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1958.
Foundations of Set Theory.
Vol. 23,
Issue. ,
p.
349.
Article contents
Axiomatic Systems, Conceptual Schemes, and the Consistency of Mathematical Theories
Published online by Cambridge University Press: 14 March 2022
Extract
Lately, an increased interest in formal devices has led to an attempt on the part of some mathematicians to do without those aspects of mathematics which require intuition. One consequence of this movement has been a new conception of pure mathematics as a science of axiomatic systems. According to this conception, there is no reality beyond an axiomatic system which the statements of mathematics are about; the fact that a statement is a theorem in the system is all that is of interest. This is a sort of nominalist position, since the contrary position seems to be committed to a belief in the existence of a domain of mathematical entities, which are just as suspect as the universale commonly discussed in metaphysical treatises. More often it is called formalism.
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Footnotes
1
This paper is a reworking of some material which appeared in my Ph.D. thesis, “On Establishing the Consistency of Systems” (Harvard, 1951). The paper was completed while I was doing research sponsored by the Office of Naval Research.
References
2 However this position is not identical with the position of N. Goodman and W. V. Quine in their “Steps toward a Constructive Nominalism,” Journal of Symbolic Logic, vol. 12 (1947), pp. 105–122. They wish to reject any theory with certain objectionable features, even if it is given in the form of an axiomatic system.
3 E.g., if there are infinitely many numbers, but only finitely many ideas in the mind, how can we say that numbers exist only in the mind?
4 The interest in the question of whether mathematical truth relates to something beyond axiomatic systems is quite widespread among mathematicians and philosophers. I. Copi seems to assume the affirmative and A. R. Turquette the negative in their recent debate. See Copi's “Modern Logic and the Synthetic A Priori,” Journal of Philosophy, vol. XVLI (1949) pp. 243–245; Turquette's “Gödel and the Synthetic A Priori,” ibid., vol. XLVII (1950), pp. 125–129; and Copi's “Gödel and the Synthetic A Priori: a Rejoinder,” ibid., Vol. XLVII (1950), pp. 633–636.
5 The word “mechanically” here is meant to be synonymous with “effectively”.
6 If the rules of inference permit the derivation of all logical consequences of the axioms, inconsistency in the sense indicated will imply that all statements are theorems.
7 This manner of truth specification is quite standard. Cf. A. Tarski, “Der Wahrheitsbegriff in den formalisierten Sprachen,” Studia Philosophica, vol. 1 (1936), pp. 261–405; and, perhaps more interesting as background for this paper, R. Carnap, “Ein Gültigkeitskriterium für die Sätze der klassischen Mathematik,” Monatshefte für Mathematik und Physik, vol. 42 (1935), pp. 163–190.
8 We can assume that an inconsistent conceptual scheme, in this sense, will specify all statements as both true and false, since the ordinary meaning of logical words is adhered to.
9 The paradoxes of set theory, which were discovered around 1900, could be thought of as contradictions discovered in a sort of conceptual scheme for what has often been called (since 1900) naive set theory. Some mathematicians were disturbed by the paradoxes, but as far as I can see there is no record of any systematic account of such a conceptual scheme. Moreover, Cantor, who in important respects was the founder of set theory, was not shocked by the paradoxes, and even claimed to have an explanation for them. (See Georg Cantor Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, réd. par Ernst Zermelo, Berlin, 1932, pp. 443–450, and p. 470.)
10 K. Gödel argues a similar point on pp. 151, 152 in his article “Russell's Mathematical Logic,” in The Philosophy of Bertrand Russell, Schilpp ed., Chicago and Evanston, 1946. 11 See, e.g., Part IV of The Discourse on Method.
12 This is the heart of the intuitionists' objection to mathematics as based on classical logic, and especially the law of the excluded middle upon which the argument of this paragraph is based.
13 K. Gödel, “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme,” Monatshefte für Mathematik und Physik, vol. 38 (1931) pp. 173–198.
14 Namely, Gödel's theorem, which says, in effect, that no axiomatic system has as theorems all and only all those statements which are true in the conceptual scheme for arithmetic; the same, therefore, holds for any conceptual scheme which is at least as rich as arithmetic. Since all known reasoning processes are capable of being carried out in some axiomatic system or other, it seems that there will always be statements of higher mathematics whose truth or falsity is unknown.
15 ibid.
16 If the system B is proved consistent in a system C, and C in D, etc., eventually this regress must terminate in a system X. Now the Gödel theorem on consistency is sufficient to establish the fact that X cannot be a system so simple as to be obviously consistent.
17 This conclusion could be construed as part of an argument for the position of Goodman and Quine, since they seek to do away with all systems, as well as conceptual schemes, which require an infinite number of entities. (See footnote 2.) To complete the argument, however, one would have to show that the human intuition is in no way capable of grasping an infinity.
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