On the Foundations of Mathematics

Abstract

Mathematics.-The notion within a given domain of defining the objects of consideration rather by a body of properties than by particular expressions or intuitions is as old a5 mathematics itself. And yet the central importance of the notion appeared only during the last century-in a host of researches on special theories and on the foundations of geometry and analysis. Thus has arisen the general point of view of what may be called abstract mathematics. One comes in touch with the literature very conveniently by the mediation of Peano's Revue des Mathematiques. The Italian school of Peano and the Formulaire Mathematique, published in connection with the Revue, * Presidential address delivered before The American Mathematical Society at its ninth annual meeting, December 29, 1902. This content downloaded from 50.193.31.206 on Sat, 21 Dec 2013 10:21:01 AM All use subject to JSTOR Terms and Conditions [N. S. VOL. XVII. No. 428. are devoted to the codification in Peano's symbolic language of the principal mathematical theories, and to researches on abstract mathematics. General interest in abstract mathematics was aroused by Hil-mathematics was aroused by Hilbert's Gauss-Weber Festschrift of 1899: 'Ueber die Grundlagen der Geometrie,' a memoir rich in results and suggestive in methods; I refer to the reviews by Sommer,* Poincare,t Halsted,j Hedrick? and Veblen. 11 We have as a basal science logic, and as depending upon it the special deductive sciences which involve undefined symbols and whose propositions are not all capable of proof. The symbols denote either classes of 'elements or relations amongst elements. In any such science one may choose in various ways the system of undefined symbols and the system of undemonstrated or primitive propositions, or postulates. Every proposition follows from the postulates by a finite number of logical steps. A careful statement of the fundamental generalities is given by Padoa in a paperT? before the Paris Congress of Philosophy, 1900. Having in mind a definite system of undefined symbols and a definite system of postulates, we have first of all the notion of the compatibility of these postulates; that is, that it is impossible to prove by a finite number of logical steps the simultaneous validity of a statement and its contradictory statement; in the next place, the question of the independence of the postu* Bull. Amer. Math. Soc. (2), vol. 6 (1900), p. 287. t Bull. Sciences Matherm., vol. 26 (1902), p. 249. t The Open Court, September, 1902. ?Bull. Amer. Math. Soc. (2), vol. 9 (1902), p. 158. 11 The Monist, January, 1903. [ 'Essai d'une theorie algebrique des nombres entiers, priced6 d'une Introduction logique a une thdorie deductive quelconque,' Bibliotheque du Congres International de Philosophic, vol. 3, p. 309. lates or the irreducibility of the system of postulates; that is, that no postulate is provable from the remaining postulates. Padoa introduces the notion of the irreducibility of the system of undefined symbols. A system of undefined symbols is said to be reducible if for one of the symbols, X, it is possible to establish, as a logical consequence of the assumption of the validity of the postulates, a nominal or symbolic definition of the form X A, where in the expression A there enter only the undefined symbols distinct from X. For the purpose of practical application, it seems to be desirable to modify the definition so as to call the system -of undefined symbols reducible if there is a nominal definition X -A of one of them X in terms of the others such that in any interpretation of the science the postulates retain their validity when instead of the initial interpretation of the symbol X there is placed the interpretation A of that symbol. If the system of symbols is reducible in the sense of the original definition it is in the sense' of the new definition, but not necessarily conversely, as appears for instance from the following example, occur-ring in the foundations of geometry. Hilbert uses the following undefined symbols: 'point, ' line,' 'plane,' ' incidence' of point and line, 'incidence' of point and plane, 'between,' and 'congruent.' Now it is possible to give for the symbol 'plane' a symbolic definition in terms of the other, undefined symbols-for instance, a plane is a certain class of points (as Peano showed in 1892), or again, a plane is a certain class of lines; while the notion 'incidence' of point and plane receives convenient definition. It is apparent from the fact that these definitions may be given in these two ways that Hilbert's system of undefined symbols is not in Padoa 's sense irreducible, at least, in so far as the symbols 'plane,' 'incidence' of point and plane are 402 SCIENCE. This content downloaded from 50.193.31.206 on Sat, 21 Dec 2013 10:21:01 AM All use subject to JSTOR Terms and Conditions