Comment by dist-epoch
Comment by dist-epoch 5 days ago
Math is done backwards.
We know what kind of results we want to be true, and then we search for the minimum number of axioms which can deliver that.
Comment by dist-epoch 5 days ago
Math is done backwards.
We know what kind of results we want to be true, and then we search for the minimum number of axioms which can deliver that.
There is another goal in addition to minimizing the number/complexity of axioms. Some "axioms" like induction actually introduce an infinite family of assumptions, a so-called schema. So in addition to working backwards from our (incomplete) knowledge, we find certain axioms let us make arguments that are obviously valid but would be formally very tedious without them.