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Definition of Mathematics

classes, relations, class, entity, sense, proposition, theory, propositional, geometry and conditions

DEFINITION OF MATHEMATICS It has now become apparent that the traditional field of mathe matics in the province of discrete and continuous number can only be separated from the general abstract theory of classes and relations by a wavering and indeterminate line. Of course a dis cussion as to the mere application of a word degenerates into the most fruitless logomachy. But on the assumption that "mathe matics" is to denote a science well marked out by its subject matter and its methods, and that at least it is to include all topics habitually assigned to it, "mathematics" is employed in the general sense' of the "science concerned with the logical deduction of consequences from the general premises of all reasoning." Geometry.—The typical mathematical proposition is: "If x, y, z . . . satisfy such and such conditions, then such and such other conditions hold with respect to them." By taking fixed conditions for the hypothesis of such a proposition a definite department of mathematics is marked out. For example, geometry is such a department. The "axioms" of geometry are the fixed conditions which occur in the hypotheses of the geometrical propositions. The special nature of the "axioms" which constitute geometry is considered in the article GEOMETRY : Axioms. It is sufficient to observe here that they are concerned with special types of classes of classes and of classes of relations, and that the connection of geometry with number and magnitude is in no way an essential part of the foundation of the science.

Classes and Relations.

We now must deduce the general properties of classes and relations from the ultimate logical prem ises. In the course of this process, some contradictions have become apparent. That first discovered is known as Burali-Forti's contradiction,' and consists in the proof that there both is and is not a greatest infinite ordinal number. But these contradictions do not depend upon any theory of number, for Russell's contradic tion' does not involve number in any form. This contradiction arises from considering the class possessing as members all classes which are not members of themselves. Call this class w; then to say that x is a w is equivalent to saying that x is not an x. Accord ingly, to say that w is a w is equivalent to saying that w is not a w. An analogous contradiction can be found for relations. It follows that a careful scrutiny of the very idea of classes and relations is required. Note that classes are here required in extension, so that the class of human beings and the class of rational featherless bipeds are identical ; similarly for relations, which are to be deter mined by the entities related. Now a class in respect to its com ponents is many. In what sense then can it be one? This prob lem of "the one and the many" has been discussed continuously by the All the contradictions can be avoided, and yet the use of classes and relations can be preserved as required by mathematics, and indeed by common sense, by a theory which denies to a class—or relation—existence or being in any sense in which the entities composing it—or related by it—exist. Thus,

to say that a pen is an entity and the class of pens is an entity is merely a play upon the word "entity"; the second sense of "entity" (if any) is indeed derived from the first, but has a more complex signification. Consider an incomplete proposition, incomplete in the sense that some entity which ought to be involved in it is represented by an undetermined x, which may stand for any entity. Call it a propositional function; and, if (/)x be a propositional function, the undetermined variable x is the argu ment. Two propositional functions /x and I,Gx are "extensionally identical" if any determination of x in 4)x which converts ci)x into a true proposition also converts fix into a true proposition, and conversely for and 4. Now consider a propositional tion Fx in which the variable argument x is itself a proposi tional function. If Fx is true when, and only when, x is de termined to be either 4) or some other propositional function extensionally equivalent to 4, then the proposition F(I) is of the form which is ordinarily recognized as being about the class deter mined by 4x taken in extension—that is, the class of entities for which 4)x is a true proposition when x is determined to be any one of them. A similar theory holds for relations which arise from 'The first unqualified explicit statement of part of this definition seems to be by B. Peirce, "Mathematics is the science which draws necessary conclusions" (Linear Associative Algebra, § i. [187o], re published in the Amer. foam. of Math., vol. iv. ). But it will be noticed that the second half of the definition in the text—"from the general premises of all reasoning"—is left unexpressed. The full expression of the idea and its development into a philosophy of mathematics is due to Russell, loc. cit.

questione sui numeri transfiniti,

" Rend. del circolo mat. di Palermo, vol. xi. (1897) and Russell, loc. cit., ch. xxxviii.

'Cf. Russell, loc. cit., ch. x.

Pragmatism: a New Name for Some Old Ways of Thinking (19o7).

the consideration of propositional functions with two or more variable arguments. It is then possible to define by a parallel elaboration what is meant by classes of classes, classes of rela tions, relations between classes and so on. Accordingly, the num ber of a class of relations can be defined, or of a class of classes, and so on. This theory' is in effect a theory of the use of classes and relations, and does not decide the philosophic question as to the sense (if any) in which a class in extension is one entity. It does indeed deny that it is an entity in the sense in which one of its members is an entity. Accordingly, it is a fallacy for any deter mination of x to consider "x is an x" or "x is not an x" as having the meaning of propositions. Note that for any determination of x, "x is an x" and "x is not an x" are neither of them fallacies but are both meaningless, according to this theory. Thus Russell's con tradiction vanishes, and the other contradictions vanish also.