Math is running out of problems

But, that does not mean math as a human activity has to stop

In a previous article, I claimed that math was running out of problems. But I gave very little attention to this statement, and some people were in disbelief over it, so I decided I should clarify what I meant.

First, when I say math, I mean research mathematics, and when I say problems, I mean new and interesting unsolved problems. And of course, there are still unsolved problems in math. The first one that comes to mind is the famous Riemann hypothesis, but there are other problems that are certainly interesting and still unsolved.

And yet, I still think that math is essentially running out of good problems. How can that be possible? Well, what I mean is that the number of problems that are at least interesting to a fair number of people is diminishing every year, and there are hardly any of those left.

To see this, here’s an exercise you can do yourself, if you have any training at all in advanced mathematics. Take a fairly generalist journal, like the Journal of Algebra (take a topic in which you have expertise — my doctoral thesis was in algebra). Look at some of the papers. How many of them are truly interesting to you?

These days, I can look at any of these journals and find at most one or two papers that are even remotely amusing, and algebra was my specialty. On the other hand, I can take a journal in biology like the Journal of Animal Behavior and still find quite a few papers in each journal that are interesting to me even though I’m not even a research biologist! Keep in mind I still like mathematics a lot, and I still enjoy algebra.

Or take a look at any undergraduate text in mathematics. How many of them will mention recent research in mathematics from the last couple decades? I’ve never seen it. Now take an undergraduate text in biology and you’ll still find quite a few citations to modern research.

The truth is, in most cases, modern-day research mathematics is confined to the interest of a handful of people for any given research topic, and that is becoming more and more exaggerated every day.

Even graduate students are often put off by the extreme specialty and abstruseness of modern day math problems — I’ve seen this happen many times. Yes, math is different than biology in that it’s a “depth” subject compared to the more broad nature of biology, but even so, does that justify its continued funding? At what point can we still say with a straight face that it makes sense to pour millions of dollars into mathematics research when its only objective seems reaching the next highest peak of hyper-specialization? Is all this worth it produce solutions that will be interesting to about twenty people per paper? It is like the public funding climbers to reach the peaks of Everest or K2.

None of this means that there isn't cool stuff left to do in math. The existing body of mathematics is a bit of a mess in places in the sense that some proofs would be very hard to write down in detail from start to finish. This is simply because the results they depend upon are scattered so far and wide, and often written down very tersely. Take the classification of finite simple groups or any results in geometric Langlands. This stuff could and should be simplified.

But research-wise, the era where novel and truly interesting problems were plentiful is long past. The era where most graduate students could work on problems that were interesting to more than a dozen researchers is gone. Is that a bad thing? No. Most bodies of knowledge have to come to terms with their own maturity — and it’s time mathematics did the same.

So what this means is that mathematics must shift its priorities. It cannot remain healthy with its incredible publication rate today of mostly useless generalizations. Such a rate is encouraged simply because having a large number of publications was something with which mathematicians measured their worth in the past.

Mathematics needs to slow down emphasize other activities such as simplifying the existing body of knowledge. Graduate students in the future should no longer have to prove something totally new. Instead, their main aim could be simplifying an elegant and truly interesting result of the past.

Mathematics above all is an art form, and it is beautiful. But the artist only creates when they can create something beautiful. A culture that encourages endless new research is like pushing an artist to paint dozens of new paintings, even when they have nothing to paint. The resulting ugliness is worse than useless, and goes against the human spirit.

My only hope for mathematics is that it finds to to become more honest, because its current level of specialization is hideous and absurd.