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I think this would be extremely valuable: “We need to focus far more energy on understanding and explaining the basic mental infrastructure of mathematics—with

• segfaultex 10mo ago
  Yeah, I don't want to be uncharitable, but I've noticed that a lot of stem fields make heavy use of esoteric language and syntax, and I suspect they do so as a means of gatekeeping. I understand that some degree of formalism is required to enable the sharing of knowledge amongst people across a variety of languages, but sometimes I'll read a white paper and think "wow, this could be written a LOT more simply". Statistics is a major culprit of this.
  • bell-cot 10mo ago
    Gatekeeping, or self-promotion? You don't get investors/patents/promotions/tenure by making your knowledge or results sound simple and understandable.
    • segfaultex 10mo ago
      Why not both? And that's a good point, there are a LOT of incentives to make things arbitrarily complex in a variety of fields.
    • master-lincoln 10mo ago
      Is that really the case or are you just assuming so? Seems counter-intuitive to me.
  • bncndn0956 10mo ago
    3blue1brown proves your point. The saying, "What one fool can do, another can," is a motto from Silvanus P. Thompson's book Calculus Made Easy. It suggests that a task someone without great intelligence can accomplish must be relatively simple, implying that anyone can learn to do it if they put in the effort. The phrase is often used to encourage someone, demystify a complex subject, and downplay the difficulty of a task.
    • gjulianm 10mo ago
      3blue1brown, while they create great content, they do not go as deep into the mathematics, they avoid some of the harder to understand complexities and abstractions. Don't take me wrong, it's not a criticism of their content, it's just a different thing than what you'd study in a mathematics class. Also, an additional thing is that videos are great are making people think they understand something when they actually don't.
    • jules 10mo ago
      3blue1brown actually shows the usefulness of formalism. The videos are great, but by avoiding formalism, they are at least for me harder to understand than traditional sources. It is true that you need to get over the hump of understanding the formalism first, but that formalism is a very useful tool of thought. Consider algebraic notation with plus and times and so on. That makes things way easier to understand than writing out equations in words (as mathematicians used to do!). It is the same for more advanced formalisms.
  • aleph_minus_one 10mo ago
    > I understand that some degree of formalism is required to enable the sharing of knowledge amongst people across a variety of languages, but sometimes I'll read a white paper and think "wow, this could be written a LOT more simply". OK, challenge accepted: find a way to write one of the following papers much more simply: Fabian Hebestreit, Peter Scholze; A note on higher almost ring theory https://arxiv.org/abs/2409.01940 https://arxiv.org/abs/2409.01940 Peter Scholze; Berkovich Motives https://arxiv.org/abs/2412.03382 https://arxiv.org/abs/2412.03382 --- What I want to tell you with these examples (these are, of course, papers which are far above my mathematical level) is: often what you read in math papers is insanely complicated; simplifying even one of such papers is often a huge academic achievement.
    • auggierose 10mo ago
      These papers are actually great examples of what is problematic with mathematics, just as what is problematic with papers in any other specialised field: how do you judge if this could be ever useful to you? If you want to understand what is going on there, what is the most effective way to build a bridge from what you know, to what is written there? If you are in a situation where the knowledge of these papers could actually greatly help, how do you become aware of it? I think if AI could help solve these two issues, that would be really something.
  • gjulianm 10mo ago
    > I suspect they do so as a means of gatekeeping I'm surprised at how could you get at this conclusion. Formalisms, esoteric language and syntax are hard for everyone. Why would people invest in them if their only usefulness was gatekeeping? Specially when it's the same people who will publish their articles in the open for everyone to read. A more reasonable interpretation is that those fields use those things you don't like because they're actually useful to them and to their main audience, and that if you want to actually understand those concepts they talk about, that syntax will end up being useful to you too. And that a lack of syntax would not make things easier to understand, just less precise.
  • MangoToupe 10mo ago
    > I suspect they do so as a means of gatekeeping. What, as opposed to using ambiguous language and getting absolutely nothing done?
  • locknitpicker 10mo ago
    > Yeah, I don't want to be uncharitable, but I've noticed that a lot of stem fields make heavy use of esoteric language and syntax, and I suspect they do so as a means of gatekeeping. I think you're confusing "I don't understand this" with "the man is keeping me down". All fields develop specialized language and syntax because a) they handle specialized topics and words help communicate these specialized concepts in a concise and clear way, b) syntax is problem-specific for the same reason. See for example tensor notation, or how some cultures have many specialized terms to refer to things like snow while communicating nuances. > "wow, this could be written a LOT more simply" That's fine. A big part of research is to digest findings. I mean, we still see things like novel proofs for the Pythagoras theorem. If you can express things clearer, why aren't you?
    • zozbot234 10mo ago
      Statistics is a weird special case where major subfields of applied statistics (including machine learning, but not only) sometimes retain wildly divergent terminology for the exact same concepts, for no good reason at all except the vagaries of historical development.
  • beng-nl 10mo ago
    My opinion on this is that in mathematics the material can be presented in a very dry and formal way, often in service of rigor, which is not welcoming at all, and is in fact unnecessarily unwelcoming. But I don’t believe it to be used as gatekeeping at all. At worst, hazing (“it was difficult for me as newcomer so it should be difficult to newcomers after me”) or intellectual status (“look at this textbook I wrote that takes great intellectual effort to penetrate”). Neither of which should be lauded in modern times. I’m not much of a mathematician, but I’ve read some new and old textbooks, and I get the impression there is a trend towards presenting the material in a more welcoming way, not necessarily to the detriment of rigor.
    • TimPC 10mo ago
      If it's actually in the service of rigor then it's not unnecessaryily unwelcoming. If it's only nominally in the service of rigor than maybe, but Mathematics absolutely needs extreme rigor.
    • zozbot234 10mo ago
      The upside of a "dry and formal" presentation is that it removes any ambiguity about what exactly you're discussing, and how a given argument is supposed to flow. Some steps may be skipped, but at least the overall structure will be clear enough. None of that is guaranteed when dealing with an "intuitive" presentation, especially when people tend to differ about what the "right" intuition of something ought to be. That can be even more frustrating, precisely when there's insufficient "dry and formal" rigor to pin everything down.
  • fragmede 10mo ago
    In this modern era of easily accessible knowledge, how gate keepy is it though? It's inscrutable at first glance, but ChatGPT is more than happy to explain what the hell ℵ₀, ℵ₁, ♯, ♭, or Σ mean, and you can ask it to read the arxiv pdf and have it explain it to you.
  • ncfj76 10mo ago
    I say the same thing about the universe. There is some gate keeping going on there. My 3 inch chimp brain at the age of 3 itself was quite capable of imagining a universe. No quantum field equations required. Then by 6 I was doing it in minecraft. And by 10 I was doing it with a piano. But then they started wasting my time telling me to read Kant.
• Someone 10mo ago
  > I’ve long thought that more of us could devout time to serious maths problems if they were written in a language we all understood. That assumes it’s the language that makes it hard to understand serious math problems. That’s partially true (and the reason why mathematicians keep inventing new language), but IMO the complexity of truly understanding large parts of mathematics is intrinsic, not dependent on terminology. Yes, you can say “A monad is just a monoid in the category of endofunctors” in terms that more people know of, but it would take many pages, and that would make it hard to understand, too.
  • voidhorse 10mo ago
    Precisely. Think of mathematics like a game. Players of magic the gathering will say a creature "has flying" by which they mean "it can only be blocked by other creatures with reach or flying". Newcomers obviously need to learn this jargon, but once they do, communication is greatly facilitated by not having to spell out the definition. Just like games, the definitions in mathematics are ethereal and purely formal as well, and it would be a pain to spell them out on every occasion. It stems more from efficient communication needs then from gatekeeping. You expect the players of the game to learn the rules before they play.
    • racl101 10mo ago
      Well said. I'd say the ability to take complicated definitions and to not have to through a rigorous definition every time the ideas are referenced are, in a sense a form of abstraction, and a necessary requirement to be able to do advanced Math in the first place.
    • matheme 10mo ago
      My entire being is anthithetical to this type of gatekeeping. > You expect the players of the game to learn the rules before they play. TFA is literally from a 'player' who has 'learned the rules' complaining that the papers remain indecipherable. > You expect the players of the game to learn the rules before they play. Actually, I expect to have to teach rules to new players before they play. We are different.
• dr_dshiv 10mo ago
  See Brett Victor’s: Kill Math https://worrydream.com/KillMath/ https://worrydream.com/KillMath/ He separates conceptual understanding from notational understanding— pointing out that the interface of using math has a major impact on utility and understanding. For instance, Roman numerals inhibit understanding and utilization of multiplication. Better notational systems can be designed, he claims.