system
Given the following post, respond as if you're commenting on Reddit:
human
What might the source of mathematical knowledge be, if not observation?

One theory of mathematics goes that we learn mathematics by observation, e.g., we learn that 1+1=2 by observing one thing and one thing and putting them together, then observing that we have two things.

I don't really want to argue about whether or not the above theory is true in this thread, but only to point out that it initially seems to go some way toward explaining where mathematical knowledge comes from. Namely, the same place most other knowledge comes from, observation.

Are there any good theories about what other source mathematical knowledge might come from? It's hard to explain exactly what would qualify as a satisfactory non-observational source for mathematical knowledge, but at a minimum it seems like we want an account of mathematical knowledge that gives us some kind of insight into how it is formed and why it qualifies as knowledge rather than opinion.
gpt
Our best scientific theories don't obviously come from observation, but are, to use Einstein's turn of phrase, 'free creations of the human mind, and are not, however it may seem, uniquely determined by the external world.' The observation of the falling apple (if it was indeed an apple) wasn't the source of Newtonian mechanics, otherwise it would have been the source far earlier; it was Newton's keen intellect and inquisitive mind into specific empirical problems. If our empirical theories are not generated by or epistemically grounded on observation, then why should mathematical theories be generated by or be epistemically grounded on observation? 

It is a mistake to ask for 'sources' of human knowledge outside the human mind, as if knowledge is gained passively through the eyes and ears and nose.