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 consequently less energy on the most recent results.”
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.
> 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.
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.
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.
Many mathematicians do in fact teach the rules of the game in numerous introductory texts. However, you don't expect to have to explain the rules every time you play the game with people who you've established know the game. Any session would take ages if so, and in many cases the game only become more fun the more fluent the players are.
I'm not fully convinced the article makes the claim that jargon, per se, is what needs to change nor that the use of jargon causes gatekeeping. I read more about being about the inherent challenges of presenting more complicated ideas, with or without jargon and the pursuit of better methods, which themselves might actually depend on more jargon in some cases (to abstract away and offload the cognitive costs of constantly spelling out definitions). Giving a good name to something is often a really powerful way to lower the cognitive costs of arguments employing the names concept. Theoretics in large part is the hunt for good names for things and the relationships between them.
You'd be hard pressed to find a single human endeavor that does not employ jargon in some fashion. Half the point of my example was to show that you cannot escape jargon and "gatekeeping" even in something as silly and fun as a card game.
The article does not complain about notation. It describes how the different fields of mathematics are so deep and so abstract that it’s hard to understand them as a professional mathematician in a different field. That’s a hard problem worthy of discussion, but as the article says, it’s not as much a problem of notation or of explanations, rather than it’s just intrinsically difficult and complex because these are abstract and deep fields.
The only thing that sentence says is that it’s impossible to understand math without understanding the language of math and how it is constructed. Not sure how that is controversial or gatekeeping. If you are annoyed at that comment saying “learn” instead of “be taught”, I think that’s a pedantic argument because the argument wasn’t about that at all.
Again, learning notation is part of the process of learning math. No one is gatekeeping anything, at no point you need to do an exam or magically be aware of notation that you never saw. Every book and every class will define new notation at the beginning, in most cases they will do so even when there’s no new notation. I am not sure what your argument is.
That’s a very good gate to keep. Some things are just meant to be gatekept so that the cranks and dilettantes that wastes everyone’s time can stay far outside.
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.
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".
> 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?
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.
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.
>
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
A little off topic perhaps, but out of curiosity - how many of us here have an interest in recreational mathematics? [https://en.wikipedia.org/wiki/Recreational_mathematics]