Comment by woopwoop | Hacker News Reader

Comment by woopwoop | Hacker News Reader

I think mathematics is in a good place with regards to tolerance of self-promotion. I do not think that we should put up with excessive hype in the name of "humanizing" papers. I do think that a lot of mathematicians do not provide enough detail or motivation for their arguments. Not necessarily motivation in the sense of "why is this important", but motivation in the sense of "we are beginning a three page proof. Let me give you a paragraph to give you the outline so that you can fill in the details yourself, rather than having to read all of the details just to reconstruct the outline."

I do have a pet peeve about mathematical exposition. At some point, phrases like "obviously" and "it is easy to see" became verboten, or at the very least frowned upon. The problem is that it didn't become verboten to skip details (this would be impossible in general), and those phrases actually do contain information. Namely they contain the information that there actually is some detail remaining to fill in here. Often in papers there will be some missing detail which is not so hard to verify, but whose presence is so ghostly in the exposition that I think I've missed somewhere where it was stated explicitly, and have to go back. I feel like this is the case of someone excising an instance of "it is easy to see that" and replacing it with... nothing.

> Not necessarily motivation in the sense of "why is this important", but motivation in the sense of "we are beginning a three page proof. Let me give you a paragraph to give you the outline so that you can fill in the details yourself, rather than having to read all of the details just to reconstruct the outline."

In graduate school this was the most frustrating aspect of paper reading (and writing). It makes sense why it exists however. Papers on mathematics in particular are laser targeted to a particular niche. As the science progresses you need more and more bespoke knowledge of previous work to even start the paper you're reading. There's an implicit assumption you've done your homework, so to speak, and authors likely feel there is no need to provide such a summary. Since, of course, if you don't have the pre-requisite knowledge the paper isn't targeted at you anyway.

Some of it of course is simply a pride thing. There have been many times I've felt the lack of exposition was a way to say "I'm better than you". I have no evidence this is the case but it would not surprise me.

culture war aside, there are many other more accurate ways to say "details omitted for brevity" than "obviously" and "it is easy to see that"

this is also something that makes me want a more interactive publishing format, though i understand the good reasons to stick to the static quo. if it's easy to see, it shouldn't be too hard to write out in a collapsible sidebar for those interested

Like everything in mathematics, "obvious" is a term of art. Broadly speaking, it refers to a fact, proof, consequence, which is necessary for the proof to advance, but which is already established elsewhere, so it does not in itself aid in understanding the proof being presented.

A proof is either providing a new result, or is proving an established result in a new way. Almost always, a proof will need other results, in a way that isn't "interesting" (another term of art). The point of introducing these results as "obvious" is basically to say "here is something which isn't proven by the proof I'm presenting, we need it, but there's no need to derive it to follow this proof", ideally, with a footnote. As language, it is a bit sly: if something is obvious in the normal sense, it will be left out.

It's a problem that the modern style is to elide anything obvious in this sense, rather than in the sense of "anyone who might reasonably read this paper may be expected to know this". But labeling these things as obvious isn't meant in the sense "if you don't know this, you're stupid or uninformed", or in fact "this will be instantly clear as soon as I mention it", it's meant to mean "if you were to follow the breadcrumbs and check up on the 'obvious' thing, it wouldn't help you much in following my proof, so take my word for it. Or, y'know, knock yourself out if this step is interesting to you".

> Like everything in mathematics, "obvious" is a term of art. Broadly speaking, it refers to a fact, proof, consequence, which is necessary for the proof to advance, but which is already established elsewhere, so it does not in itself aid in understanding the proof being presented.

"Trivial" is a term of art (and doesn't mean exactly this) but I'm not sure that "obvious" is.

I think that something already proved is called an immediate corollary if it's a very small step, theorem if it's not that small but well known, a previous result (with a citation!) if not well known, or if you can't cite anything, at least say it's part of "mathematical folklore".

As a working mathematician, I would say "obvious" is most often used for omitted arguments whose analogues are widely established elsewhere.

As an elementary example, in an analytic number theory paper one might need to know that for some constant C we have

5x + x^3 + ln(x) < Ce^x

for all positive x. (In practice, one would not need to determine such a C, just know that one exists.)

Such a claim would usually be described as "obvious" or else stated without any justification. I doubt you could find precisely this claim in the literature, but one can find plenty of very similar arguments in textbooks, and anyone doing research in analytic number theory would find this "obvious".

This might actually be nice, and its probably not that difficult to set up in PDFs or similar.

Would be cool, just because you could see that the details were actually "omitted for brevity" and not "omitted because they're sketchy". And if you rrrreally want to look through the details, then they're fairly easily available.

Downside, it might have a chilling effect on papers because the scale of writing necessary.

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