In lieu of an abstract, here is a brief excerpt of the content:

CONCERNING SOME VIEWS ON THE STRUCTURE OF MATHEMATICS I lose my patience, and I own it too, When works are censur'd, not as bad but new. -POPE. I. INTRODUCTORY MATHEMATICS has always had for civilized peoples an appeal sometimes practical, often fanciful, at times even theoretical and philosophical. The empirical mensuration of the ancient Egyptians, the mystical numerology of the Pythagoreans, the precision, to a degree, of the Grecian geometers, the highly mathematicized structure of modern science -all these, and many more, are but varied colors in the Joseph's coat of interest in mathematics. So it is not surprising that we read in Plato's Republic: Mathematicians only dream about being, but never can they behold the walking reality so long as they leave the hypotheses which they use unexamined, and are unable to give an account of them. For when a man knows not his own first principle, and when the conclusion and intermediate steps are constructed out of he knows not what, how can he imagine that such a conventional statement will ever become a science? This was a challenge of prime importance to all mathematicians of future ages, a challenge which the modem mathematician has eagerly taken up. Within recent years mathematicians have been intensely busy in a critical study of the foundations of mathematics. A casual glance at the current mathematical literature will testify to this-a fact which is indeed one of the crowning glories of twentieth century mathematics. Today there are fearless and tireless workers in mathematics whose every ambition is to make as stable and :firm as their talents allow the basis upon which the extensive development of modern mathematics rests. 431 EVERETT H. LARGUIER They are determined to establish for mathematical theory a genuinely consistent structure. Profoundly critical is the current effort, with the almost inevitable result that the mass of its literature is scarcely within reasonable bounds. As a consequence , when we try to present a critique of this vast enterprise we find it almost impossible. It should be remembered that the problem is, in the most literal sense imaginable, an infinitely complex one; it is broader, deeper, and more ambitious than any other program as yet undertaken in scientific circles. A brief study, therefore, of some points in what many consider one of the more interesting and significant parts of mathematics should prove welcome. In fact, the intense preoccupation of the mathematician with the foundations on which the whole intricate superstructure of modern mathematics rests has begun to awaken interest in other quarters. At the very outset, however, I should like to make it dear that the reader cannot, and obviously should not, look for an exhaustive and compendious treatment of all questions connected with the foundations of mathematics. Besides requiring more genius than I possess, such a pretentious program would most certainly go beyond the more modest task which I have set for myself--remarks concerning some views on the structure of mathematics. Indeed, if this study will give only a cursory glance at this field of scientific endeavor-a veritable universe of thought-it will have accomplished something. Even an estimate of the far-reaching importance of all these varied efforts toward a critical evaluation of the fundamental principles of mathematical thought has some value. More than this is beyond the scope of these pages. II. SoME Vmws To get a general view of the scope of these efforts before entering into the criticism of some points, we present the following bird's-eye view of the program of each of the important schools of modern mathematical thought. Although the easily discernible lines of demarcation between the original SOME VIEWS ON THE STRUCTURE OF MATHEMATICS 438 foundation programs in mathematics have become somewhat blurred in the process of time through the introduction of variations, compromises, and the like, still the original division is best adhered to for the sake of clarity. We must admit, however, that there exist at present much confusion and irreconcilable differences of opinion on the whole problem of the foundations of mathematics. a. The Postulational School. It has been said that from the postulational point of view mathematics is a collection of...

pdf

Share