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Comment by jeff76 | Hacker News Reader

Disagree. Some proofs are very straightforward and standard. Those are easily checkable with a solution guide.

Yes, some are straightforward. Most are not. For example, suppose a student shows that T: V --> W is bijective and linear on the way to proving a solution to an exercise. The textbook solution shows dim(V) = dim(W) and T is injective instead. Another student shows that dim(V) = dim(W) and T is surjective, and still another student simply uses the rank-nullity theorem directly

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.

Yes, which is why you include "several proofs of this could include..."

Sure. But now you're talking about half a page to a full page of solution commentary for each exercise. Math textbooks at the upper-undergrad and grad level usually have 10-20 exercises per chapter. Those exercises will usually require a student to fill out a quarter to a half page for any correct solution. Even if an author was decidedly economical in how they did it, writing comprehensive solutions for each exercise would constitute another book's worth of material.

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.

When you hit upon a proof like that, you know it. If you're struggling and need feedback, then your solution is probably more complicated than the elegant one in the book, and the question you're asking is, okay, my solution isn't great, but is it even a solution?

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.

Think you missed my entire point of including solutions for the beginner, i.e., someone that lacks math maturity and the reasoning of how easily such a person could fool themselves into thinking their thought process is valid.

Someone who is a beginner will be working on easy material with a textbook that has very mild expectations of "mathematical maturity." Textbooks mostly (with a few intentional exceptions) calibrate the mathematical maturity they require according to how much work a student will have done in order to be ready to tackle the book's subject matter. Some people might think they can be ready to do problem sets in an advanced math textbook while skipping the work that would have given them the maturity it requires, but they are kidding themselves. They can't advance on the spectator track and then make a horizontal move over to actually doing the math. There's nothing wrong with people passively reading the material if they enjoy it, but if they want to tackle the problems, they are going to have to work through the earlier material by solving problems as well, and if they do that, they'll find that their sophistication increases along with the demands on it.

The way you were arguing makes it sound like you are against beginners having access to solutions, or that you completely misunderstood my initial argument. I am not arguing about anything other than that. If you want to address that in particular sure. I am not saying anything about people skipping material and getting bitten by it later.

Even the first textbooks that start introducing abstract math and developing mathematical maturity in beginning college students often (usually?) have problem sets without solutions, and I'm defending that. I think it's better than providing solutions, even for people working alone.

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.

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