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I'm a little confused when I read that mathematical logic is actually a very recent field (1800's - 1900's), regarding the foundations of mathematics and so on.

By this I refer to writing proofs in classical propositional logic systems (Hilbert-style, natural deduction, sequent calculus, etc), extending up to first-order logic, using that to define theories like ZFC which (to my knowledge) is largely seen as a modern-day foundation for mathematics. Likewise for Peano axioms, Peano arithmetic, and so on.

I get confused on this because haven't we been doing mathematics for thousands of years? How were we doing proofs before? How were we doing mathematics with no "foundation"?

Were we just blindly using arithmetic and real numbers and so on without really defining what they were or how they worked? Did Euler and Gauss do all their advanced number theory stuff on these informal foundations? What about Newton and Leibniz inventing calculus? All of this years before we start talking about logic and model theory? How did people convince each other that this stuff actually worked especially once we start getting into concepts like infinity?

I don't really understand the timeline of it all or why logic was such a late subject. Did we choose to formalize it so late for some specific reason? What were we trying to solve or achieve? How was it being done before? Why did it take so long before we started asking questions about mathematical foundations?

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    $\begingroup$ This question may be better-suited to the History of Science and Mathematics Stack Exchange. $\endgroup$ – Blue Oct 29 '18 at 3:33
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    $\begingroup$ See en.wikipedia.org/wiki/Entscheidungsproblem Much of formal mathematics was motivated by this very problem. $\endgroup$ – John Douma Oct 29 '18 at 3:33
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    $\begingroup$ This question isn't appropriate for this site, I think - I think it's a very good question, but HSMse would be a better fit - but let me just mention that your final phrasing "What were we trying to solve or achieve? How was it being done before? Why did it take so long before we started asking questions about mathematical foundations?" is sort of the point. Foundational questions were asked extremely early on; however, foundational concerns didn't really "catch fire" until rather late, precisely because there wasn't any obvious issue they would be needed to solve. $\endgroup$ – Noah Schweber Oct 29 '18 at 3:47
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    $\begingroup$ It may be worth simply reflecting on your own experience learning/studying mathematics. For example, you were presumably told you were working with "real numbers" for most of your mathematics education while never being given a definition of "real numbers" (or polynomials or continuous functions etc.) until much, much later on (if ever...) How much did this seem to matter to you? You were almost certainly not taught much formal logic early on, and yet your educators were presumably often able to convince you some times (and not just via authority). $\endgroup$ – Derek Elkins Oct 29 '18 at 3:50
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    $\begingroup$ (That said, of course we're right and they're wrong. And you can trust me, because I never lie and I'm always right.) Anyways, I am voting to close this question as off-topic because it belongs on HSMse. (Sadly, that site isn't one of the "migrate to" options!) But let me restate: I think this is a very good question, just not necessarily fitting this specific site. $\endgroup$ – Noah Schweber Oct 29 '18 at 3:55

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