The real reason is twofold:

1. Authors write textbooks for universities and professors, not students. They write for the academic paradigm they’re familiar with, in which the professor is primarily teaching a student. The textbook is simply used to arrange a course, and professors would rather not have solutions available in the textbook.

2. Authors don’t earn much money from textbooks and textbooks aren’t weighed very heavily for tenure. There isn’t much of an incentive for them to write textbooks with solutions for the reason stated above, and it’s actually significantly more work to include solutions to your exercises.

To put it very bluntly, there are extremely few non-students in the market for math textbooks, which is already a high-effort and low-return market for authors as it is. To get such a textbook you’d need someone who: knows the material extremely well, is very good at exposition, has a lot of time to invest in this as a passion project and doesn’t mind if it’s not used much or at all in well-known universities.

I completely agree that any book with exercises without solutions is completely useless. I didn't even have access to professors or any teaching assistants even DURING my undergraduate years. So the assumption that you can discuss the questions with your professor doesn't even hold water.

I always wondered why there weren't supplement books/websites of practice problems that were separate from the text book. Seems like an easy way to make money. I guess because it makes it harder to pass on to the students.

It’s true that these are most common for big, intro-to-intermediate classes, and less common elsewhere, but there are a lot kicking around....

    > So, why would we want to write mathematics any differently?

I haven't looked at the book in question, but generally, you don't just "know if" your solution is correct, you prove it, or at least consider examples that can demonstrate if your solution works in some conceivable cases. That's how "unit tests" work in mathematics.

If the book has armed you with enough knowledge to work though a problem to the end, it has done it's job. The real test will come as you build on that knowledge later on in the book or in real-life applications.

Moreover, it's not like this stuff is obscure knowledge. You can find help online or by looking at other books or connecting with others.

In any case, since the study of mathematics is cumulative. Even if the student can't prove or verify that their answer to a specific problem is correct they will eventually reach a point, often within the same problem-set, where inability to solve a problem will force them to encounter their gaps/misconceptions.

The most important thing in a mathematics text, as with any other text, is lucidity of writing and the preparation of the student.

Test cases are not proofs. I don't want to write mathematics, I want to write code and maybe draw some fuzzy and fallible human inspiration from mathematics.

I did find this: https://github.com/pim-book/exercises but it doesn't seem to have taken hold quite yet. Maybe if you and others are so inclined, a nice community effort can solve this issue together and maybe that interactive discussion helps mediate some of the other reasons people give for not providing the solutions up front.

All of these solutions are equivalent (or at least, may be if there are no other mistakes). There are probably other equivalent formulations that I can't immediately think of. But each of these students may see the solution and not realize that they're equivalent. Or worse, they may understand the equivalence of the foregoing statements but make a critical flaw elsewhere in the proof, and still think they have the correct solution.

I'm not arguing against solutions in textbooks. I'm just saying that for most proof-based mathematics they wouldn't be useful as solution checkers.

Can it be done? Yes, absolutely. It is likely to be done? Almost certainly not. This is why we have solutions books - you can usually write a full book consisting only of solutions to another book's exercises.

There's a reason these books declare some level of "mathematical maturity" as a prerequisite. By the time you get to certain content you have to either be comfortable checking a proof (maybe set it aside for a few days like you would any other writing) or have human help.

Of course it isn't ideal to work alone; it's helpful to have other people to point out errors in your proofs and to present their own proofs to you. I just don't think providing solutions helps at all in that way. What gives you the best approximation of that experience is spending a long time working on problems. Sometimes you'll realize you've proved something false and have to debug your own proof, so you'll be collaborating with yourself. Solutions will only make it harder for you to force yourself to have those experiences.

That’s completely fine. Don’t advertise the books for self-learners or “programmers”.

I would like to see actual books targeted at these demographics that provide solutions and aren’t concerned about being adapted in a classroom.

There is a market for those that graduated university a long time ago and would like to learn math outside of a classroom.

These sort of people don’t want to see solutions because they have to turn their work in for a grade, but because they want to check their work and get feedback. Outside of having to consult with TAs or professors they don’t have access to.

It does not provide answers to every exercise--maybe 10% tops--but a lot of the exercises are small and should not be any problem. These are often in a group, where he takes something that would be one hard exercise in another book and breaks it down almost to the level it would be if were part of the main text, leaving just small thing for you to fill in as exercises.

There are a few recurring themes throughout the exercises, where he applies the material of the chapter to some specific application in several exercises (e.g., error correcting codes if I recall correctly), and subsequent chapters continue with those themes in their exercises.

[1] https://www.amazon.com/Book-Abstract-Algebra-Second-Mathemat...

For Linear Algebra, "Linear Algebra Done Right" -- way better than my college lin alg course.

For "general mathematics", I like books that read almost like novels to really grasp the "why" of mathematics, so these are more to embellish your general understanding of "what is the point" type questions -- so things here like Euclid's Window come to mind, and "An Imaginary Tale: The story of square root minus one" will help explain complex numbers in more detail than you ever cared. Reading about the history of mathematics and the writings of some of the greats, like Rene Descartes, Lehonard Euler, Gottfried Wilhelm Leibniz, Issac Newton (the amazing thing is all these greats lived within a century of each other)

http://joshua.smcvt.edu/linearalgebra/

I definitely sympathize with your position. But I think a few things are worth pointing out.

Firstly, it's useful to remember that the purpose of a mathematics book is to teach mathematics, and conversely the purpose of studying one is to learn mathematics, not to solve its exercises. That is, the exercises are not a goal in themselves — they are an additional gift offered by the author in addition to the mathematical content. (Many famous mathematical books don't even have exercises... writing a book is already a lot of effort, coming up with good exercises is additional work, and including solutions is a bit more on top of that.) Moreover, even if a book has exercises, they are never enough, and it is the reader's job to make up many more of their own. (I think readers' giving so much weight to exercises that happen to be written in the book come from school/college mathematics education and testing.)

[1]: https://academia.stackexchange.com/questions/56739/why-dont-...

[2]: https://math.stackexchange.com/questions/57889/hardy-wrights... ("A random sample of 10 maths books I have handy shows that 4 doesn't have exercises.")

[3]: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm (search for "no exercises")

To put it in programming terms: the exercises are the tests, the way to verify that your understanding of the mathematics has no bugs in it. If you're trying to verify that your solutions to exercises are correct, you're in a position akin to writing tests for your tests: it can be useful, but it's a second-order concern. Just as tests should be as “dumb” and “obvious” as possible, if there's any doubt at all that you've solved an exercise correctly, that itself is an indication that your understanding of the mathematics isn't complete, and you need to go back and engage with the material (think deeply, write your own exercises/tests, etc) until it becomes obvious. Of course this takes a lot more effort than plowing through exercises and verifying one's solutions.

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. As computers require a level of precision greater than human readers, this will force your understanding to become really sharp, as you have to deal with all the corner cases as well. (No doubt there are areas of mathematics for which this is hard to do, but most of the undergraduate curriculum can fit, and if you find something that's hard then maybe making it “programmable” can be your unique contribution.)

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, and though there's a cost here (if you needed to be told your solution is correct then maybe you should actually spend more time understanding the mathematics instead of going to the next section — but then again maybe not everyone really wants to understand the mathematics that well), I think that's probably the main thing that's missed when books don't have solutions. As a self-learner, motivation is most of the challenge, and solutions can definitely help there.

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Personally, if I look back at books I went through a good fraction of as a self-learner (and loved), there's an even mix:

• Concrete Mathematics and Generatingfunctionology have complete solutions.

• Burton's Elementary Number Theory has “Hints”, and “Answers” to selected exercises (i.e. those where the answer was a number, not a proof — these are not much work for an author to include).

• The Art of Computer Programming (obviously I've read only a tiny bit of it) has problem ratings and sometimes very terse solutions / outlines of solutions.

• Uspensky and Heaslet's Elementary Number Theory (which I read before Burton) and Analytic Combinatorics have no solutions at all.

I don't think solutions to exercises have made a substantial difference in my engagement with these books, but who knows.

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.