The difficulty in formulating my question lies in defining what I mean by "unique."
What I mean by "uniqueness": For example, the concepts that
- 1 + 1 = 2
- 5 + 2 = 7
- 6 x 3 = 18
- 6 - 9 = -3
only represent one unique class of concepts, movement about a number line of integers. Perhaps a second unique concept would be the extension of this number line to include all real numbers, complex numbers, etc. Other unique concepts be may the idea of limits, geometric perimeters, etc.
I suppose what I mean by "uniqueness" of a concept is a concept that cannot be derived or is an extension of other concepts. Instead, there must be some intellectual ingenuity that perhaps a human may be able to come up with but not a calculator.
I understand that my definition of "uniqueness" may be imperfect, but I do not believe an imperfect definition prevent meaningful discussion. We lack perfect definitions for what truth is, but can we still discuss truth in a meaningful way?
Considering this, are there infinitely many unique mathematical concepts? Is the landscape of mathematical knowledge in this sense infinite or is it finite? If humans of limitless intellect could live for any arbitrary amount of time, would we run out of ideas?
And if one says yes to my question because one could propose infinitely many axioms that are "unique," then how is this possible in a finite world? To conceptualize these "unique" axioms, one should have some sort of corresponding neurological structure. Aren't there finitely many neurological structures?