Definitions do sometimes end up being justified by their consequences though, eg 0^0 is just defined to be 1 because it results in nicer equations. For that matter, you can sometimes do mathematics with only rough definitions and then make it well-defined later- umbral calculus "worked" before anyone actually set it on a solid mathematical grounding:
(you can see in that Wikipedia article the umbral calculus was originally invented and used because it seemed to work, long before anyone found a set of definitions to justify it to modern standards)
The axiom of choice is accepted because it's useful and required for certain results to hold that mathematicians aren't willing to give up, etc. If ZFC were to be somehow found to be inconsistent it would probably be patched up by altering the axioms rather than wholesale tossing out all the theorems.
I guess what I mean is that mathematics uses deduction in proofs but that's really not the end (or the beginning) of the story.
https://en.wikipedia.org/wiki/Umbral_calculus
(you can see in that Wikipedia article the umbral calculus was originally invented and used because it seemed to work, long before anyone found a set of definitions to justify it to modern standards)
The axiom of choice is accepted because it's useful and required for certain results to hold that mathematicians aren't willing to give up, etc. If ZFC were to be somehow found to be inconsistent it would probably be patched up by altering the axioms rather than wholesale tossing out all the theorems.
I guess what I mean is that mathematics uses deduction in proofs but that's really not the end (or the beginning) of the story.