draft started January 4th; been thinking about this for a while
When government funding of math was under threat, Terry Tao made a mathstodon post justifying the importance of math for society. The central evidence he provided was the following anecdote (emphasis mine):
One well-known example that I was involved many years ago was the 2004 program https://www.ipam.ucla.edu/programs/long-programs/multiscale-geometry-and-analysis-in-high-dimensions/ on Multiscale geometry and analysis in high dimensions [….] I participated extensively in this program, and in particular interacted quite a bit with one of the organizers (Emmanuel Candes) as well as Justin Romberg, leading to several foundational papers in the field now known as “compressed sensing”, which permits (in certain circumstances) the rapid acquisition of high-resolution images or other information from a relatively small number of measurements. (Perhaps the most well known applications of the compressed sensing algorithms that came out of the work of Emmanuel, myself, Justin, David Donoho, and others was the ability to speed up the time required for a medical-grade MRI scan by up to an order of magnitude.)
So, because of the math work of Terry & friends at IPAM, MRI scans are ten times faster. Am I the only one who finds this obviously underwhelming? Terence Tao is the undisputed greatest living mathematician, and yet his largest impact on the world is that MRIs can be done a bit faster? A decent software engineer can get a 10x speedup from a stern glance in the general direction of a codebase; half-clever ideas give 10x speedups all the time.1: I realize there is a difference between the compressed sensing results, which reduce “shot count” and therefore e.g. impart less radiation on their patients. Some orders of magnitude are harder than other orders of magnitude, and some are more important than others. I would say Math seems pretty useless if the greatest contribution from the greatest mathematician is so marginal: not even a new paradigm, a new possibility, a new invention, but a modest improvement on an axis I’m not sure anyone cares about.
This story becomes even sadder with the context that this algorithm he introduced and analyzed, compressed/compressive sensing, is generally understood to be a case of over-promising and under-delivering in the applied math community. The argument for this can be found (in a polite form) in this review of Joel Tropp, but the upshot is as one would expect: it is a beautiful theory which is extremely influential in applied math, and yet it has surprisingly few applications to practice.2: Joel also appears to be a tad skeptical of the “order of magnitude” claim, as the improvement claimed by him is smaller, but I’ll grant it for the purposes of this post. The reason for the lack of applications is that this method is more shot-efficient for structured signals, but usually the number of samples matters less than you’d think, and the signals are not that structured. You see this everywhere in applied math topics: randomized numerical linear algebra and quantum computing are obvious examples where you can throw a stone at a conference and hit a researcher whose life’s work is to prove the existence of amazing algorithms which are too structured and noisy and hard to make sense for any practical purpose.3: For what it’s worth, compressed sensing is probably less impractical than these other fields I name.
Well, then, what is the point of funding math research? This question also keeps coming up in the context of the automation of math by AI. Every researcher I know is saddened by the progress, including some of the people at the companies leading the charge. We’ve all been upstaged by a chatbot in one way or another. I don’t go into the office anymore. Simons used to be buzzing, vibrant, and hopeful; now everyone there is going through the motions, waiting for the final blow.
Every so often someone decides to post a lamentation, make public this sentiment that we normally keep private. The response is typically mixed: why should we care about some professor feeling less useful? Why should we care if the enigmatic beauty of a difficult proof is rent prosaic by abundance? Does my tax money go to your grants so that you can produce “enigmatic beauty” that I will never possibly fathom? I thought it went to you to produce things that are useful.
So there’s a contradiction: we claim that our work is meant to be useful, when in fact we are optimizing for something entirely different. Mathematicians don’t often try to reconcile these two things in public. I certainly am fine holding these two beliefs in my head at once, and I am perfectly capable of saying one thing and doing another. But I also thing that the correct way to reconcile these things is a little tougher to justify.
My honest belief is that:
- The point of proof-based research is to produce beautiful ideas within the ‘medium’ or ‘aesthetic’ of mathematics;
- Math results in themselves are rarely useful for anything; what is important is building up “mathematical aesthetic” in a variety of domains, because the culmination of this work can be appreciated by outsiders.
Consequently, I think math is essentially art with applications, and it is a fundamentally human pursuit like art is. And I’m willing to swallow the bullet that “math is as useless as art”, because I don’t think art is useless.
Most of my math opinions run downstream of 1 and 2.
For funding, I believe we should fund math, not because it is directly useful to MRI machines or whatever, but because it pushes the boundaries on the beauty we can find in the mathematical world. I want math funded in the same way I want my government to fund noise music made from recordings of whales, performance art that only a few dozen people will ever see, and indie games which are so experimental that they can’t be understood. Moreover, what math is good or bad, deserving or undeserving of funding, is an aesthetic judgment, one of artistic merit and potential. These aren’t fully subjective judgments, but they are not directly tied to application. I wouldn’t fund an English professor to write a paint-by-numbers romance, and I wouldn’t fund a math professor to prove a generalization of an existing theorem that an LLM could probably one-shot.
For AI, math is not dead if an AI can produce beautiful mathematics, in the same way that visual art was not killed by the photograph. Math in the future may look very different from the math of today, but it was never about the game of problem-solving—painters spend all day on the technical details of painting, but the point of it was never to make the most technically proficient painting, to paint something no one else could paint. Who honestly cares about the details, how clever you had to be, how many nights you stayed up, how many wives left you because of your devotion to the craft. The point is to do something beautiful.
Ok, I actually do believe that math should probably be funded more than art. So, if you like, you can supplement the above with the additional points which make math research more fundable.
- For whatever reason, scientifically minded people are inspired and motivated by mathematical beauty and the grand project of mathematics.
- For whatever reason, having a mathematical aesthetic is useful in many technical fields.