A beginner could easily fool herself onto thinking she has correct understanding, but actually her proof is buggy. Without feedback, she wouldn’t know.

BTW, I didn't mention this earlier, but for verifying solutions to exercises which ask to prove something, I've never found it useful to see a solution that is simply a proof. The only way to get more confidence that you've proved something correctly is to find someone adversarial (e.g. a friend with similar or greater mathematical maturity) and try to prove it to them while they keep say “why” or “I'm not convinced”. (A proof is a social process; you cannot get feedback on your proof from a static, non-interactive source like a book.) One's proof may be superficially similar to what's in the book and yet not be a proof, while it can be very different and still be fine.

What provided proofs can help with is getting you unstuck when you've been unsuccessful at proving something. So they help with frustration and motivation, as I mentioned earlier (feedback not so much).

There's probably a reason for that. It might be good if people made a sincere effort to actually understand that reason, instead of just dismissing it and trying to explain it away.

Programmers have another advantage over other mathematics students: you can test your understanding of the mathematics by writing actual programs. For example, if you're reading an elementary number theory book and learn about the Chinese Remainder Theorem, you can write an actual program/function for finding solutions to a system of modular congruences.

Agreed. But I personally don't find that to be sufficient justification for not including solutions to exercises. But at least those of us who are programmers do have this option to help supplement our understanding.

All that said, it's definitely comforting to have solutions to exercises: there's a boost in motivation from being told that your answer is correct and you can progress to the next section

I guess everybody approaches this different, but for me, it's not a question of comfort or motivation... I'm literally blocked from proceeding to the next section until I know I understand the current section. I mean, yeah, it's a self-imposed limitation, but it's no less real as a result.

Of course, at the end of the day, it's always possible to make progress, because you can check your answer other ways (using math.se, /r/cheatatmathhomework, the physicsforums.com boards, etc.) but for my money, a good textbook is one which provides solutions, or for which a solutions manual is available (to students, not just to teachers).

• Level 1: You're not seeking much understanding, just reading the book like a novel, maybe glancing at the exercises and trying to solve one here or there. (As we're talking about solutions to exercises, we can ignore this demographic for purposes of this discussion.)

• Level 2: You're seeking some amount of understanding, the kind that would be enough to pass a test. You read through the theorems, then you try each exercise (or most of them), want to be sure you've got it right and could solve it again in a test, etc. Most of the “work” (the kind with pencil and paper) you do with any section, you do when you're solving the exercises.

• Level 3: You're trying to really understand the mathematics, not just what would be enough to get an A if you'd been taking it in a classroom somewhere. You read the book carefully, you treat every theorem as an exercise (i.e. try to prove it yourself before reading the proof given in the book), ask lots of questions (see the Halmos quote starting with “Don't just read it; fight it!”), try lots of examples and counterexamples. Conversely you may even ignore the details of a proof that's written out in the book, if you have one of your own and can see at a glance that there's nothing mathematically interesting going on in what's in the book. Or you may read the book's sections in a different order, after getting a feel for what it's trying to do (see Thurston's quote about “the best psychological order”: https://books.google.com/books?id=5irlDQAAQBAJ&pg=PR9). Either way, by the time you reach the part labelled “exercises”, you've filled up more pages of paper than you will by working through all the exercises. And despite all this, there may still be parts that you realize you only half-understand (or not at all), but you're excited and want to read ahead anyway and know you'll come back to this.

Part of “mathematical maturity” is going from 1 to 2 to 3. Of course, we're all at different levels when it comes to different areas of mathematics, and different life situations. (Frankly for many areas of mathematics I'm at level 0, actively trying to avoid reading anything about it because I find it so painful.)

Anyway, to talk more concretely, when it comes to exercises, there are two kinds:

• those that ask you to find or compute an answer, something which satisfies some equation(s) or inequalities or some property or whatever. These are the “hard to find but easy to verify” kind, and can have one-line answers at the back of the book, and frankly there's no reason authors shouldn't include them.

• those that ask you to prove something. Here a solution given in the book can show you one way to prove it, and can be extremely useful when you're stuck and frustrated that you cannot solve the exercise and would really like to do so before you proceed. (So I always appreciate one when it's present.) What it cannot do in general is give you feedback on whether your proof is correct: there can be many ways of proving the same thing, and many incorrect ways that look very similar to what the book may have.

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> it's not a question of comfort or motivation... I'm literally blocked from proceeding to the next section until I know I understand the current section. I mean, yeah, it's a self-imposed limitation, but it's no less real as a result.

My two comments to that are: Firstly, reminding oneself that the goal really is “I know I understand the current section”, not “I know my solution to this exercise is correct” — if you have doubts about the latter then almost certainly the answer to the former is no, you don't yet understand the current section (even if you can solve all the exercises) and would benefit from more time spent on it. But secondly, the opposite: from http://pi.math.cornell.edu/~hubbard/readingmath.pdf

> This may contradict what you have been told—that mathematics is sequential, and that you must understand each sentence before going on to the next. In reality, although mathematical writing is necessarily sequential, mathematical understanding is not: you (and the experts) never understand perfectly up to some point and not at all beyond. The “beyond,” where understanding is only partial, is an essential part of the motivation and the conceptual background of the “here and now.” You may often (perhaps usually) find that when you return to something you left half-understood, it will have become clear in the light of the further things you have studied, even though the further things are themselves obscure. Many students are very uncomfortable in this state of partial understanding, like a beginning rock climber who wants to be in stable equilibrium at all times. To learn effectively one must be willing to leave the cocoon of equilibrium. So if you don’t understand something perfectly, go on ahead and then circle back.

For what my goals are, I don't necessarily need to get to level 3 (although I might like to if I had infinite time available to indulge in every hobby/interest that I have).

Anyway, my feeling is that exercise solutions are very useful up to, or just around, what you call "level 2." And I expect that a lot of other HN users making this same complaint about solutions, are probably also working at or near that same level. For getting to level 3, I would agree with you that solutions are less useful (although even there I expect they would have some value, at least for self-learners).

It really is a different thing when you're trying to learn this stuff on your own, instead of being in the context of a classroom where you have an instructor and/or TA's that you can get to check your answers. For those of us in that mode, anything that adds friction to the process is extremely frustrating, at least from my subjective perspective.

This may contradict what you have been told—that mathematics is sequential, and that you must understand each sentence before going on to the next. In reality, although mathematical writing is necessarily sequential, mathematical understanding is not: you (and the experts) never understand perfectly up to some point and not at all beyond.

I expect this is another of those things where it kinda depends on where you are. Getting up to and through, say, Multi-Variable Calculus does seem to depend on a lot of earlier subjects that depend on earlier things, in a roughly sequential stream. But beyond that, I can see that things start to diverge and become not-strictly-sequential.