# MathOverflow to the rescue — not in the usual way

Every now and then we get a post to meta about ways in which MathOverflow has improved the quality of someone's research. One way I haven't seen mentioned is one that just now happened to me, and (though my memory is a little vague) probably not for the first time:

In the course of trying to devise a homework problem in game theory (!), I stumbled onto an extremely complicated looking expression involving multiple values of multiple polygamma functions. Mathematica told me that my expression, evaluated at any positive integer, is always rational. This was useful for my application, but it appeared to be a sheer miracle. I've spent the better part of the past week trying to demystify that miracle.

Finally I decided it was time to unleash the power of MathOverflow, and carefully composed a question that I then hesitated to post, for fear that I'd missed something obvious. So I spent the better part of the past day going over and over my draft of that question, just to make sure.

And tonight, I had one of those "Aha! It is obvious!" moments. I now understand exactly what's going on. I also see, beyond any shadow of any doubt, that it really should have been obvious days ago, and I have no doubt that if I'd posted it, any number of people would have been quick to see what it took me a week to decipher, and I'd have thanked them profusely, and I'd have felt glad to have my answer, and only a little sheepish. So that would have been a good outcome. But getting it on my own means I had a series of minor insights I will never, ever, ever forget, and that's an even better outcome.

Moreover---and this is really the point---if MathOverflow hadn't been there, I might have decided three days ago that this problem was just too hard for me, and given up on it. Knowing that I could almost surely get my answer by posting ruled out giving up. And then fear of embarrassing myself ruled out posting. The only remaining option was to solve my problem.

So MathOverflow just made me a little smarter by making me feel just a little intimidated about revealing how un-smart I might be (and in fact, in this instance, turned out to be). One more reason to be glad it's there.

Edited to add: Speaking of things that should have been obvious but weren't, I've only just realized that this is not a question and hence perhaps inappropriate. If that's a problem, I won't object to a deletion.

• If you are looking for similar stories, my present research path has been highly influenced by the presence of MathOverflow. I can't claim any immediate success stories, but I have found things mathematical that I have seen nowhere else, and posting them on MathOverflow (and getting little response) seems to confirm that these findings are original. Gerhard "Perhaps Also Of Some Interest" Paseman, 2020.03.17. – Gerhard Paseman Mar 18 '20 at 5:13
• I suppose you could make it into a question by asking "Has anything like this ever happened to you?" – Gerry Myerson Mar 18 '20 at 6:19
• It might be "sharing knowledge policy" which is welcome at stackoverflow in general , but un-welcome at mathoverflow - post question and answer by yourself should be nevertheless applied to that case - I mean I would suggest to post a question and answer by yourself to mathoverflow. – Alexander Chervov Mar 27 '20 at 16:56
• Thank you for this post. Would not it be appropriate to still post that prepared question together with an answer? – მამუკა ჯიბლაძე Apr 7 '20 at 19:28

This is the very essence of the Feynman Algorithm for solving problems:

1. Write down the problem.
2. Think real hard.
3. Write down the solution.

None of these steps is easy, but we tend to skip the first step in many cases. MathOverflow does indeed force you to take it more seriously.

But all in all, well done!

In programming, this is known as rubber duck debugging: you often figure out the solution during the process of explaining your problem in detail, even if you are explaining it just to an inanimate object like a rubber duck. It is a well-known side-benefit of Q&A sites. One of the founders of Stack Exchange described it in a blog entry.

You may remember Stack Exchange adding a Clippy-like rubber duck to their websites in their 2018 April fool's day joke; it was a reference to this.

This story resonates with me a lot. For me, when grappling with a new problem, it's really become an integral part of my process to try to formulate sub-problems and related questions as MO questions. Only in a minority of cases does this result in actually asking an MO question (in which case the answers are typically helpful, of course), but the other cases are often like this -- the process of formulating a question properly leads me to solve it or to realize I should be asking a different question.

Sometimes I even get to the point of writing down my MO question in the actual question box on the website before understanding the answer myself, and usually this is because the question-writing process has gotten me there. This is because the writing process forces me to ask questions like

• Have I framed this properly?
• Have I followed up on the obvious leads in the literature?
• Have I crunched through the most straightforward, obvious first steps?
• What will the first comment be? Can I address it ahead of time to advance the starting point of the conversation?
• ...

Sometimes this shows -- for example, this question is a case where the process of formulating an MO question led me to answer it myself, but in this case I had just posted the question before I realized that in writing out the question properly, I had already answered it. It was a bit embarrassing, but looking back I think it offers a window into how one can use MO to bring some order to one's thought process.

Even when it works out differently, sometimes there are spin-off questions which make it onto MO as being of independent interest even as I realize they're not as central to my research as I originally thought. This one and this one are recent examples for me.

• In that question about factorization systems, shouldn't you post an answer to your own question? I thought that this sort of this is encouraged (but not done often enough). – Toby Bartels Mar 29 '20 at 5:08
• @TobyBartels You're probably right. Sometimes I feel like I do this too often -- for me it's often a symptom of exactly this phenomenon of not having thought through things completely. But in the end, it's probably best to have some sort of answer recorded! – Tim Campion Mar 31 '20 at 11:32
• You can also wait a while for other answers, and if none are forthcoming, then you can write your own. (But you have to remember to check.) – Toby Bartels Mar 31 '20 at 14:57

On top of forcing you to state the problem clearly, composing a question for MathOverflow has the bonus advantage of the suggested related questions being much better than the usual search results. More than half of the questions I've started drafting on MathOverflow have been answered this way before I posted them.