I think that mathematics are the internal ground rules by which the brain organizes relationships. That is to say, pure mathematics doesn’t describe properties intrinsic to external phenomena, but rather describes the terms by which the mind organizes its internal representation of the phenomenal world.
The only argument I have to offer to support this view is the case of Daniel Tammet, the English prodigy who performs truly gigantic mental calculations (including pi to 22,514 places). The reason I think this has a bearing on this question is that Tammet does not do these calculations algorithmically. He doesn’t, in fact, do the calculations; his role is more one of passive observer as the calculations perform themselves in his mental space.
He explains that he experiences number as shape, and that calculations unfold as dynamic transformations of those shapes. This happens without any exertion on his part; he simply reads the resulting shape to arrive at the answer to the calculation.
I can’t reconcile Tammet’s experience with any other explanation than that he is observing, in a far more lucid way than can most of us, the very mechanism of mathematics at work in his own brain.