6 ms·

Math is running out of problems

62 points by vouaobrasil 2y ago

seanhunter 2y ago
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.
- vouaobrasil 2y ago
  Well, it is difficult to obtain data on this because it would require a survey of mathematicians, or a rigorous study of citation figures but maybe I'll do it if I have time. Incidentally, I am a mathematician, or at least was a practicing one for some years and have published in the field...and I can say that from personal experience, math is exceptionally fragmented. Yes, you point out a valid criticism but I don't really have the time to collect data on this. Edit: I also wonder if the immediate denial of my statement is due to the emotional attachment that some people have to the purity of and beauty of mathematics. I also think mathematics is beautiful but that doesn't mean it doesn't have problems.
  - g15jv2dp 2y ago
    > Yes, you point out a valid criticism but I don't really have the time to collect data on this. You did have time to make a blog post, though. Apparently, it's more important to you to broadcast your opinion than to make sure it is correct?
    - vouaobrasil 2y ago
      > You did have time to make a blog post, though. Apparently, it's more important to you to broadcast your opinion than to make sure it is correct? Well, I know it's true from experience as a researcher.
      - g15jv2dp 2y ago
        And I know it's false from experience as a researcher. What now?
        vouaobrasil 2y ago
        I guess nothing. I'm trying to convince people of one thing, you can do another.
        hobofan 2y ago
        You don't believe him and that's it. It's a blog post and not a published research paper for a reason. And reality also exists outside the scope of what's published in journals.
- j16sdiz 2y ago
  I think you missed the point.
- sebstefan 2y ago
  I'm not sure if that makes it obviously untrue or obviously not backed up by something solid
  - vouaobrasil 2y ago
    I will do some serious data analysis on this topic and get back here with part two when I have time.
- nihzm 2y ago
  While I also disagree with the post, my take on it was it is more a critique of the increasing hyperspecialization of mathematical problems. Of course the number of mathematical problems is endless, but I also feel that they are becoming more and more inaccessible. In my current university (and others) undergraduate math students do not even have to write a thesis anymore because of the number of people who came at the end without any new results to show. Instead, to obtain their degree they are tasked with things like rewriting existing papers into chapters for lecture notes (what the author suggests). For graduate students theses are still a thing but the situation is not much better. Talking to my peers I got the impression that many (but not all of course) in their theses are either generalizing something they themselves already tought was too general or are overanalyzing an obscure problem using tools that can only be found in maze of unreadable papers. I'm not saying that papers should be easy, but if even most graduate students can't read the research material there is probably an underlying problem.
- deleted 2y ago
  [deleted]
ThereIsNoWorry 2y ago
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.
- vouaobrasil 2y ago
  Well, that is what mathematicians do, which is why I said "interesting" problems. I mean, I can think of several problems, like "classify projective modules over rings of global dimension 4 that have no zero divisors". Even popular problems in Langlands like "explicitly find a trace formula for theta groups" will appeal to the fifteen people in the world that can understand what I'm even talking about.
Chinjut 2y ago
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.
- vouaobrasil 2y ago
  No way, if you take a generalist algebra journal from 50 years ago, chances are many more graduate students interested in abstract algebra will be able to understand and appreciate the nature of the research. I mean, the best and most interesting papers from algebra were published between 1950-1990, IMO.
  - Chinjut 2y ago
    Is the test about ability to understand or about finding it very interesting? Of course it's easier to understand what has already become old hat and popularized in the field.
    - vouaobrasil 2y ago
      I would say both are dwindling. Few people can understand modern results and few people find them interesting.
      - Chinjut 2y ago
        Again, it's not clear to me that if you asked 1950s mathematicians to look at math journals in the 1950s, they would react very differently. You haven't presented any evidence of this. It's certainly true that math is a more fragmented specialized field today than it was in the eras of Euler or Gauss, but without some more concrete evidence or objective claim, I don't know what is so bad about math journals today compared to 1950.
surfingdino 2y ago
"Math is running out of problems you can get famous for solving" FTFY.
- vouaobrasil 2y ago
  I would replace "famous" with "known beyond thirty researchers".
  - g15jv2dp 2y ago
    I routinely attend conferences where way more than thirty people take several hours out of their days to listen to the new problems and their solutions. In a single field of math. In my little corner of the world. With people who managed to free themselves that week and find travel money.
0003 2y ago
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.
- vouaobrasil 2y ago
  Well, I think my statement is not clickbait because it is just what I believe to be true. I was even at a talk where a senior mathematician said this also, that "math is running out of problems". I could have said "You won't believe what these mathematicians are doing to obtain new problems WAAH! -- Now that would be clickbait :)
  - RunawayGalaxy 2y ago
    The title is far more general than what you actually mean. What's the reason for that if not click bait?
jaza 2y ago
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!
- vouaobrasil 2y ago
  My definition of "running out of problems" as I stated was "running out of problems that more than a handful of people care about", and I think this is definitely true in math.
  - zehaeva 2y ago
    I don't want to be snarky, but I do seriously wonder how many people really cared about calculus when Newton/Leibniz developed it. It honest couldn't have been more than a handful because Newton slept on it for the better part twenty years. I honestly think Math as a field has always been defined by "problems that only a handful of people care about". The only exception I can think about is maybe basic addition and multiplication.
    - pixl97 2y ago
      Yes, exactly. A whole lot of math problems were not interesting until we got relatively fast digital computers as they would have taken till the end of the universe if done by hand. Then suddenly it becomes a product you can implement in a library and perform simulations of reality on. Now suddenly a lot more people are interested in the math because improving the algorithm could lead to millions in power saving, or it could lead to far higher accuracy. Simply put, it's very hard to predict the usefulness of math at the time it's created.
Aardwolf 2y ago
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.
- vouaobrasil 2y ago
  I'd say it's kind of a cool topic and would probably lead to some interesting recreational math but it probably won't lead to new, significant research, which means a new theoretical understanding of some abstract structure or another.
sebstefan 2y ago
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?
- olddustytrail 2y ago
  Maybe not the average butcher or baker, but to engineers and scientists it was incredibly important. The slide rule was invented just a few years later based on Napier's work and was used continuously for the next 350 years, until the invention of the modern calculator/computer. I don't think I disagree with your overall point I just think you chose the worst example :)
- djmips 2y ago
  So average people from the 2000s care about John Napier and his treaty on logarithms? Or even just logarithms?
  - sebstefan 2y ago
    Ah shit, I guess not Maybe the average 17 year old but that's it
voidUpdate 2y ago
> 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
- vouaobrasil 2y ago
  Well, first few undergraduate texts mention the Einstein tile, and secondly, my point was that almost no modern results are mentioned in undergraduate texts, whereas many undergraduate biology texts mention modern results.
  - voidUpdate 2y ago
    Probably because it was too recent a discovery to be in undergrad texts yet
booleandilemma 2y ago
As long as people are getting paid to be mathematicians I'm sure this will never be the case.
- vouaobrasil 2y ago
  That is true, and that was part of my point.
axegon_ 2y ago
I have only one [1] thing to say here [1] https://en.wikipedia.org/wiki/Charles_Holland_Duell#Everything_that_can_be_invented_has_been_invented https://en.wikipedia.org/wiki/Charles_Holland_Duell#Everythi...
- vouaobrasil 2y ago
  Well, a general statement about technology and invention is not exactly the same as a highly specific branch of knowledge becoming mature and not having anything innovative left to add to it.
  - axegon_ 2y ago
    If there's one thing that history has taught us, regardless of which field, it's that anything we manage to answer raises at least half a dozen other questions. Saying that math, of all fields, is running out of problems is one of the most absurd statements in this day and age. I was blown away by the capabilities of dumb SVMs in university and today SVMs look like a child's play - just over a decade later. For as long as humanity has existed, whenever we feel like we've reached our peak, something happens and completely shatters our understanding. Say we have prime numbers and their occurrence is completely unpredictable - the same way Aristotle was convinced that objects come to a rest because they get tired. It's not a specific branch of knowledge, it's an outrageous claim like many have already pointed out. Could it be that math is running out of problems? Yes - in the same way that the universe might vanish tomorrow. Both of those claims are equally absurd.
    - vouaobrasil 2y ago
      What about organic chemistry? Do you think there are a lot of open and interesting problems there?
  - halostatue 2y ago
    I am sort of surprised that no one has countered with the quote often misattributed to Lord Kelvin. More interesting was the advice given to Max Planck: One of Jolly's students at the University of Munich was Max Planck, whom he advised in 1878 not to go into theoretical physics.[5] Nevertheless, Planck's later work lead to the discovery of quantum mechanics.[5] Later in life Planck reported:[2][6] As I began my university studies I asked my venerable teacher Philipp von Jolly for advice regarding the conditions and prospects of my chosen field of study. He described physics to me as a highly developed, nearly fully matured science, that through the crowning achievement of the discovery of the principle of conservation of energy it will arguably soon take its final stable form. It may yet keep going in one corner or another, scrutinizing or putting in order a jot here and a tittle there, but the system as a whole is secured, and theoretical physics is noticeably approaching its completion to the same degree as geometry did centuries ago. That was the view fifty years ago of a respected physicist at the time. https://en.wikipedia.org/wiki/Philipp_von_Jolly https://en.wikipedia.org/wiki/Philipp_von_Jolly Related: https://en.wikipedia.org/wiki/Timeline_of_geometry#20th_century https://en.wikipedia.org/wiki/Timeline_of_geometry#20th_cent... Math is not running out of problems just like physics didn't stop advancing (or merely become more precise measurement) in the 19th and 20th centuries. Something new an innovative might be around the corner — it might not. It might result in a new field entirely. It might not.
empath75 2y ago
Meanwhile, the economy is investing billions of dollars into researching and using neural networks, an almost purely mathematical construct that nobody yet really understands.
- vouaobrasil 2y ago
  They do it because people believe there is an immediate economic advantage: namely, to strangle ordinary people and concentrate all that is worth trading in the hands of the already rich.
  - mturmon 2y ago
    You’re changing the subject! There is a really good problem in there. Why does this stuff work so well, for what classes of problems, and can we say anything about the fundamental energy/accuracy limits of this approach? I think of it as an engineering problem, but there are mathematical rephrasings of this problem, as Michel Talagrand’s work has shown. Don’t let the greed/stampede obscure the very interesting technical principles here.
xiaodai 2y ago
one of the funniest titles.
math_dandy 2y ago
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.
markgall 2y ago
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
- vouaobrasil 2y ago
  I will say after looking at hundreds of journals and papers, it's also true of the others. -Another algebraist.
  - markgall 2y ago
    I think if you look at a new algebra paper in a good journal, it's as likely to be interesting as a random algebra paper in a good journal from 1980. (Of course neither is anywhere near 100%, there were many boring papers back then too.) If you look at a new algebra paper in J Algebra, of course it's not going to be interesting, what do you expect.
    - vouaobrasil 2y ago
      Dude, I spent seven years doing nothing but that in hundreds of journals. I still maintain my view.
g15jv2dp 2y ago
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.
- esd_g0d 2y ago
  I second this last paragraph. I want to add my pet theory that it is not a purely psychological phenomenon. It's like that thing where we only remember the good movies (the "classics"), and the others are forgotten. When you first get started on the field, you're reading just the "classics". When you get to the current state of the art, the classics have not been sorted out yet, and so you get a lot of trash.
- vouaobrasil 2y ago
  > I'm sure that if you had more than passing knowledge in animal behavior, you would also find most of the papers dull Disagree. I read papers in biology and ecology regularly, and the relevance of them is much greater and at least easier to appreciate them. (I do some science popularization now and people care much more about recent research in biology than math graduate students care about the latest papers in math in neighbouring fields.)
shrimp_emoji 2y ago
I thought math was obliterated at the foundations by Gödel's incompleteness theorems. Did they fix that?
- jraph 2y ago
  There being some limits and some improvable stuff doesn't mean the whole domain is doomed or broken and isn't huge, or even infinite. Throwing away math because of those limits would be throwing the baby with the bathwater. What would you like to see fixed?
1970-01-01 2y ago
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.
- vouaobrasil 2y ago
  My claim is that the number of people who find professional level problems interesting in any given field is only a few dozen and hence doesn't deserve public funding.
golol 2y ago
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.
- vouaobrasil 2y ago
  Okay: "Integral p-adic Hodge Theory" [1]. Go ahead. [1] https://link.springer.com/article/10.1007/s10240-019-00102-z https://link.springer.com/article/10.1007/s10240-019-00102-z
  - golol 2y ago
    I am not an expert on this so I can obviously not give a good judgement but this is about Cohomology and p-adic numbers. Whatever it is they may prove, they are clearly furthering our understanding of these subjects. When you want to apply a theory it really helps if the theory is well-developed, people know different approaches to define things, what are the standard results, what things are equivalent, etc. These kinda of theoretical results work in the background to allow people to effectively and comfortably apply their linear algebra or calculus, for example. So now I will argue that Cohomology and p-adic numbers are interesting and useful. Cohomology and hodge theory are about geometry and partial differential equations. This can have applications in AI for example, since data lies on manifolds. I saw some paper a while ago that layers in neural networks fold the data manifold onto itself to reduce its topological complexity and this can be measured by computing some "Betti numbers", which are related to homology and thereby also cohomology. Now is this really true or useful? I don't know but having a mathematical theory makes it possible to even start thinking about such ideas. Also, partial differential equations have obvious applications. By the way, most theories have finite-dimensional/discrete analogues, for example discrete Hodge theory exists, and usually when you understand something about the smooth version of a theory then there are some equivalents for the discrete theory. So if you have data as a graph then you may want to investigate some discrete forms of homology/cohomology on that and may wonder how different types are equivalent etc. P-adic numbers are related to modular arithmetic and therefore pretty useful just because of computers and cryptography and these things.
sabas123 2y ago
> 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.
jimhefferon 2y ago
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.
- vouaobrasil 2y ago
  That's not really a "lot" of problems though.
josefritzishere 2y ago
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.
Xen9 2y ago
The author of the Medium article is running out of imagination!
Log_out_ 2y ago
We now go life to our expert on running out of problems https://en.m.wikipedia.org/wiki/Georg_Cantor https://en.m.wikipedia.org/wiki/Georg_Cantor
hoseja 2y ago
t. Kelvin
oglop 2y ago
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.
dxbydt 2y ago
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.
Someone 2y ago
> 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 https://en.wikipedia.org/wiki/Galen) or Hippocrates (https://en.wikipedia.org/wiki/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 https://en.wikipedia.org/wiki/Alfréd_Rényi#Quotations) it runs on coffee.
- practicemaths 2y ago
  > It runs on coffee Or for Erdös, amphetamine. Also I find it ironic to fro them to gripe about alleged "hyper-specializatiin" given that part of the beauty of Math is discovering how seemingly unrelated areas are in fact connected AND discovering generalizations that can be easily applied. Caveat, I am no where close to being a mathematician.
hintymad 2y ago
> 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.
dougdimmadome 2y ago
Might I suggest adding Mo Money
rldjbpin 2y ago
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).
bbkj 2y ago
[flagged]

62 points by vouaobrasil 2y ago

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.

vouaobrasil 2y ago
Well, it is difficult to obtain data on this because it would require a survey of mathematicians, or a rigorous study of citation figures but maybe I'll do it if I have time. Incidentally, I am a mathematician, or at least was a practicing one for some years and have published in the field...and I can say that from personal experience, math is exceptionally fragmented. Yes, you point out a valid criticism but I don't really have the time to collect data on this. Edit: I also wonder if the immediate denial of my statement is due to the emotional attachment that some people have to the purity of and beauty of mathematics. I also think mathematics is beautiful but that doesn't mean it doesn't have problems.
- g15jv2dp 2y ago
  > Yes, you point out a valid criticism but I don't really have the time to collect data on this. You did have time to make a blog post, though. Apparently, it's more important to you to broadcast your opinion than to make sure it is correct?
  - vouaobrasil 2y ago
    > You did have time to make a blog post, though. Apparently, it's more important to you to broadcast your opinion than to make sure it is correct? Well, I know it's true from experience as a researcher.
    - g15jv2dp 2y ago
      And I know it's false from experience as a researcher. What now?
      - vouaobrasil 2y ago
        I guess nothing. I'm trying to convince people of one thing, you can do another.
      - hobofan 2y ago
        You don't believe him and that's it. It's a blog post and not a published research paper for a reason. And reality also exists outside the scope of what's published in journals.
j16sdiz 2y ago
I think you missed the point.
sebstefan 2y ago
I'm not sure if that makes it obviously untrue or obviously not backed up by something solid
- vouaobrasil 2y ago
  I will do some serious data analysis on this topic and get back here with part two when I have time.
nihzm 2y ago
While I also disagree with the post, my take on it was it is more a critique of the increasing hyperspecialization of mathematical problems. Of course the number of mathematical problems is endless, but I also feel that they are becoming more and more inaccessible. In my current university (and others) undergraduate math students do not even have to write a thesis anymore because of the number of people who came at the end without any new results to show. Instead, to obtain their degree they are tasked with things like rewriting existing papers into chapters for lecture notes (what the author suggests). For graduate students theses are still a thing but the situation is not much better. Talking to my peers I got the impression that many (but not all of course) in their theses are either generalizing something they themselves already tought was too general or are overanalyzing an obscure problem using tools that can only be found in maze of unreadable papers. I'm not saying that papers should be easy, but if even most graduate students can't read the research material there is probably an underlying problem.
deleted 2y ago
[deleted]

ThereIsNoWorry 2y ago

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.

vouaobrasil 2y ago
Well, that is what mathematicians do, which is why I said "interesting" problems. I mean, I can think of several problems, like "classify projective modules over rings of global dimension 4 that have no zero divisors". Even popular problems in Langlands like "explicitly find a trace formula for theta groups" will appeal to the fifteen people in the world that can understand what I'm even talking about.

Chinjut 2y ago

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.

vouaobrasil 2y ago
No way, if you take a generalist algebra journal from 50 years ago, chances are many more graduate students interested in abstract algebra will be able to understand and appreciate the nature of the research. I mean, the best and most interesting papers from algebra were published between 1950-1990, IMO.
- Chinjut 2y ago
  Is the test about ability to understand or about finding it very interesting? Of course it's easier to understand what has already become old hat and popularized in the field.
  - vouaobrasil 2y ago
    I would say both are dwindling. Few people can understand modern results and few people find them interesting.
    - Chinjut 2y ago
      Again, it's not clear to me that if you asked 1950s mathematicians to look at math journals in the 1950s, they would react very differently. You haven't presented any evidence of this. It's certainly true that math is a more fragmented specialized field today than it was in the eras of Euler or Gauss, but without some more concrete evidence or objective claim, I don't know what is so bad about math journals today compared to 1950.

surfingdino 2y ago

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

vouaobrasil 2y ago
I would replace "famous" with "known beyond thirty researchers".
- g15jv2dp 2y ago
  I routinely attend conferences where way more than thirty people take several hours out of their days to listen to the new problems and their solutions. In a single field of math. In my little corner of the world. With people who managed to free themselves that week and find travel money.

0003 2y ago

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.

vouaobrasil 2y ago
Well, I think my statement is not clickbait because it is just what I believe to be true. I was even at a talk where a senior mathematician said this also, that "math is running out of problems". I could have said "You won't believe what these mathematicians are doing to obtain new problems WAAH! -- Now that would be clickbait :)
- RunawayGalaxy 2y ago
  The title is far more general than what you actually mean. What's the reason for that if not click bait?

jaza 2y ago

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!

vouaobrasil 2y ago
My definition of "running out of problems" as I stated was "running out of problems that more than a handful of people care about", and I think this is definitely true in math.
- zehaeva 2y ago
  I don't want to be snarky, but I do seriously wonder how many people really cared about calculus when Newton/Leibniz developed it. It honest couldn't have been more than a handful because Newton slept on it for the better part twenty years. I honestly think Math as a field has always been defined by "problems that only a handful of people care about". The only exception I can think about is maybe basic addition and multiplication.
  - pixl97 2y ago
    Yes, exactly. A whole lot of math problems were not interesting until we got relatively fast digital computers as they would have taken till the end of the universe if done by hand. Then suddenly it becomes a product you can implement in a library and perform simulations of reality on. Now suddenly a lot more people are interested in the math because improving the algorithm could lead to millions in power saving, or it could lead to far higher accuracy. Simply put, it's very hard to predict the usefulness of math at the time it's created.

Aardwolf 2y ago

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.

vouaobrasil 2y ago
I'd say it's kind of a cool topic and would probably lead to some interesting recreational math but it probably won't lead to new, significant research, which means a new theoretical understanding of some abstract structure or another.

sebstefan 2y ago

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?

olddustytrail 2y ago
Maybe not the average butcher or baker, but to engineers and scientists it was incredibly important. The slide rule was invented just a few years later based on Napier's work and was used continuously for the next 350 years, until the invention of the modern calculator/computer. I don't think I disagree with your overall point I just think you chose the worst example :)
djmips 2y ago
So average people from the 2000s care about John Napier and his treaty on logarithms? Or even just logarithms?
- sebstefan 2y ago
  Ah shit, I guess not Maybe the average 17 year old but that's it

voidUpdate 2y ago

> 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

vouaobrasil 2y ago
Well, first few undergraduate texts mention the Einstein tile, and secondly, my point was that almost no modern results are mentioned in undergraduate texts, whereas many undergraduate biology texts mention modern results.
- voidUpdate 2y ago
  Probably because it was too recent a discovery to be in undergrad texts yet

booleandilemma 2y ago

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

vouaobrasil 2y ago
That is true, and that was part of my point.

axegon_ 2y ago

I have only one [1] thing to say here [1] https://en.wikipedia.org/wiki/Charles_Holland_Duell#Everything_that_can_be_invented_has_been_invented https://en.wikipedia.org/wiki/Charles_Holland_Duell#Everythi...

vouaobrasil 2y ago
Well, a general statement about technology and invention is not exactly the same as a highly specific branch of knowledge becoming mature and not having anything innovative left to add to it.
- axegon_ 2y ago
  If there's one thing that history has taught us, regardless of which field, it's that anything we manage to answer raises at least half a dozen other questions. Saying that math, of all fields, is running out of problems is one of the most absurd statements in this day and age. I was blown away by the capabilities of dumb SVMs in university and today SVMs look like a child's play - just over a decade later. For as long as humanity has existed, whenever we feel like we've reached our peak, something happens and completely shatters our understanding. Say we have prime numbers and their occurrence is completely unpredictable - the same way Aristotle was convinced that objects come to a rest because they get tired. It's not a specific branch of knowledge, it's an outrageous claim like many have already pointed out. Could it be that math is running out of problems? Yes - in the same way that the universe might vanish tomorrow. Both of those claims are equally absurd.
  - vouaobrasil 2y ago
    What about organic chemistry? Do you think there are a lot of open and interesting problems there?
- halostatue 2y ago
  I am sort of surprised that no one has countered with the quote often misattributed to Lord Kelvin. More interesting was the advice given to Max Planck: One of Jolly's students at the University of Munich was Max Planck, whom he advised in 1878 not to go into theoretical physics.[5] Nevertheless, Planck's later work lead to the discovery of quantum mechanics.[5] Later in life Planck reported:[2][6] As I began my university studies I asked my venerable teacher Philipp von Jolly for advice regarding the conditions and prospects of my chosen field of study. He described physics to me as a highly developed, nearly fully matured science, that through the crowning achievement of the discovery of the principle of conservation of energy it will arguably soon take its final stable form. It may yet keep going in one corner or another, scrutinizing or putting in order a jot here and a tittle there, but the system as a whole is secured, and theoretical physics is noticeably approaching its completion to the same degree as geometry did centuries ago. That was the view fifty years ago of a respected physicist at the time. https://en.wikipedia.org/wiki/Philipp_von_Jolly https://en.wikipedia.org/wiki/Philipp_von_Jolly Related: https://en.wikipedia.org/wiki/Timeline_of_geometry#20th_century https://en.wikipedia.org/wiki/Timeline_of_geometry#20th_cent... Math is not running out of problems just like physics didn't stop advancing (or merely become more precise measurement) in the 19th and 20th centuries. Something new an innovative might be around the corner — it might not. It might result in a new field entirely. It might not.

empath75 2y ago

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

vouaobrasil 2y ago
They do it because people believe there is an immediate economic advantage: namely, to strangle ordinary people and concentrate all that is worth trading in the hands of the already rich.
- mturmon 2y ago
  You’re changing the subject! There is a really good problem in there. Why does this stuff work so well, for what classes of problems, and can we say anything about the fundamental energy/accuracy limits of this approach? I think of it as an engineering problem, but there are mathematical rephrasings of this problem, as Michel Talagrand’s work has shown. Don’t let the greed/stampede obscure the very interesting technical principles here.

xiaodai 2y ago

one of the funniest titles.

math_dandy 2y ago

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.

markgall 2y ago

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

vouaobrasil 2y ago
I will say after looking at hundreds of journals and papers, it's also true of the others. -Another algebraist.
- markgall 2y ago
  I think if you look at a new algebra paper in a good journal, it's as likely to be interesting as a random algebra paper in a good journal from 1980. (Of course neither is anywhere near 100%, there were many boring papers back then too.) If you look at a new algebra paper in J Algebra, of course it's not going to be interesting, what do you expect.
  - vouaobrasil 2y ago
    Dude, I spent seven years doing nothing but that in hundreds of journals. I still maintain my view.

g15jv2dp 2y ago

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.

esd_g0d 2y ago
I second this last paragraph. I want to add my pet theory that it is not a purely psychological phenomenon. It's like that thing where we only remember the good movies (the "classics"), and the others are forgotten. When you first get started on the field, you're reading just the "classics". When you get to the current state of the art, the classics have not been sorted out yet, and so you get a lot of trash.
vouaobrasil 2y ago
> I'm sure that if you had more than passing knowledge in animal behavior, you would also find most of the papers dull Disagree. I read papers in biology and ecology regularly, and the relevance of them is much greater and at least easier to appreciate them. (I do some science popularization now and people care much more about recent research in biology than math graduate students care about the latest papers in math in neighbouring fields.)

shrimp_emoji 2y ago

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

jraph 2y ago
There being some limits and some improvable stuff doesn't mean the whole domain is doomed or broken and isn't huge, or even infinite. Throwing away math because of those limits would be throwing the baby with the bathwater. What would you like to see fixed?

1970-01-01 2y ago

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.

vouaobrasil 2y ago
My claim is that the number of people who find professional level problems interesting in any given field is only a few dozen and hence doesn't deserve public funding.

golol 2y ago

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.

vouaobrasil 2y ago
Okay: "Integral p-adic Hodge Theory" [1]. Go ahead. [1] https://link.springer.com/article/10.1007/s10240-019-00102-z https://link.springer.com/article/10.1007/s10240-019-00102-z
- golol 2y ago
  I am not an expert on this so I can obviously not give a good judgement but this is about Cohomology and p-adic numbers. Whatever it is they may prove, they are clearly furthering our understanding of these subjects. When you want to apply a theory it really helps if the theory is well-developed, people know different approaches to define things, what are the standard results, what things are equivalent, etc. These kinda of theoretical results work in the background to allow people to effectively and comfortably apply their linear algebra or calculus, for example. So now I will argue that Cohomology and p-adic numbers are interesting and useful. Cohomology and hodge theory are about geometry and partial differential equations. This can have applications in AI for example, since data lies on manifolds. I saw some paper a while ago that layers in neural networks fold the data manifold onto itself to reduce its topological complexity and this can be measured by computing some "Betti numbers", which are related to homology and thereby also cohomology. Now is this really true or useful? I don't know but having a mathematical theory makes it possible to even start thinking about such ideas. Also, partial differential equations have obvious applications. By the way, most theories have finite-dimensional/discrete analogues, for example discrete Hodge theory exists, and usually when you understand something about the smooth version of a theory then there are some equivalents for the discrete theory. So if you have data as a graph then you may want to investigate some discrete forms of homology/cohomology on that and may wonder how different types are equivalent etc. P-adic numbers are related to modular arithmetic and therefore pretty useful just because of computers and cryptography and these things.

sabas123 2y ago

> 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.

jimhefferon 2y ago

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.

vouaobrasil 2y ago
That's not really a "lot" of problems though.

josefritzishere 2y ago

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.

Xen9 2y ago

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

Log_out_ 2y ago

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

hoseja 2y ago

t. Kelvin

oglop 2y ago

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.

dxbydt 2y ago

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.

Someone 2y ago

> 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 https://en.wikipedia.org/wiki/Galen) or Hippocrates (https://en.wikipedia.org/wiki/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 https://en.wikipedia.org/wiki/Alfréd_Rényi#Quotations) it runs on coffee.

practicemaths 2y ago
> It runs on coffee Or for Erdös, amphetamine. Also I find it ironic to fro them to gripe about alleged "hyper-specializatiin" given that part of the beauty of Math is discovering how seemingly unrelated areas are in fact connected AND discovering generalizations that can be easily applied. Caveat, I am no where close to being a mathematician.

hintymad 2y ago

> 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.

dougdimmadome 2y ago

Might I suggest adding Mo Money

rldjbpin 2y ago

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).

bbkj 2y ago

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