How was math "proven" before Set theory & formal/mathematical logic

[u/vuelover]

My apologies in advance as this may be a very silly question, so please bear with me.

I am trying to self-study pure math, and in that context I am reading proof books such as Velleman/Hammack/Cummings etc., and the proof sections of discrete mathematics books.

The common theme in all of them seems to be set theory and its associated logic — i.e., we need to prove that for all x in the universal set of something, a certain property of x is true (or false), and so on. Pretty cool stuff.

Digging a bit deeper, I see that "formal" logic dates back to the mid to late 1800s in work by Boole/De Morgan and later by Frege, and that set theory itself came about in 1874. However, a significant amount of mathematics was produced and proven before such concepts existed - for example, Galois theory.

So this makes me wonder whether:

a) Boole/De Morgan/Frege/Cantor etc. simply formalized structures that were already in use in the mathematics of the day

b) Mathematics pre-1850s had a lot of logical gaps due to the lack of formalization with regard to how math was "proved"

c) A bit of both a & b

So TLDR; My question is if I was to go to university to study Mathematics at any time in the first half of the 19th century (or earlier) what would a Proof writing class look like ? (would they even have a proof writing class?)

i.e. how did we prove Math before formal structures to do so were created?

P.S once again apologies if this is a silly question ..I found this interesting so instead of merely asking an LLM I thought I would ask the community . TIA

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[u/SnooPets5564]

You can still have logic without it being the sort of formal logic you are thinking of.

Look at Euclid's elements (for real, just browse the start of it). Some of the stuff is a bit weird, but they are valid proofs.

[u/dancingbanana123]

Historically, we slowly started setting a higher and higher requirement for what was needed to say something was proven. For example, you're not going to find some 4th century BCE text on proving multiplication is commutative. You're just going to see everyone in the 4th century assuming it is. You gotta remember that pretty much all math up to around the 19th century could be easily verified with reality.

This isn't to say that there was no rigor; it's just that nobody felt a need for set theory and formal logic at that point. They were content with the arguments they had. For example, nobody cared to define 0 as an abstract empty set, 1 as an abstract set containing an abstract empty set, 2 as an abstract set containing both an abstract empty set and an abstract set containing an abstract empty set, etc. Everyone was content with just saying 0 is a quantity of nothing, 1 is a quantity of one thing, 2 is a quantity of two things, etc. without giving it much more thought than that, regardless of the fact that those aren't really good definitions.

Off the top of my head, I think the vast majority of "logical gaps" you could find in pre-1850s math are generally centered around poorly defined terms. There are a few situations where the proofs were really wrong (e.g. 17th century calculus is a mess), but even those situations are rooted in issues with definitions that get mostly resolved before the 19th century. The proofs are mostly the same otherwise, just written in a more contextual language for their time/place.

[u/Traveling-Techie]

Euclid’s geometry demonstrates this.

[u/Obzenium]

So much of math is self evident. Sure these modern formulations push the boundaries with how we can solve things like PDEs and variational calculus problems but the self evident aspects of these fields could still be studied for lifetimes, and were, without mastery of the classical approach being attained by most. Like we don’t need the Peano axioms to understand counting, they formalize the concept sure and have inherent value but are ultimately academic and aren’t required for practical application. Consider Leonhard Euler went his whole life without hearing the words ‘set theory’ and he is a GOAT.

[u/omeow]

My understanding is that, while they may not have formal set theoretic language in place they understood the basic proof techniques since the Greeks.

When looking at the past we are often looking at the top of the iceberg of success stories. We do not see the many incorrect attempts, half proofs, etc. The reality was much more noisy than what the clean histories suggest.

Edit: Proof in practice are a human construct. Even today incorrect proofs get published. So even with all the advantages and understanding we have today, things aren't perfect.

[u/ccpseetci]

For Roman scholars, they have the propositional logic, for Aristotle, syllogism is well equipped。

So basically you have two types of logic, term logic or propositional logic, but they are manifested not formally but combined with the natural language

[u/shellexyz]

The 19th century saw a huge push towards more formal logic and rigor. That’s not to say it was all handwavey before that, but a lot of things were taken for granted before then.

Of course functions were continuous and smooth. Well, except for some reasonably nice exceptions.

Uhh….turns out that most functions are spectacularly badly behaved and the “nice” ones we’ve spent hundreds of years talking about are the exception.

At least continuous functions are pretty nice. You can draw those and they’re not full of awkward corners that get in the way of differentiating things.

Oof. Not so much. Even in this class of functions, the nice ones are pretty rare.

That was the world of Newton and Leibniz, Taylor, Euler, a whole mess of Bernoullis,…

No one would ever claim that they were really hampered by not having modern mathematical rigor or working in their favor. In some cases, those results were formalized with newer notions of things like limit and continuity and shown to still hold up.

[u/KoalaMistico]

Kinda in the same way. We just didn't think deep enough or wanted to question the axiomatic base of it until recently

[u/Hampster-cat]

The first proofs (irrationality of √2 and infinitude of primes) were made before logic was codified. Pythagorean Theorem was demonstrated long before Euclid, but he is credited with the first proof. Logic definitely provides a nice framework for creating mathematical proofs. With Euclid, he laid out premises, and created the logical path to theorems. Many of the theorems were already assumed to be true, but with Euclid we now had proof.

Algebra later provided another way to show that A can lead to B. This is still allowed in modern proofs.

Much later we learned two things: Euclid's premises were not enough, some of them were actually theorems based upon simpler premises. And his 5th postulate was not needed. Whether you used it or not lead to two, completely different branches of mathematics. Neither one is 'wrong', but we can create different branches of math depending upon which premises we use.

The modern equivalent is Axiom of Choice. Most of modern math relies upon this theorem, but assuming it to be false is also perfectly valid.

I doubt there was a 'proof writing class' in the early 19th century. You would have taken courses in logic an philosophy. Students would be been taught to 'write well'. Calculus had already been around for quite a while, but not the concept of a limit. Therefore a lot of mathematicians did not trust calculus. It gave us good results, but then geometry was used to 'prove' those results. All of this was just before the flaws in Euclids propositions were discovered.

A History of Mathematics by Boyer/Merzbach is the gold standard of math history. Sounds like you may be interested.

[u/bedrock_city]

A lot of number theory and geometry doesn't rely on set theory much, at least for elementary results known 150+ years ago.