Foundations of Mathematics for the Working Mathematician

I am very grateful to the Association for Symbolic Logic for inviting me to give this address-an honor which I am conscious of having done very little to deserve. My efforts during the last fifteen years (seconded by those of a number of younger collaborators, whose devoted help has meant more to me than I can adequately express) have been directed wholly towards a unified exposition of all the basic branches of mathematics, resting on as solid foundations as I could hope to provide. I have been working on this as a practical mathematician; in matters pertaining to pure logic, I must confess to being self-taught, and laboring under all the handicaps that this implies; and if, after no little selfquestioning, I am speaking here today, I am doing so chiefly in order to enjoy the benefit of your professional advice and criticism, by which I hope to correct my views before I venture into print with them. Whether mathematical thought is logical in its essence is a partly psychological and partly metaphysical question which I am quite incompetent to discuss. On the other hand, it has, I believe, become a truism, which few would venture to challenge, that logic is inseparable from a coherent exposition of the broad foundations on which mathematical science must rest. On the true function of logic in this connection, however, there may still be room for some difference of opinion. The history of mathematics would perhaps throw not a little light on this subject; and a detailed study of the pattern according to which the feeling for rigor at times is emphasized and at times recedes, and the foundations of our science as a whole, and of its various branches, are at times scrutinized and then again neglected, would indeed offer a fascinating topic of investigation for a historian more concerned with ideas than with bare facts. Such a study has not yet been attempted, and would perhaps be premature until some crucial periods in the history of our science be more thoroughly examined. And it is doubtful whether extant documents will ever enable us to draw valid conclusions about those decisive centuries in early Greek science when the need for proofs first reached the level of consciousness and a technique was slowly and laboriously worked out to satisfy that need. Proofs, however, had to exist before the structure of a proof could be logically analyzed; and this analysis, carried out by Aristotle, and again and more deeply by the modem logicians, must have rested then, as it does now, on a large body of mathematical writings. In other words, logic, so far as we mathematicians are concerned, is no more and no less than the grammar of the language which we use, a language which had to exist before the grammar could be constructed.