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

Inhaltsverzeichnis

1. Frontmatter

2. Chapter 1. Introduction

   Vitaly Kiryushchenko

   Abstract

   Mathematicians apply visual diagrams 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?

3. Chapter 2. Meritocratism, Errors, and The Community of Inquiry

   Vitaly Kiryushchenko

   Abstract

   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:

4. Chapter 3. Logic and Mathematics

   Vitaly Kiryushchenko

   Abstract

   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 general, and his research into the psychology of great men (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” (MHFC Peirce to James, 07/16/1907).

5. Chapter 4. Peirce’s Transcendental Deduction and Beyond: Categories, Community, and the Self

   Vitaly Kiryushchenko

   Abstract

   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 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).

6. Chapter 5. Sign Relation

   Vitaly Kiryushchenko

   Abstract

   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.

7. Chapter 6. One, Two, Three

   Vitaly Kiryushchenko

   Abstract

   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.

8. Chapter 7. Iconicity, Novelty, and Necessity

   Vitaly Kiryushchenko

   Abstract

   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, if not entirely realized, part in our visual integration.

9. Chapter 8. The General and the Particular

   Vitaly Kiryushchenko

   Abstract

   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.

10. Chapter 9. Diagrams Between Images and Schemata

    Vitaly Kiryushchenko

    Abstract

    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.

11. Chapter 10. Existential Graphs

    Vitaly Kiryushchenko

    Abstract

    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 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.

12. Chapter 11. Iconicity, Similarity, and Habitual Action

    Vitaly Kiryushchenko

    Abstract

    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 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.

13. Chapter 12. Mapping Philosophy: Peirce’s Quincuncial Projection

    Vitaly Kiryushchenko

    Abstract

    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:

14. Chapter 13. L’image-Mouvement, Mathematically Sublime, and the Perception of Totality

    Vitaly Kiryushchenko

    Abstract

    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:

15. Chapter 14. The Metaphysics of Continuity

    Vitaly Kiryushchenko

    Abstract

    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.

16. Chapter 15. Conclusion

    Vitaly Kiryushchenko

    Abstract

    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 of his own personality and 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.

17. Backmatter