant in my state of mind is there cognition or representation, but in the relation of my states of mind at different instants there is. (W2: 227)

Whatever we think “about” is, by definition, always already in the past by the time we think about it, but whatever “meaning” we ascribe to it is in the possible future effects of our interaction with it. A movement-as-image (whether understood logically or cinematographically) contains no privileged moments because what makes it an image is the sum total of equally important relative moments in the overall continuous flow. What such an image is is in direct contradiction to the way “image” was understood in the period of classical modernity (and by Kant in particular). In Kant’s view, an image is something predetermined by a “conceptual scheme”—a set of a priori notions that define perception according to a set of non-­ pictorials “transcendental conditions.” Meanwhile, what happens on the cinematic screen, or on the Peircean sheet of assertions, cannot be deconstructed into a set of moments one necessarily must see, as opposed to those one may omit. A sequence of graphs scribbled on the sheet of assertions, just as a sequence of frames projected to the cinematic screen, does not supply our vision with any particular ideology that is aimed to make us selective in a particular way or to make us sensitive to this moment rather than that. It simply shows what needs to be shown. A motion picture is an experience of free uninterrupted perception. Likewise, visual logic is an experience of free uninterrupted thought. Both keep us open to all possibilities we are able to envision, anticipate, and use.

Chapter 11

Iconicity, Similarity, and Habitual Action

Three Formulations of the Maxim We already know that, according to Peirce, signs refer to their objects through habitual action. Peirce’s maxim tells us that all our general idea of an object consists in is an account of our would-be responses to the changes resulting from our experiments with this object. What ideas mean to us lies in whatever we are prepared to do with the objects those ideas are about. Peircean sign systems, therefore, act like any language, with the proviso that signs, as Peirce understood them, refer to their objects not through arbitrary conventions, but through patterns of adaptive habitualized behavior. According to Peirce, then, the meaning of every sign ultimately depends on what habits of conduct it is going to bring about. Peirce believed that one way to establish this fact is to provide an explanation of the relationship between his pragmatism and his semiotics—an explanation he considered one of the ways to “prove” his version of pragmatism. The most notable attempt at such proof is provided by Peirce in MS 318, where he integrates his pragmatism within his semiotics by reconciling his pragmatic maxim and his general definition of a sign (EP2: 398–433). Pragmatically speaking, the meaning of a concept is not an immaterial platonic entity to which the concept refers but consists in conceivable practical outcomes of our possible interactions with the object of the concept (the meaning of an “orange” is all of the things we can realistically expect to do with it if it truly is an orange). Semiotically speaking, a sign cannot be reduced to an arbitrary relation between a material signifier and an immaterial signified but involves a triadic relation, i.e., represents something that stands for something else, to someone, in some respect or capacity. The claim “this is an orange” is meaningful only as addressed to someone with a certain expected reaction to the claim in view of some adopted hypothesis. For instance, in applying the word “orange,” I bring someone’s attention

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to an object the word stands for, in order to communicate my intention of eating it, dissuade someone from thinking it is a plum, begin quoting “Oranges” by Gary Soto, or hint at the Cockney origins of the phrase “Clockwork Orange” as the title of Anthony Burgess’s famous novel. Peirce’s proof of pragmatism shows in detail how and why the two definitions (of meaning as the sum total of practical consequences and of a sign as a triadic entity) are translatable into one another. Peirce attached extreme importance to this task, as he believed that such reconciliation would allow him to incorporate the two parts of his philosophy (pragmatism and semiotics) into a wider unified architectonic framework, thereby connecting his normative theory with his evolutionary metaphysics and his logico-mathematical doctrine of categories. Overall, the proof Peirce presents in this manuscript is a complex, multilayered affair, an analysis of which is beyond the scope of this chapter. All we need here is to note that the proof suggests that every conception (as it is defined in the maxim) is a sign (and, therefore, is a triadic relation) and that our capacity to interpret signs assumes our capacity to predict the outcomes of our possible future actions. This, in turn, means that we always need to have some sort of a plan in advance, which is a part of what we “mean.” The final interpretant of a sign, therefore, should necessarily be of the nature of a habit: in addressing our thoughts and practical decisions to possible future, we need some circularity, a loop that can provide us with feedback. The goal of the feedback is to reconcile our expectations about our actions with the status quo that we face in performing those actions. Signs form habits and bring them about. At the same time, they also play the role of the catalysts that cause those habits to be changed, reinforced, or abandoned. A practice of sign interpretation that follows the maxim of pragmatism is thus necessarily self-corrective, i.e., characterized by self-controlled, habit-driven action. Apart from MS318, the hints about how Peirce’s pragmatic maxim and his theory of signs are mutually interpretable are scattered in various definitions Peirce gives to the pragmatic maxim (see, e.g., Turrisi 1992; McCarthy 1990). Hookway (2005) lays special stress on the following three of Peirce’s formulations, the relationship between which more or less recapitulates the idea of Peirce’s “proof”: 1. The initial formulation from “How to Make Our Ideas Clear” (1878): “Consider what effects that might conceivably have practical bearings we conceive the object of our conception to have. Then our conception of those effects is the whole of your conception of the object” (W3: 266). This formulation gives us a pragmatist explanation of the meaning of a concept in terms of the possible consequences of our actions involving an object to which the concept properly applies. According to this formulation, the meaning of a concept depends on our capacity to predict practical outcomes of our interaction with the object. ­Meanings do not lie in what either rationalists or empiricists called “ideas” but are grounded in the (realistically expected) outcomes of purposeful human action. Our interactions with the object of a concept result in certain “effects”;

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these latter have “practical bearings” on us, and our account of those bearings is the principal ingredient of the meaning of an appropriate concept. For example, our knowledge of certain properties of vinegar amounts to our knowledge of what to expect when we perform a variety of actions involving a portion of it. In dipping a piece of litmus paper into it, we expect the paper to turn certain color; in putting foods into it, we anticipate these foods to get pickled; in trying to make more vinegar, we consider initiating the fermentation of ethanol by acetic acid bacteria, etc. 2. The formulation from the first of Peirce’s Harvard Lectures (1903): “Pragmatism is the principle that every theoretical judgment expressible in a sentence in the indicative mood is a confused form of thought whose only meaning, if it has any, lies in its tendency to enforce a corresponding practical maxim expressible as a conditional sentence having its apodosis in the imperative mood” (CP 5: 18; EP2: 354). This formulation gives us a language-related version of the initial one above. It says that, by endorsing the expression “vinegar is an acidic liquid,” we acquire the habit of using a variety of conditionals the expression entails, each of which would be a maxim of action that entails practical consequences if acted upon: “Were you to dip a piece of litmus paper into this liquid, the paper would turn red”; “Were you to put foods into it, these foods would eventually get pickled”; “Were you to decide to make more vinegar, you would have to initiate the fermentation of ethanol by acetic acid bacteria”; etc.1 3. The formulation from “Issues of Pragmaticism” (1905): “The entire intellectual purport of any symbol consists in the total of all general modes of rational conduct which, conditionally upon all the possible different circumstances and desires, would ensue upon the acceptance of the symbol” (1905, EP2: 346). This formulation uses a semiotic vocabulary in explaining meaning. It tells us that the meaning of what Peirce calls a “symbol” consists in our conditional resolutions to act in case we believe that what the symbol conveys is true. In other words, a sign proper is always related to its object by means of possible human conduct.

 To rephrase those conditionals more in the spirit of the formulation, which requires apodoses to be in the imperative mood, “If a piece of litmus paper is to turn red, dip it into this liquid,” “If more vinegar is to be produced, initiate the fermentation of ethanol by acetic acid bacteria,” etc. All these formulations vary in wording, but not in essence. All of them lay stress on the fact that the meanings of descriptions consist of sets of conditional propositions appropriately indicating what to expect from interactions with objects to which these descriptions apply. All of them suggest that to mean something is to think over possible practical effects of what we do in order to adapt to our environment and to pursue our goals in a variety of relevant hypothetical situations. 1

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What Is Likeness? The question now is whether diagrams are any different from other kinds of signs with respect to their reference to patterned behavior. We are told that, as iconic signs, diagrams are related to their objects by means of likeness. But what does this mean? What does “likeness” revealed in vision (and, more broadly, in perception) amount to? Peirce himself recognized that there is a quandary here: “I myself happen, in common with a small but select circle, to be a pragmatist, or ‘radical empiricist,’ and as such, do not believe in anything that I do not (I think) perceive. … Only, the question arises, What do we perceive?” (CP 7.617–618; emphasis added). If we confine iconicity to the domain of vision only, we face a problem. For instance, in comparing what we see in the portrait of a mathematician Christiaan Huygens by Caspar Netscher, with what we would see if we lived in the seventeenth century and actually met Huygens, we juxtapose a two-dimensional image and a real person. In none of its parts the portrait is “like” Huygens himself. And the smaller the parts that we compare get, the more evident the lack of likeness becomes. No matter how far we go into detail in trying to explain what particular feature or quality serves as a ground for our comparison, the “what” of similarity always escapes analysis. For instance, we might agree that the shape of the nose is the same. But again, while the nose of the real Huygens had three dimensions, that of his portrait has only two; the painting has no apertures, is smoother than its referent, and has a black dot corresponding to Huygens’s left nostril. What is the correspondence, exactly? If we insist on defining a unit of similarity, then, after as many attempts as we can bear, we will ultimately find ourselves taking notes that “this” is similar to “that,” where both terms are purely denotative and have no real content whatsoever (cf. Eco 1992, pp. 191–216). In his Theory of Semiotics (1979), Eco claims that the recognition of the similarity of some one thing with some other thing is based on cultural convention; that it “does not concern the relationship between the image and its object, but that between the image and a previously culturalized content” (Ibid., p.  204; cf. Nöth 1995, p. 127). The aim of Eco’s critique is to replace the idea of a sign as a unit of fixed correlation with a classification of the modes of codification. In explaining the likeness that connects signs and their objects, he claims, we can appeal to codes and conventions that determine the ways we can read those signs. Codes provide a framework within which signs make sense. They reproduce selected conditions applicable to our perceptions of the object based on a set of established cultural imperatives. But does this really make the problem go away? I think not. If we define effective communication as one that results in using similar codes, we seem to face the same issue all over again, only elevated one level up. What is the “unit of similarity” between different codes? Alternatively, if we turn from conventionalist to naturalistic explanations of likeness, in order to explain how this quality is similar to that, we will need something no simple quality contains, and, therefore, something more than a naturalistic explanation of likeness will ever be able to offer. In explaining what similarity is, we cannot appeal either to codes or to some sort of

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Adamic language whose descriptions are justified by their immediate appeal to the very essences of things they are applied to. Is there a way out? I think there is. First, we have to admit that codes should represent a broad interpretive framework used to encode and decode our messages; in other words, that likeness can properly be established not between particular things, but only holistically and only between different sets of practices that together form a system of relations. As Wittgenstein has it in a famous passage from On Certainty, “when we first begin to believe anything, what we believe is not a single proposition, it is a whole system of propositions. (Light dawns gradually over the whole)” (Wittgenstein 1969, p. 141). Second, we have to admit that, when we are talking about similarity, we cannot appeal to vision simpliciter, but rather to some sort of basic perceptual schematism. As we saw in Chapter 9, according to Kant, our perception has necessary conditions that define what schemata are at work in each particular case. The ultimate question is whether there is an approach that can unify the two explanations of likeness between a sign and its object, and which, accordingly, can provide a link between the schematic character of visual perception and perceptual practices as the ground for “likeness.” As we have seen, Peirce’s principal suggestion is that what underpins our perceptions of things as being alike, and what reveals the true conditions of their being so, is the isomorphism not of substances, but of relations (see also EP2: 13; Stjernfelt 2007, pp. 50–77; Paavola 2011); and he claims that maps, charts, geometric diagrams, and mathematical equations are the primary examples of such isomorphism (NEM4, p. xv; CP 4.530). Thus, for instance, the relation between points on a map of Toronto is isomorphic with the relation between the corresponding places on the Earth’s surface. And the relation between neighboring candles on a NASDAQ chart is isomorphic with the moves of the index’s price over the relevant periods of time. Similarly, the mathematical function “__is a square of__,” or f(x) = x2, is a mapping rule for a set of ordered pairs, in which one element is mapped onto the other, so that is followed by , which is followed by , etc. (cf. Bradley 2004, pp. 71–73). What all these cases have in common is that, in interpreting each of them, we pay attention to a certain form of a relation. Viewed from this perspective, imaging is no different than lining up mathematical eqs. A feature, with respect to which a portrait is like its object, is always dynamic. Like a map of Toronto or a NASDAQ chart, it is a result of mapping one set of relations onto another that reveals a character of the portrayed person based on the schematization of an anticipated facial change. What we know in all these cases, including the portrait, is how to go on (by looking at the chart, I know that the NASDAQ trend is up; by looking at the map, I know that I need to turn right on Bloor Street to get to Robarts Library; and by looking at the portrait, I see that Netscher’s Huygens is unhappy, likely because, as I happen know from reading his biography, he was sick with a flu,—a rather serious inconvenience at the time). Thus understood, diagrams are iconic signs precisely because the similarity they convey reflects not the way their objects look, but the way they behave, their modus operandi. The relational (schematic) character of similarity is thus closely connected to habituality. Viewed relationally, diagrammatic iconicity is about how we read charts, use maps, do math, and

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interpret a facial expression on a portrait. Diagrams, including mathematical ones, copy their objects, but the schematisms that underlie the copying process are based neither on mental imagery, nor on any explicitly formulated rules, but on a mode of action. And, as far as our basic visual intuitions are, in some important sense, part of our natural endowment, the imagery used in mathematical diagrams shows how our reasoning capacities are grounded in our ordinary visual experience. The diagrams copy the way their objects behave and, therefore, just like other signs, ultimately refer to habits of action; in this particular case—to the habits that form our ordinary visual integration. It must be noted, however, that there is a general characteristics Peirce ascribed to all iconic signs, which prima facie seems to contradict the conclusion above: Icons are so completely substituted for their objects as hardly to be distinguished from them. Such are the diagrams of geometry. A diagram, indeed, so far as it has a general signification, is not a pure icon; but in the middle part of our reasonings we forget that abstractness in great measure, and the diagram is for us the very thing. So in contemplating a painting, there is a moment when we lose the consciousness that it is not the thing, the distinction of the real and the copy disappears, and it is for the moment a pure dream,—not any particular existence, and yet not general. At that moment we are contemplating an icon. (EP1: 226; emphasis added)

This contradiction, however, turns out to be no contradiction at all. What, according to Peirce, happens “in the middle part of our reasoning” only proves that the likeness is neither in the sign itself, nor in the object as it is represented by the sign, but in the way the two are brought together by an interpreting mind relative to some practical purpose the mind has (sf. Parker 2017, p. 68). The ground for the relational similarity, or for the iconicity of a diagram, is the relational isomorphism between an expected experimental change in the relations between its parts, on the one hand, and the corresponding observed transformation in its object, on the other hand. It is in this and no other sense that Peirce insisted that diagrammatic (and, more broadly, visual) experience allows for the perception of generality: in observing a diagram, we directly grasp a general pattern of the expected change. The advantage of explaining the nature of iconicity in this way is that it allows us to steer our way between the Scylla of conventionalism and Charybdis of naturalism. On the one hand, we can retain the idea of a code as a mode of sign production. On the other hand, we can also salvage the very idea of similarity and its cognitive relevance both for perception and for habitual action (cf. Sebeok 1979, pp. 112–120; Sjöberg 1972; Tversky 1977). Again, as Peirce puts it, “in the middle part of our reasonings” the distinction between a sign and its object vanishes, and, in dealing with the sign, we “lose the consciousness that it is not the thing” (EP1: 226). In a similar manner, in the middle of our routine, we lose the consciousness of particular things we use and see (in some sense of “see”) only an image of our own habit. A moving image. Diagrammatic aspects of the link Peirce saw between the relational character of perception and habitual action have some deep implications. If the above interpretation of Peirce’s view is correct, it should be clear that, for Peirce, the possibility of visual representation not only of linguistic but also of all semiotic relations was essential for making the most of his pragmatist approach to meaning. And it is not

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surprising that by the mid-1900s Peirce began to consider using his diagrammatic logic as a tool for “proving” his version of pragmatism. But long before EG was introduced, while working at the US Coast and Geodetic Survey, Peirce came up with an interesting case of how a real, geographical map can demonstrate relational similarity with its object by replicating the object’s modus operandi. This early case is important because it proves that Peirce’s pragmatist and semiotic ideas find support in his early practice as a scientist and an applied mathematician. Peirce’s map projection represents an intricate amalgamation of applied mathematics, Peirce’s early semiotic ideas and his metaphysics of continuity, thus providing an intriguing example of the intersection of real scientific practice and armchair metaphysical speculation.

Chapter 12

Mapping Philosophy: Peirce’s Quincuncial Projection

Language and Maps From very early on in his career both as a scientist and as a philosopher, Peirce paid close attention to the role played in cognition by maps. For Peirce, a map can serve as a metaphor applicable to such major philosophical concepts as those of the self and self-consciousness (CP8: 122–125). He sometimes mentions mapping as an operation that helps one organize and clarify his thinking process (see, e.g., CP1: 364; CP4: 533). “Prolegomena to an Apology for Pragmaticism,” one of Peirce’s papers on diagrammatic logic, begins with the following imaginary dialogue, which reveals an important comparison between maps and diagrams: ‘But why do that [use maps] when the thought itself is present to us?’ Such, substantially, has been the interrogative objection raised by … an eminent and glorious General. Recluse that I am, I was not ready with the counter-question, which should have run, ‘General, you make use of maps during a campaign, I believe. But why should you do so, when the country they represent is right there?’ Thereupon, had he replied that he found details in the maps that were so far from being ‘right there,’ that they were within the enemy’s lines, I ought to have pressed the question, ‘Am I right, then, in understanding that, if you were thoroughly and perfectly familiar with the country, as, for example, if it lay just about the scenes of your childhood, no map of it would then be of the smallest use to you in laying out your detailed plans?’ To that he could only have rejoined, ‘No, I do not say that, since I might probably desire the maps to stick pins into, so as to mark each anticipated day's change in the situations of the two armies.’ To that again, my sur-rejoinder should have been, ‘Well, General, that precisely corresponds to the advantages of a diagram of the course of a discussion. Indeed, just there, where you have so clearly pointed it out, lies the advantage of diagrams in general. Namely, if I may try to state the matter after you, one can make exact experiments upon uniform diagrams; and when one does so, one must keep a bright lookout for unintended and unexpected changes thereby brought about in the relations of different significant parts of the diagram to one another. Such operations upon diagrams, whether external or imaginary, take the place of the experiments upon real things that one performs in chemical and physical research. Chemists have ere now, I need not say,

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described experimentation as the putting of questions to Nature. Just so, experiments upon diagrams are questions put to the Nature of the relations concerned’ (CP4: 530).

Can diagrams, then, according to Peirce, be maps of thought used “to experiment upon” and “to stick pins into” in order to both mark anticipated changes in planning or thinking about something and keep “a bright lookout” for changes unintendedly brought about by our experimentation? They can, of course, be the intermediary conclusions (or results) of thought, which can play a role in further thought. And, while thinking is a dynamic process,1 a map is an accomplished result, a given static object. Whether Peirce would agree with this or not, we shall see. Peirce’s interest in maps was not purely metaphysical. As a mathematician, he was interested in mappings as applied in topology. He was directly involved in calculating map projections (W4: 68–71) and solving problems related to geological maps (CP7: 85). He also attempted to prove the so-called four color theorem, which states that, given any map representing a set of contiguous regions, no more than four colors are required to cover the map in such a way that no two adjacent regions have the same color (CP2: 105; CP5: 490; NEM4: 216–222). Still, maps were extremely important for Peirce for the reasons of a more general character. Peirce, for instance, considered maps the most accurate schematic representations of the basic spatial relations of similarity and contiguity, the two kinds of associative relations upon which, he believed, linguistic meanings were heavily dependent (“The Critic of Arguments,” The Open Court, 1892; CP 3.419). Being a language user, Peirce claims, necessarily presupposes having some essential diagrammatic experience. Because a natural language is a kind of algebra, its syntax is a diagram. But a diagram cannot actually point at “the where and the when of the particular experience” unless we construe it as a map whose mathematical law of projection we understand. Therefore, in using language, we can have an idea of what we are talking about (can recognize and acknowledge contiguous things as sharing some quality or other) only if we understand the law of the projection. Obviously, as Peirce extrapolates the idea of mapping to the relationship between language and the world, “the law of projection” he has in mind here should be understood not just as a set of linguistic paradigmatic relations, but rather in a Kantian vein, as a set of conditions of the possibility of experience in general. These conditions need a schema that maps language onto the world. With this requirement unfulfilled, as knowers and language users, we have little more than associations by similarity, which leaves unexplained the aboutness of our knowledge and our language. How “schemata” are actually defined in this case is irrelevant to the point. It boils down to taste depending on one’s theoretical background, i.e., a schema can be “an a priori rule” for a Kantian, “logic” for a positivist, “a practice” for a Wittgensteinian, or “a cultural imperative” for a Marxist, etc. What such a schema does is it justifies this particular language use in this particular situation—just as a law of projection justifies the way we see the surface of this particular map as referring to a portion of the Earth’s surface.  Or, to use William James’s metaphors, a “flow” or a “wave” (see, e.g., James 1977: 70).

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According to Peirce, then, understanding our capacity to refer to things meaningfully when using language is akin to understanding the basic visual schematism encoded by the mathematical laws of map projection. Such basic diagrammatic experience is at the core of our linguistic capacities. The projection of a map builds a relation between the mathematical function that transforms pairs of coordinates from the curved surface of the Earth to the plane of the map and our recognizing one thing as a sign of another (say, a twisting blue line on a map of Toronto as Don River). Similarly, a law of cognitive projection is necessary for us to acknowledge that a given description corresponds to the status quo described. This law is what makes a language a language. To make a metaphor of a map a bit more pronounced, we might say that, what Peirce’s extrapolation seems to imply is that certain basic visual schematisms build a relation between the text printed on the flat surface of a page and things my imagination creates while reading it. From this, it follows that if I know (or am able to use) the rule of interpretation, any experience involving interpretation can be diagrammatized. In his early paper “Validity of the Laws of Logic” (1869), Peirce makes an interesting claim related to the point above. For a map to count as a representation of the Earth’s surface, some parts of the surface should not be represented on this map. Likewise, for a language user to know what they are talking about is inseparable from their not being able to tell everything. Based on this, Peirce distinguishes between the ideas of simply being something and representing something, by pointing at an analogy between maps and syllogisms: It is as though a man should address a land surveyor as follows: ‘You do not make a true representation of the land; you only measure lengths from point to point. … So, you have to do solely with lines. But the land is a surface; and no number of lines, however great, will make any surface, however small. You, therefore, fail entirely to represent the land.’ The surveyor, I think, would reply, ‘Sir, you have proved that my lines cannot make up the land, and that, therefore, my map is not the land. I never pretended that it was. But that does not prevent it from truly representing the land, as far as it goes. It cannot, indeed, represent every blade of grass; but it does not represent that there is not a blade of grass where there is. To abstract from a circumstance is not to deny it.’ Suppose the objector were, at this point, to say, ‘To abstract from a circumstance is to deny it. Wherever your map does not represent a blade of grass, it represents there is no blade of grass. Let us take things on their own valuation.’ Would not the surveyor reply: ‘This map is my description of the country. Its own valuation can be nothing but what I say, and all the world understands, that I mean by it. Is it very unreasonable that I should demand to be taken as I mean, especially when I succeed in making myself understood?’ … Now this line of objection is parallel to that which is made against the syllogism. It is shown that no number of syllogisms can constitute the sum total of any mental action, however restricted. This may be freely granted, and yet it will not follow that the syllogism does not truly represent the mental action, as far as it purports to represent it at all (W2: 249).

All these functional parallels and interchanges between the symbolic structures of language and maps possibly influenced some of the terminological choices Peirce made when describing the role of visual experience in mathematics. Peirce was a knowledgeable etymologist and paid close attention to what he called “the ethics of terminology” (CP2: 219–226). Being extremely careful with both introducing new

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terms and tracing the provenance of the existent ones was, for Peirce, not a matter of displaying his erudition, but an essential condition of successful research and scientific collaboration (Haack 2009; Ketner 1981; Oliver 1963). Thus, one of Peirce’s observations in “Notes on Symbolic Logic and Mathematics” (1901) is that, although German Abbildung is usually translated by English “imaging,” this has not always been the case in mathematics. In particular, Peirce stresses the fact that mathematicians often used Abbildung in a much wider sense, to mean simply “representation.” The only exception he quotes is the use of this term by Carl Friedrich Gauss, who narrowed it down to mean specifically a map projection. While Peirce deems Gauss’s interpretation of Abbildung to be unjustifiably narrow (CP3: 609), his own uses of the word “map” varied. For instance, he hesitated whether a sheet of assertion—a surface on which existential graphs are scribed— should be called a map or a photograph (CP4: 513), and once used a phrase “a map of a labyrinth.2” With all these aspects of Peirce’s interest in maps in view, let us now turn to Peirce’s map projection and see whether it can serve as an illustration for the mode of copying of the modus operandi of its object which, according to Peirce, is specific to diagrams.

Quincuncial Map Peirce’s careers as a scientist and as an academic philosopher overlapped in 1879. That year Peirce started teaching at Johns Hopkins University, while continuing his pendulum experiments at the U.S. Coast and Geodetic Survey as well as his spectroscopic studies. Three months prior to his appointment to Johns Hopkins, in May 1879, Peirce presented a paper at the annual meeting of the National Academy of Science, which contained his original mathematical calculations aimed to amplify the use of diffraction spectroscopes. That same year, Peirce also published a short paper in the American Journal of Mathematics describing his new map projection, which he called “quincuncial.” The term comes from the combination of Latin words quinque and uncia, which stand for a Roman coin that contained five-twelfths of a Roman bronze libra. The bronze content of the coin was signified in the quincunx manner, with four dots at the corners and the fifth at the center of the coin— hence the name of the map3 (Figure 12.1):

 “When I was a boy, my logical bent caused me to take pleasure in tracing out upon a map of an imaginary labyrinth one path after another in hopes of finding my way to a central compartment” (CP4: 533; see also Peirce’s picture of Minotaur’s labyrinth as in Brent 1998, p. 310). 3  Peirce’s terminological choice in this case is also not a sheer coincidence. Just after his appointment at Johns Hopkins ended abruptly, in the mid-1880s, Peirce became a member of several government commissions on weights and measures and closely collaborated with the US Mint in Denver and Philadelphia. 2

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Figure 12.1 The quincuncial pattern

The first brief description of the map appears in the Report of the Superintendent of the US Coast and Geodetic Survey: Among several forms of projection devised by Assistant Peirce, there is one by which the whole sphere is represented upon repeating squares. The projection, as showing the connection of all parts of the surface, is convenient for meteorological, magnetological, and other purposes. The angular relation of meridians and parallels is exactly preserved; and the distortion of areas is much short of the distortion incident to any other projection for the entire sphere (as quoted in Eisele 1979, p. 148).

Prior to the publication, Peirce also mentions the map in one of his letters to Superintendent Patterson: I have found a graphical method of drawing any projection of the sphere which preserves the angles as soon as the North and South poles are given. The method throws a fine light on the general run of mathematical functions and is chiefly useful for that. … It turns out that the parallels have the same shape as the Coel surfaces about centres of attraction or repulsion inversely proportional to the distance and the meridians have the shape of the lines of force (Ibid., p. 155).

In his letters, Peirce acknowledged that the principal inspiration of the idea of his new projection came from the work of a German mathematician Karl Hermann Schwarz (1843–1921). Schwartz is known for the improvements he made to the proof of the Riemann mapping theorem and his own proof of the fact that a sphere has less surface area than any other body of equal volume. He is also praised for his original studies in complex analysis, the advancements of which ultimately led to the development of techniques for conformal mapping, where points of a flat surface are handled as numbers on the complex plane. According to Schwartz, for any positive integer n, the integral z

z

0 n

1



1  zn



2

dz

transforms the interior of a circle into a polygon with n sides. In a similar manner, a sphere is transformed into a regular polyhedron. Peirce applied Schwartz’s work to cartography to produce a conformal map projection with n = 4. In fact, Peirce’s map represents a transformation of conformal stereographic projection (Figure 12.2) and one of the first maps created with an application of the theory of functions of a complex variable.

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Figure 12.2  Stereographic projection of a sphere from the North Pole in 3D (a) and 2D (b)

The minimal requirement for the conformality of a map (a sign) with the globe (an object) is the consistency of the azimuthal angle between the two (an interpretant), which allows maintaining shapes from the globe to the map locally. The principal effect of the conformality is that, although the scale is unstable and varies throughout the entire surface of the map, it is the same in all directions at any particular point of the map so that small areas on it are always represented by a correct

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shape. Quincuncial mapping is similar to other conformal projections in that it preserves the angles at which curves cross each other on the globe and thus favors correct representation of the shapes on the globe over isometry. It also shares with standard stereographic projection another important feature; namely, one of the poles in this map is situated in the center, while the other one is at infinity, being distributed in an infinite number of points along the perimeter of the map (as shown on Figures 12.2 and 12.3). At the same time, Peirce’s map deviates from the standard version of stereographic projection in two respects. First, its segments are arranged not in a circle (Figure 12.2), but in a square (Figure 12.3), hence the quincuncial pattern, with one pole in the center and another distributed between the four corners of the square. Second, it is a projection not of an entire sphere from the north pole onto a plane below the sphere (as shown in Figure 12.2a), but of two hemi-spheres, where one occupies the central position in a square and the other is split into four triangles surrounding the equator, which is represented on the map as the perimeter of the smaller square inscribed in the bigger one (Figure 12.3). This, again, reproduces the quincuncial pattern. Put slightly differently, the entire surface of the map amounts to four squares overlain by a fifth in the form of a diamond. This preserves the

Figure 12.3  Peirce’s quincuncial map: The Earth’s surface conformally in a square

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angles between the curves throughout the entire face of the map except at the corners of the centrally inscribed square. These are points a, b, c, and d (Figure 12.3), which are the right-angle turns of what should be a smooth curve of the equator but appears on the map as a perimeter of this same centrally inscribed square. At these four points (and at these points only), the conformality of Peirce’s map with the globe fails. These are also the points, the area around which is characterized by the maximal scale distortion. The map can also be tessellated by iteration of its parts, with each copy’s features exactly matching those of all its immediate neighbors (Figure 12.4). And the tessellation may be continued in any direction ad  infinitum, until we have every part and every point of the imaginary sphere connected with every other part and point (Figure 12.5). The continuity of tessellation is a principal feature of this kind of projection. Due to the possibility of endless tessellation, the map has no edges. It is like a wallpaper, endlessly reproducing a pattern—only the pattern it reproduces is supposed to cover (and ideally match) a spherical surface. And because the map has no edges, we can

Figure 12.4  Peirce’s quincuncial map tessellated

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Figure 12.5  Peirce’s quincuncial map fully tessellated

lay a continuous route of any length on it—which is not the case with many other projections. In the case of other maps, having this feature always comes at a price. Consider, for instance, the Mercator map, another conformal projection with which an average person is most familiar. It is perfectly capable of showing the entire globe repeatedly in recurring strips so that the surface of the map reappears after a set of iterations like a sequence in a film rolling from one reel to another (Figure 12.6). But in preserving this feature, Mercator maps inflate the size of objects away from the equator, with the distortion increasing progressively towards the poles and finally reaching infinity at them (that is, the poles are represented as two infinite lines that mark the upper and the lower edges of the map). This distortion makes a continuous representation of the Earth’s surface possible, but it does so at the expense of the scale, and we can move continuously along the map only within the limits of its poles represented by the edges of the map. On the opposite side of the spectrum, we find Fuller’s Dymaxion map (Figure 12.7). Just like Peirce’s map, it has a flat surface heavily interrupted in order to preserve shapes and sizes with

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Figure 12.6  Mercator projection

Figure 12.7  Buckminster Fuller’s Dymaxion map

minimal distortion. However, quite unlike Peirce’s map, it does not allow infinite tessellation, i.e., none of its areas can be produced more than once. As will become clear shortly, it is the capacity to keep both of the features mentioned above (preserving shapes and being capable of infinite tessellation) that makes the quincuncial map a perfect diagrammatic interpretant of the Earth’s surface and a virtual picture of the globe—precisely in the sense that Peirce ascribed to these terms. Imagine now that we are moving along Peirce’s tessellated quincuncial map— say, from the North Pole to the North Pole along a meridian line (Figure  12.8). Effectively, we will thereby make a full turn around the imaginary sphere. Meanwhile, laying the same route on the map, we will end up at a different spot. And if we continue moving along the meridian line, we will, as it were, enter another imaginary spherical surface. In fact, depending on the length of the route and on how far we want to go on with the tessellation, the flat surface of the map, if tessellated, will represent an iteration or a certain number of spherical surfaces. The

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Figure 12.8  Peirce’s map as an interpretant of the Earth’s surface

surface of a sphere is continuous in the sense that every point and every part of it is connected with every other point and part, and so the shortest distance between any two points is always possible. The shortest distances between points on Peirce’s map are also possible, but the shortest distances between any and every point of the surface are distributed along different parts of the map through tessellation. A two-­ dimensional quincuncial map reproduces this feature of the three-dimensional globe by iterating all curved segments of it projected onto the plane in such a way that all immediately neighboring areas match. An important proviso here though is that any route that might look like a straight line on Peirce’s map (say, any of the four sides of a smaller square on Figure 12.3, which stands for one-fourth of the length of the equator, or any of the lines representing a Prime Meridian) would still correspond to a curve on a surface of the Earth. It is important to be aware of the fact that any route on the quincuncial map is a curve, even though it is represented as a straight line.

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It is remarkable that, in laying out the foundations of EG, Peirce provides comparisons (both direct and indirect) of the sheet of assertion with a map, based on the idea of continuous tessellation. This comparison allows us to see quite clearly an important semiotic analogy between Peirce’s early work as a geodesist and a cartographer, on the one hand, and his studies as a logician and a mathematician, on the other hand. The analogy, again, is based on the idea of continuity. Thus, when describing the Gamma part of the Graphs, Peirce first calls the sheet of assertions a photograph, but then changes his mind and, “so as not to overwhelm [the reader] with all the difficulties of the conception at once” calls it “a map of such a photograph” (CP 4:513). He then goes on to describe maps in general as representations of connection between all the points of one surface and corresponding points of another surface in such a manner as to preserve the continuity of the former unbroken, “however great may be the distortion” (Ibid.). Given this description, Peirce sees the quincuncial principle of projection as the best representation of the sort, given that maps created according to this principle: …show the whole earth over and over again in checkers, and there is no arrangement you can think of in which the different representations of the same place might not appear on a perfectly correct map. This accounts for our being able to scribe the same graph as many times as we please on any vacant places we like. Now each of the areas of any cut corresponds exactly to some locus of the sheet of assertion where there is mapped, though undeveloped, the real state of things which the graph of that area denies. In fact it is represented by that line of the sheet of assertion which the cut itself marks (Ibid.).

Now, even a brief glance at the map will reveal right away that just one segment is not enough to show the interconnection of all points and parts the way this interconnection is established on the globe. A single segment of the map depicts the full surface of the globe, i.e., it shows, in accordance with a law of projection appropriate to the case, every point of the sphere (Figure 12.3). A single segment of the map is thus a picture of the Earth’s surface, but it is neither a diagram nor a proper sign of it, in the sense Peirce attached to the terms. A single segment translates some aspects of simple visual resemblance that allow us to recognize some elements of the map as corresponding to the Earth’s surface based on what we know about how this surface should look like. Yet it does not truly represent the Earth’s surface. To become not just a picture but also a true representation of the globe, a two-­ dimensional map should demonstrate an essential feature of the globe qua three-­ dimensional object; namely, the interconnectedness of all its points and parts, or the true continuity of its surface. In order to do that, a projection of the Earth’s surface should be capable of both preserving the shapes representing the corresponding objects and features of the surface and being capable of infinite tessellation of the entire surface. In summary, Peirce’s map cannot represent the whole surface of a sphere in any one of its fragments, but it represents a number of such surfaces as a sum of those fragments arranged in a set of interconnected quincunx patterns, in order to properly represent the spherical continuity. It is easy to see now that the role tessellation plays in the constitution of Peirce’s map is analogous to the role his definition of the community of inquiry plays both in his semiotics and in his pragmatism. Recall that,

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as noted earlier in Chapter 2, Peirce defines this community in his “Some Consequences of Four Incapacities” (1868) as “a community without definite limits, and capable of indefinite increase of knowledge” (W2: 239). Also recall that, in Chapter 3, we discussed Peirce’s interpretant as the last step in bringing the manifold of impressions to the unity of a proposition. An interpretant tells us that, although the end of our inquiry is unachievable, we need it as an ideal in order to go on, because the reference to possible future interpretations is necessary for grasping the ideas we share as ours. By analogy, as the tessellation of the quincuncial map can go on ad infinitum, the total sum of the fragments can never be accounted for, and yet the very idea of endless tessellation still serves as a necessary condition for a proper construction of the map so that it could represent spherical continuity. In Chapter 6, we used an example with a map of an island which, if laid down anywhere upon the ground of the island, will contain an infinitely small spot that represents a point on the map as the same point on the island. According to this example, there is always some one point on the map, laid down upon the territory, of which it is a map, which is also a point on the surface of this territory. But this point is immaterial, i.e., it does not represent any particular locus of the island. Now imagine that a map of an island, laid down on the surface of the island, itself has another map located somewhere on its surface, which has, in turn, a map of itself on its surface, etc., ad infinitum. The point connecting the surface of the island with all the maps on it is a purely denotative “this.” It does not represent any territory, and yet it does represent a hypothetical spot in which the map and its object coincide, representing one and the same thing. With this in mind, if the object of a map is a sphere, and if the map can be tessellated so that the spherical continuity is preserved, then, after an infinite number of tessellations, we must be able to identify every point on the map as a point on a sphere. To put it differently, in this case, the whole of a single sphere is correctly represented by an infinite continuous plane constructed in accordance with an appropriate projection rule. And just as, according to Peirce’s description of the quincuncial projection, particular fragments of a two-dimensional map add up to form the concept of a sphere through further tessellation of those fragments, according to Peirce’s NL, we grasp the unity of our ideas through their further correlation to the ideas of others. Two-dimensional maps and the three-dimensional objects those maps aim to represent, like polygons and spherical bodies in Edwin Abbot’s novel Flatland: A Romance of Many Dimensions, belong to totally different worlds. Yet Peirce’s two-dimensional map, due to the combination of its conformality and its capacity for the infinite iteration of its parts, preserves the continuity of the spherical surface. Put simply, infinitely continued tessellations two-dimensionally reproduce a travel one can undertake around and around on the surface of the earth. By virtue of tessellation, the map’s fragments represent the true relations between points and parts of the map, thus becoming a true interpretant of the Earth’s surface.

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The Virtual As discussed in the previous chapter, Peirce’s idea of a “proof” for his pragmatism amounts to finding a way to make his pragmatic maxim and his semiotics mutually interpretable. It appears that the quincuncial map projection can be of some help in this respect. Semiotically speaking, the map is a diagram, a skeletal image of its object visualizing the features of the object that are most relevant to the case at hand. Pragmatically speaking, the visual schematism that supports the analogy between the map and its object is grounded not in how good of a picture of the Earth’s surface the map is, but in how the map can be used given the basic properties of the object it represents. What I see on a map depends on what I, pragmatically speaking, expect to happen given the things I can do with the map based on my familiarity with the ways to use it. We suggest that one way to unpack this connection between the (semiotically approached) visuality and the (pragmatically approached) habituality is to invoke Peirce’s idea of the virtual. This idea will clarify another important aspect of the relationship between Peirce’s philosophical theory and his scientific practice and thereby help us better appreciate Peirce’s incessant desire to build a strong connection between the semiotic idea of interpretation and the pragmatic idea of a habit of action. According to Peirce, the term “virtual” was first used in Duns Scotus’ logic (CP6: 372). Although Scotus never gives any explicit definition of it, throughout his works he provides various descriptions of one idea “virtually containing” another. In his Man’s Natural Knowledge of God, Scotus writes: “No object will produce a simple and proper concept of itself and a simple and proper concept of another object unless it contains this second object essentially or virtually” (Wolter 1987, p. 23). Esposito (1999) claims that Scotus’ short descriptions have some genetic and generative aspects. For instance, to use one of Scotus’ own examples, if we regard a sphere, we have to regard a circle, because our mind is under the influence of an intelligible being (sphere) that contains a virtual being (circle) with the power to thrust itself upon the mind and be recognized for the essential nature it is. … Reasoning occurs as a linking process because every thought contains an actual and virtual dimension, the virtual dimension serving as a vehicle or link to further actualization of thought (p. 1).

Peirce was an expert in medieval logic and on numerous occasions confessed that the works of Scotus had a deep influence on his semiotic ideas (CP1: 6–29; CP2: 161–162, 398; CP4: 28, 38, etc.). Therefore, it would not be a long shot to suppose that Peirce’s idea of possible practical effects echoes Scotus’s “further actualization of thought.” Since Scotus, the notion of virtuality underwent several remarkable shifts. Prior to Peirce, it was used by Machiavelli, who defined political life by the interaction of virtù and fortuna, or skill and luck. Nietzsche used the term in his Beyond Good and Evil in order to ridicule the Kantian explanation of the possibility of a priori synthetic judgments by reference to “faculties” or “capacities” and translated the Latin virtus as Vermögen (Nietzsche 2008, I, 11). Peirce gives his

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own, most commonly quoted definition, in an entry on the term “virtual” for Mark Baldwin’s Dictionary of Philosophy and Psychology: A virtual X … is something, not an X, which has the efficiency (virtus) of an X. This is the proper meaning of the word; but (2) it has been seriously confounded with ‘potential,’ which is almost its contrary. For the potential X is of the nature of X, but is without actual efficiency. A virtual velocity is something, not a velocity, but a displacement; equivalent to a velocity in the formula, ‘what is gained in velocity is lost in power.’ (3) Virtual is sometimes used to mean pertaining to virtue in the sense of an ethical habit. (Baldwin 1902: 763).

According to this definition, a sign is considered a virtual unit just in case its meaning ultimately lies not in any sort of abstract idea or a mental copy of its object, but in its object’s conceivable practical applications, or in certain habits that it would produce given the circumstances. Consequently, a belief caused by such a sign always has some pragmatic content and is consistently interpretable as both an intelligible object and a source for action. Given Peirce’s definition of a virtual X as “something, not an X, which has the efficiency (virtus) of an X,” Peirce’s quincuncial map may be considered an example of a two-dimensional picture that virtually is a three-dimensional figure. The two-dimensional map, in this case, possesses a feature that allows us to use it in a way we can use the three-dimensional object it represents. The map is characterized by a continuity specific to its object, that is— has a virtus of it.4

 The fact that it is the combination of tessellation-induced continuity and the capacity to preserve shapes at the expense of isometry that makes Peirce’s map a virtual representation of the globe finds support in the ways the map has been actually applied in practice. Most of the map’s applications have been astronomical. In one instance, it was used to locate the umbral limit lines relative to the surface of the globe for solar and lunar eclipses during certain periods of time. As shown in Taylor and Bell (2013), in this case, tessellation is especially important. Since the umbral limit lines always correspond near to a pole, the use of a tessellated map is crucial, as otherwise some of the lines, as they appear on the map, would be broken. Another use of the projection is for making a map of the night sky as an observer sees it looking up from the ground with the naked eye. In this instance, the map’s shape-preserving capacity is especially important. If we want to show the track of a comet on the map, we have a choice of positioning it away from the areas most distorted in the map—the areas near the four points at each 90° bend in the equator (Taylor and Bell 2013: 14–15). 4

Chapter 13

L’image-Mouvement, Mathematically Sublime, and the Perception of Totality

Synthetic Unity vs. Dynamic Totality As has been discussed in Chapter 3, mathematicians are often reluctant to build bridges over the canyons filled with formal complications. They want neither to examine every element of the logical truss nor to dissect the process of reasoning into its simplest steps. What they want is to discover the fastest and the most efficient way to prove (or discard) their current assumptions. As has been suggested, seen from this perspective, the distinction Peirce aims at here is the one between the mathematical practice of making inferences and the logical theory that has those inferences as objects of study. Of course, mathematical intuition should not be perceived as being simply at odds with the capacity to produce long strings of formal logical proofs. Yet the distinction between the two is salient. For instance, when it comes to a computerized rewrite of a solution to a non-trivial mathematical problem, the absence of intuitive guidance, which initially paved the way to the solution, leads to the exponential growth in the number of possible rewrites that are, at times, too much to handle even for a powerful AI (Kulpa 2009, p. 76). Mathematicians tend to make shortcuts in their demonstrations, and this habit is nothing new. It was just as much in use among mathematicians back in Peirce’s day. As Henri Poincaré once wrote, “If it requires 27 equations to establish that 1 is a number, how many will it require to demonstrate a real theorem?” (2009, p. 178). Yet this distinction still raises a problem. According to Kulpa (2009), the problem, in summary, is this: [M]athematicians usually produce informal proofs using much intuition and informal leaps of imagination, but still maintaining a certain discipline and rigor that convinces them that the result in principle can be formalized if need be. However, it is hard to hear a convincing answer to the question what exactly makes them so sure of that possibility (p. 76).

How are mathematical “intuition and informal leaps of imagination” to be justified? What exactly is the law that makes such leaps legitimate? How do we know that, for whatever solution a mathematician comes out with, it will be possible to © Springer Nature Switzerland AG 2023 V. Kiryushchenko, Diagrams, Visual Imagination, and Continuity in Peirce’s Philosophy of Mathematics, Mathematics in Mind, https://doi.org/10.1007/978-3-031-23245-9_13

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check it by dissecting the process of reasoning into its simplest logical steps? In order to answer these questions in this chapter, we need to revisit Kant’s categories and Peirce’s diagrammatic version of the Kantian schematism of the imagination in light of Gilles Deleuze’s Bergsonian interpretation of continuity. As has already been established, mathematicians approach problem situations by creating relational images whose very construction, along with the transformational grammar implied by the construction, prompts certain changes that lead to the discovery of new relations among the parts of the images. The images they create are sets of skeletonized diagrams in which not just particular moves, but certain general pattern dynamics is anticipated. What experimenting with diagrams thus allows mathematicians to immediately grasp is not lengthy successions of discrete visual objects following one another, but rather something that Gilles Deleuze, in trying to integrate Peirce’s theory of signs into his interpretation of Henry Bergson’s metaphysics of time, describes as l'image-mouvement (Bergson 2007; Deleuze 1986). A movement-image, according to Deleuze, reflects not just a relation between different positions that the objects involved take at different moments but a relation between the general patterns of the behavior those objects reveal and the corresponding qualitative changes of the image as a whole (Deleuze, 1986, pp. 56–70). In commenting on Bergson’s Matter and Memory (1896), Deleuze says that “[when] one constructs a Whole, one assumes that ‘all is given,’ whilst movement only occurs if the whole is neither given nor giveable. As soon as the whole is given to one in the eternal order of forms or poses, or in the set of any-instant-whatevers, … there is no longer room for real movement” (ibid., p. 7). Here, a static, monolithic whole, complete in itself and expressed by Peirce’s category of quality (or firstness) is opposed to a dynamic whole of a law or thirdness (or, to use familiar terminology from Chapter 3, logical abstraction as opposed to mathematical generalization). As has been discussed in Chapters 4, 5 and 6, a quality is complete in itself and self-sufficient whole represented by a one-valence relation. It is such as it is regardless of anything else; a Pythagorean unity of a single whole or a Hegelian unreflective unity of the manifold of whatever is given in a single act of attention. Whatever is perceived, constitutes the whole of the content of the perceiver’s consciousness, which leaves no place for the recognition of the very act of attention. In this case, then, I am what I perceive. This Pythagorean unity is also expressed numerically by 1. We do not define it by relating it to something else, but simply point at whatever is here, now. It is a qualitative abstraction or a mere logical possibility. A law, on the contrary, is a whole consisting of an infinite set of parts. It is present in the general way in which the events that fall under it interact with each other. Just as the idea of a quincuncial map cannot be reduced to a definite number of iterations, the meaning of a law cannot be entirely defined by the total sum of concrete events and interactions that exemplify it. Knowledge of a quality amounts to the awareness of being hic et nunc. Knowledge of a law amounts to being able to predict how things will play out in general; what it represents is esse in futuro. A Deleuzean movement-­ image may be considered a reconciliation of the two. It is an image that cancels the dichotomy between the static whole and changes in its parts because, according to Deleuze, it is actually expressed as a whole by the very continuity of the transformation of its parts. Thus, a movie scene cannot be reduced to a sequence of separately

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described events but is constructed as a whole by the narrative continuity as a progression of anticipations. What Deleuze’s analysis in Cinema implies is that a movement-image is a movement (to use the terminology of Peirce’s maxim, conceived or anticipated movement), enacted by the mathematical imagination. A diagram or a graph, as Peirce saw it, unlike a string of discrete linguistic symbols, presents its meaning as accessible in its totality at any given moment during its transformation. We do not read it linearly, but see it in its entirety, with some stages already in the past, but still controlled in retention (or immediate memory), and some other stages in the future, but already grasped in protention (or immediate anticipation). Likewise, Deleuzean image-movement is an image neither of a final state of affairs nor of a passing instant. It is wholly determined by the Bergsonian durée, in which the past, the present, and the future are grasped together in the dynamic totality of a graph as the continuity of change that takes place in accordance with a certain law or regularity. The parts of this image are in space, but the whole of it is in duration and change. What I think both Peirce and Deleuze have in mind here is Kant’s definition of time as the form of all our sensible intuitions of objects and as a schema that shares the nature of a universal concept and a phenomenon (as discussed in detail in Chapter 9 above). Both Peirce and Deleuze aim to reinterpret the Kantian idea of time as a form of sensibility into the idea of time as the most basic representation of continuity. Although Deleuze, whose philosophy in many respects draws upon historical developments in mathematics from Leibniz to Riemann and Weyl, does not say this directly, it seems obvious that his description of movement-image refers to the idea of a mathematical function (see esp. Deleuze 1968 and Deleuze 1988). Represented in a graph, a mathematical function shows us how certain elements of a Cartesian product are combined in a pattern, and as a result, we know how to go on in interpreting the underlying changes, calculate derivatives, etc. The mathematical idea of totality has already been discussed in some detail with reference to Kant’s table of categories in Chapter 9. As we have seen, Kant associates the triad of the universal concepts related to quantity with the idea of time series. While unity is just a moment in the series, and plurality is a manifold of such moments piling up indefinitely, totality represents all moments of the series—a certain number of them. Totality is thus thought of by Kant to be a result from the combination of unity and plurality achieved by the application of the idea of number, which, according to Kant, serves as the schema of quantity. But while the Peircean/Deleuzean totality is dynamic, the Kantian totality is fixed1. According to Kant, totality is a universal concept that needs a time-related schema to be applied to the manifold of intuition. Kant describes totality as a “plurality considered as unity” (KRV: B 111) and as a mathematical category of quantity. Peirce agrees with

 In “Transcendental Logic” Kant uses the term Allheit (“allness”), which he qualifies in brackets with the Latin universitas. However, the comparison of the Peircean totality and Kant’s allness is justified for two reasons. First, throughout the text of the Critique of Pure Reason, Kant often uses interchangeably Allheit and Totalität (see, e.g. B 111; A 322/B 379). Second, Peirce uses “totality” as a translation of Kant’s Allheit, and, in describing this Kantian category, unequivocally refers to thirdness (see, e.g., CP1: 302 and CP1: 563). 1

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Kant as to the fact that the totality of a logical or a mathematical argument differs both from the unity of the parts of the argument and from the generality of its conclusion. But Peirce’s overall interpretation of totality differs essentially from that of Kant. It is based on the claim that anything general is by definition relational and that a relation is something that can always be represented visually in its dynamics. Therefore, it might be claimed that Peirce’s reinterpretation of the Kantian category of totality is a recap of Peirce’s criticisms of the shortcomings of Kant’s schematism of the imagination. This reinterpretation demonstrates once again that, in developing his diagrammatic logic, Peirce pursues the same goal as in developing his early list of universal categories. NL reinterprets the Kantian schematism of the imagination as a continuous act of reference to the possible future. Using the mathematical idea of relation, Peirce unwinds the Kantian synthetic unity of self-consciousness as an infinite set of interrelated triadic structures mediating between conceptual material and sensible intuitions. In a similar fashion, Peirce’s diagrammatic logic unwinds the Kantian synthetic a priori by providing a link between the external, image-based visuality of perception and the immanent, schematic visuality of thought. Peirce does not deny the role our human capacity of imagination plays in Kant’s architectonic. He simply adds an important proviso, which is that this capacity has to have a set of instruments that help us find a proper connection between vision and intellect. Recall that, according to Peirce, visual perceptions are inferential. Simply seeing something as “red” requires the capacity to apply the concept “red,” and acquiring such a concept involves a long history of piecemeal adjustments and gradually habitualized intakes and responses to various objects in various circumstances. Perception, therefore, requires mastering some inferential skills. Peirce’s mathematical diagrams represent a move in the opposite direction: with the help of simple graphic conventions, they demonstrate how inferences themselves can become a matter of visual perception. On the one hand, therefore, we have perceptual images supported by inferential ties that hold together our linguistic competences and through that support our visual integration. On the other hand, vice versa, we have inferences encoded visually. And it is due to the exchange between the inferentially supported imagery of ordinary perception and the diagrammatic imagery of mathematical thought that the schematism-based Kantian synthetic unity becomes the Peircean diagram-based dynamic totality. What an ordinary perception simply relies upon non-consciously, mathematical thought deliberately uses as an instrument of reflection.

The Mathematically Sublime In “Transcendental Analytic,” Kant acknowledges some difficulties with his first triad of mathematical categories: For the combination of the first and second concepts [plurality and unity], in order that the third [totality] may be produced, requires a special act of the understanding, which is not

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identical with that which is exercised in the case of the first and the second. Thus the concept of a number (which belongs to the category of totality) is not always possible simply upon the presence of concepts of plurality and unity (for instance, in the representation of the infinite). … Obviously, … a separate act of the understanding is demanded (KRV: B 111).

In the case of a continuous infinite series, the idea of which bears so much importance for Peirce on many levels, plurality and unity do not immediately result in a representation of number, but require, as Kant puts it, a further mediation by “a separate act of the understanding.” Kant had a fondness for architectonic symmetry, and this problem appears to be a specialized example of a more general problem of the overall relationship between Kant’s three Critiques. This general problem is at the core of Kant’s overall architectonic argument that explains the relationship, and a brief exposition of it is now in order. While this exposition initially is going to briefly take us away from Kant’s philosophy of mathematics, it will ultimately help us see how Peirce’s treatment of mathematical diagrams in the context of the idea of dynamic totality is rooted in Kant’s theory of aesthetic judgment. Apart from this, with reference to the earlier discussion in Chapters 2 and 3, such exposition will also shed more light on Peirce’s distinction between mathematical practice and logical theory and help us better understand how exactly fast mathematical thinking works in tandem with the pedestrianism of logical analysis. In Critique of Pure Reason (and in the “Transcendental Analytic” in particular), Kant’s goal is to show how we appreciate the unity of intellectual form in a sensible variety or what the conditions are that allow us to partition conceptually the manifold of sensual experience. The pure concepts of the understanding (or categories), due to their non-empirical origin, cannot connect immediately to the objects of experience. According to Kant, to subsume sensible intuitions, these concepts must be “schematized” (mediated by the transcendental schemata). Schemata are mediating representations that establish a connection between categories and appearances because they share the natures of both and determine sensible contents by a priori concepts in relation to the continuity of time. In the Critique of Practical Reason (Kant 1997), Kant offers a similar description, this time not of our cognitive capacities, but of our action. Just as the categories of the understanding impose an intellectual form on a variety of sensible data so that one could judge about the world, human reason imposes its own laws on nature so that one could make practical decisions and act in the world in accordance with those laws. In other words, in Kant’s second Critique, reason imposes those laws on nature not theoretically, but practically. In this case, a law determines not our understanding, but our will. Just like the categories of the understanding, our practical maxims are formal. They tell us what to do, but they do not tell us how to apply them. “Thou shalt not kill” is a strong imperative, but as it stands, it does not help us understand why exactly we should always act in accordance with what it prescribes. Because of this, it might seem that, by analogy, something like a schema is needed that could mediate between the universality of our practical maxims and particular effects of our actions. Yet, according to Kant, there cannot be such a schema, and here is why. My will is determined by a set of purposes that may be seen in a hierarchical manner: some of them I pursue for the sake of other purposes, some of those for

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others still, etc. At the very top, there is the highest purpose which is not conditioned by anything else and which commands that all my acts are to aim only at my own virtue as well as happiness for every rational agent. The categorical imperative (the “golden moral rule”), which conditions this highest purpose and which I use as a standard for assessing my own maxims, is an a priori rational principle. The first definition of the imperative is given by Kant in “Groundwork of the Metaphysics of Morals”: “Act only according to that maxim whereby you can at the same time will that it should become a universal law” (Kant 1993 [1785], p. 30). Unlike a hypothetical imperative, which tells me what I need to do in order to attain a certain end, it tells me to always treat other actors as ends, not as means. I apply it simply because I am in possession of rational will, without reference to any particular lower-level purpose that I might have. The imperative is abstracted from all sorts of subject matter of the will in favor of its universal form (I do not treat others as ends for the sake of some external practical purpose). Because of this, whenever, facing a particular practical choice, I question this or that particular resolution to act, I can never answer the question why I have to follow the imperative based on my current experience. As Peirce might probably add to this Kantian line of arguments, all I can do is improve my understanding of the highest end by an infinite approximation (which, in Peirce’s case, is expressed in terms of a statistically interpreted maxim of conduct). At the same time, I also know that the final purpose is achievable in principle. Therefore, the imperative has only a regulative significance, and no schema can effectively mediate between it and my actions in the way it mediates between categories and experience. According to Kant, the gap between the theoretical self of nature (knowledge) and the practical self of freedom (action) is inherent in practical consciousness, which seeks to reconcile the two selves in the imaginative reinterpretation of the schematism of understanding in aesthetic judgment. In §59 of the Critique of Judgment (“Of Beauty as the Symbol of Morality”), Kant claims that aesthetic judgments, just like moral judgments, are characterized by universality. In both cases, the subjective principle of judging is represented as valid for every man. But unlike moral judgments, aesthetic ones are not based on knowledge of any explicitly formulated rule; in Kant’s own words, they are “not cognizable through any universal concept” (Kant 1986, p. 2522). As the oft-quoted saying by Johann Silesius goes, “The rose is without ‘why.’ She blooms because she blooms.” Kant shows that aesthetic judgments nevertheless help the agent to work their knowledge into a finite set of practical habits in appealing to what Kant calls sensus communis, the “communal sense”—a sense, he says, that we are “compelled to postulate […] to account for the agreement of men in their appreciation of beautiful objects” (KU: xxii). Just like the moral agent cannot provide the ultimate reason why this particular action is morally valid, the aesthetic agent cannot rationalize why this particular something appears beautiful to them. In the latter case, however, no explanation is actually needed: no one, says Kant, is able to live in an aesthetically deficient world, so the

 Henceforth, KU followed by paragraph number and then page number.

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changes one makes as an aesthetic agent always conform to the law that one cannot formalize. I need no more justification to call a rose beautiful that I do to call Euler’s identity beautiful. They just are. As Hannah Ginsborg summarises it nicely, aesthetic judgment: …can function independently of the understanding in situations where the relevant rules or concepts are not already specified. To exercise judgment in this independent way is to judge particulars to be contained under rules or concepts which are, so to speak, not already in understanding, but rather made possible by those acts of judging, themselves. (Ginsborg 2011: 253).

Accordingly, although Kant’s second and third Critiques make two different cases for our capacity to judge, they also represent two series of arguments, each forming an integral part of his overall architectonic: aware of the categorical imperative, I cannot work this awareness into a finite set of practical habits (it is possible that, as a practical agent, I act contrary to my moral beliefs); and conversely, not aware of the law of aesthetic judgment, I always have an idea of how to rearrange my setting (i.e., as an aesthetic agent, I always act on my beliefs). In other words, because the normative force of the principles of practical reason may not always work in the same way as the normative force of logical forms, Kant offers a reinterpretation of logically structured knowledge into practically sound conduct by an aesthetic idea that confers meaning to moral actions. It is the power of aesthetic judgment, according to Kant, that provides the possibility to correlate an object of freedom with an object of nature, thus rendering the world surrounding us a symbol of the moral, i.e., as something that converts my “not understanding why” into a symbolic experience of freedom. In the Critique of Judgment, Kant introduces two basic types of aesthetic experience: the beautiful and the sublime. An experience of the beautiful concerns the form of an object. The form of a beautiful object is predetermined for our power of judgment, and our aesthetic delight results from the harmony between the category-­ constrained lawfulness of our understanding and the free play of our imagination. When I say, “This is a beautiful sunset,” these two capacities agree with one another, and the harmony between them pleases me. An experience of the sublime, on the contrary, is formless; its object is ill-matched and does violence to my imagination. In considering the sublime mathematically, Kant ties it directly to the ideas of magnitude and infinite quantity. The mathematically sublime characterizes our encounter with something overwhelming in size; we experience it when we are confronted with something so large that it entirely overwhelms the capacity of our imagination to comprehend it. Among Kant’s examples in this case are high mountains, the ocean, and colossal buildings. When, for instance, a spectator for the first time has a chance to enter St. Peter’s Basilica in Rome and stand before its majestic proportions, he tries to comprehend the massive size of what surrounds him. But the attempt of his imagination to grasp the interior of the basilica as a whole fails, and the only result, Kant says, is perplexity at this failure and realization of the inadequacy of his aesthetic comprehension. As Kant puts it, “[h]e has the feeling that his imagination is inadequate for exhibiting the idea of a whole, [a feeling] in which

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imagination reaches its maximum, and as it strives to expand that maximum, it sinks back into itself, but consequently comes to feel a liking [that amounts to an] emotion” (KU: §26, 108). Kant further adds that: [i]n the logical estimation of magnitude, the impossibility of ever arriving at absolute totality by measuring the things in the world of sense progressively, in time and space, was cognized as objective, as an impossibility of thinking the infinite as given, and not as merely subjective, as an inability to take it in. For there we are not at all concerned with the degree of the comprehension in one intuition, [to be used] as a measure, but everything hinges on a numerical concept. In an aesthetic estimation of magnitude, on the other hand, the numerical concept must drop out or be changed, and nothing is purposive for this estimation except the imagination’s comprehension to form a unity (KU: §27, 116–117).

What Kant means here is that we can always express certain magnitude algebraically (we can always measure things like the walls of St. Peter’s basilica or, to take another Kant’s example, the Earth’s diameter), but when the sublime is at play, imagination cannot comprehend the measured magnitude in one intuition, and, as a consequence, “the numerical concept must drop out” (ibid.). At the same time, while our imagination strives to progress toward infinity, our reason always demands the absolute comprehension in one intuition even for the objects we can never so comprehend. Due to this demand, “comprehending a multiplicity in a unity … and hence comprehending in one instant what is apprehended successively, is a regression that in turn cancels the condition of time in the imagination’s progression and makes simultaneity intuitable” (KU: §27, 116). Mathematically sublime thus marks our inability to grasp an object in its totality and, at the same time, “cancels the condition of time in the imagination’s progression” or, in other words, makes us realize our capacity to think about the infinite presented in the object as given. Even though we fail to cognize the totality of an object in intuition, our reason insists that we ought to seek it as the end of a (potentially infinite) series. Because of the mismatch between our failed imagination and our reason’s demand for totality, our mind is attracted by the object and, at the same time, repelled by it as well. This inadequacy between the feeling of being confined within certain bounds and the idea of the unattainable absolute whole brings about the feeling of elevation, which Kant characterizes as a negative liking, or “a pleasure that is possible only by means of a displeasure” (KU: §27, 117). This feeling, in turn, according to Kant, makes us realize that our capacity of thinking infinite continuous series as a whole is the evidence of “a mental power that surpasses any standard of sense” (KU: §26, 111). When we call something large in every respect and beyond any comparison, “we do not permit a standard adequate to it to be sought outside it, but only within it. It is a magnitude that is equal only to itself. The sublime, therefore, must not be sought in things of nature, but must be sought solely in our ideas” (KU: §25, 105). Granting this, whenever we characterize as sublime any sensible object, be it the Egyptian Pyramids or Mount Everest, as a matter of fact, we refer only to certain ideas of our own reason, and our sensibility being overwhelmed by those objects only leads our minds to those ideas (KU: §27, 114).

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The Architectonic Role of Mathematics Let us summarize what has been achieved in this chapter. According to Kant, the relationship between reason, understanding, and imagination in aesthetic judgment helps us close the gap between theoretical knowledge and action-guiding competence—or, in Kant’s own terms, between category-constrained experience (as described in the Critique of Pure Reason) and practical laws (as described in the Critique of Practical Reason). In the judgment of the mathematically sublime, the schematism of the imagination is replaced with the idea of the totality of our comprehension of an object—the idea which the imagination fails to deliver but which reason demands from us all the same. The mathematical representation of an infinite series is thus supplemented by the aesthetic representation of the ultimate whole. In this case, there is no schema (i.e., number) that would determine our sensibility (e.g., as presupposed by our ability to measure objects) in relation to time. However, reason allows us to glance beyond the limits of experience due to our aesthetic bewilderment at sublime objects. Kant claims that what we characterize as sublime “cannot be contained in any sensible form but concerns only ideas of reason” and that even those ideas “cannot be exhibited adequately” but “are aroused and called to mind by this very inadequacy, which can be exhibited in sensibility” (KU: §23, 99). It seems obvious that what Kant says about the sensus communis and the mathematically sublime is somewhat similar to the way Peirce describes his idea of a community of inquiry—at the very least in the sense that both the former and the latter are postulated in order to account for a peculiar kind of agreement, which is not possible to account for in terms of social contract. It is no less obvious, however, that Peirce aims to make a substantial change to Kant’s argument. As we have seen, it is imperative for Peirce to find a way to represent totality not as an a priori concept, but as a sensible form. Kant’s first two Critiques leave a gap between theoretical and practical reason. Likewise, there is a split in Peirce between the idea of infinite time series and a representation of the totality of a given object. However, while Kant resorts to aesthetics, Peirce deliberately stays within the realm of mathematical concepts. According to Peirce, the infinite, through the notion of statistical approximation, serves as a mathematically framed regulative ideal of a future community. This ideal, according to Peirce, expresses the intellectual hope which science necessarily requires in order to account for the fallibility of its results. But the totality of an object that corresponds to this ideal is represented not as a conflict between reason and imagination that results in the aesthetic perplexity of the elevated mind but, diagrammatically, as a dynamic image of anticipated pattern variation. Given Peirce’s expertise in Kant, and with due regard for the crucial role the relationship between logic and ethics plays in Kant’s critical philosophy, Peirce could not have overlooked the solution Kant set forth in terms of aesthetic judgment in the Critique of Judgment. Meanwhile, he never criticized this solution directly. The only exception is a few remarks Peirce makes in his “Basis of Pragmaticism in

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the Normative Sciences” (1906) on the fact that Kant and some other German thinkers, as he put it, “limit it to taste, that is, to the action of the Spieltrieb from which deep and earnest emotion would seem to be excluded” (EP2: 378)3. In the same paper, Peirce also defines aesthetics as a theory aimed at describing the deliberate formation of “a habit of feeling which has grown up under the influence of a course of self-criticisms” (EP2: 378). It has been suggested that Peirce paid little attention to Kant’s third Critique due to his “fixation” on logic and mathematics (Kaag 2005: 517; Anderson 1995: 21). This may well be the case; Peirce does at times describe his passion for logic in terms of addiction and at least once mentions that this passion has something to do with him being “more or less addicted to self-­ observation” (CP2: 186; emphasis added)4. As the current analysis demonstrates, if the fixation was indeed the case, it had some important theoretical implications. If the above explanation of these implications makes sense, it may also shed some light on the fact that the phenomenon of genius, which is the topic of Kant’s third Critique immediately following “Analytic of the Sublime,” interested Peirce preeminently in terms of mathematical statistics. According to Kant, “genius is the innate mental predisposition (ingenium) through which nature gives the rule to art […] From this it may be seen that genius (1) is a talent for producing that for which no definite rule can be given, and not an aptitude in the way of cleverness for what can be learned according to some rule” (KU: §46). In defining genius in this way, Kant is making a connection between the creative powersthat allow to endow nature with meaning and moral value, and the laws that nature itself mechanically obeys. Accordingly, while the rest of us cannot but simply agree about the beauty of something, a genius creates the very object of this agreement. Peirce’s approach to the subject is different. While teaching at Johns Hopkins University in 1879–1883, he created a research group with his students involved in studying the psychology of great men. The group prepared a set of statistical questionnaires that covered ancestry, character traits, physical characteristics, type of intelligence, and social environment of approximately 300 greatest geniuses in all human history. Although the project was never completed, it was unique at the time and differed methodologically from both Galton’s eugenics and Lombroso’s anthropological criminology. As noted in Nathan Houser’s introduction to W4, “Peirce wanted to demonstrate that statistical analysis could be fruitfully applied even in situations where the primary data is impressionistic (based on impressions). This study may have been the first extended application of statistical methods to comparative biography” (W5: lxii)5. According to this group research, the reason that brings forth an “outstanding” personality amounts to a sophisticated combination of facts, circumstances, traits and qualities, each of which is not unique by itself,  For detailed discussions of Peirce’s pragmatism in the context of Kant’s idea of architectonic relevant to the present account, see Atkins 2008; Gava & Stern 2016; Kiryushchenko 2011. 4  See also Peirce’s letter to James quoted in the beginning of Chapter 3 above. 5  For some of the results of this research included in Writings of Charles S. Peirce. A Chronological Edition, see W5: 26–106); as well as Peirce’s paper “The Century’s Great Men in Science” published in the Annual Report of the Smithsonian Institution for the year 1901 (Peirce 1901). 3

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whereas a “genius” is always a result of the uniqueness of the elements that make up the combination. The two cases, taken together, are intended to demonstrate how fortuity operates at two different levels and yields two different kinds of results (an outstanding personality and a genius). Importantly, the research showed some parallels between how fortuity works its way into the character of a genius and how it plays out in the process of scientific discovery. For instance, both Michelangelo’s personality and Faraday’s law of electromagnetic induction may be recorded as a set of statistical facts within the framework of probability theory and thus may be represented as elements of the design that regulates the exchange between order and chaos. In this context, the Peircean conception of dynamic totality proves to be an important milestone in the development of Peirce’s ideas about diagrammaticity. It allows us to connect his diagrammatic reinterpretation of the Kantian schematism of the imagination with several other key topics discussed in this book. In particular, it allows us to see the mathematical relationship between Peirce’s conceptions of statistical approximation and totality as his response to Kant’s purely aesthetic resolution of the conflict between theoretical and practical ideals of reason. It also allows us to establish a somewhat similar architectonic relationship between Peirce’s idea of diagrammatic logic and the statistically interpreted maxim of pragmatism, where the latter can be read as an expression of Peirce’s leisured pedestrianism, while the former—as an expression of the impatient immediacy of mathematical understanding. A synthesis of the two yields a non-contradictory union between the logical understanding as a guidance towards a desirable consistency and completeness of a given theory and the mathematical demand for understanding as an initial meritocratic condition for our very capacity to create it.

Chapter 14

The Metaphysics of Continuity

Reality, Generality, and Continuity In Chapter 6, we discussed Peirce’s realism. It is based on the distinction between law-like thirdnesses expressed in a series of conditional statements and fact-like secondnesses understood as concrete instantiations of the laws related to each other by virtue of brute causal force (CP1: 420). A law is real in the sense that it is always present in the way things interact, but it cannot be defined entirely in terms of those interactions alone. The reality of a law cannot be exhausted by the existence of facts that confirm its truth. A law, if it is truly a law, exceeds all its effects. There is, one might say, a “normative residue” that defines a law even when all the facts that instantiate it are gone. According to Peirce, we can define reality either in pragmatic or in semiotic terms. Pragmatically, the aforementioned “normative excess” finds its expression in our capacity to predict future events based on our idea of the possible effects of our involvement with the world. The excess lies in the way a law virtually affects our decision-making and our actions. Whatever line of action we ultimately decide to adopt is influenced by what we think would happen if we did. In other words, the law is real in the sense that it affects the choices we make presently in enabling us to correct our conduct by virtue of appealing to a possible future (the “would-bes”). As far as a law represents a general rule that relates our current state of knowledge to possible future interpretations of this knowledge, it necessarily goes beyond any denumerable collection of facts. A law, therefore, represents an infinite set whose cardinality exceeds that of the set of all current facts. Similarly, the Peircean community of inquiry goes beyond any finite number of actual members of it. Just as a law does not exist in the facts that represent it, but only through its virtual effects, the community, as a regulative ideal, does not exist in concreto, but only through our appeal to the ultimate future “We.” Just as a law is virtually (not actually) present in

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the predictive powers of the mind, the community of inquiry, as an ultimate ideal, is virtually (not actually) present every time we are involved in interpretation of signs. Semiotically, the normative excess that makes a law a law finds its expression in sign’s interpret-ability, or its capacity to establish a connection with some other sign. According to Peirce, the link between the actuality of facts and the reality of laws is provided by thirdnesses, or triadic relations. These relations are capable of expressing generality graphically. Moreover, as discussed earlier in Chapter 5, genuine (non-degenerate) triadicity, according to Peirce, has something to do with making the process of interpretation continuous and, in virtue of this, as both Peirce’s NL and his logic of relatives show, enables this process to result in the production of concepts and general ideas. The special power of mathematical diagrams in particular, Peirce believed, consists in their being capable to show not just a continuity, but a necessary connection between two general ideas by visualizing the very process of deductive inference from one to another. Without this visually available link between facts and laws, all that a given number of dyadic relations represents is just a collection of individual facts, with no normative bond between them. From a semiotic point of view, no number of dyadic relations can yield a genuine triad— and, by extension, can yield no genuine continuity and normativity. Therefore, no finite number of dyadic relations can either represent a true continuum or result in the formation of a concept. Peirce thus saw a tight link between reality, generality, and continuity. As he claims in the last of his 1898 Cambridge conference lectures, “continuity is shown by the logic of relations to be nothing but a higher type of that which we know as generality. It is relational generality” (CP6: 190; emphasis added). In other words, continuity, according to Peirce, is what generality becomes in his logic of relations through various ways of visualizing genuine triadicity. Visualization is what creates a link between generality and continuity. In the Dictionary of Philosophy and Psychology (1902), Peirce attempts to explain the connection between the two ideas using the distinction between a collection and a set: A collection, or system, is an abstraction or abstract ens. … If we conceive an object to be a collective whole, but to be so in such a way that it has no part which is not itself a collective whole in the same way, then, if the collection is of the nature of a sorite, it is a general, whose parts are distinguished merely as having additional characters; but if the collection is a set, whose members have other relations to one another, it is a continuum. The logic of continua is the most important branch of the logic of relatives, and mathematics, especially geometrical topic, or topical geometry, has its development retarded from the lack of a developed logic of continua (CP3: 642).

Applying this distinction to the topic of our discussion in Chapter 5, a collective whole, in Peirce’s terminology, represents a case of a degenerate triadicity, where members of a collection “are distinguished merely as having additional characters” (ibid.). A collection, in other words, is merely a sum of its parts. Taking an example from Peirce’s “Fixation of Belief” (Chapter 4), one might claim that “collection” properly describes the outcome of the application of the second method of fixing beliefs—that of authority—and thus refers to “the social” in general. A social order,

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as Peirce implies in FB, signifies the world of relations which, although already formalized, need to be constantly reconfirmed over time. Accordingly, what we do as social creatures differs essentially from the innovative modes of conduct that, according to Peirce the meritocrat, we adopt as scientists. The scientific vs. the social, Peirce says, is the rigorous logic and openness to criticism vs. conformism and conventionality, and a free, unimpeded communal effort vs. the lack of creativity and personal weakness. The members of a set, unlike the members of a collection, in addition have “other relations to one another” (ibid.) which differ from the one that makes them into a collection. In his 1889 entry on continuum for the Century Dictionary, Peirce brings forth the following geometrical clarification: A true continuum is something whose possibilities of determination no multitude of individuals can exhaust. Thus, no collection of points placed upon a truly continuous line can fill the line so as to leave no room for others, although that collection had a point for every value towards which numbers, endlessly continued into the decimal places, could approximate; nor if it contained a point for every possible permutation of all such values (CP6: 170).

Just as no collection of individual facts truly represents a law that can explain those facts, no collection of points represents the true continuity of a line on which those points can be marked. A general idea (e.g., the law of gravitation), with reference to the discrete representations of its expected effects (e.g., objects in a free fall), is a continuum, just as a line, with reference to the discrete marks on it, is a continuum. Peirce notes that, in some sense, it is possible to talk about a certain length as a spatial arrangement of points, if only as something a pointer necessarily has to cross when passing from the inside to the outside of the curve of, say, an oval. However, even in this context, a reference to all the points on the curve that might be crossed, just as a reference to all members of the community of inquiry, would be too vague to have any meaning (CP6: 165). A reference to all members of the community of inquiry would make no sense because this would mean the actual (i.e., not purely regulative) end of the process of interpretation and, consequently, the actual existence of signs that lack interpret-ability. The analogy between a general idea and the continuum of a geometrical line is a corollary of Peirce’s claim that continuity cannot be approached analytically. Two paragraphs earlier in the same entry for the Century Dictionary quoted above, Peirce gives an important comment on the disadvantages of understanding the continuity of a number line as a collection of points. This understanding, he says, leads us to the idea of infinite divisibility, while the real definition should imply that a truly continuous line contains no points. In this case, if we really need to use the term, “point” should refer not to something individual, but rather to a mere possible place, which is indivisible, which does not exist unless there is something there to mark it as such, and which by this very marking interrupts the continuity. He goes on to criticize the analytic conception of continuity and introduce the Kantian idea of it, with which he generally agrees: I certainly think that on any line whatever, on the common sense idea, there is room for any multitude of points however great. If so, the analytical continuity of the theory of functions,

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which implies there is but a single point for each distance from the origin, defined by a quantity expressible to indefinitely close approximation by a decimal carried out to an indefinitely great number of places, is certainly not the continuity of common sense, since the whole multitude of such quantities is only the first abnumeral multitude, and there is an infinite series of higher grades. On the whole, therefore, I think we must say that continuity is the relation of the parts of an unbroken space or time. … [A] continuum, where it is continuous and unbroken, contains no definite parts; that its parts are created in the act of defining them and the precise definition of them breaks the continuity. In the calculus and theory of functions it is assumed that between any two rational points (or points at distances along the line expressed by rational fractions) there are rational points and that further for every convergent series of such fractions (such as 3.1, 3.14, 3.141, 3.1415, 3.14159, etc.) there is just one limiting point; and such a collection of points is called continuous. But this does not seem to be the common sense idea of continuity. It is only a collection of independent points. Breaking grains of sand more and more will only make the sand more broken. It will not weld the grains into unbroken continuity (CP 6.168; emphasis added).

In this paragraph, Peirce does several things. First, he uses his doctrine of the categories in order to show the pitfalls of interpreting the real continuity of the number line (or thirdness) as a collection of points understood as mere possibilities (or firstnesses) actualized on the line as discrete marks (or secondnesses) that interrupt the line. Second, he points at the fact that Kant’s definition of continuity as “the property of magnitudes by which no part of them is the smallest possible, that is, by which no part is simple” (KRV: A 170/B 212) does not necessarily involve infinite divisibility. The notion of infinite divisibility represents the analytical approach that, Peirce believed, betrays the very idea of continuity. Third, Peirce insists that we should understand continuity as “the relation of the parts of an unbroken space or time” (CP 6.168). The goal now is to understand what Peirce means by this definition. One way to begin is to determine what is this analytic approach that is based on the conception of continuity Peirce is arguing against.

The Analytic Approach to Continuity: Cantor and Dedekind One example of the analytic approach at issue is represented by the idea of continuity introduced by Georg Cantor (1845–1918). According to Cantor, each point on a line is supposed to mark a real number as a coordinate equal to that number. To define a point on a line (and the corresponding real number), Cantor describes the distance between the points as a limit of a sequence, or an infinitely decreasing value. In addition to that, he introduces the concepts of a point set, a condensation point of a point set, and a neighborhood. These three ideas, according to Cantor, are interrelated. A neighborhood is defined as any interval on the line that contains a condensation point. A condensation point, in turn, is defined as such a point that every neighborhood, of which it is a part, contains uncountable many points of a point set. The set of all such condensation points, each of which contains an uncountable infinity of points, is further defined as the one whose cardinality is that of a

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continuum. This allowed Cantor to establish an axiomatic connection between points on a line and real numbers and, at the same time, to reconcile the essential discreteness of a point and a continuity of a line―or, in Kantian terms, to reconcile our spatial (geometric) and temporal (arithmetic) intuitions. This approach to continuity also helped Cantor formulate the so-called “diagonalization argument” that later played a significant role in the development of set theory. The argument shows that the set of real numbers is uncountable, i.e., that real numbers cannot be organized in a sequence that can be put into a list (cannot be counted). Suppose we make a list of the following infinite fractions, which would reflect the one-to-one correspondence of those fractions to natural numbers:

1  x1  0.2 56143



2  x 2  0.68 4517



3  x 3  0.876 431



4  x 4  0.600010 



5  x 5  0.925418 6  x 6  0.42600 7 



 



n  x n  0.b1 b 2 b3 b 4 b 5 b6  bn

Now, let us construct a number, each of whose consecutive decimal places equals the corresponding decimal place in x1, x2, x3, x4 … xn, plus 1. In this way, none of the decimal places of this number could appear in the same numerical position as those of the corresponding positions of the numbers in the list. Obviously, then, 0.397128 is a number that cannot occur in the list. Even if we include this number in the list, we can still construct, in the exact same diagonal way, another number that, again, will not occur in the list, etc. This, according to Cantor, proves that there cannot be a one-to-one correspondence between natural numbers and real numbers and that, therefore, the cardinality of the set of real numbers is bigger than the cardinality of the set of natural numbers, even though both sets are infinite. Consequently, the set of real numbers is uncountable and—however paradoxical it may sound—the set of natural numbers and the set of real numbers represent infinities of different sizes. The cardinality of an infinitely enumerable set is smaller than the cardinality of the continuum represented by real numbers. That being established, to determine whether the continuum of real numbers is the true and uninterrupted one, according to Cantor, we need to prove the “continuity hypothesis,” which amounts to the claim

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that there are no infinite sets of real numbers of intermediate size. In other words, we need to prove that there are no infinite sets of real numbers that could be put both into one-to-one correspondence with the natural numbers and into one-to-one correspondence with the real numbers. If we proved that such a set does not exist, the reality of the continuity of the number line would become a matter of fact. Cantor was unable to provide a decisive argument in favor of the hypothesis, and it remains to be proved up to this day. Richard Dedekind (1831–1916) reasoned along similar lines. According to Dedekind, as the set ℚ of rational numbers contains gaps at irrational numbers such as π, √2, and the like, in order to preserve the continuity of the number line, we need to fill those gaps by finding a way to represent all irrational numbers on the line. But how do we do that, given that there are infinitely many irrational numbers between any two rational numbers? How can we supplement the multitude of rational numbers in such a way that the resultant range of numbers could be as complete and continuous as a geometrical line? This, according to Dedekind, is done by introducing a “cut.” A cut is a split on the number line, performed with the condition that every number on the left side of the split is smaller than every number on the right side of the split. Whenever such a cut is performed, there is one and only one number that makes the cut. Thus, for x=∛2, the cut may be represented as follows:

Here, the horizontal line represents the set of rational numbers, and the vertical line marks the cut representing the unique irrational number for the cube root of two. According to Dedekind, the set of all such cuts is the set of all real numbers (all points on a line), which represents the true continuity of a line. In Peirce’s terms, a cut virtually represents a given irrational number, but it cannot precisely locate it on the line as a point: the number is the cut itself. In other words, the idea of a point represents thirdness, while a point on the line represents secondness that instantiates the idea by directing our attention to a place on the line. But the only way to actually locate the point is through a cut, which is firstness, or a simple quality defined as “being smaller than a and bigger than b.” The diagram above is, thus, a sign of a general idea, ∛2, which represents its object by marking on the number line an actual interruption dividing the line into two segments, with only one possible coordinate in-between.

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Continuity and Infinitesimals Even though, as proposed above, it is quite possible to apply Peirce’s semiotic terminology in describing Dedekind’s cut, Peirce’s approach to continuity differs essentially from the analytic treatment of it by both Cantor and Dedekind. The principal point of disagreement between the way Cantor and Dedekind treated continuity and Peirce’s understanding of it is that both Cantor’s and Dedekind’s arguments are based on the arithmetization of it. From the Peircean point of view, whenever we explain continuity through the application of the discrete units, we cannot help seeing it as containing ultimate parts. Recall that, as early as 1868, in “Some Consequences of Four Incapacities” Peirce writes that “no present actual thought (which is a mere feeling) has any meaning, any intellectual value; for this lies not in what is actually thought, but in what this thought might be connected with in representation by subsequent thoughts … At no one instant in my state of mind is there cognition or representation, but in the relation of my states of different instants there is (EP1: 42).” According to Peirce, cognition is not a succession of separate ideas (just as a line is not a succession of points), but a true continuum, and one specific instant in a “state of mind” (however defined) contains no knowledge, but relations between such “states” do. Generality, then, is not a characteristic of an abstract content in one’s individual mind, but is, essentially, a mediating capacity. This mediation is triadic when the relation is the one of a non-degenerate sign to its object through its interpretant. As discussed above, only general ideas can be embodied in such mediation, as normativity that enables interpretation pertains to proper (non-degenerate) triadicity exclusively. In NL, an early paper that lays grounds for Peirce’s later semiotics (Chapter 4), Peirce also shows that mediating representation is a continuous process that leads us from unorganized manifold of sense impressions to concepts. Discreteness and particularity belong to the level of secondness, where neither continuity nor generality or normativity applies. It is important that in connecting his mathematical ideas about continuity to his wider philosophical and scientific views, Peirce resuscitates the notion of infinitesimals. The latter was widely abandoned in philosophy by the time it was first mentioned in Peirce’s writings. In mathematics, it was used to explain the continuity of a function. The view was that a function is continuous when infinitesimal changes in the value of its argument bring about similar changes in the value of the function. It prevailed in mathematics until the beginning of the nineteenth century, albeit Newton, in the third edition of his Principia, already used the concept of limit instead. In a draft of a lecture that was never delivered (c. 1897), Peirce explains that, on the view he propounds, every “continuous expanse” (i.e., any example of continuous growth, a function, a number line, etc.) contains continuous lines that are infinitely short. The property that distinguishes such infinitesimal distances is that the way we are accustomed to think about finite quantities is not applicable in their case. Peirce offers an example:

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Namely, mark any point on the line A. Suppose that point to have any character; suppose, for instance, it is blue. Now suppose we lay down the rule that every point within an inch of a blue point shall be painted blue. Obviously, the consequence will be that the whole line will have to be blue. But this reasoning does not hold good of infinitesimal distances. After the point A has been painted blue, the rule that every point infinitesimally near to a blue point shall be painted blue will not necessarily result in making the whole blue. Continuity involves infinity in the strictest sense, and infinity even in a less strict sense goes beyond the possibility of direct experience (CP1: 166; emphasis added).

In another piece, dated the same year as the one above and called “On Multitudes,” Peirce gives another example: Across a line a collection of blades may come down simultaneously, and so long as the collection of blades is not so great that they merge into one another, owing to their supermultitude, they will cut the line up into as great a collection of pieces each of which will be a line, —just as completely a line as was the whole. This I say is the intuitional idea of a line with which the synthetic geometer really works, –his virtual hypothesis, whether he recognizes it or not; and I appeal to the scholars of this institution where geometry flourishes as all the world knows, to cast aside all analytical theories about lines, and looking at the matter from a synthetical point of view to make the mental experiment and say whether it is not true that the line refuses to be cut up into points by any discrete multitude of knives, however great. (NEM3(1): 96).

Again, according to Peirce, infinitesimals are not points, but continuous lines infinitely short. And it is those lines that are the ultimate parts of a true continuum—the parts that are, as such, also continua. Just as discrete entities are made up of indivisibles, continua are made of infinitesimals that are themselves divisible. Therefore, we can never understand the nature of continuity either by extrapolating the ways of reasoning applicable to discrete finite quantities nor by explaining continuity analytically, i.e., by beginning with particulars, whether we define those particulars as “cuts” or as “condensation points” on the number line. Following Peirce’s approach, first, we need to abandon the notion that distinct individuals have independent existence; in his view, for something to be continuous is for it not to be precisely defined in all its parts. Second, we must begin, not with particulars, but with a continuum postulated as a synthetic fact. Such facts, according to Peirce, are represented by triadic relations only, as distinguished from monadic and dyadic relations. In using the term “synthetic” in this context, Peirce, of course, refers to Kant’s synthetic unity of apperception. Recall that, according to Kant, the unity of my conscious life cannot be analytically inferred from my knowledge of external facts (see Chapter 4). This unity is a synthetic fact that I need to postulate in order to make sense of my experience. The generality of the idea of such synthetic unity of the self cannot be reduced to a collection of facts. In Peirce’s terms, it should have a virtual element to it, i.e., the capacity to represent, or to make one thing stand for another to something or someone else (see Chapter 12). Thus, instead of the Kantian self-consciousness, Peirce proposes the relation of reason between a sign and the thing signified. This relation requires the regulative idea of a community as the ultimate end of interpretation. According to Kant, time is the foundation of the transcendental schematism, which makes the application of both the mathematical and the dynamical categories

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of the understanding to phenomena possible. Time, therefore, Kant says, is a form of consciousness (see Chapter 9). In NL, Peirce changes Kant’s vocabulary and, instead of time as a form of consciousness, talks about continuity as a form of interpretation, with a community of inquiry as its ultimate end. One way to legitimize this shift in terminology is to show that the idea of infinitesimals, as applied to time, explains its continuity. Thus, for instance, in “The Law of Mind” (1892), Peirce writes: It has already been suggested by psychologists that consciousness necessarily embraces an interval of time. But if a finite time be meant, the opinion is not tenable. If the sensation that precedes the present by half a second were still immediately before me, then, on the same principle, the sensation preceding that would be immediately present, and so on ad infinitum. Now, since there is a time, say a year, at the end of which an idea is no longer ipso facto present, it follows that this is true of any finite interval, however short. But yet consciousness must essentially cover an interval of time; for if it did not, we could gain no knowledge of time, and not merely no veracious cognition of it, but no conception whatever. We are, therefore, forced to say that we are immediately conscious through an infinitesimal interval of time. This is all that is requisite. For, in this infinitesimal interval, not only is consciousness continuous in a subjective sense, that is, considered as a subject or substance having the attribute of duration, but also, because it is immediate consciousness, its object is ipso facto continuous (CP6: 110).

As follows from the above analysis, it is the idea of infinitesimals that helps us explain our knowledge of the past. It allows us to reconcile the fact that any conscious experience has duration and, therefore, presupposes change, with the fact that we are directly aware of the immediate past. The present connects with the past by a series of infinitesimal steps, but the steps themselves belong to the present. Because these steps are infinitely small, in experiencing them, we do perceive the temporal sequence, but, as Peirce puts it, “not … in the way of recognition, for recognition is only of the past, but in the way of immediate feeling” (CP6: 111). Peirce further explains how the mathematically grounded idea of mediation is introduced in this conceptual explanation of continuity: Now upon this interval follows another, whose beginning is the middle of the former, and whose middle is the end of the former. Here, we have an immediate perception of the temporal sequence of its beginning, middle, and end, or say of the second, third, and fourth instants. From these two immediate perceptions, we gain a mediate, or inferential, perception of the relation of all four instants. This mediate perception is objectively, or as to the object represented, spread over the four instants; but subjectively, or as itself the subject of duration, it is completely embraced in the second moment (ibid; emphasis added).

Mediated perceptions are, to some degree, present in mediating ones. Therefore, mediation is woven into the fabric of both our perception of time and, as Peirce shows further in “The Law of Mind,” our conceptual understanding of it as a relation between the past, the present, and the future. The continuum of the present is infinitesimal in duration and contains innumerable infinitesimal parts. As this continuum is, due to its very nature, unlimited, in it “a vague possibility of more than is present is directly felt” (CP6: 138). That opens the possibility for the continuum that leads from the past to the present. It is based on causal connection and reflects how things are from one moment to another. The third kind of continuum is the one that

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represents “relational generality” (CP6: 190) and is based on a habit. In “The Law of Mind,” Peirce describes all three by introducing the concept of the insistency of an idea: The insistency of a past idea with reference to the present is a quantity which is less the further back that past idea is, and rises to infinity as the past idea is brought up into coincidence with the present. Here we must make one of those inductive applications of the law of continuity which have produced such great results in all the positive sciences. We must extend the law of insistency into the future. Plainly, the insistency of a future idea with reference to the present is a quantity affected by the minus sign; for it is the present that affects the future, if there be any effect, not the future that affects the present. Accordingly, the curve of insistency is a sort of equilateral hyperbola (CP6: 140).

According to this diagram, just as a present idea is affected by a past one because it is causally connected with it, a future idea is affected by a present one because it is brought into the present by the bond that is established by a habit while this idea is still in futuro. At the same time, as the maxim of pragmatism tells us, my deliberation over the present idea depends on conceivable effects that accepting this idea would bring at some point in the future (Chapter 2). On this view, although Peirce clearly denies it in the passage quoted above, the relationship is the other way around. The future affects the present and thus makes my deliberation self-­corrective. Reference to the possible future thus both makes this future come true and introduces some changes to what my ideas mean to me currently. Now recall that, according to Peirce, the law-like behavior of something defines its real nature and not merely its existence from one point in time to another as a member of some causal chain (Chapter 6). With this view on the reality of laws and general ideas, as opposed to the actuality of individual objects in mind, Peirce’s conception of continuity may serve as one of the representations of Peirce’s philosophical idealism, with mathematics at its basis.

Chapter 15

Conclusion

This study brings together bits of Peirce’s biography that reflect his mathematical cast of mind, some important outcomes of his work as an applied mathematician at the US Coast and Geodetic Survey, his interpretations of Kant’s critical philosophy, and some of the major themes in his philosophy of mathematics pertaining to diagrammatic reasoning. Such admixture of data, as unrelated as they prima facie might seem, is, in fact, more than appropriate to the case. Peirce was a highly original individual and a polymath who applied himself in many areas. By Peirce’s own admission, mathematics played a unifying role in that it served as a keystone holding together different parts of his overall theoretical edifice and, at the same time, provided useful hints as to how some of his idiosyncrasies and personality traits are relevant to his idea of what it means to be a mathematician. He used mathematical statistics to explain the nature of genius and took his own mathematical mindset and his personal aptitude for visual representation to be responsible for certain traits both of his own personality and of his practical scientific accomplishments. With all this in mind, the principal assumption of the current study is that finding patterns of interconnection between life, armchair philosophical speculation, and real scientific practice can help us better understand Peirce the mathematician and tell us something important about the role of vision in mathematical reasoning in general. Accordingly, the current study shows that there is a set of complex correlations between Peirce’s meritocratic stance towards education and his statistical approach to observational errors, his logical pedestrianism and the idea of perfect language, “fast” intellect and mathematical demonstrations, Peirce’s left-handedness and his overall attitude towards everyday social practices, logic and chemistry, the ideas of relational likeness and patterned behavior, geographic maps and natural languages, etc. In Peirce’s case, all these beliefs, personality traits, mathematical ideas, philosophical theories, and scientific achievements turn out to be interconnected into an intricate web by the idea of diagrammatic representation.

© Springer Nature Switzerland AG 2023 V. Kiryushchenko, Diagrams, Visual Imagination, and Continuity in Peirce’s Philosophy of Mathematics, Mathematics in Mind, https://doi.org/10.1007/978-3-031-23245-9_15

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This overarching idea clarifies the explanation of meanings in terms of conceived behavior as it is formulated in the maxim of pragmatism and, at the same time, sheds light on how the mathematical turn of mind can enforce certain psychological predispositions, form social ideals, initiate new theories, and influence life decisions. In thus pointing at a reciprocal relationship between life and theory, this idea expresses the very thrust of Peirce’s pragmatist methodology shaped by the unceasing dialectics of the practical and the ideal, the particular and the universal, words and images, and action and thought. As shown by our analysis of Peirce’s diagrammatic interpretation of Kant’s categories, which form the central part of the current study, a similar reciprocity is observed in the relationship between the external, inference-based imagery of ordinary perception and what we called the immanent visuality (the diagrammatic imagery of mathematical thought). What is at issue in the former case is our inferential capacities that support our visual experience. We have to practically master the inferential relations of an empirical concept to a variety of other descriptive concepts in order to use it properly. We have our vision enabled by inferences. In the case of immanent visuality, vice versa, we have diagrammatic visualizations that do not just support but directly embody our inferences. In applying diagrams as mathematicians, we use the particulars of visual experience in order to cash out our general inferential capacities. We have inferences visualized. An ordinary person is content with the passive exercise of external perception only: We see what we see because we use this particular language and are initiated into these particular customs. Mathematicians can do more than that. Along with passive perception, they make a good use of the active interplay between the external visuality of objects and the immanent visuality of inferences by mapping one onto another. Maps constitute an important theme in Peirce’s mathematical thought. On the one hand, the idea of mapping helps us better understand Peirce’s semiotic notion of iconicity as a similarity between relations. On the other hand, studying maps makes it possible to see parallels between the work of an applied mathematician and the metaphysical speculations of a philosopher. In our chosen example, the quincuncial map projection is a relational image that allows us to use a two-dimensional object as if it was a three-dimensional one. We chose this example as an illustration of the relationship between different modes of signification and what a pragmatist calls a “habit.” It represents a particular case of likeness which consists in visualizing the modus operandi of one object as a sign of another. Throughout the current study, we have seen that diagrams may be understood mathematically as mediators between the general and the particular, images and arbitrary linguistic symbolisms (or algebraic formalisations). Like a Kantian schema, a diagram, according to Peirce, combines some characteristics of a particular image and the generality of a concept. It acts as a meaningful unit of a language and, at the same time, it is based on the most fundamental visual schematisms that underlie our ordinary perception. Diagrammatic mathematical reasoning essentially amounts to using our most basic spatial and temporal intuitions in order to bring the unorganized perceptual manifold under general concepts and, at the same time, to reveal the necessary relations that are wired into our vision. Using diagrams thus

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makes it possible for us to observe that the conclusion of an argument is compelled to be true. But it also makes it possible for us to observe that the conclusion’s being so compelled owes to the very way we, as mathematicians, construct the diagrams expressing the argument. We have also seen that, according to Peirce, diagrammatic approach is useful in explaining many other features of mathematical reasoning, as well as reasoning in general. In particular, this approach provides a link between generality, triadicity, and continuity. The fact that generality can be graphically expressed by a genuine triadic relation suggests that Peirce’s system of graphical logic actually shows the same process of general concept formation that his early “New List” attempts to describe. Both the deduction of the universal concepts in Peirce’s early work and Peirce’s triadic logic have, as their end result, a representation of thought as continuous act of correlation. And this result proves to be in direct contradiction with both Kant’s logic and his philosophy of mathematics, which serves as the starting point for Peirce’s analysis. As has been shown, apart from all that, Peirce’s diagrammatic approach helps us better understand why mathematical reasoning is deductive and yet characterized by novelty, thus enabling a meaningful discussion of the role of generalization and imaginative experimentation in mathematics. We have also learned that what we see when manipulating diagrams in order to arrive at a desired conclusion, is a generalized image of all potential moves, an array of possible transformational patterns that guide us through our reasoning process. The idea of a generalized image then led us to introducing what we called the “dynamic totality” and Bergsonian durée, the two notions that bring together Kant’s definition of time as the most fundamental schema of all concepts, and Peirce’s definition of sign as the most fundamental triadic relation of meaning. This further led us to the introduction to Peirce’s system of existential graphs and ultimately—to his synthetic conception of continuity as opposed to the analytic conceptions of it in Cantor and Dedekind. Due to the fact that diagrams exhibit a number of phenomena not occurring in algebraic formalizations, diagrams enable us to create structures that are more versatile and more efficient in solving mathematical problems than strings of arbitrary language-based symbols. They represent the synthesis of perceptive selectiveness and conceptual generalization, of imagination and passive perception. They bring together the visual grasp of the dynamic totality of an argument and the perception of particular changes in its structure. The means by which we construct diagrams are our means, yet where this construction ultimately leads us is beyond our individual whims. Diagrammatic reasoning is thus the means to introduce into our perception something that, we would normally think, could not be perceived. It shows us that there is certain logic to the arrangement of mathematical diagrams, such that, by following it, we can visualize the very continuity of our thought.

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