What is Mathematics?

This has been the subject of debate for centuries, so I will only give my opinion and some competing perspectives.

To me, mathematics is the study of emergent phenomena in iterative deductive systems. Say you start with some set of axioms A, and this all lives inside of some ambient language L (leaving “language” ambiguous enough to capture nonstandard, yet static, collections of valid inputs). Think of the axioms as generators of a subspace of the language, where the generating mechanism is determined the logic we work in. The axioms are “zero order” statements, and we can combine axioms to find first order statements. There may be torsion in the language, so that some first-order statements are actually equivalent to some axioms. In general, however, this game of iteratively applying the deduction rules over statements of a fixed order produces increasingly complex true statements. Eventually we begin describing families of high-order statements as theories unto themselves. We may even find ways to abstract away from the original system (L,A) by collating expressions and formulating embedded systems (for example, choosing to atomically describe category theory rather than relying on an underlying set theory).

This game of producing higher-order internal truths and abstracting is what I call mathematics. I am a Platonist, though, because I think the most fundamental objects in “reality” are relatively consistent systems (ones which are consistent with respect to an ambient system), and that our universe is one such object.

Others maintain that mathematics is something dependent on human language and evolutionary psychology. There are some people that believe mathematics is constrained by what is physically computable or what is possible to express linguistically. Some say that mathematics is looking into the eye of God.

Whatever mathematics “is,” we can say that it can be shared and communicated between thinking agents, and that the “truths” are only dependent on a static collection of assumptions.

Give your opininons..........................