Axiomatic Systems, Conceptual Schemes, and the Consistency of Mathematical Theories

Lately, an increased interest in formal devices has led to an attempt on the part of some mathematicians to do without those aspects of mathematics which require intuition. One consequence of this movement has been a new conception of pure mathematics as a science of axiomatic systems. According to this conception, there is no reality beyond an axiomatic system which the statements of mathematics are about; the fact that a statement is a theorem in the system is all that is of interest. This is a sort of nominalist position, since the contrary position seems to be committed to a belief in the existence of a domain of mathematical entities, which are just as suspect as the universale commonly discussed in metaphysical treatises. More often it is called formalism

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