I do not get it. How was a+b=b+a or a (b +c) = ab + ac proved by ancient mathematicians before set theory or Peano axioms? If you cannot logically prove the laws of arithmetic without Peano or set theory then how did ancient mathematians have confidence in claiming laws of math?
Ive seen a video (on youtube) proving ab=ba using intuitive graphical methods etc, such as imagine a rectangle with sides the units of the the two numbers being multiplied and then rotate it and youve proved ab=ba. Which i consider a proof because how else did they prove it. But commentors say this isnt a proof it doesnt use set theory etc. If math is based on proof and all about being able to prove your assertions then what was math prior to set theory? A bunch of no proven guesses? How did mathematicans have any confidence in ab=bc by way of mathematical proof? Surely they had a way of proving it (ab=ba) 2000 years ago without set theory or peano axioms as neither set theory or peano existed then? And if so how did they prove it?
@Andrés E. Caicedo's comment below: correct. So what axiomatic system did they develop to prove basic laws of arithmetic? Thats my question. And if they didnt how was what they were doing math it was conjecture only as they lacked a proof and it didnt come until 2000 years later with peano and zfc
Im getting the idea they had no proof as we know it today and accepted them as axioms or true because you can tell something like ab = ba is true in your mind even if you cannot necessarily write down exactly why deductively at the level of zfc or peano and this is what they did to establish much of mathematics.