Asking as a complete layman that does not have a background in math:

Is there a concern that there will be a point in the future where the barrier to entry of math (due to how far it's progressed) will make it very difficult for humans to come up with new research? ie will there be a point in which it will take a human many years to ramp up on existing knowledge before they can even begin to contribute new ideas.

I ask because there seems to be a lot of very complicated and esoteric research out there, that only has a small handful of experts that understand the theory. Reading about the Inter-universal Teichmuller theory debacle comes to mind - without commenting on the validity of the theory, it seems to be generally considered an extremely dense theory understood by few. Are there other theories like this at the forefront of math, and what happens when proofs like this become the prerequisite to further research?

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4 Answers 4


I don't expect that to happen. The barrier to entry at the leading edge of math has always been high. New math ideas are continually being introduced widening the field.

It isn't necessary, nor even possible any more, to know "all" of mathematics in order to do research in some specialized area. Many of us are much more proficient in some area than others and can guide newcomers in that sub sub sub field.

It is said that the last person to know "all" of mathematics was probably Henri Poincaré at the start of the 20th century. But a lot has obviously happened since then.

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    As a mathematician, I sometimes also ask myself this question. Another point of concern is that people sometimes "rediscover" previous results from a few decades ago. Maybe people will have to specialize more and earlier as times goes on, and only be specialists on very narrow fields.
    – Albert
    Commented Jun 8 at 17:45
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    Also isn't math expanded by mathematicians? Meaning... math couldn't get more difficult than mathematicians can make it - just logically. Math is advanced by mathematicians, they couldn't advance it beyond their own skills. It would be maximum hubris to think that some one or few mathematicians could advance it with their skills and then no one after them could reach the same skill level. Commented Jun 9 at 16:08
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    @ToddWilcox, generally true but there is also serendipity that accounts for some advances.
    – Buffy
    Commented Jun 9 at 16:14

There is a factor you are disregarding in your analysis: entry-level math becomes more accessible as knowledge progresses. Theorems that were novel in the 1800s are now first-year undegraduate material, with more streamlined, simplified proofs; and introducing the right definitions and basic concepts such as "functions" and "vector spaces" early makes things easier to understand.

At the same time, some math that was once useful became obsolete and is driven out of the standard syllabus: pencil-and-paper calculation techniques, complicated integrals, how to use a slide rule.

Also, sometimes new techniques and formalisms make old results simpler; for instance, differential geometry and differential forms.

This trend counterbalances the fact that there is more math to learn in the first place: new math enters, but old, unnecessary math is being shed out at the same time.

And very rarely does a mathematician need to read something written more than, say, 70 years earlier: new books reprocess old material and present it with modern notation, making it simpler to understand at the same time.

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    I learned Calculus in college but it is now a common subject in secondary school. Not then.
    – Buffy
    Commented Jun 9 at 16:17
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    @Buffy, calculus keeps bouncing back and forth between the two. My father learned calculus in high school as part of a reasonably routine course of study; I only learned it before college because I made a special effort to do so.
    – Mark
    Commented Jun 9 at 21:13

There's a legitimate concern not just in math but in all fields that as the fields advance more the needed knowledge base will get larger. There is a really good short story about this.

But in terms of math, we're nowhere near that point in general. While there are some subfields where the basic background needed is really massive and keeps getting more (algebraic geometry and some parts of number theory are good examples), there are other subfields where there's very little background. Elementary number theory and graph theory are both areas where there's a lot of room. For example, my own first paper was a number theory paper when I was in high school. That's not a brag; that is pretty common. There are also entire journals devoted to publications of genuine research by undergrads and high school students. Involve is one of the more prominent ones.

And in the short-term, math research is becoming more accessible. With the rise of the internet, and putting almost all preprints on the arXiv, people can the world over access math research and share their ideas to an extent which could not be done previously. Since math does not require big equipment or the like, math is one of the fields which benefits the most from this sort of broad access. So at least for math, the short-term trend is the exact opposite.

  • I upvoted, but like to point out that I disagree that the linked short story were good. It mentions a few nice points somewhere in the first half (although I'd say that it does so in a rather flat way) - but towards the end it gets extremelly weird. It conveys a very unrealistic picture of how knowledge gets actually transferred from one generation to the next one. Commented Jun 9 at 10:30
  • @JochenGlueck Yes it is weird. And sure, in real life knowledge isn't transmitted exactly that way. But there's some approximation of that. If a proof took 150 years worth of ideas to be fully understood and integrated, humans aren't going to come up with it now. As human knowledge increases, this sort of thing in the limit will be more of an issue, maybe not now, but possibly 200 or 300 years from now.
    – JoshuaZ
    Commented Jun 9 at 12:33
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    @JochenGlueck (Also I understand seeing the story as feeling flat. This is apparently something a fair number of people have said about that author's stories. One person said they read like lectures wrapped in stories. So there is some taste preference aspect here probably.)
    – JoshuaZ
    Commented Jun 9 at 12:34
  • The short story also ignores that as we understand a topic better, we can teach it better. And for today's age, math online can reach millions more people than any traditional book or lecture.
    – qwr
    Commented yesterday
  • @qwr, I think the story does acknowledge that? They talk about improving teaching. They discuss improving redaction and condensation and teachers teaching better teacher.
    – JoshuaZ
    Commented yesterday

If there is enough interest in a subfield, usually people will find ways to present the core ideas of the field in a way that is accessible enough to draw graduate students. Otherwise, the subfield does not grow and will in the medium to long term die out/become inactive/go out of fashion. This does happen from time to time. In general, barriers to entry are hardly ever absolute, but they can get high enough relative to the draw of a subfield that people do not bother surmounting them.

Barriers to entry can also certainly from the outside seem higher than they really are. Many papers are adding just a few simple ideas to an existing toolset (which is not a bad thing - finding a simple but useful and novel idea that extends an existing set of tricks just by a little bit, and working out all the details and thereby making it work can make for an extremely useful and quite innovative contribution sometimes, and the little added idea will seem obvious only in retrospective), and often these could be described in isolation even to an audience that has only a general understanding of the toolset. Some mathematicians think that only the finished, fully worked out product is worth publishing, and then the publication buries the simple underlying idea in a mountain of technical details. Works of this kind can seem impossible to understand to an outsider, but people who directly interact with the authors will often find that in more informal settings (research seminars, sessions with graduate students, etc.) the authors can and will clearly express the basic idea and their point of departure. Research-level math does become more accessible (but of course remains hard) if one's access includes the word-of-mouth, student-of-X side channels.

I think accessibility to wider audiences will improve in the future. Language models and their descendants will get better at mathematical reasoning. At some point, they will be able to serve as expert-level math tutors that have internalized the whole body of published mathematical writing. Even if general reasoning ability remains below human expert level, I expect such systems will be great at breaking down communication barriers between subfields, making results and intuitions from other subfields accessible, explaining the high-level results of mathematical papers at the level appropriate for the person who needs the explanation, and even at deriving new results that have a simple proof if you know all of the literature.

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