Philosophy of Mathematics: Selected Writings
336
Philosophy of Mathematics: Selected Writings
336eBook
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| ISBN-13: | 9780253004697 |
|---|---|
| Publisher: | Indiana University Press |
| Publication date: | 08/19/2010 |
| Series: | Selections from the Writings of Charles S. Peirce |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 336 |
| File size: | 607 KB |
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Philosophy of Mathematics
Selected Writings
By Charles S. Peirce, Matthew E. Moore
Indiana University Press
Copyright © 2010 Indiana University PressAll rights reserved.
ISBN: 978-0-253-35563-8
CHAPTER 1
[The Nature of Mathematics]
[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 frames and then traces to their consequences." The mathematician is not, however, concerned with whether or not these hypotheses are true — that is a matter for the empirical scientist who makes use of the mathematician's results. A further point of contrast is the mathematician's minimal use of observation: he "observes nothing but the diagrams he himself constructs." This relative independence of observation sets mathematics apart, not just from empirical science, but also from logic and metaphysics, which both rely more heavily on observation than mathematics does. Mathematics is distinguished from other practices (such as poetry) that "frame hypotheses" by its exclusive concern with deducing the consequences of its hypotheses.
Much of the selection is devoted to the criticism of competing definitions, many of them due (or at least heavily indebted) to figures who deeply influenced Peirce himself: Aristotle, Kant and his own father, Benjamin Peirce. Throughout this critical discussion Peirce continues to emphasize the mathematician's indifference to the facts.
He begins with a dismissive treatment of the traditional definition of mathematics as the science of quantity. His rejection of this definition is deeply rooted in his mathematical heritage. The first of the mathematical chapters (pp. 183–193) in Murphey (1961) summarizes the developments in nineteenth century mathematics that did the most to undermine the traditional definition; these are all developments with which Peirce was intimately acquainted, in some cases through his own direct involvement. Algebra, which played a major undermining role, ran in Peirce's family. Benjamin Peirce's Linear Associative Algebra (Peirce 1870) is an important contribution to the field, which opens with a definition of mathematics (see note 10) that greatly influenced his son; that definition opens in turn, as it happens, with a dismissal of the traditional one, just as Peirce's does here (and elsewhere). Peirce himself, of course, is a great figure in the algebraic tradition in logic. Murphey also reviews developments in geometry that cast doubt on the identification of mathematics with the science of quantity; and here, as in other discussions of that definition, Peirce adduces projective geometry as a fatal counterexample. He did not just discard the older definition altogether, however; he takes a more irenic attitude in later writings: see especially selections 3, 13, and 14.
Peirce is equally dismissive of the suggestion, taken up from Kant by De Morgan and Hamilton, that mathematics is the science of space and time. But even here there turns out to be a grain of truth: in the classification of the sciences with which this selection concludes, Peirce assigns space and time to "the most abstract of the special sciences"; so there is after all a close affinity between the sciences of space and time, and mathematics, the most abstract science of all.
The last definition to receive extended treatment is that of Peirce's father Benjamin: "the science which draws necessary conclusions." Peirce argues that it follows from his father's definition that "mathematics must exclusively relate to the substance of hypotheses," but he rejects his father's claim that the framing of hypotheses for mathematical study is a logical and not a mathematical task. He counters by denying "that everybody who reasons skilfully makes an application of logic." This sounds a major theme of Peirce's philosophy of mathematics: the independence of mathematics from logic. As he frequently does in this connection, Peirce notes here that metaphysics, by contrast with mathematics, does depend very closely upon logic. In discussing his father's definition, he focuses narrowly on mathematical hypotheses, whose formulation requires no logic because those hypotheses are not answerable to the facts, and therefore not open to logical criticism. But later in the selection he touches on another, deeper reason for the independence of mathematics: that "in the perspicuous and absolutely cogent reasonings of mathematics ... appeals [to logic] are altogether unnecessary." The ensuing diatribe against publishers and teachers is more an application than an explanation of this dictum. Peirce does drop an important clue when he insists that mathematics is more, not less, abstract than logic; he has hinted at the reasons for this earlier on in the selection, with the remark that "logic rests upon observations of real facts." But this account of the independence of mathematics is incomplete at best: Peirce will do better in selection 2.
The selection ends with the first few levels of a classification of the sciences adapted from Comte. Sciences higher up in the tree are more abstract, and independent of those lower down, though they may "[borrow] data and suggestions from the discoveries" of those below. Mathematics accordingly winds up at the top. It is noteworthy that some kind of observation plays a role at every level.
§2. The nature of mathematics
Art. 2. As a general rule, the value of an exact philosophical definition of a term already in familiar use lies in its bringing out distinct conceptions of the function of objects of the kind defined. In particular, this is true of the definition of an extensive branch of science; and in order to assign the most useful boundaries for such a study, it is requisite to consider what part of the whole work of science has, from the nature of things, to be performed by those men who are to do that part of the work which unquestionably comes within the scope of that study; for it does not conduce to the clearness of a broad view of science to separate problems which have necessarily to be solved by the same men. Now a mathematician is a man whose services are called in when the physicist, or the engineer, or the underwriter, etc. finds himself confronted with an unusually complicated state of relations between facts and is in doubt whether or not this state of things necessarily involves a certain other relation between facts, or wishes to know what relation of a given kind is involved. He states the case to the mathematician. The latter is not at all responsible for the truth of those premises: that he is to accept. The first task before him is to substitute for the intricate, and often confused, mass of facts set before him, an imaginary state of things involving a comparatively orderly system of relations, which, while adhering as closely as possible or desirable to the given premises, shall be within his powers as a mathematician to deal with. This he terms his hypothesis. That work done, he proceeds to show that the relations explicitly affirmed in the hypothesis involve, as a part of any imaginary state of things in which they are embodied, certain other relations not explicitly stated.
Thus, the mathematician is not concerned with real truth, but only studies the substance of hypotheses. This distinguishes his science from every other. Logic and metaphysics make no special observations; but they rest upon observations which have been made by common men. Metaphysics rests upon observations of real objects, while logic rests upon observations of real facts about mental products, such as that, not merely according to some arbitrary hypothesis, but in every possible case, every proposition has a denial, that every proposition concerns some objects of common experience of the deliverer and the interpreter, that it applies to that some idea of familiar elements abstracted from the occasions of its excitation, and that it represents that an occult compulsion not within the deliverer's control unites that idea to those objects. All these are results of common observation, though they are put into scientific and uncommon groupings. But the mathematician observes nothing but the diagrams he himself constructs; and no occult compulsion governs his hypothesis except one from the depths of mind itself.
Thus, the distinguishing characteristic of mathematics is that it is the scientific study of hypotheses which it first frames and then traces to their consequences. Mathematics is either applied or pure. Applied mathematics treats of hypotheses in the forms in which they are first suggested by experience, involving more or less of features which have no bearing upon the forms of deduction of consequences from them. Pure mathematics is the result of afterthought by which these irrelevant features are eliminated.
It cannot be said that all framing of hypotheses is mathematics. For that would not distinguish between the mathematician and the poet. But the mathematician is only interested in hypotheses for the forms of inference from them. As for the poet, although much of the interest of a romance lies in tracing out consequences, yet these consequences themselves are more interesting in point of view of the resulting situations than in the way in which they are deducible. Thus, the poetical interest of a mental creation is in the creation itself, although as a part of this a mathematical interest may enter to a slight extent. Detective stories and the like have an unmistakable mathematical element. But a hypothesis, in so far as it is mathematical, is mere matter for deductive reasoning.
On the other hand, it is an error to make mathematics consist exclusively in the tracing out of necessary consequences. For the framing of the hypothesis of the two-way spread of imaginary quantity, and the hypothesis of Riemann surfaces were certainly mathematical achievements.
Mathematics is, therefore, the study of the substance of hypotheses, or mental creations, with a view to the drawing of necessary conclusions.
Art. 3. Before the above analysis is definitively accepted, it ought to be compared with the principal attempts that have hitherto been made to define mathematics.
Aristotle's definition shows that its author's efforts were, in a general way, rightly directed; for it makes the characteristic of the science to lie in the peculiar quality and degree of abstractness of its objects. But in attempting to specify the character of that abstractness, Aristotle was led into error by his own general philosophy. He makes, too, the serious mistake of supposing metaphysics to be more abstract than mathematics. In that he was wrong, since the former aims at the truth about the real world, which the latter disregards.
The Roman schoolmasters defined the mathematical sciences as the sciences of quanta. This definition would not have been admitted by a Greek geometer; because the Greeks were aware that the more fundamental branch of geometry treated of the intersections of unlimited planes. Still less does it accord with our present notion that as geometrical metrics is but a special problem in geometrical graphics, so geometrical graphics is but a special problem in geometrical topics. The only defence the Romans offered of the definition was that the objects of the four mathematical sciences recognized by them, viz: arithmetic, geometry, astronomy, and music, are things possessing quantity. It does not seem to have occurred to them that the objects of grammar, logic, and rhetoric equally possess quantity, although this ought to have been obvious even to them.
Subsequently, a different meaning was applied to the phrase "mathematics is the science of quantity." It is certainly possible to enlarge the conception of quantity so as to make it include tridimensional space, as imaginary quantity is two-dimensional and quaternions are four-dimensional. In such a way, this definition may be made coextensive with mathematics; but, after all, it does not throw so much light upon the position of mathematics among the sciences as that which is given in the last article.
De Morgan and Sir William Rowan Hamilton, influenced indirectly, as it would seem, by Kantianism, defined mathematics as the science of Time and Space, algebra being supposed to deal with Time as geometry does with Space. Among the objections to this definition, the following seem to be each by itself conclusive.
1st, this definition makes mathematics a positive science, inquiring into matters of fact. For, even if Time and Space are of subjective origin, they are nevertheless objects of which one thing is true and another false.
The science of space is no more a branch of mathematics than is optics. That is to say, just as there are mathematical branches of optics, of which projective geometry is one, but yet optics as a whole is not mathematics, because it is in part an investigation into objective truth, so there is a mathematical branch of the science of space, but this has never been considered to include an inquiry into the true constitution and properties of space. Euclid terms statements of such properties postulates. Now by a postulate the early geometers understood, as a passage in Aristotle shows, notwithstanding a blunder which the Stagyrite here makes, as he often blunders about mathematics, a proposition which was open to doubt but of which no proof was to be attempted. This shows that inquiry into the properties of space was considered to lie outside the province of the mathematician. In the present state of knowledge, systematic inquiry into the true properties of space is called for. It must appeal to astronomical observations on the one hand, to determine the metrical properties of space, to chemical experiment on the other hand to determine the dimensionality of space, and the question of the artiad or perissid character of space, or of its possible topical singularities (suggested by Clifford) remain as yet without any known methods for their investigation. All this may be called Physical Geometry.
2nd, this definition erroneously identifies algebra with the science of Time. For it is an essential character of time that its flow takes place in one sense and not in the reverse sense; while the two directions of real quantity are as precisely alike as the two directions along a line in space. It is true that + 1 squared gives itself while -1 squared gives the negative of itself. But there is another operation precisely as simple which performed upon -1 gives itself, and performed upon +1 gives the negative of itself. Besides, the idea of time essentially involves the notion of reaction between the inward and outward worlds: the future is the domain over which the Will has some power, the past is the domain of the powers which have gone to make Experience. The future and past which are essential parts of the idea of time cannot be otherwise accurately defined. Yet algebra does not treat of Will and Experience.
3rd, this definition leaves no room for some of the chief branches of mathematics, such as the doctrine of A-dimensional space, the theory of imaginaries, the calculus of logic, including probabilities, branches which it would be doing great violence to the natural classification of the sciences to separate from algebra and geometry.
(Continues...)
Excerpted from Philosophy of Mathematics by Charles S. Peirce, Matthew E. Moore. Copyright © 2010 Indiana University Press. Excerpted by permission of Indiana University Press.
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Table of Contents
Preface vii
Introduction xv
1 [The Nature of Mathematics] (1895) 1
2 The Regenerated Logic (1896) 11
3 The Logic of Mathematics in Relation to Education (1898) 15
4 The Simplest Mathematics (1902) 23
5 The Essence of Reasoning (1893) 37
6 New Elements of Geometry (1894) 39
7 On the Logic of Quantity (1895) 43
8 Sketch of Dichotomic Mathematics (1903) 57
9 [Pragmatism and Mathematics] (1903) 67
10 Prolegomena to an Apology for Pragmaticism (1906) 79
11 ['Collection' in The Century Dictionary] (1888-1914) 85
12 [On Collections and Substantive Possibility] (1903) 89
13 [The Ontology of Collections] (1903) 99
14 The Logic of Quantity (1893) 107
15 Recreations in Reasoning (1897) 113
16 Topical Geometry (1904) 119
17 A Geometrico-Logical Discussion (1906) 129
18 ['Continuity' in The Century Dictionary] (1888-1914) 135
19 The Law of Mind (1892) 141
20 [Scientific Fallibilism] (1893) 155
21 On Quantity [The Continuity of Time and Space] (1896) 159
22 Detached Ideas Continued and the Dispute between Nominalists and Realists (1898) 165
23 The Logic of Continuity (1898) 179
24 [On Multitudes] (1897) 189
25 Infinitesimals (1900) 201
26 The Bed-Rock beneath Pragmaticism (1905) 207
27 [Note and Addendum on Continuity] (1908) 211
28 Addition [on Continuity] (1908) 217
29 Supplement [on Continuity] (1908) 221
Notes 227
Bibliography 263
Index 281
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Focuses on the major writings Peirce produced that are of greatest significance for a correct appreciation of his larger philosophical agenda.