What brought about the need for real analysis and formal logic in recent years?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.