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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!

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  • $\begingroup$ You can also say that some mathematical structures such as the integers, the Turing machines are implicitely present in every possible mathematical theories, since they accurately represent how the theorems of a mathematical theory are "generated" by the chosen axioms and rules of inference $\endgroup$
    – reuns
    May 8, 2016 at 17:25
  • $\begingroup$ I think you should read "Mathematics: The Loss of Certainty" by Morris Kline. It's exactly what you're looking for (and more) and doesn't require a formal background in mathematics. $\endgroup$
    – jazzinsilhouette
    May 8, 2016 at 17:27

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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

A famous axiom is Euclid's Parallel postulate.

You can not proof it, so now there is a choice to consider it true or false. Either choice leads to different geometries which are not contradictory within themselves. True: Euclidean geometry, false: non-Euclidean geometry.

You can not have them true at the same time because you would have contradictions, like the above axiom being true and false at the same time, which seems not useful.

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