Should I really just "shut up and calculate"? On learning at a good pace without sacrificing rigour

Question

This question is about getting realistic expectations for how a university student actually does and should learn maths. I'm becoming increasingly suspicious that my approach is detrimental, but I don't see a way out of it. I know that screams "off-topic personal question", but I think the themes of this question should appeal to a broad audience and can be answered without personal knowledge of who I am and how I learn. Please don't automatically close this; I will endeavour to edit this into a more answerable shape if necessary.

Tl;Dr - skipping the context, my question is more or less what it says in the title. I learn things very carefully but there is a massive time overhead involved. I would like to find some sort of compromise position, but don't know how to change. I could restate the question as: "how to take better notes?" since my current note-taking strategy is rather intense.

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Mathematics is supposed to be a rigorous subject. As long as you don't delve into foundational issues - as long as you can believe in a bare minimum of basic set theoretic constructions and basic logic - you can, in principle, understand and prove beyond all doubt any statement you learn in your journey. This is what draws me to mathematics beyond any other subject; anything I read, I am empowered to question and (attempt to) check on my own. Despite my relative ignorance, I can point out mistakes in the author's work and decide for myself what the correct statements should be and I can know with certainty (ignoring foundational issues) that it is right.

When I started getting into higher maths (i.e. beyond school level) I was aware that "in 'real' maths, we prove things" so I made an effort to learn how to read and write proofs; every new thing I learnt I would either try to prove for myself or look up a proof that I could understand. There is considerable additional effort spent in expanding upon and annotating proofs with further explanations - Lord knows authors leave a lot of missing gaps for the readers to fill. I kept this attitude up; it has only been in exceptionally rare cases that I have noted down a statement without noting its proof. I have only done so when the statement was a side remark invoking a theorem that was way above my paygrade at the time. Even then, I would have tried (and failed) to understand the relevant papers.

As you can imagine, this is a very, very slow process. I've somehow managed to cover a lot nonetheless, but it can sometimes take me hours or even weeks to get notes on a single paragraph - there's something I don't understand, so I spend time trying to check it, then I find I maybe need to learn some new things before moving on, etc. and before you know it, what feels like an eon has passed.

• On the one hand, this is a good way to expose myself to new maths and new ideas, and by thoroughly checking everything written and filling in the gaps I get experience in "research", overcoming frustration and doing my own proof writing, problem solving. Moreover, by writing down annotated and fully explained proofs I have a reliable reference so that whenever I need to revisit a theorem and its proof, I can have more confidence that (a) the result is correct, because my younger self checked it and (b) I don't need to waste extra energy checking mysterious steps in the proof, since my younger self has hopefully explained them
• On the other hand, this is absurdly slow (I check and take the time to write down absolutely everything) and everyone I've talked to on (mathematical) social media seems to not at all follow the same procedure. I get the impression most maths students - very successful ones - 'get away' without painstakingly examining and cross-examining their textbooks. It seems many people don't even read the proofs of many of the statements they might learn in a textbook, preferring to mostly focus on the ideas, the intuition; this seems to work wonderfully for them.
• On the third hand, authors are capable of mistakes and omissions. It is not common but it is not uncommon either that I encounter sloppily stated theorems/proofs or false/grossly misleading statements. One benefit of my approach is that I can usually detect this and correct it; had I just had pure faith in my learning source, I may have learnt a false thing and missed out on the opportunity to learn/discover new mathematics necessary to make the corrections. This phenomenon gets worse as the maths gets harder and more abstract, I've found - which motivates to be more fastidious, essentially making me learn slower and slower

The 'fear' of learning false statements and blindly accepting mistakes from authors is with me, and it's why I struggle to 'grow out' of this approach of mine. However when I read things like this I really get the sense that I've been doing it all wrong, very wrong. I start university this year and I should probably start on the right foot; I don't want to be left behind in the dust by less fastidious - but faster - students.

Questions:

• Is this slow approach genuinely problematic, or am I just being dramatic?
• How can one get through content more quickly but still manage to learn proof techniques and feel like they understand everything that's going on? I fear that if I started, say, skipping the proof of every other theorem, I would quickly lose track of how things are working and I would be afraid of failing to catch mistakes/sloppy statements

Perhaps the question becomes more answerable if I state it like this:

• In your experiences of teaching and learning, what are some good strategies for learning mathematics at a reasonable pace, all the while maintaining a high (ideally perfect?) standard of rigour without getting bogged down?

In case this context is needed; I’m not following any course (not at university yet). I’m entirely self studying; at the moment this has been investigations into enriched and model category theory and simplicial homotopy theory, following a few different textbooks.

Answer 1

I looked at some of your posts on MSE before answering. Well, I wouldn't say that you are a "Jack of trades" yet, but you are certainly way above what one would expect from somebody 2 years out of high school even by the 21-st century standards. So, if I were you, I wouldn't complain much about "going too slowly": you go many times faster than most university students I have seen already and, being young, are going only to accelerate in the next 10-20 years if your brain doesn't fizzle at some point.

There is one thing you should understand however: one cannot learn and become proficient in all mathematics. So, you will always have some areas in which you'll have solid rock knowledge, some areas in which your knowledge is fuzzy and some areas in which you are totally ignorant. Of course, you can choose any particular fuzzy or even unknown stuff and turn it into the solid rock one by putting enough time and effort into it, but while staying omnivorous is an excellent strategy, you'll not be able to swallow and digest everything you can bite. So, get comfortable with the idea that some fuzziness somewhere is OK.

On the other hand the only way to build the solid rock part of your knowledge is the way you are following: go into everything, scrutinize every detail that is still not obvious to you (some of your posts on MSE show that you already treat a lot of standard stuff as "trivial", which is good for you since you haven't made any mistakes there that I would notice, so I assume you, indeed, absorbed that stuff completely, but bad for your readers to the extent that they often wait for someone else to provide a more detailed explanation even after you have provided a full answer), try to prove everything by yourself, and so on. There isn't and there will never be any shortcut here. And yeah, that solid rock part grows like a stalactite with just small mineral deposits from a huge flow of information you work through. Fortunately you will realize soon (if you haven't realized that already) that a lot of that information is repetitive and then, if you want accelerate, you'll have to sift it for really new ideas, techniques and tricks and ignore the stuff that is fairly familiar.

What I'm advocating for is not abandoning that solid rock building strategy (God save you from that!), but complementing it with some fuzzy area building one. You should always understand that that fuzzy area is fuzzy, so you'll see those things in a kind of light (or even moderate) fog, but the advantage is that you'll be able to see much more and way farther and it can be expanded much faster than the solid rock one, which may be useful even if the crystal clarity of the vision you've got so accustomed to is sacrificed. However, if something in that fuzzy area attracts your attention and arouses your interest, you always have an option to jump upon it and add it to your solid rock part. Big fuzzy area will just give you more options and choices of what to do next (and yeah, that means that you'll have to select and prioritize).

As to "this" that you've read, the actual advice there was merely a) to solve problems, not just to read theory (which you are apparently doing anyway, so everything is fine here) and b) not to waste time on reading and verifying slow-paced repetitive expositions that have never been intended for anyone reasonably smart in the first place (or, more blatantly said, to choose the books that match your level or slightly exceed it). Both are reasonable pieces of advice and I absolutely do not see why you should get frustrated about them.

The last piece of advice: find somebody to work and study with. Solitude is fine, but the life gets way more fun when you can share the excitement and compete with others. Also, as many people said already, a good mentor (at the university professor level, say) may benefit you too.

Unfortunately, I can say nothing about category theory and such because I'm a total ignoramus there, so I hope somebody else will provide some pertinent comments related to your current studies (like suggesting well-written and reasonably fast-paced books, etc.)

Just my 2 cents.

Answer 2

Take a class, and ask the professor.

As the class starts, try to get an idea for what kinds of statements the professor is assuming to be true without justifying them, that you do not immediately see how to justify. Then pick one or two specific ones (not three or ten) and take them to office hours in the second week or so. Tell the professor that you don't immediately see how to justify these statements, and ask the professor if it would be a good use of time for you to work through the details.

You will get one of these answers, probably:

• No, this is too foundational for you to be worrying about right now; you will not be able to work on the material that is the purpose of this class if you spend time on this issue, or
• Yes, it would be good for you to work out those details so that you are comfortable with them, or
• This statement is obvious; there is nothing to prove here. (You would need to then explain the specific situation you are worried about that makes you think it is complicated, or retreat and regroup and think about why this was the answer.)

The first two answers are clearly good for you. The third one is actually also good for you because you asked for the expectations of the class and got the answer. If you get the third answer, then yes, you should "shut up."

In summary, ask the person leading your class, and try to respect the answer you get.

Answer 3

I did the exact same thing you did, except in physics instead of math; I would in fact rederive the relevant equations from scratch every time I used them so as to really hammer into my head where they were coming from, what their assumptions were, and when they made use of approximations or might break down.

In college, I, like you, found myself somewhat overworked and overwhelmed and working frantically to catch up. It didn't help that I had a very good physics teacher in high school with a very clear understanding of the material and a gift for transmitting it, whereas the professors in college were great researchers but were absolutely terrible at conveying their understanding. So, asking questions to clarify my understanding quite often plain didn't work, and however much time I spent on a confusion it would never really get dissolved.

I gave up on some of my attention to detail as a result, just taking a few things for granted and moving on. But I've pretty consistently regretted it since, because inevitably it comes up later; I can see where my ability to understand a topic is limited by my shoddy foundation. Shoddy understanding produces shoddy understanding, and my ability to take advanced courses that ought to have built on earlier knowledge is diminished. Perhaps it is only diminished in a way that my classmates would say is normal, what they're used to and how they've learned everything until now, but I'm certainly not satisfied with it and I certainly don't want to perpetuate that cycle any further than it's already gone.

So no, I would absolutely say that as much as possible, you should stick to your habits. I don't know what your peers and classmates are planning to do with their future careers, and likely you don't either; but if you intend to go on to grad school and research, you want as firm a foundation as you can get. Your insight later in life will largely be determined by the strength of your foundation now.

That's not to say you can't, for example, look up new fields on Wikipedia and use places like Quanta for a high-altitude overview of a topic, as a way of orienting yourself or looking for existing work in a field; nor is it an argument against reading natural language summaries of a topic for a general idea of where you're going. But once you've decided to learn something for real, then...

Well, okay, one exceptional case: there's a thing that sometimes happens, where you "shut up and calculate", reach the end of the proof or chapter, then go back to where you lost the plot and try to understand from there, and this is often easier than trying to figure out a specific difficult step and the overall direction of the proof at the same time. But that should never have more than about a chapter or so of 'understanding debt' between the point where you move on and the point stop and return.

Other than that, I'll go so far as to say that in my opinion, there's no point in studying a topic at all in math if you're not willing to be rigorous about it!

Answer 4

Mathematics is supposed to be a rigorous subject. As long as you don't delve into foundational issues - as long as you can believe in a bare minimum of basic set theoretic constructions and basic logic - you can, in principle, understand and prove beyond all doubt any statement you learn in your journey.

I find this statement interesting because I personally enjoy dabbling in the foundational issues. One doesn't have to, of course, but your wording suggests that this mindset shapes your view of what mathematics is.

You say mathematics should be rigorous, but what does that mean? Consider that until 1922, there was no ZFC to root your rigor in. Rigor was defined differently before that point. There was, in my opinion, a great deal of persuasion involved, convincing the reader that one's logic was sound, and that persuasion may not have the luxury of being founded in formal notations.

I point this out because I don't believe this approach fundamentally changed just because we developed fundamental tools like set theory. Sure, a lot shifted there, because mathematicians in general agree with the results that come from set theory, but there's still a part of mathematics that isn't rigorously proven with formal logic.

Were mathematicians shackled to such rules, logicism would still be a big thing. There would be no value in a thing that is not proven with a formal proof. For me, the spark of mathematics lies buried under these lines. There's some value in having a hypothesis which isn't quite proven formally and exploring it. Using it. Seeing whether it leads you in a direction of sufficient value as to warrant taking the time to pen a formal proof.

In math class, they are teaching you dead math. They're teaching you things that are substantially proven and agreed upon. When one starts looking at live math, one lives on the hairy edge between that which is proven and that which is still seeking proof.

I like to look at the history of calculus as an example. The calculus of Calc I and Calc II are dead math. But at one point, they were alive. When the mathematics of calculus were being constructed, they did not rely on epsilon-delta proofs (the current gold-standard for arguing the provability of calculus). Indeed, epsilon-delta proofs were not invented for over 100 years. There was enough of a spark there to drive mathematicians for a century before the formal proofs finally nailed down the roots of calculus.

It's great to get to learn all of the little formal proofs behind every statement. It attracted me to mathematics as well. But there's also something to be said for being comfortable working with things that aren't quite formally proven yet -- or at least not formally proven to you. The skill in working which such potential truths will be very helpful as you approach the living edge of mathematics.

Answer 5

...yes.

1. If your approach is holding you back, you need to change it. Consider foreign language study. If you stop and try to translate every word, research its etymology, etc., you will go much slower than someone who does more immersion. (And probably give up!) The issue is NOT one of the structure (of language or of math) but of HUMAN LEARNING. This doesn't mean that you'll never learn all those words (or research the foundations). But for now, you need to acquire key familiarity, first.

2. There is SIGNIFICANT benefit in building your algebraic muscles from the "just calculate" mode. And by the way, there's a buttload of algebra in epsilon-delta or in series convergence, etc. It's not just divided by zero function domain kvetching.

3. And even purist Hardy was very good at the calculate part of math (taking joy in Cleo-like solution of definite integrals). And so was Ramanujan. And so is Andrew Wiles.