One very interesting experience for me was recently relearning set theory from a textbook. It talked a bit about the history of the Schröder–Bernstein theorem, a fundamental (and relatively "simple" sounding) result in Set theory.
What I found interesting was that this originally stated, but not proved, by Cantor. Then a few semi-flawed attempts at proofs were made (flawed in the sense that they relied on other axioms like choice, or that they had errors).
These are leading mathematicians, including Cantor, who basically invented Set Theory from scratch. And they went through many years of struggle to arrive at a proof of a result. Which you now learn in an introductory class.
It really put into perspective for me just how hard some of these things are, and made me feel less bad when I struggle with some mathematics.
The reasons such proofs are non-obvious is because they are proofs about the relative “size” of hypothetical infinite objects of a type we can never actually grapple with in any physical way even in principle, but only posit as a thought experiment, based on invented axioms in an invented logical system. There’s no concrete computation involved (or even possible) in this kind of context, and no practical examples. So everything must be done in a purely abstract and formal way. Without any examples to test, it’s hard to notice gaps in logic.
According to Wikipedia this particular theorem was proved almost immediately (but not published) by Dedekind, and then 10 years later proved by a 19 year old student in Cantor’s seminar. It’s not clear whether more than a handful of people cared or worked on it in between.
As for Cantor being a leading mathematician of the day: most of the other mathematicians of his time regarded this whole mess to not be mathematics at all. Poincaré called Cantor’s set theory a “disease”. It wasn’t until several decades later that a broader group of mathematicians decided that roping in set theory allowed them to conveniently hand-wave away (“oh, the set theorists will take care of that part”) a lot of thorny questions, letting them get on with their work as they had before, but now with an deflective answer whenever anyone asked such questions. ;-)
Which even amplifies the observation. Not only is the process of proving a theorem more complex than it looks in retrospect, but the whole process of deciding whether a theorem or even a whole field of math is even useful or not can be driven by chance over a time span of decades. What is now "topology" was kicked around for many years with many approaches before a general agreement that open sets formed the most useful basis (heh). Advanced textbook authors regularly have to cut topics from subsequent editions of the book, because what seemed promising 30 years ago turned into a dead end. So math is far from the discrete list of topics and answers that it seems to be when you're picking out classes in your syllabus.
If something that looks obviously true is hard to prove using some given language, couldn’t that be a sign that we’re using the wrong language to construct the proof?
Could there exist a language where these sorts of proofs become easier to write, because that language captures the problem in a better way?
What I found interesting was that this originally stated, but not proved, by Cantor. Then a few semi-flawed attempts at proofs were made (flawed in the sense that they relied on other axioms like choice, or that they had errors).
These are leading mathematicians, including Cantor, who basically invented Set Theory from scratch. And they went through many years of struggle to arrive at a proof of a result. Which you now learn in an introductory class.
It really put into perspective for me just how hard some of these things are, and made me feel less bad when I struggle with some mathematics.