Every time there is a post here about a mathematics book, or even only only tangentially connected (as here), someone complains about the lack of solutions to exercises.

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.