9

I study mathematics and sometimes I have seen book that contains problems but not their solutions. Why are there no solutions available anywhere?

For example, the books I have read are:

  • Neukirch: Algebraic number theory,
  • Lang: Algebra,
  • Liu: Algebraic geometry and arithmetic curves.
  • 1
    Would you please tell us what level of mathematics you are studying? – scaaahu Oct 23 '15 at 7:09
  • 1
    Sometimes textbooks are accompanied by solutions manuals, one of which your instructor possibly has. – user38309 Oct 23 '15 at 7:12
  • 3
    This question might be more appropriate at Mathematics Educators Stack Exchange. – Joel Reyes Noche Oct 23 '15 at 7:46
  • 5
    I'm really not sure why this is closed, and have voted to reopen. Just because it would be on-topic elsewhere doesn't make it off-topic here. – jakebeal Oct 23 '15 at 17:25
  • 1
    Re-opened. If something is on-topic both here and elsewhere, it should stay here... being on-topic elsewhere is not a reason to close here. – eykanal Oct 23 '15 at 19:45
25

For graduate-level math books, the answer is typically not a value but a complete proof---typically of a related but relatively uninteresting topic. For example, one of the first exercises in the Neukirch book you reference is:

Show that the ring Z[i] cannot be ordered.

For such problems, the proof itself is typically much less educational than the process of struggling with the concepts in order to produce said proof.

By the time that students are taking graduate-level mathematical courses, they are expected to have already mastered the general skills of constructing proofs. Seeing how somebody else has proved a point is thus not expected to be particularly educational, whereas struggling to prove something oneself forces a student to engage deeply with the material at hand.

Finally, examples of working with the concepts in the exercises are typically already given in the chapter, in the proofs of the main results, so adding extra examples by working proofs for the exercises would typically be of only incremental benefit, but undermine the value of students having to work through the proofs themselves.

8

Writing a graduate math textbook is a large effort that usually takes several years. The author typically has a vision of what material he/she wants to cover. After writing all the chapters and polishing everything, the exercises are probably the last part he/she works on. They are often meant as a pointer to additional more advanced topics in the literature that expand on the main content of the chapter, and adding solutions could require an effort comparable to writing an entirely new chapter (or several) to present that material in a polished, readable form. So, by that point the author feels that he/she is ready to move on to new projects and in any case the community is best served by releasing the book without exercise solutions. Solutions are sometimes added in later editions if the book is successful and the author is still passionate about the project.

Source: personal experience as a textbook author.

Edit: Another thought that occurs to me is that adding exercise solutions can substantially increase the book's size. If the book is already of a good length (say 300 pages or more) then doing this could make the publisher very unhappy, and could potentially make the book less appealing to readers, who would start being intimidated/turned off by the book's length.

TL;DR: Real Artists Ship; Less Is More.

  • 4
    "the exercises are probably the last part he/she works on" Really? I think most authors would probably collect them over all the years of writing the text. Coming up with a good exercise on a subject is only moderately easier than remembering a joke when prompted to, and if I were to write a book this is the one thing that I would certainly not procrastinate on. (That said, I fully agree with the rest of the answer.) – darij grinberg Oct 24 '15 at 2:28
  • @darijgrinberg yes, well, I was thinking of my own experience but it may very well not generalize to that of others. Also by "works on" I meant "brings to a presentable format", i.e., I certainly wrote up working versions of various exercises long before the end, but they were the last things I worked on cleaning up and polishing. In any case that remark was not an essential part of my answer. – Dan Romik Oct 24 '15 at 5:00
5

It is widely believed that there should be no solutions available, even privately, since this somehow ruins the game. This presumes that there should be "exercises" of the traditional sort in advanced mathematics courses, which is already partly dubious, since (as is often visible in commercially successful texts) it leads to make-work exercises often of questionable interest. I'd agree that there do exist significant, meaningful questions that may not fit into a small book... but would argue that then good write-ups of their solutions/resolutions should be available somewhere as models. Otherwise, all the students ever see is their peers' solutions... which in principle could be fine, but, observably, in practice, often overlook (through misunderstanding) ideas (from the text or otherwise) that make the resolution far more graceful and persuasive. That is, without good solutions, the only models anyone ever sees are "iffy".

(E.g., my abstract algebra text originally aimed to work a large fraction of the traditional significant questions as "examples", exactly to overcome the inertia of traditional-not-so-good alleged solutions of them, and have no "exercises" whatsoever. However, the publisher, who'd already made surprising concessions about intellectual property stuff, really-really wanted "exercises". So I made some near-clones of the worked exercises... And I've received several comments that I'm an anarchist for making those good solutions public!)

So, indeed, I think it's a bad idea to try somehow to suppress "good solutions". People will still grasp at bad solutions, and will be learning deficient versions of things to the extent they learn anything.

By the way, it is certainly not the case that the standard graduate mathematics texts provide means to resolve all their exercises. Often there is a considerable disjunction. Typically, the disjunction is that the theorems in the chapters do not at all suggest any quasi-algorithmic devices for doing computations in any particular case. E.g., abstract Galois theory usually disregards Lagrange resolvents, so does not hint at how to solve equations even when they can be proven solvable by radicals...

Nor is it the case that beginning math grad students are adepts at writing... so there is considerable feedback among them of marginal write-up style, marginal technical viewpoint, too much attention to secondary and tertiary details (often strictly demanded by in-my-opinion misguided texts or instructors), and needlessly distorted ambient language. Good writing models would help people "get over" this.

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    It is widely believed that there should be no solutions available. Really? I'd argue it is just as "widely believed" that having solutions is just great. E.g., Richard Stanley's Enumerative Combinatorics and Donald Knuth's The Art of Computer Programming, surely among the most successful graduate-level math textbooks out there, both have solutions. As another example, my own book was criticized for not having solutions in a review of it published in MAA Reviews. – Dan Romik Oct 23 '15 at 22:54
  • @DanRomik, I myself would agree that there should be "solutions", or, really, "worked examples", but out of dozens of often-used higher-level textbooks, very few have any reasonable "solutions". Atiyah-MacDonald? Lang's books? Any book on "analysis"? It may be that an MAA reviewer is more in tune with pedagogical reality than some... – paul garrett Oct 24 '15 at 22:10
  • I didn't say it is widely believed there should be solutions, only that it is good to have them. Indeed most books don't include them, but I suspect it's not because of some taboo or a negative view of their worth (although who knows, maybe some authors really think that way) but more for the reasons I speculated on in my answer. Certainly those reasons applied to my own experience. – Dan Romik Oct 24 '15 at 23:39
2

One possible reason is that the author foresees that students will be using the book as a textbook, with professors assigning homework questions from it. Thus, they omit solutions from it.

Another equally possible reason is the lack of space (printing limited by number of pages).

Yet another possible reason is it takes too much time/effort on the part of the author. (Edit: changed the adjective)

-3

It's a commercial decision driven by the wishes of professors (who assign textbooks). Not having the answers makes them more important. I know of one author who wrote a very well regarded textbook with all the answers and had to remove them in second edition because his publisher said it wasn't selling as well.

Note that textbooks from 100 years ago commonly had all the answers ESPECIALLY in college level math. So we have actually become less liberal, more restrictive over time.

I recommend people consult Schaum's, Kahn Academy, or look for books like Stroud or Granville that contain all the answers. It very much helps self study or even directed study (since the major learning comes NOT from the professor, but from working problems on your own).

  • 3
    Your comment seems more relevant to undergraduate textbooks. The question asked here is about graduate level mathematics textbook which have very different commercial and intellectual priorities – Yemon Choi Jul 15 '17 at 14:28

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