The other night I saw some old friends from college. They had both taken Math 230, the super hard introductory class for math majors: it covered linear algebra and some real analysis, but apparently manifolds also made an appearance. I mean they’d both tried to take it; one of them didn’t make it to the second semester. It was taught by Yair Minsky, who incidentally did lots of incredibly ground breaking work on the curve complex I keep talking about. At some point one of them said “[the difficulty] wasn’t Minsky’s fault; it just went over my head.”

This stuck with me because I’ve heard something similar before: a few months ago I gave an Ignite! style talk in a course full of scientists. My feedback afterwards was excellent, but at some point the professor said “your explanation was great; I just couldn’t keep up with it.”

Here’s the common thread, as far as I understand it: “*if you don’t understand something a mathematician says, it’s not their fault for communicating poorly, but yours for being too stupid/slow/dense to get it”.*

My friend and this professor are EXTREMELY far from being stupid/slow/dense. And I’m a pretty approachable person. In my mind, there was no reason for this professor to think that the math I was doing was above his head (as a test I explained the exact same topic to my friend)- if you don’t understand something, you can always ask a question. **But to these two (and many others), math is an inaccessible topic maybe composed of magic.** Like we probably sit in our offices with cauldrons a-simmer to come up with and understand these fantastic abstract ideas. And outsiders can’t do so because they haven’t taken the holy vows of mathematics.

I wonder if this phenomenon is specific to math: I very often hear mathematicians saying similar things, blaming themselves rather than the speaker/writer for not understanding something. I do the same thing.

If I explain something to you, blog reader, and it’s not clear/you don’t understand despite making your best or even reasonably good efforts, then I am doing a poor job. Math is hard, but if you can learn other academic-y things, and if you want to, you can do math too. I sincerely believe you don’t have to be a super genius to take some joy in say, Cantor’s diagonalization argument (will blog it sometime), or that Ramsey theory proof I wrote about earlier this summer.

I would/do love it when people tell me that something I wrote was unintelligible, and exactly where they got lost (my super excellent roommate does this all the time). And then I fix it/write more/use better wording. So if you read my math posts, please comment or email me or tell me in person that this or that was hard to follow. There’s no need for self-effacement (“it was written well but I’m just too stupid at this to get —-“). **I believe in you. You can understand this. I just need to explain it better**.

OK that got sort of sentimental. Thinking more about the issue of accessibility in mathematics, and I think about girls and mentorship and STEM a lot, naturally leads to wondering if we should make this stuff more accessible. What’s the point of having more people understand math? For that matter, what’s the point of us doing math? I mean, what net good are we doing for the world by thinking about and solving these problems?

I strongly believe that basic research is important and essential. And it’s often (and I think should be) undertaken with no preconceived ideas of future applications. But I suspect that the vast majority of mathematical knowledge out there today will *never* have practical applications, or lead to mathematics that does so, or inspire mathematicians that do so. Yes, definitely, someone (many people) should do math. But do we all/I need to be doing it? Would our problem-solving super-analytic brains be better spent attacking more immediate real-world problems?

You guys, I have zero answers here. The people I know do math for fun, for intellectual challenge, and to further human knowledge (ish). Sometimes it seems too hedonistic to me- I asked someone on a train what she thought, and she said “we live in a world where we have this luxury now.” It’s a luxury to be able to spend my days thinking about these hard, divorced-from-the-real-world problems. It’s a luxury to play these games. Sometimes I feel guilty for indulging in that luxury.

That’s it that’s where this post ends. I’m a little down now. Sorry if you are too. I’ll include some puppy pictures from the internet to make us feel better.

Loved this post! Going to share!

Thanks so much, Kathy! I was afraid it was a little too pathetic so it’s good to hear you liked it!

“But I suspect that the vast majority of mathematical knowledge out there today will *never* have practical applications, or lead to mathematics that does so, or inspire mathematicians that do so.”

I take some small comfort in believing that it is impossible to know in advance whether a piece of mathematical knowledge is of the type that will lead to results. Timothy Gowers makes a strong case for this fact in his 2000 address to the Clay Mathematics Institute. For empirical evidence, one may read A Mathematician’s Apology, in which G. H. Hardy wrote of his work in number theory, “I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.” His work ended up being extraordinarily important after his death, not only for its direct applications (some are listed at https://en.wikipedia.org/wiki/G._H._Hardy#Pure_mathematics), but for its influence on the direction of Number Theory in general.

If we believe that some mathematics will be useful but agree that it is impossible to predict which, then it stands to reason that we should worry less about applicability and more about mathematical richness and elegance, which are the qualities I believe most of us are reacting to when we call a piece of mathematics “interesting”. Approaching mathematics from this perspective has the added bonus that it brings out our best possible work, since we are intrinsically interested in the idea as an end in itself, rather than as a means to an end.

I’ll definitely check out Gower’s lecture. Thanks for your thoughtful comment. I’ve read a philosophy paper before (I’ll have to look it up…) on *how* the mathematical community dictates what is “interesting”- certainly there are trends in research, subject to the whims of the mathematical community. But the “outside” community, I believe, cannot understand what we mean by beauty and richness without an example- every non-mathematician I’ve ever spoken to wants to know the applicability of research, and to say we do it essentially for fun seems like such a letdown.

You’ve set a noble and challenging goal for yourself when you take this stance: “I believe in you. You can understand this. I just need to explain it better.” So maybe it would be helpful to set a similar challenge for explaining the “purpose” of doing math: “I believe in you. You can understand why this is fun, beautiful and rich. I just need to communicate it in a way that won’t seem like a letdown.” It won’t be easy, but from what I’ve read on this blog, I think you are up to the challenge. A good warm-up might be to imagine someone who does not understand the purpose (fun, beauty, richness) of baking or pun-making*, and think about how you could communicate that. I hope you will keep working on this, because I’m really interested in seeing the results.

*Sorry. I know it’s kind of disturbing to imagine such a person, and their almost certainly joyless existence!