Mathematics?

Let us consider the view that mathematics is discovered. This does not necessarily mean that mathematics is the language of nature (or reality) or that abstract (mathematical) objects exist, but it can also mean that mathematics is discovered not from reality but from our psychology.

Consider two possibilities. First Platonism in one of its forms may be true and mathematics is just discovered truths about some of these abstract objects. This cannot be proven or disproven I suppose.

The other alternative is that mathematics is a discovery of our psychology, of our rational nature. This can be illustrated by a particular example. Suppose mathematics started with counting, cavemen had a bunch of apples and they started to invent words to count, one, two, three and so on and hence there was numbers and summation, and maybe most mathematics proceeded from this. If this is the case it may seem like mathematics is an invention of these cavemen, but It would still not be an invention per se. Prior to the invention of words to describe quantities and a system of operations, there exist independent ideas in the cavemen's mind which is not derived from observation. Discreteness, hence an apple can be said to be one, categorization, hence we can add apples (things of the same category/kind), are just two examples of such ideas which exists in human's psychology (mind).

Could mathematics be the truths regarding the relationships of these ideas in human's psychology, and by discovering mathematics, we discover our psychology?