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The metaphysician of mathematics attempts to construct a system that explains the nature of mathematical objects (for example, sets, numbers, functions, properties, and relations). The process involved in the development and construction of such a metaphysics, however, is very different from that of mathematics. In mathematics, the metaphysical significance of a new mathematical theory must be proven, often by detailed use and comparison with older theories. Mathematics is, however, not the only subject with significant metaphysics; the study of philosophy, for example, is another field with a metaphysics, as well as the study of language, history, biology, and others.

Metaphysics is not without controversy. While some object that metaphysics is entirely useless, others maintain that it is the study of the most important phenomena of the world. However, the philosophical tradition favors the latter at least initially, and this background has formed the concept of metaphysics as metaphysical philosophy. The metaphysician of mathematics might be said to be a metaphysician of philosophy, and philosophers of a variety of backgrounds attempt to build metaphysics to the level where the metaphysics of mathematics.

In a metaphysics that is compatible with metaphysics of mathematics, it is crucial to connect and examine principles in both mathematics and philosophy. Therefore, metaphysics must be compatible with both mathematics and philosophy. In particular, it must be possible to translate, with slight modifications, mathematical theorems into metaphysical slogans and postulates. Metaphysical slogans must not, however, be derived from metaphysics; they must be directly translatable into mathematical language, and the given metaphysical axiom is not sufficient to render completely rigorous mathematical proofs or statements. For example, the axiom of choice does not imply the existence of a set of all sets, and the existence of a set of all sets does not imply the possibility of a choice function for all sets. Of course, just as one can think of coming to understand the difference between numbers and sets in mathematics, one can think of the difference between numbers and sets in philosophy. d2c66b5586