# Philosophy of Mathematics

Selected Writings

Publication Year: 2010

The philosophy of mathematics plays a vital role in the mature philosophy of Charles S. Peirce. Peirce received rigorous mathematical training from his father and his philosophy carries on in decidedly mathematical and symbolic veins. For Peirce, math was a philosophical tool and many of his most productive ideas rest firmly on the foundation of mathematical principles. This volume collects Peirce's most important writings on the subject, many appearing in print for the first time. Peirce's determination to understand matter, the cosmos, and "the grand design" of the universe remain relevant for contemporary students of science, technology, and symbolic logic.

Published by: Indiana University Press

#### Cover

#### Contents

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pp. v-vi

#### Preface

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pp. vii-xiii

The purpose of this book is to make Peirce’s philosophy of mathematics
more readily available to contemporary workers in the field, and to students of his thought. Peirce’s philosophical writings on mathematics are reasonably well represented—despite the shortcomings detailed in Dauben (1996, 28–39)—in the *Collected Papers*, much more so in *The New Elements of* ...

#### Introduction

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pp. xv-xli

Charles Sanders Peirce’s philosophy of mathematics plays a vital role in his mature philosophical system, and ought to play a more vital role in contemporary philosophical discussions of mathematics. The main business of this introduction will be to flesh out and defend these claims, and to exhibit their interdependence. The interdependence is important: the force of the claims is ...

#### 1. [The Nature of Mathematics] (1895)

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pp. 1-9

*[Peirce 1895(?)b]* Our first selection is an extended discussion of the nature of mathematics. Proceeding on the general principle that the definition of a science should be based on the function its practitioners perform within science as a whole, Peirce identifies as the “distinguishing characteristic of mathematics . . . that it is the scientific study of hypotheses which it first ...

#### 2. The Regenerated Logic (1896)

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pp. 11-14

*[Peirce 1896]* The security of mathematical reasoning, which was raised but not explained in selection 1, receives a much more expansive treatment here. That security manifests itself in the history of mathematics, in the absence of prolonged disagreement over any properly mathematical question. The explanation for that, according to Peirce, is that the objects of mathematics ...

#### 3. The Logic of Mathematics in Relation to Education(1898)

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pp. 15-21

*[Peirce 1898c]* The opening paragraphs of this selection cover the same ground as selection 1, with important variations. Kant receives separate treatment, and his authority is invoked in opposition to the definition of mathematics as the science of quantity, and in support of the diagrammatic nature of mathematical reasoning. The Hamilton/De Morgan definition of ...

#### 4. The Simplest Mathematics (1902)

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pp. 23-36

*[Peirce 1902]*^{1} The last of our selections on the nature of mathematics is the fullest and latest; it comes from a logic text (the Minute Logic) and its date of 1902 makes it roughly contemporaneous with the Harvard Lectures on pragmatism
and the Lowell lectures excerpted in selections 9, 12, and 13. Peirce
varies and enlarges upon themes from the preceding selections: the relationship ...

#### 5. The Essence of Reasoning (1893)

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pp. 37-38

*[Peirce 1893a]* The ideality of mathematics is the center of attention in this brief but very rich excerpt from the *Grand Logic*. Peirce compares arithmetic, a branch of pure mathematics, with the applied fields of logic and geometry. Logic is “intermediate” between the other two: unlike arithmetic,
it is concerned with “questions of fact,” but unlike geometry, it “knows nothing ...

#### 6. New Elements of Geometry (1894)

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pp. 39-41

*[Peirce 1894]* Like the previous selection, this is an excerpt from a textbook: Peirce’s revision of his father’s geometry text, which was rejected by Ginn and Company because “the mathematical philosophy [in it] would meet with hardly any sale” (Brent 1993, 242). Peirce begins by observing that the mathematician’s diagrams have a tendency to take on a life of their own: to ...

#### 7. On the Logic of Quantity (1895)

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pp. 43-56

*[Peirce 1895(?)c]* The manuscript from which this selection is taken contains fragments of more than one version of the same treatise; hence the multiple treatments of diagrams, and of the place of probable reasoning in mathematics. The discussion of diagrams contains, along with material by now familiar, a strong affirmation of their visual character. At the same time ...

#### 8. Sketch of Dichotomic Mathematics (1903)

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pp. 57-66

*[Peirce 1903(?)]* Like selection 4, this text is a philosophical preliminary to a treatment of “the simplest possible mathematics.” It is Peirce’s explanation to the reader of the terminology he will use to organize his presentation. That terminology is taken largely from Euclid’s *Elements*, and Peirce’s explanations are partly a commentary on the traditional vocabulary, partly an exposition ...

#### 9. [Pragmatism and Mathematics] (1903)

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pp. 67-77

*[Peirce 1903c]* Peirce wrote this selection as one of his 1903 Harvard lectures on pragmatism, but did not deliver it. Given his aims and audience, this was probably the right choice; but this is nonetheless one of his richest and most comprehensive treatments of the philosophy of mathematics. The definition of mathematics with which Peirce begins is more or less ...

#### 10. Prolegomena to an Apology for Pragmaticism (1906)

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pp. 79-83

*[Peirce 1906]* Peirce opens this defense of his “pragmaticism” (so named to distinguish it from the pragmatisms of James and others)^{1} with an imaginary debate over the diagrammatic nature of exact thought. He insists very literally on the importance of *experimentation* upon diagrams, and is undeterred by an apparent disanalogy between the diagrammatic experiments of the ...

#### 11. [‘Collection’ in *The Century Dictionary*] (1888–1914)

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pp. 85-88

*[Whitney 1889; Ms 1597; Peirce 1888(?)–1914(?); Whitney 1909]* The original definition of ‘collection’ in *The Century Dictionary* was written, not by Peirce, but by an anonymous contributor; it is reproduced here for reference. Peirce’s critical notes on that definition, written in his own copy of the dictionary, date from some time around the turn of the century; they are followed ...

#### 12. [On Collections and Substantive Possibility] (1903)

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pp. 89-98

*[Peirce 1903g]* Peirce delivered two major series of lectures in 1903: the Harvard Lectures on Pragmatism, and a series at the Lowell Institute on “Some Topics of Logic bearing on Questions now Vexed.” This rejected draft of the third Lowell Lecture begins with a polished review of Peirce’s definition of mathematics, and of the classification of mathematical specialties ...

#### 13. [The Ontology of Collections] (1903)

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pp. 99-105

*[Peirce 1903d]* There is a good deal of overlap between selection 12 and this one, which is taken from the fifth of the Lowell Lectures. The substantial differences, even in the areas of overlap, show that Peirce’s views were evolving as he wrote. A noteworthy difference between the two presentations is that Peirce’s categories play a much more explicit role in this attempt ...

#### 14. The Logic of Quantity (1893)

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pp. 107-112

*[Peirce 1893c]* Kant and Mill, the principal objects of Frege’s critical scrutiny in *The Foundations of Arithmetic*, were of great importance for Peirce as well, and in this selection he locates his epistemology for mathematics relative to theirs.^{1} He takes Kant to task for holding that analytic truths can be ascertained by “a simple mental stare,” and proceeds to rewrite Kant along ...

#### 15. Recreations in Reasoning (1897)

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pp. 113-118

*[Peirce 1897(?)b]* Along with selection 10, this mathematico-philosophical treatment of the natural numbers is one of the strongest pieces of evidence in support of the claim that Peirce held to a kind of mathematical structuralism.^{1} It is also, as discussed in the introduction (xxxix), an important source of information about Peirce’s views on the metaphysics of mathematical ...
structure.
The selection culminates in a Dedekindian axiomatization of the natural ...

#### 16. Topical Geometry (1904)

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pp. 119-127

*[Peirce 1904(?)b]* Peirce took a serious interest in what even in his own day was known as topology: he worked hard on the problem of map coloring, and also on what he held to be an improved formulation of Listing’s Census Theorem.^{1} The title of this text, intended for publication in *Popular Science Monthly*, is one of Peirce’s preferred designations for topology; he introduces ...

#### 17. A Geometrico-Logical Discussion (1906)

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pp. 129-133

*[Peirce 1906(?)]* The ontological status of points is a central, and troublesome, issue in Peirce’s philosophy of continuity. In this late text he attempts to address that issue in the setting of his general account of mathematical objects as *entia rationis*. Peirce introduces this term of art with a near-translation (“creations of thought”) that highlights its mentalistic overtones, but ...

#### 18. [‘Continuity’ in *The Century Dictionary*] (1888–1914)

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pp. 135-139

*[(Whitney 1889), (Peirce 1888(?)–1914(?))]* Peirce’s definition of ‘continuity’ for the *Century Dictionary* was written around 1884; its significance for his own theory of continuity is mainly negative. In his dictionary entry he declares for Cantor’s definition, of which he gives a largely, but not completely, accurate account. Take particular note of Peirce’s use of time as the ...

#### 19. The Law of Mind (1892)

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pp. 141-153

*[Peirce 1892]* In the early 1890s Peirce began to insist on the central importance of continuity in his philosophical system. This selection is taken from one of his first attempts to explain at length what continuity is and why it matters philosophically. The title of the piece highlights the role of continuity in relating ideas in the mind, but this turns out to involve the continuity of ...

#### 20. [Scientific Fallibilism] (1893)

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pp. 155-158

*[Peirce 1893d]* In the summer of 1893 Peirce wrote a lecture on “Scientific Fallibilism,” which has only recently been reconstructed (De Tienne 2001) from a number of scattered manuscripts; one of these, the source of this selection, was excerpted separately in volume 1 of the *Collected Papers* under the editor-supplied title “Fallibilism, Continuity and Evolution.” The ...

#### 21. On Quantity [The Continuity of Time and Space](1896)

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pp. 159-164

*[Peirce 1896(?)]* Around the beginning of 1896 Peirce discovered a version of what has become known as Cantor’s Theorem (for one of his proofs, see selection 22). A corollary of that theorem—that there is no greatest multitude—transformed his theory of the continuum. His quarrel with Cantor over “all the points” comes to take the form of a requirement of what he would ...

#### 22. Detached Ideas Continued and the Dispute between Nominalists and Realists (1898)

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pp. 165-178

*[Peirce 1898a]* Not quite two years after writing selection 21, Peirce is in a position to give a more systematic account of continuity. This excerpt from his Cambridge Conferences Lectures of 1898 contains a very elegant argument for his version of Cantor’s Theorem: for any collection C of distinct elements, the collection S(C) of its subcollections is itself a collection of distinct ...

#### 23. The Logic of Continuity (1898)

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pp. 179-188

*[Peirce 1898b]* The last of Peirce’s Cambridge Conferences Lectures complements the collection-theoretic definition of continuity given in selection 22 with a geometrical analysis based on Listing’s contributions to topology. In proving his Census Theorem, Listing identifies a number of topological invariants, which Peirce uses in the latter part of this lecture to classify such ...

#### 24. [On Multitudes] (1897)

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pp. 189-200

*[Peirce 1897(?)a]* Peirce tries to come to grips, in this mathematically and philosophically abundant manuscript, with some of the difficult questions about continuity left open by the Cambridge Conferences Lectures. Here we find, in the opening summary of his theory, a full exposition of his subtly fallacious “proof” that a collection both supermultitudinous and discrete would ...

#### 25. Infinitesimals (1900)

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pp. 201-205

*[Peirce 1900]* Josiah Royce’s *The World and the Individual* contained a long “Supplementary Essay” (Royce 1899, 473–588) on “The One, the Many and the Infinite,” in which he correctly criticized Peirce for failing to recognize Cantor’s opposition to infinitesimals. In this letter to the editor of *Science*,
Peirce does not offer anything close to an adequate defense against this ...

#### 26. The Bed-Rock beneath Pragmaticism (1905)

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pp. 207-210

*[Peirce 1905a]* In this footnote to an unfinished article for the Monist, Peirce promises, but does not deliver, an improved definition of continuity which begins with a clarification of the part/whole relation. According to his definition of ‘material part’ it makes sense to talk of the material parts of what we would call a time-slice of a jack-knife (a jack-knife at a specific time). A ...

#### 27. [Note and Addendum on Continuity] (1908)

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pp. 211-216

*[Peirce 1908e]* In an important footnote to an installment in his “Amazing Mazes” series for the *Monist*—the first half of this selection—Peirce formulates a potentially fatal objection to his supermultitudinous theory of continuity. The objection has to do with linear orderings. Peirce shows how to impose such orderings, not just on countably infinite collections, but also on ...

#### 28. Addition [on Continuity] (1908)

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pp. 217-219

*[Peirce 1908a]* There are two manuscripts among Peirce’s papers whose content links them closely to selection 27, and we know that the unpublished texts were written two days before the published one, because Peirce very considerately dated all three. He did not go so far as to note times of day, but the apparent progression of thought in the two unpublished manuscripts ...

#### 29. Supplement [on Continuity] (1908)

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pp. 221-225

*[Peirce 1908f]* This is the second of Peirce’s fragmentary attacks on the complex of problems he raises in selection 27. There is much more explicit attention to the part/whole relation in this version, including an elaborate taxonomy of the different kinds of parts, which Peirce never gets around to actually using in a definition of continuity. His general account of a part, as ...

#### Notes

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pp. 227-261

#### Bibliography

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pp. 263-279

#### Index

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pp. 281-290

E-ISBN-13: 9780253004697

E-ISBN-10: 0253004691

Print-ISBN-13: 9780253355638

Page Count: 336

Illustrations: 3 b&w illus.

Publication Year: 2010

Series Title: Selections from the Writings of Charles S. Peirce