[Ntrails]

> I stopped enjoying math for about this reason. My university tried to teach with proofs, which to me is about as far from an intuitive understanding as possible

Interesting, because until I had the proofs for eg differentiation, or heck even the quadratic formula - they were meaningless rote learnings. The proof is the intuition imo

[aschismatic]

Exactly! Proofs are why I love mathematics so much. There's nothing quite like the great "ah-ha!" moment when you find that one leap of logic that topples the rest of the dominoes in a proof. Of course, I understand that some people are better at teaching proofs by involving students in the discovery. Maybe the parent commenter was taught proofs in that same rote manner. I've had professors that have done that, and it's like someone sucked all the fun and learning out of the subject.

From the preface of Computation: Finite and Infinite Machines by Marvin L Minsky:

> The reader is therefore enjoined not to turn too easily to the solutions; not unless a needed idea has not come for a day or so. Every such concession has a price—loss of the experience obtained by solving a new kind of problem. Besides, even if reading the solutions were enough to acquire the ability to solve such problems (which it is not), one rarely finds a set of ideas which are at once so elegant and so accessible to workers who have not had to climb over a large set of mathematical prerequisites. Hence it is an unusually good field for practice in training oneself to formalize ideas and evaluate and compare different formalization techniques.

[Fr0styMatt88]

For proofs, could you recommend any good resources for a beginner? Is there a 'beginner proof' that's great to start with?

I figured out I actually like maths waaaay after I'd left uni. From that time at uni I have a vague memory of proofs being something like a whiteboard full of equations that I got lost somewhere in.

I have a vague feeling that what I'm thinking of is 'formal proofs', but I'm not sure.

[gumby]

Euclid. He tried to prove theorems of basic plane geometry (hence "euclidean geometry). Since we all have an intuitive understanding of (at least the basics of) plane geometry you can look at the work and not have to also learn the domain.

People recommending the classics can come off as pretentious so I will add that I am serious: a modern book of Euclid's methods should be quite accessible.

As a followon bonus: Minsky's and Papert's 1967 book "Perceptrons" (the one that said you can't do XOR with a single-layer network, though you can with a multilayer one) that lead to 25 years of lack of interest in neural networks is entirely about using neural networks on Euclid. So you can go from one to the other!

[aschismatic]

The first upper-division course I took in college concerned itself solely with learning the art of mathematical proofs. I had an excellent professor, so I can't really say how much this book helps with the learning process when used by itself, but we were assigned An Introduction to Mathematical Reasoning: Numbers, Sets and Functions, by Peter J. Eccles. Might be a good place to start!

[ghufran_syed]

I really like "Mathematical proofs - A transition to advanced mathematics" by Polimeni, Chartrand and Zhang If you're looking for a free option, "Book of proof" by Hammack also looks good, but I have less experience with it ( https://www.people.vcu.edu/~rhammack/BookOfProof/ )

[wnewman]

Both proof and explanation seem to be important. Two experiences with explanation made a particular impression on me.

First I found it hard to get much out of a Real Analysis course when I was a grad student. Only partly because of lack of explanations, admittedly. Probably even more of a problem was another kind of cultural mismatch between physical scientist me and the math prof that taught the course. My interest was mostly that I was actually doing path integral Monte Carlo calculations (for my Chemistry thesis) and wanted to make sure I understood the fundamentals. The prof, like many (most?) mathematicians seemed to be more interested in investigating ingeniously weird boundary cases. So the course didn't seem to teach me much about the gotchas that might come up in actual statistics or numerical analysis, and instead more about the ingenuity of mathematicians in constructing absurdly farfetched abuses of e.g. the axiom of choice. But besides that cultural mismatch, lack of an explanatory framework sure didn't help. Thus, I was very happy decades later when I ran across Terence Tao's book on measure theory (available as a free manuscript online), which had a lot of the same kind of material with quite a good framework of motivation and explanation wrapped around its proofs.

I also like Vapnik's _The Nature of Statistical Learning Theory_ which as I understand it is highly parallel to a much longer proof-heavy version of most of the same material. I much preferred this book to the approach in my undergraduate course in statistics. Again, the difference wasn't only lack of explanations (also, e.g., not enough enough grounding in proof or plausibly provable propositions, and too narrowly based in a frequentist worldview assumed without any explicit justification), but the lack of explanatory framework sure didn't help, so later I welcomed Vapnik's explanations of his approach. I have never been motivated to read the proof-heavy parallel book by Vapnik, but I do find it reassuring that it exists in case I ever work with problems that were weird enough to start raising red flags about breaking the comfortable assumptions in my informal understanding of the statistics.