Daheane: The Number Sense

Excerpt from the above book, pp. 242-45 are below. I'd say whether or not one believe in Platonic math, both it and formal math are abstract with a priori axioms divorced from concrete reality. The intuitionist  or constructivist math he notes below, while accepting our innate categories of thought, are not the same as the image schema and basic categories of cognitive linguistics (and in fact are not referenced). But it bases this similar idea on the relation of math to our embodiment.

"Twentieth-century mathematicians have been profoundly divided over this fundamental issue concerning the nature of mathematical objects. For some, traditionally labeled 'Platonists,' mathematical reality exists in an abstract plane, and its objects are as real as those of everyday life. [...] For an epistemologist, a neurobiologist, or a neuropsychologist, the Platonist position seems hard to defend—as unacceptable, in fact, as Cartesian dualism is as a scientific theory of the brain."

"A second category of mathematicians, the 'formalists,' view the issue of the existence of mathematical objects as meaningless and void. For them, mathematics is only a game in which one manipulates symbols according to precise formal rules. Mathematical objects such as numbers have no relation to reality: They are defined merely as a set of symbols that satisfy certain axioms. [...] Though the formalist position may account for the recent evolution of pure mathematics, it does not provide an adequate explanation of its origins."

"A third category of mathematicians is thus that of the 'intuitionists' or 'constructivists,' who believe that mathematical objects are nothing but constructions of the human mind.
In their view, mathematics does not exist in the outside world, but only in the brain of the mathematician who invents it.  [...] Among the available theories on the nature of mathematics, intuitionism seems to me to provide the best account of the relations between arithmetic and the human brain. The discoveries of the last few years in the psychology of arithmetic have brought new arguments to support the intuitionist view. [...] These empirical results tend to confirm Poincare's postulate that number belongs to the 'natural objects of thought,' the innate categories according to which we apprehend the world. [...] Intuition about numbers is thus anchored deep in our brain. Number appears as one of the fundamental dimensions according to which our nervous system parses the external world."