Argument from mathematical realism
Various mathematicians have observed that mathematics is very often useful in modelling observed reality.
- "The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. "
Some mathematicians argue that mathematics exist metaphysically:
"Might mathematical entities inhabit their own world, the abstract Platonic world of mathematical forms? It is an idea that many mathematicians are comfortable with. In this scheme, the truths that mathematicians seek are, in a clear sense, already there, and mathematical research can be compared with archaeology; the mathematicians' job is to seek out these truths as a task of discovery rather than one of invention."
- — Roger Penrose 
Apologists argue that usefulness of mathematics and mathematical realism implies God exists:
- "Once thiesm is dropped, it is difficult for realism to explain (1) where objective mathematical truths exist and (2) how we have access to them. [...] The existence of eternal, ideal mathematical thoughts seems to require the existence of something actual in which they exist. [...] If mathematics is merely a human invention, why is it that relatively simple mathematical theories yield such accurate representations of the physical world? "
- "Mathematical reality — if indeed it exists — is, admittedly, mysterious. But invoking God does not dispel this puzzlement; it is an instance of "The Fallacy of Using One Mystery to Pseudo-Explain Another." The mystery of God's existence is often used, by those who assert it, as an explanatory sink hole. "
An axiomatic basis
Not all mathematicians accept mathematical realism. Most mathematicians use ZFC (Zermelo–Fraenkel set theory) axioms as a basis for their work. However, there are many other axiomatic systems that are used for special purposes or for the sake of simplicity. There are an infinite number of possible mathematical systems but most of them are not useful. The choice of an axiomatic system is based largely on its utility in that it corresponds with observed reality.
- "The traditional realist view of mathematics as a description of how the world was had to be superseded by a more sophisticated view the recognized mathematics to be an unlimited system of patterns that arise from the infinite numer of possible axiomatic systems that can be defined. Some of these patterns appear to be made use of in nature, but most are not."
The question "why is mathematics so useful" becomes "why is this set of axioms, and their implications, so useful among the other possible choices?" and can be answered simply by "because this set of axioms was chosen because they were useful". There is really nothing to explain since "Mathematics is useful" is a tautology - there is no branch of mathematics that is not useful while still supporting this argument.
Problems of interaction
If mathematics exists metaphysically, it is unclear how the physical world interacts with this seemingly separate mathematical reality. (This problem is also suffered by dualism.)
- "What sort of mechanism could convey information of the sort bodily movement requires, between ontologically autonomous realms? To suppose that non-physical minds can move bodies is like supposing that imaginary locomotives can pull real boxcars. "
- ↑ Eugene Wigner , The Unreasonable Effectiveness of Mathematics in the Natural Sciences, 1960
- ↑ Roger Penrose, What is Reality?, New Scientist, November 18-24 2006
- ↑ 
- ↑ Rebecca Newberger Goldstein, 36 Arguments for the Existence of God: A Work of Fiction, 2011
- ↑ John D. Barrow, Godel and Physics, 2006
- ↑