I am a mathematician turned cognitive scientist/AI researcher. I wrote a short essay on the philosophy of mathematics that I think will be of interest to many people here. I combine platonist and formalist ideas and frame mathematics as an experimental activity, extremely similar to physics and other hard sciences with computation playing a parallel role to experimentation, and the creation of definitions and axiomatic frameworks being similar to the creation of physical theories like Newtonian gravity or Quantum Mechanics.

This perspective also stresses the non-formalistic aspects of mathematical research by gesturing at a strong division of intellectual activity into semantics and syntactics. Although this is by far the least developed part of the essay, it is the secret motivation behind me writing the essay in the first place, and my core interest. I am secretly using mathematics as a test case to probe the nature of general cognition.