Math is running out of problems

Math is mind-boggingly huge. We've been at it for thousands of years. The area of our knowledge is enormous. But so is its boundary. We aren't running out of problems any time soon. Including "interesting" problems.

> 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.

I'm sure that if you had more than passing knowledge in animal behavior, you would also find most of the papers dull. Learning completely new things is of course always a blast. Learning about the latest bleeding edge advances in a field where you already know a lot is not as exciting. I'm not sure what point you were trying to make there. When I read papers in physics I'm always thinking "holy shit electrons are so cool and crazy" because I'm always discovering something new at basically every paragraph. But for an expert the novelty eventually wears off.

The article suggests this test to establish that math is running out of interesting research: "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?".

But it seems to me fairly likely that applying the same test to math journals from 100 or 200 years ago would produce similar results. Most published papers will not be of great interest to any particular one person.

I think this note also misses that there are idiosyncratic factors related to the Journal of Algebra. This used to be a quite good generalist journal focused on algebra -- the Tits Alternative appeared there in the 70s, for example. Elsevier greatly increased the page count in the ensuing decades and it's now mostly dreck. These are papers that might be good to have in print for the sake of completeness of the literature, but nobody is going to send an actual interesting result in algebra there anymore. - An algebraist

I don't think the average people from the 1600s cared much about John Napier and his treaty on logarithms published in 1614.

But we care a lot about logarithms now.

Maybe people never cared about current mathematics. Maybe that's just the pace of progress.

If most of our current problems are solved by results from 50 years ago, could it just be that our future problems will be solved by results from right now?

Firstly: this can probably be neither proven nor disproven, but my intuition tells me that it's by definition impossible for mathematics to run out of problems.

Secondly:

> It cannot remain healthy with its incredible publication rate today of mostly useless generalizations.

So the issue isn't that mathematics is running out of problems. The issue is that there are more publications than there are new problems being discovered / solved, and, ergo, the majority of publications are of limited value / interest. And that isn't an issue unique to mathematics, that's just how academic research is in the 21st century!

Which problems people work on is dictated to a large extent by the need to publish to keep your job. There is a lot of incentive to work on publishable low-hanging fruit problems. Hence the abundance of “write-only” journals in mathematics.

I don’t think there is by any means a shortage of hard, interesting problems. But working on them directly comes with significant career risk.

I have only one [1] thing to say here

[1] https://en.wikipedia.org/wiki/Charles_Holland_Duell#Everythi...

> 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.

That’s because, in the natural sciences, a lot of what was considered knowledge long ago has been found out to be incorrect.

If you study Galen (https://en.wikipedia.org/wiki/Galen) or Hippocrates (https://en.wikipedia.org/wiki/Hippocrates), or Newton’s works on alchemy, you aren’t studying medicine or chemistry, but the history thereof.

On the other hand, look at the Pythagorean theorem. There has been a bit of chipping at its corners when non-Euclidean geometry was discovered/invented, but it remains true in large branches of mathematics.

And this isn’t a matter of centuries. A lot of genetics work that predates the discovery of the structure of DNA isn’t worth studying anymore.

> 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?

Luckily, lots of mathematics research is fairly cheap. As Alfréd Rényi said (https://en.wikipedia.org/wiki/Alfréd_Rényi#Quotations) it runs on coffee.

> 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?

The Einstein Tile was discovered in 2022, and that's received a decent amount of press

My Theory of Computation text mentions a lot of unsolved problems. Some are in CS of course, but I was just editing an example about whether for all n there is a prime between n-squared and (n+1)-squared. That seems like a problem a US sophmore could appreciate.

If you run out of (solvable!) problems in your given logic space, just start branching out your space. Until you find yourself in such esoteric spheres, not even your best math co-researcher knows anymore what's happening and vice versa.

What about tetration with arbitrary real or complex bases and exponents, or is that too fringe?

Anyway, if mathematicians indeed have less to do, perhaps they could start working on standardizing tau over pi, to make radian angles less confusing for everyone.

While I agree with the author broadly, he is definitely overselling his thesis. Going from 'fewer grad students now care about problems in Journal of Algebra' to 'Math is running out of problems' is quite a stretch.

Take any period of time - some subfields will run hot & others will be fallow. Doesn't mean we have run out of problems.Trace formula for theta groups will appeal to only fifteen people - ok so what's the issue ? Math isn't some popularity contest. We have a ballroom at the university which is reserved for talks from visiting professors. When we have an economics lecture, usually it is jampacked. All 100 seats are taken, not even standing room. Then the next talk is by some topologist. The room practically empties out in real time. If you watched it live, you would be shocked at how fast people are rushing out of the room - you would think some stinkbomb was thrown. Finally, nobody is left other than the topologist himself & 5 grad students, 4 of whom look like they literally jumped out of bed & grabbed a coffee mug on the way. That's math for you. That's how its always been.

mathematicians work on problems which are interesting to other people, usually mathematicians. Sone people dont seem to be aware of this, but this is a graph with a general direction trend from the pure to the applied. Show me any math paper and I can tell you how this could potentially help solve real world problems.

Speaking as a math amateur, I find good problems are entirely subjective. If math isn't scratching some itch, go play in another science or do some engineering for awhile and it will come back.

Meanwhile, the economy is investing billions of dollars into researching and using neural networks, an almost purely mathematical construct that nobody yet really understands.

The author of the Medium article is running out of imagination!

Should have started with the last paragraphs first, and used the top paragraphs to support it's argument -- Of course; this approach would not be clickbait.

> How many of them will mention recent research in mathematics from the last couple decades

I think computer science, especially TCS, will mention recent research from the last couple of years. Technically, TCS is a branch of maths too.

If you’ve ever read Harvest and Sowings in conjunction with Pursuing Stacks then you’re already aware of this problem.

Possibly the greatest intellectual troll of all time. Rip to a real one. Miss you Grothy baby.

> 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.

Can't you also say this is directly disproving his point as well? It might be that there are so many open interesting problems that we can become highly picky what problems get solved to the point these preferences are shares between less people. Indicating an expansive set of problems instead of an exhausted one instead.

Math is an art form? On art/science demarcation I'd definitely rate math a science. I'm unsure what they're going for in that statement.

nowhere capable of solving them, but my understanding is that we still have a sea of unchartered territories when it comes to research here. but it has become apparent now more than ever that what seems alien or niche today might become seminal in the future.

long story short - we just need the link between theory and real life. you will find plenty problems, interesting even (at least for someone).

As long as people are getting paid to be mathematicians I'm sure this will never be the case.

Might I suggest adding Mo Money

We now go life to our expert on running out of problems https://en.m.wikipedia.org/wiki/Georg_Cantor

"Math is running out of problems you can get famous for solving" FTFY.

one of the funniest titles.

t. Kelvin

This is one of the most obviously untrue blog posts I've read on medium which is some sort of achievement I suppose.

The null hypothesis has to be that the number of interesting and important open research problems in mathematics is expanding without limit. If the author thinks that's not the case it's up to them to actually justify their position rather than just blandly state it with a "No true Scotsman" addition that the number of problems that are interesting to "a fair number of people" is diminishing on the basis that they find "The Journal of Algebra" to contain things that are not interesting to them.

Most mathematicians I know seem accutely aware that the field of mathematics as a serious intellectual endeavour is over 3000 years old at this point and therefore are aware of its maturity as a field.

I thought math was obliterated at the foundations by Gödel's incompleteness theorems. Did they fix that?