Hey I am a high school student who is very interested in the philosophy of mathematics. I was watching this talk by Stephen Wolfram about whether or not mathematics is invented or discovered. In it he pointed out that we use one of many different types of mathematics; ie. All of our math is built from certain axioms, and if we were to choose and accept different axioms we would have a completely different structure to build a mathematics from.
My question is this:
Really what I want to know is what are the implications of having more than one accepted mathematical structures. Are they independent of one another and thus not contradictory, or must we begin to question the validity of our current mathematical framework since we can choose axioms to make facts either true or false at our whim.
I would really like any reading that you could recommend on this subject! Thank you!