Public Displays of Mathematics

Let me start by saying that I’m in favor of making the effort to both educate the public about mathematics (as well as science more generally) and to convey to them a sense of the excitement of our discipline. But the science always has to come first, and should never be twisted for the purposes of sensationalism. I understand that I have more antipathy than most towards this sort of thing, but I wanted to discuss a few examples in this post of things which I have seen recently that have particularly annoyed me.

Examples of “math in the news” which didn’t quite live up to the hype have been around for a while. There was the whole E8 thing. I’m not sure how these things start, but it has to involve some unholy trinity of sensationalist journalism, self-promoting universities (or institutes), and complicit [or merely naive] authors. It’s not so easy to untangle the web of blame in any particular situation, but, at the very least, let me recommend to anyone to avoid AIM when it comes to matters of publicity. Not even wrong had an discussion on this particular case, but there’s also an interesting follow up on the E8 article here, where, to give credit where it is due, Oliver Roeder somewhat redeems himself for his previous crimes. (To be clear, the math behind the E8 story — removed from any breathless claims about how it will change the word — is pretty interesting.)

Second, let me also admit that I could be completely wrong about all of this. Obviously I’m not the target audience for popular articles on mathematics, and maybe, even when the truth is stretched beyond recognition or even just a little in that direction, such publicity is good for mathematics. I really don’t know. I acknowledge that I can be out of touch on some issues. I am, after all, someone who gets annoyed by the idea of NPR discussing non-classical music. But (at least in that case) I am sufficiently self-aware to appreciate that my opinion on the matter can safely be disregarded. Maybe that’s true in this case as well. Still, I feel that one can write with excitement about mathematics to a general audience and still be faithful to the mathematics. I grew up with a collection of articles reprinted from Scientific American in the ’60s and ’70s, and they were never afraid to challenge their audience with difficult technical concepts in order to elucidate some often difficult but important idea. One can also find such writings nowadays, although almost always in blog form rather than in print. (I don’t really read popular math blogs that much, but this blog is an example of how one can demonstrate the delights of our subject without being super-technical and yet still being honest about the underlying mathematics.)

I have little faith that university publicists or (even worse) journalists have much interest in being accurate. But I save the most opprobrium for mathematicians who make unreasonable claims about their own work. To make it clear, I was previously a little critical and cheeky about the publicity surrounding the LMFDB here, but that wouldn’t even rise to the level of a small misdemeanor compared to the crimes outlined below. I’m taking here about levels of intellectual dishonesty which would be more appropriate at a business school.

Plimpton 322: I was first alerted to this by a recent post on facebook linking to this article. A few sample sentences are as follows:

“This means it has great relevance for our modern world. Babylonian mathematics may have been out of fashion for more than 3,000 years, but it has possible practical applications in surveying, computer graphics and education. This is a rare example of the ancient world teaching us something new.”

At first, I interpreted this as merely the type of quality rubbish one expects from the Guardian. But then, to my horror, I clicked on the accompanying video to discover that this nonsense was coming directly from the authors (mathematicians, not historians) themselves. Due to ignorance, I will leave aside here the historical claims of the authors (which to be fair are the main point of the article, although they also have been met with skepticism by actual historians) and merely comment on a few of their mathematical claims, specifically, that “Perhaps this different and simpler way of thinking has the potential to unlock improvements in science, engineering, and mathematics education today.” This is so patently absurd that it’s not worth spending much time discussing it, but here goes. There is nothing (and indeed far less) in the “exact” nature of the tablet that isn’t immediately a consequence of the usual rational parametrization of the circle. (Yes, you can use slopes instead of angles!) The “sexagesimal” aspect of the tablet is also a red herring. If you take any base B which is not a prime power, it admits a similar tablet comprised of Pythagorean triples with a side length one and the two other sides given by terminating decimals whose ratios (for large enough triples) are dense in any interval. Here’s a fragment of the corresponding decimal version:

(The column consists of triples \((\delta,a,c)\) with \((a,b,c)\) a Pythagorean triple, b an S-unit with S=10, and \(\delta = (c/b)^2\) given as an exact decimal. The triples are ordered by \(\delta\) and limited by some height restriction.) Since 60 has more distinct prime factors than 10, the size of the entries in this table is little larger, but that’s about it. Of course, even if you wanted to base a primitive trigonometric system on exact ratios, you would much prefer to use rational points on the unit circle of small height, rather than insisting that the ratios involved had finite expansions, which is very restrictive. (I’m not denying that this may have been convenient historically for numerical computation, I’m only addressing the absurd claim that we have something mathematically to learn from this tablet.) I would stop short of saying that the claims of the authors are fraudulent, but I would go way further than to say they are simply overreaching. Let’s stick with saying that they are vastly overstated purely in order to drum up public interest for their own professional enhancement. And this type of irresponsible behavior leads, inevitably, to this:

OK, enough of that nonsense, let us move on.

Numberphile: OK, perhaps this will be a little bit more controversial. Perhaps the correct thing to bear in mind with this rant is to recall my comment about NPR and classical music above: just because it really annoys me doesn’t mean that I can’t simultaneously accept that it’s probably a good idea for NPR to be somewhat inclusive (I guess). Numberphile is funded by MSRI and that’s probably a good thing, but it still (sometimes) annoys the hell out of me. Should anyone care? I’m not sure. It’s also important to note that there are better and worse numberphile videos — if they restricted themselves to the good ones I would only have very positive things to say. Readers may be aware of the infamous 1+2+3+4+5…=-1/12 video (see here for a takedown). But it gets worse. And not necessary worse in the “this is just rubbish” kind of way, but in the “this gives absolutely the wrong impression about what mathematics is and dresses it up as some ridiculously stupid parlour game instead of something with deep and profound connections” kind of way. There’s a lot of dross to draw from, but here is one typical example:

OK, so what’s the problem? A tiny bit of mathematical knowledge reveals that the (concept of) the Mills’ constant is an interesting observation about what we know concerning (upper bounds for) gaps between primes. But that’s not what one gets out of this video at all. At first, it seems as though there is some mysterious prime generating constant — perhaps you as a youtube viewer can discover a closed form and reveal the mystery of the primes! But this is just rubbish, the actual number (or smallest such number) is of little interest. It’s true that they are slightly more honest towards the end of the video, but the actual mathematics behind this story is always completely obscured. Honestly, if they had just spent a little time (maybe even a minute) saying something at least tangentially related to the real point behind Mills’ constant I would have been much happier. Is enthusiasm better than accuracy? (genuine question). To me, Numberphile can sometimes seem to be the video series that will launch a thousand cranks rather than a thousand mathematicians. It doesn’t help that there’s a bit too much emphasis on recreational mathematics in the worst sense (2 million views between them), which are to real mathematics roughly what eating play-doh is to molecular gastronomy. (Hmmm, maybe a bad analogy, I can totally imagine a play-doh course at Alinea.)

It’s not easy to get it right when it comes to publicizing mathematics (and mathematicians), but it can be done (here, here, and here to name three recent examples). But it helps to start with something serious and try to explain how interesting it is.