On the Synthetic Aspect of Mathematics

Abstract

In the most recent edition of Language, Truth and Logic, Professor A. J. Ayer still maintains that pure mathematics is analytic, being in fact merely a vast system of tautology. He is much more confident about this than are most contemporary professional mathematicians who have investigated the foundations of their subject. Following the breakdown of the efforts both of Frege and of Russell and Whitehead to derive pure mathematics from logic, i.e. to prove that the denial of any one proposition of mathematics would necessarily be self-contradictory, Hilbert attempted to prove the more modest thesis that pure mathematics is consistent, i.e. that no two propositions of mathematics can contradict each other; but in 1931 Gödel discovered that even this thesis was undecidable according to the “rules of the game.” As Weyl has recently lamented, “From this history one thing should be clear: we are less certain than ever about the ultimate foundations of (logic and) mathematics.”The sense in which Ayer uses the terms analytic and tautology implies also that in his view the activities of pure mathematicians lead to nothing new. It is true that he remarks that “there is a sense in which analytic propositions do give us new knowledge. They call attention to linguistic usages of which we might otherwise not be conscious, and they reveal unsuspected implications in our assertions and beliefs.” But, he continues, “we can also see that there is a sense in which they may be said to add nothing to our knowledge. For they may be said to tell us what we know already. ” This denial of novelty in mathematics is as typical of contemporary positivism as the prophecy of Comte that the composition of the stars would never be revealed to us and the objections of Mach to the atomic hypothesis were characteristic of nineteenth century positivism. Indeed, one wonders why the term positivism should have been appropriated by successive philosophers whose common outlook could be so much more fittingly described as negativism.