# Is it acceptable to include footnotes in mathematical proofs?

When writing a technical article that includes mathematical proofs: Is it acceptable to have footnotes within the body of a formal proof (to elaborate/clarify non-essential points) or is the use of footnotes in proofs considered bad practice?

• Footnotes should not be used to explain part of the proof, but are sometimes acceptable for asides. Many authors don't seem to like using footnotes, however. Apr 7, 2015 at 12:22
• Bear in mind that footnote markers might be confused with exponentiation or other mathematical operators, in particular if somebody just wants to look up a part of your proof and doesn't look all that closely at the surroundings, so keep your footnotes well away from equations. Apr 7, 2015 at 18:59
• I've seen it, but largely in older works. It's not as common these days, I think. I used a brief one in a paper of mine, but not as a way to clarify, just as an aside which was too long to mention in the proof as it would have derailed it. Apr 8, 2015 at 1:42

Like almost anything else in writing, style should follow function. In other words, if you think that a footnote makes it easier for your reader to follow a text (for example, because it explains an aside that is too long for a parenthetical remark), then it is appropriate to use one. There is no general guideline whether footnotes are acceptable or not. It all depends on what you want to say and how you want to say it -- choose whatever means you think are appropriate to tell the story of the proof.

I personally find footnotes exactly as acceptable or unacceptable in a proof as in other parts of a paper. A well-written proof often contains quite a lot of explanatory prose, and there is nothing special or magical about a proof that prevents one from having an "aside" comment within it.

I would, however, find it very strange to have a footnote in the statement of a theorem, just as I would find it strange to have a footnote in an equation.

• A footnote in an equation --- as normally written, with a superscripted number, asterisk, or similar symbol --- would just be begging to be misunderstood. Apr 7, 2015 at 13:44
• If i remember correctly Kunith addressed footnoting equations, But I can't remember his solution from the top of my head. Apr 7, 2015 at 16:46
• @MarkMeckes I couldn't resist: xkcd.com/1184
– yo'
Aug 17, 2015 at 5:18

I cannot remember ever having seen a footnote in a proof. My instinct would be to put elaborations or clarifications into a remark after the proof.

In an actual math paper, you could actually have a "Proposition 1", followed by its "Proof", then a "Remark 2". For instance:

Remark 2. Note that \$X\$ in the preceding proof does not have property \$y\$, which would have allowed us to use the technique from Foo & Bar (2015).

However, this may well depend on your field, on your journal, on the editor and on the referees. Some of these may well frown on footnotes, while others may be fine with them.

I'd say you'd be safest with putting additional material into Remarks.

• One thing that comes to mind is that the particular example of a remark you've given here is something that does not add any value to the proof itself. So it makes sense to leave it for afterward. For different kind of remark, which actually clarifies some point in the proof that might otherwise be unclear to a few readers, it makes more sense to reference it directly from the relevant point in the proof, e.g. by footnote. Apr 8, 2015 at 4:47
• (Shudder) Numbered remarks make the baby Jesus cry. Just say it. Jul 28, 2015 at 1:39
• @JeffE I don't see why you make this remark (no pun intended). I see it the opposite way: If you remark something but don't plan to refer to it later, you don't need the word Remark. at all, you can simply make it part of the article prose.
– yo'
Aug 17, 2015 at 5:21
• @yo' My point exactly. Even if you refer to it later, just work the remark into normal prose. ("See our remarks after the proof of Lemma 23.9.") Aug 18, 2015 at 12:58
• @JeffE Works in some cases better than in other ones. But I get your point and I agree, this is one of the ways :-)
– yo'
Aug 18, 2015 at 12:59

Footnotes within proofs are fine and to be recommended.

I've read tons of (multipage) proofs (in signal-processing, or machine learning) where the author's train of thought was impossible to follow and obscured by unnecessary steps and diversions. Twenty pages of matrix algebra and higher-order derivatives in order to establish some underwhelming pseudo-result which follows directly from well-known basic theorems familiar to people in the field; or else can be sketched out in one or two paragraph of paraphrase, to articulate the necessary motivation before wading into verbose proofs.

A proof [in most academic domains] should be aggressively made as compact as it can be, without removing anything essential. Shunt all non-essentials, sidebars and footnotes into footnotes or remarks.

One frequent paradigm (common in signal processing): assume an independent normal distribution on multiple variables (where this is known to be a ridiculously invalid assumption), waste 8+ pages deriving a non-result based on that assumption, before instantly discarding and invalidating that result and introducing a new (but much weaker) derivation with the proper assumptions that should have been used in the first place, but render the subsequent result very basic and uninformative. Finally fall back on showing a few graphs or experimental results to discuss how the process behaves in reality .

• I don't think they should be recommended. If your writing style is such a disaster that it necessitates footnotes, perhaps you shouldn't be the one writing the paper (instead a colleague should be writing it). Apr 8, 2015 at 1:45
• @CameronWilliams it's nothing to do with "disasters". My background is EE not math, but I've often seen authors in both domains do unnecessarily overly long proofs just as a form of showboating.
– smci
Apr 8, 2015 at 10:18
• Downvoted for the sentence in bold. Concision per se is not a virtue. What you should aim for is clarity. Jul 28, 2015 at 1:36
• 1-page proof with a 19-page appendix is still a 20 page proof. What's the point? Jul 28, 2015 at 5:37
• If a 1-page proof requires a 19-page appendix, then the 1-page proof is not clear. Jul 28, 2015 at 22:39