# What is the recommended level of detail for published mathematical proofs?

Suppose I am writing a Mathematics paper to a peer review reputed journal. I have to go from step 1 to step 2 in the paper which needs the knowledge of some well-established theorem X. Can I go from step 1 to step 2 without saying anything about X?

For example, this page contains some identities related to Fibonacci numbers, which (I assume that) may be well known to reviewers. So, instead of explicitly showing the identity, I want to use it implicitly from going one statement to another statement.

Does it give a poor impression on my paper? If I need to include all supporting identities, the paper may become too long!

• If all your readers know that identity, go ahead without citing it. If none of your readers know that identity, then cite it. If (as usual) the situation is somewhere between, it is a judgment you have to make. But a small notation "by [9, Equation (3), page 22]" is not adding a poor impression. – GEdgar Aug 15 '18 at 13:49
• For your specific example, I doubt even the foremost experts on Fibonacci numbers would have that entire long list of identities memorized. – user37208 Aug 15 '18 at 14:44
• If in doubt, I'd write everything out in detail, then ask a more experienced person to check how much of that detail is unnecessary. – Faheem Mitha Aug 15 '18 at 17:24
• I recently got a request from reviewer to give a reference for hamming distance. this implicit knowledge may harm the presentation of tour paper to the broad audience. – seteropere Aug 16 '18 at 1:59

What is obvious to you may not be obvious to others.

Let me repeat a story I told in another answer here. I'll just repeat it, but also note that user Stella Biderman commented that the story may be apocryphal and not from KU at all.

Here is a story (real occurrence) from the University of Kansas about 50 or so years ago. A prof, quite well known and respected, was lecturing in Topology. This meant writing proofs on the board. Occasionally a step wouldn't be filled in with the statement "Of course it follows, trivially, that ...". On one such statement, a student didn't see the obvious connection and so asked. The prof looked at the board and the developing proof for a few minutes. Then walked over to the corner of the (chalk) board and started making notes to himself in a tiny script. He went into totally abstract thinking mode, ignoring the class. After a bit of writing and erasing, etc, he wandered out of the room toward his office. The students followed along, gathering outside the office. The prof started pulling books off of his shelf and consulting them - several books - several more minutes.

Then he seemed to find enlightenment and returned to the classroom. When everyone was again seated he announced. "Yes, of course. It's trivial."

The conclusion is that you should make the connections yourself, and not rely on reviewers grokking it. If they don't make the connection they will comment on it and at least slow down acceptance. It is also possible, as the story above, indicates that you might actually be making too big a step or even an incorrect one.

• I've also heard this story, except the lecturing professor was one G. H. Hardy, with the details slightly different. – Theoretical Economist Aug 15 '18 at 16:25
• I've heard multiple versions of this urban legend as well, but never about anyone in particular. – JeffE Aug 15 '18 at 19:28
• A few weeks ago I noticed that an arxiv preprint had been edited. At first it included a paragraph-long proof of a theorem. The author edited the paper by removing the proof and saying that it's obvious! – user74089 Aug 15 '18 at 20:55
• @CoffeeBliss, Fermat did something similar as I recall ;-) – Buffy Aug 15 '18 at 20:57
• There's a similar story in Surely You're Joking, Mr. Feynman. It ends with the onlookers deciding that "trivial" means "proved". – Ray Aug 15 '18 at 22:38

You need to write your paper so that a reasonable reader is able to verify everything. (Here, "reasonable" reader depends on your audience. The level of detail is not the same if you write a course for high school, a paper for a top generalist journal, a specialized journal, a popularization magazine, slides for a talk, etc.)

You also need to keep in mind that the reader cannot read your mind, only what you have written. You are not writing for yourself; you are writing for your reader.

When you go from one step to the next in a computation, only you know what identity you have used if you do not write it. Even if the identity is well-known, it is not necessarily obvious for someone other than you which one it is. In some cases it's possible to determine easily what was used (e.g. if you write \sum_{n\ge0} 1/2^n = 2 then hopefully everyone will be able to tell what's going on). But in other cases (especially if it's in the middle of a long computation), this may not be the case. So you need to seriously ask yourself whether someone who is not in your shoes can tell what's going on. If necessary, ask a colleague/friend to read it.

Because what will happen to your paper is even worse than giving a bad impression. It will be unconvincing. And an unconvincing math paper is basically worthless.

• +1. But keep in mind that the appropriate model "reasonable reader" is very much a function of the target audience venue. The "same" proof should include very different details in a submission to the Annals, to PNAS, to a specialized journal on algebraic geometry, to a computational geometry conference, to a robotics journal, or to the American Mathematical Monthly. – JeffE Aug 15 '18 at 19:34
• @JeffE For sure. I'll edit my answer. – user9646 Aug 16 '18 at 8:11

Look at what level of detail other papers in the area use.

With journals moving more and more online, length isn't really a problem, these days. Most people would probably prefer to read a 40-page paper that spells everything out than read a 20-page paper where they have to scribble down 20 more pages of calculations and 20 pages of failed attempts at calculations.

Finally, if it's just a matter of replacing "Therefore, Z" with "Z now follows from Theorem X", why not just do it?

If I need to include all supporting identities, the paper may become too long!

As others have noted, there's a middle ground here: you don't have to explain (let alone prove) every identity you use, but you can simply mention it ("by X, we have ...") to save the reader having to guess.

Looking at it from an economic perspective, there's a small cost to you in mentioning X (it takes you that little bit closer to the journal's word/page limit, so you might have to tighten your wording elsewhere), but a potentially huge cost to the reader who has to work out how you got from step 1 to step 2 if you don't mention X at all (I'm sure as students we've all cursed authors who stole hours from our lives in this manner). And there are many readers vs one of you so that multiplies the cost even further.

Thinking of this cost/benefit trade-off can help you determine the right level of detail. When you're wondering how explicit you need to be, you can ask yourself "what are the chances that a reader of this journal will know what I'm doing, and how much time will it cost them to work it out if they don't?"

• Indeed... and a novice may have great difficulty confidently answering all the questions about the readership, and the cost/benefit analysis... so a more experienced person needs to be consulted. – paul garrett Aug 16 '18 at 0:04