Symposium on the foundations of mathematics

Abstract

The problem of the logical and epistemological foundations of mathematics has not yet been completely solved. One of the most important questions for the foundations of mathematics is that of the relation between mathematics and logic. Logicism is the thesis that mathematics is reducible to logic, hence nothing but a part of logic. Mathematical predicates are introduced by explicit definitions. Since an explicit definition is nothing but a convention to employ a new, usually much shorter, way of writing something, the definiens or the new way of writing it can always be eliminated. Antinomies of the second kind are known as “semantical” or “epis-temological” antinomies. They include our previous example, “heterological,” as well as the antinomy, well-known to mathematicians, of the smallest natural number which cannot be defined in German with fewer than 100 letters. The intuitionist mathematician proposes to do mathematics as a natural function of his intellect, as a free, vital activity of thought.