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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.

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    $\begingroup$ Ancient mathematicians simply assumed most of the basics. Peano and set theory came about relatively late in the history of math, - the 19th century or so - as the desire to give a strong foundation to math. Even Euclid, a partial effort to formalize geometry, was missing a fair amount of the foundations of geometry (you have to wait until Hilbert to fill in the holes.) $\endgroup$ – Thomas Andrews Feb 13 at 18:55
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    $\begingroup$ They were just axioms. You have to start somewhere. $\endgroup$ – saulspatz Feb 13 at 18:57
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    $\begingroup$ I think it is a mistake to conceive of mathematics as something that can only be developed axiomatically. This is typically not how things occur. Axiomatizations typically occur after intuitions are developed and after it is clear that there will be benefits to the formalization process. $\endgroup$ – Andrés E. Caicedo Feb 13 at 19:00
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    $\begingroup$ @Noanswersatisfies I think you would benefit from reading a book about the rise of formalism in the 19th century. Again, mathematics is something much more organic and malleable than your comments would suggest. $\endgroup$ – Andrés E. Caicedo Feb 13 at 19:35
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    $\begingroup$ Specifically, your most recent comment makes the following key assumptions: rigorous proof is what makes mathematics mathematics, and our criteria for what constitutes a rigorous proof don't change over time ("if it was a satisfying proof ... why cant we use their proofs as sufficient?"). The first criterion is arguable, and the second is simply false. I think all this will be helped by reading some of the actual primary sources, not necessarily with an eye towards the specific mathematical methods but rather focusing on the language and argument structure they employ. $\endgroup$ – Noah Schweber Feb 13 at 20:16

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