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A few years ago I became briefly obsessed with the philosophical foundations of mathematics. When I dug into it I realized that it’s actually quite simple.
Mathematics is nothing more than the consequences of axioms. An axiom is a primitive rule that we just accept as a starting point. For example, people generally just accept the rules of arithmetic as a starting point without trying to prove that they’re correct. Other common axioms make statements about the properties of sets, functions, and other higher-order mathematical objects.
Once a collection of axioms has been selected, the purpose of mathematics is to work out the consequences. Theorems are mathematical statements that are implied by the axioms. In order to establish a theorem, you have to prove its correctness by tying together statements that follow only from the axioms, or from other theorems that you have already proved.
That’s it!
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