I can't seem to find a clear, definitive, non-circular answer on this. For centuries and centuries, we've been doing mathematics in one form or another, be it geometry and pictures, or inventing symbols to represent numbers and then manipulating them using rules of arithmetic, but I don't understand where these rules were formalized for so many years.

Only in recent years did we get the Peano axioms for defining what a natural number is, or a real number in terms of Cauchy sequences, or Peano arithmetic, etc.

We even had calculus, power series, etc, before a lot of these things took off.

We didn't seem to have a "proof theory" where we all agreed what constituted a proof or what was considered a correct / incorrect proof.

I don't understand how people defined and used all these terms and techniques before we defined what these things were. Every answer I see on this feels circular to me. Did we just "do the math" without caring too deeply about possible edge cases and issues that arise in certain situations? Were proofs simply an appeal to our senses and whether we found it convincing, as opposed to showing 100% that these things always held?

How exactly did we do mathematics for so long and what necessitated the need to redefine the foundations?

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    Humans walked for millenia (and much more) before phisiologists and physicist become aware of the electro-chemical mechanisms that rules muscles and bones. – Mauro ALLEGRANZA Oct 30 at 20:16
  • @MauroALLEGRANZA Sure, but this is the question I am after. Did we just "use" things like real numbers for years and years without really asking "What is a real number, exactly?" – user525966 Oct 30 at 23:03
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    We now ask and answer: a real is a certain set (of equivalence classes of Cauchy sequences in the field of quotients of the integers, which are etc., or similar). But then in turn we “just use” sets, subject to axiomatic rules, without really asking “What are they, exactly?” Plus ça change... – Francois Ziegler Oct 31 at 0:48
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    @FrancoisZiegler But we used real numbers before Cauchy sequences, and we used sets before we had Peano axioms / first order logic rules, etc – user525966 Oct 31 at 0:54
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    The point is, AFAICT there are always things we “just use” (subject to rules) without further attempt to define “what they are” in terms of other things. Currently these primitives are (usually) sets. Earlier, it was (usually) other things. – Francois Ziegler Oct 31 at 1:09

As I explained in my answer to your other question, mathematics was always done using ordinary (non-formalized) logic. Attempts to formalize logic begin with Aristotle. (This is called "formal logic").

At the later time, the idea arose to use mathematics to formalize logic. One of the early proponents of this idea was Leibniz, and it achieved further development in the work of 19 century mathematicians, like George Boole and A. de Morgan. This new approach to logic was called "mathematical logic".

So logic which mathematicians always used in their research is something "outside mathematics". But "mathematical logic" is a part of mathematics. There is no vicious circle here: see Metamathematics.

Why did people try to formalize logic? There are at least two reasons:

a) to better understand what the rules of logic really are, and

b) the idea that logical arguments can be performed by a machine.

The second idea is much older than it may seem to modern students (it goes back to Raymond Lully (1232-1315), and apparently was an important motivation for Leibniz and Boole.

  • "mathematics was always done using ordinary (non-formalized) logic" Can you give an example of what you mean by this? I also don't see how Aristotle formalized logic, every example of his I see is a very informal "syllogism" where we say things like "All/some/no X are Y, ...etc" – user525966 Nov 3 at 17:54
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    @user525966: Almost any modern mathematical paper can serve as an example. You will find in it neither axioms, nor syllogisms. I wonder what percentage of modern mathematicians even know the complete set of ZFC axioms:-) – Alexandre Eremenko Nov 3 at 23:34
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    But aren't these modern papers using mathematical frameworks that, if you kept following their definitions and systems and all that, would eventually reduce to something like ZFC or peano axioms or first order logic or propositional logic? – user525966 Nov 3 at 23:36
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    @user525966 Just because you can make it look like they rely on that doesn’t mean they do. “If tables, chairs, cupboards, etc. are swathed in enough paper, certainly they will look spherical in the end”. – Francois Ziegler Nov 4 at 0:26
  • I don't understand what you mean by that – user525966 Nov 4 at 0:55

We didn't seem to have a "proof theory" where we all agreed what constituted a proof or what was considered a correct / incorrect proof.

Yes we did. (The Greeks theorized proof by contradiction, excluded middle, contraposition, etc.)

A short answer to the title question (with its emphasis on analysis) is that new “needs” arose from very concrete problems in exactly when and where Fourier series converge. This forced Dirichlet, Riemann, Weierstrass, Cantor, their cohort Dedekind, and others into considering more general “functions” and “sets” (not to mention “numbers”) than anyone had before; and axiomatizing those turned out to be slightly subtler, as more obvious nonsense occurs if we’re not careful.

In my opinion, the best antidote to the sort of vertigo you express — and to the general notion that the purpose of mathematical logic is to give mathematics “foundations” or “define everything” (it isn’t: that would indeed be circular) — is to read and understand Wittgenstein. It’s therapeutic.

  • Wasn't Kant's point that Aristotle's "proof theory" can not frame actual proofs even in Euclidean geometry? And wasn't Wittgenstein's point that we can have a rule governed practice without making a theory out of it? There were also many developments in 19th century unrelated to Fourier series, in algebra and geometry, for example, that exerted formalization pressures. Pasch, Frege and Dedekind were not motivated by Fourier series. – Conifold Oct 30 at 17:56
  • I feel that Fourier series were a convenient test case and application, an occasion, rather than reason for formalization, even in analysis. Bolzano was not concerned with them, and Weierstrass had many other concerns. Your opinion of Frege is very surprising to me given the reach of Principia he inspired. And according to modern studies (see e.g. Manders in Mancosu volume, p.87) Euclidean demonstrations essentially require diagram based inferences to go through. – Conifold Oct 30 at 22:31
  • Whatever the cause was it seemed to act simultaneously in different areas, geometry, analysis, algebra, logic. So I'd say it was bigger than any particular paper or person, something like a phase transition, continuation was obstructed without restructuring. In all areas the movement was towards spelling out objects studied, assumptions for working with them, and detailing arguments with more care. – Conifold Oct 31 at 0:06
  • I wish I had time to write an alternative answer, I just don't think that any "analysis oriented" explanation can work. But we can agree to disagree. – Conifold Oct 31 at 21:09
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    I just wish you removed "yes we did". The idea that Aristotle's syllogistic was of much use in mathematical proving, let alone provided its "theory", is almost universally rejected today. Euclid's rigor relied on construction based inferences, not syllogisms. Indeed, the realization of this fact after Kant was another major catalyst of the change in the standard of proof, with explicit effort to remove "intuition", i.e. construction, and replace it with vastly expanded logic. See e.g. Mumma Proofs, Pictures, and Euclid. – Conifold Nov 1 at 21:09

A different thread (in addition to one of my favorites, Fourier's extravagant claim that "every" [sic] function was "expressible" [sic] in trigonometric series (by the way, the pointwise convergence under various hypotheses is not to be purely attributed to Dirichlet, since he had seen a manuscript of Fourier's in which that proof occurred... but for some reason, political?, had its publication obstructed for several years...) ...

... is Kronecker's description of finite (hence, algebraic) field extensions of a field $k$ as $k[x]/\langle p\rangle$ for irreducible polynomials $p$. This in contrast to the prior style of "believing" that there was/is a larger realm in which all kinds of mystical numbers exist, which we can "adjoin" to the rationals, etc. Of course, that is the spirit. And a first approximation. But... wait a mo', is that legit? :)

And, similarly, the number-theoretic notion (Dedekind) that not every field extensions of $\mathbb Q$ is canonically imbedded in $\mathbb C$. Rather, the collection of such imbeddings is the object of interest...

In topology, the idea of computing simplicial homology independent of triangulations required some notions of "abstract" groups and such, which would not have made sense decades earlier.

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