The combinatory foundations of mathematical logic

E-ISSN： 1943-5886|7|2|49-64

ISSN： 0022-4812

Source： The Journal of Symbolic Logic, Vol.7, Iss.2, 1942-06, pp. : 49-64

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Abstract

In investigations of the foundations of mathematics we can distinguish two separate tendencies. On the one hand, one may seek to define his subject with greatest possible explicitness: to obtain a formulation which satisfies the most exacting demands for precision, and which is at the same time free from paradoxes and adequate for the purpose. On the other hand, besides the problem of formulation, there is that of simplification; one can seek to find systems based upon processes of greater and greater primitiveness. The reduction of a piece of mathematics to a formal system, and still further to a completely formalized system (as explained, for example, in my New York address a year ago), is a step toward the first of these objectives. But it is evident that one can formalize in various ways, and that some of these ways constitute a more profound analysis than others. Although from some points of view one way of formalization is as good as any other, yet a certain interest attaches to the problem of simplification, as is shown by the attention which some of the greatest mathematicians have devoted to it.