How detailed should proofs be in a mathematical PhD thesis?

Question

I heard this suggestion from a mathematics professor:

In writing a mathematical Ph.D. thesis, it is far more tolerable to be tediously-lengthy than having a gap in the proofs.

I think what he means is that whenever in doubt, adding more details to make the argument clearer is always better, even if sometimes doing this may make the proof too wordy.

Now if I really follow his advice literally, it seems there are too many details for me to write. For example, I don't even feel safe to write "the case n=1 is trivial" when proving by induction, or "by a direct calculation we have the following result". Another example is, whenever writing a commutative diagram (some of my diagrams are 3-D), I doubt if I need to prove that every small triangle or rectangle is commutative. (Actually I have asked this question but have received no answer so far.)

Thus my question is: How detailed should be the proofs be in a mathematical Ph.D. thesis?

Answer 1

I usually give the following advice:

If something is clear, just write it down clearly.

Of course, there is a caveat: What is "clear" clearly depends on the reader, and also what is considered a "clear explanation" certainly depends.

But the message is: If you are tempted to write "The case $n=0$ is obvious." think a minute about the explanation and how to write it down. If you then think that writing down the explanation does not add anything, leave it out.

But in general, I would say, being more verbose is the better option.

Another test you could apply is: Write down "obviously…" but note down a longer explanation somewhere else. After a week or two reread that part and try to recover the result again. If you don't have any problem then you are probably fine without a longer explanation.

Some resource that may be helpful for your proof-writing skills is the talk given by Leslie Lamport on proof-writing last year at the Heidelberg Laureate Forum: Here is a blog post about it and the whole talk is here.

Answer 2

Think about who is going to read your thesis. My thesis had four audiences: me, my advisors (2), and my mom. Unless you're Tate or Serre you probably have a pretty similar number.

I wanted all the details in it so that I wouldn't have to sweat over them again when I rewrote it into a journal article. I knew my advisors could skim past what they thought was obvious. I knew the whole thing was going to be incomprehensible to my mom and so it might as well be long to be impressive. So I wrote it long.

Answer 3

The useful distinction is not between "long-winded" and "concise", etc. Verbosity per se is not helpful, nor is succinctness bad (it's good).

Wordiness does not automatically prevent gaps in arguments. If anything, it may merely obscure them. Terseness in arguments is not the same thing as "gaps in reasoning".

Yes, discussions can be shortened by deliberately omitting the least-interesting fragments of proofs or explanations. For experienced people, who trust themselves or are trusted to be able to fill-in "standard" gaps, this doesn't matter. Perhaps a novice should doubt to some degree their own capacity to distinguish "standard" from "critical" issues, and this distinction is exactly what other people will wonder about. In fact, a typical thesis may exactly be a protracted exercise in making sure that one can carry out all those (often eventually boring, not too dramatic) routine arguments "once in one's life".

Answer 4

You must be wary of "COIK." Clear Only If Known. In a thesis (dissertation?), one is generally too close to the content to spot jumps. Too much is better than the committee sending it back for multiple edits, i.e., they get lost at point A, so they send it back, you edit, now they get lost at point B, you edit, etc.

If some of the material is too tedious for the main text, however, one option is to dump that stuff into appendices.

Answer 5

Detailed enough that you could have followed it when you were a third year grad student.

Answer 6

This, as everything else, is going to depend on context. What role is this result and proof going to play in your document? Have you already published the proof in question elsewhere? How cool is it really? These are some questions that can drive your approach.

If this is as yet unpublished material, you can't afford to leave the details out, so they're going to have to go somewhere. If your whole thesis is this one result, put your proof(s) in the main body. You can still break the more tedious parts off (say by impromptu lemmas) and put them in appendices, if that helps your flow, but all the details should appear. If the things you think are trivial really are explainable with a few words, use those words.

If the thesis is about more than the result (how the result applies to other areas, how it applies to solve some other class of problems, did you just create a new branch of mathematics), then you can try to describe the proof in a way that relies more on intuition (or give an overview), not just so that laypersons can get a better grasp of it, but so that everyone who wants/needs to read it can get a good feeling of why it must be true, and then it becomes easier to explain its importance. Then you can put the hard, dry version elsewhere, possibly in another chapter ("Chapter 9: The Nitty-Gritty") or an appendix, but the details still must appear somewhere. Contrary to Big T Larrity in Code Monkeys, this is one occasion where you never want to leave 'em guessing.

Answer 7

Whenever a proof, a theorem or any other such thing can help the reader to understand any single step of the procedure then do write them down fully and even verbosely, if necessary. On the other hand, whenever it is only about mere technicalities that do not add any deeper understanding or insight it is best practice not to be too pedantic, for the sake of readability.

"Things must be made as simple as possible, but not any simpler." (A. Einstein)

Answer 8

Saying it is "far more tolerable" does not mean it is desirable. Proofs should not be tedeously lengthy if they don't need to be. You supervisor is telling you that there is more tolerance in the assessment of a dissertation than in the assessment of a paper submission, which is true. This certainly doesn't invert the normal principles for giving a clear parsimonious proof.

Answer 9

It should be stronger than the Italian geometers who made entire theories which were false.

Answer 10

In general there are two aspects to this:

Overkilling It: You do not have all the time in the world. Therefore, you can't spend years on the thesis. As Long as you backed up the assumptions during the introduction and background section of your thesis; and you are able to connect the dots, during your contribution section, you should be fine. Because you did provide enough information (e.g., background material, citations, etc.) and then pointed out your contribution, so the reader can follow and do more readings on this based on your direction points.

Fault Assumptions With no Backup: In this case, you just wrote something to fill the papers; with assumptions that you didn't thought it through while writing your thesis. The experience academics look for these faulty parts of the thesis and will "grill" you during your defense, because they know you didn't thought it through; regardless how good the other parts of the thesis are.

So the rule of thumb here is as long as you know how to defend these 'gaps'; you should be fine and don't overkilling it because the clock is ticking.