Long ago I started to study physics at Simon Fraser University. That meant studying applied mathematics. Somehow the foundations of this seemed unclear. I first learned the differential calculus years earlier from an old textbook of my father. It was old enough that it didn’t use limits in its explanations. When I took it again at SFU, I discovered that the old explanation or proof was considered wrong, and we had to learn it in terms of limits.
This prompted me to take a pure math course, the first of several. But they seemed to abuse logic. I learned that there was an entirely different approach to mathematics, called Intuitionistic Mathematics, based on a logic which does not take the law of the excluded middle as dogma. This kind of mathematics is weaker than the ordinary kind, since it does not permit the proof of some things most mathematicians cling to as vital.
Later I learned that there were other kinds of logic, based on other axiomatizations of set theory. In other were different flavors of mathematics based on different flavors of logic. Mathematics based on ZF logic and set theory is not identical with mathematics based on VBG logic and set theory.
So, I studied logic in the philosophy department, hoping for some intuition about it. Doing so, I was exposed to Linguistic Philosophy, which was popular at the time. It attempted to argue away all philosophical questions by referring to our use of language.
To settle some of their arguments, I started studying linguistics. In the SFU Modern Languages department, where linguistics were studied, I learned the slipperiness of both syntax and semantics.
Two things came out of this. One was that mathematics could be put on a firm semantic foundation by going back to the old idea that it was simply the consequences of our definitions.
The other was that Wilhelm von Humboldt’s idea that all natural languages were the expression of culture in an underlying universal language. Eventually I came to the realization that this implied that human beings were ultimately linear. This became the topic for my master’s theses. You may read about that on another website.
It has been very difficult to make any headway with my notion of mathematics as the collected consequences of our definitions, because it doesn’t seem susceptible to mathematical proof. It comes down to the same kind of argument which went on for a while between Intuitionistic and other mathematicians.