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So I find this interesting, because it seems to confirm a suspicion I held. I "only" did some undergrad math during my computer science (but due to a lack of sp

by hannob 7y ago

So I find this interesting, because it seems to confirm a suspicion I held. I "only" did some undergrad math during my computer science (but due to a lack of specialization at my university back then it was "real math", i.e. we took the same courses as the math students). My impression often was that a lot of this stuff isn't so hard if you "get to it", but it's clouded in a lot of complicated language that makes it sound harder than it really is. I never dared to postulate that this is a more general problem of math, but your words sound like that's exactly what is happening. Do you think there's a way forward to make math more accessible by using simpler language without sacrificing correctness?

mathgenius 7y ago
Absolutely. It's a problem with academia in general. Academics don't get rewarded for explaining things in an easy to understand way. Moreover, it is often a huge amount of work to distill results down like this. And if you succeed the response you may get is "well that is obvious". Anyway, this is my counter argument to the "formalization of mathematics": why don't we incentivize people to bring some clarity to the exposition? If mathematics is suffering from faulty proofs, then I have little sympathy. Make it simpler, organize the concepts. Muscular calculations only go so far.
- ncmncm 7y ago
  In other fields, writing survey articles is (or was, recently) an appreciated activity. Survey articles are explicitly supposed to bring people up on the important results and current state of a subject.
- skela224 7y ago
  There's the idea that pure math papers should not have a "Conclusion" section, where the author summarizes what has been done and says a few informal words. This doesn't seem to help the exposition but is nonetheless prevalent: if you haven't paid attention to the proofs and derivation, they aren't going to retell what happened in plain plain language. e.g. look at this advice from the academia stackexchange, to a young aspiring mathematician: "If you're relatively young and inexperienced and hoping for best results on the rapid publication of your work in strong journals, I would stick pretty mercilessly to the format: (i) strong introduction motivating your work and explaining clearly the value added both in the results themselves and the techniques of proof and (ii) the rest of the paper contains careful proofs of all the results, in a very clear, linear, easy to follow fashion, e.g. "Section A.B: Proof of Lemma C".
I_AM_A_SMURF 7y ago
> My impression often was that a lot of this stuff isn't so hard if you "get to it", but it's clouded in a lot of complicated language that makes it sound harder than it really is. I never dared to postulate that this is a more general problem of math, but your words sound like that's exactly what is happening. I "only" have a masters degree in Math, but this sounds really wrong to me. Most advanced math concepts are complex and can take months to understand for someone with an unrelated math background (e.g., say, Lebesgue measure theory or Galois theory). I wouldn't even know how to start teaching something like that to someone who doesn't have a formal mathematical background.
- LolWolf 7y ago
  I think the GP is trying to say that, once you get past all of the definitions and proofs, the underlying idea is almost always relatively simple. I do agree that it's pretty hard to be rigorous without setting up all of this edifice, but I've also found that it's very easy (at least, as an applied mathematician) to take an idea and continue generalizing it until the original application becomes a very small corollary of the new statement—one that appears usually unmotivated and "magical." In some sense, this is a wonderful thing: you can usually prove a much stronger statement using the same tools you used to prove the weaker one, but in some sense it makes many things unbelievably difficult to understand as a first or even second-time reader. To give an example, I was reading a book on dissipative PDE-solving as an optimization problem. The book's main chapter starts with a theorem that is so general and so far removed from any one specific application that, even as someone who works on several related fields, I struggled to understand its significance. The book then spends the next several chapters showing how existence/uniqueness of solutions to a bunch of PDEs are all corollaries (with some work) of this theorem. The problem I have is that the presentation would've been much clearer going the other way. Start with the easy-to-prove existence theorems for the PDEs you care about (which mostly relied on simple tools and could've been proven in a few lines) and build up the generalization to the "main theorem." There, the reader doesn't have to guess the meaning (and/or marvel at the "ingenuity") of the original statement since the motivation is perfectly natural and clear, and they certainly don't have to struggle through a proof without a clear ending. Plus, you usually get the nice side-benefit that, by the time you present the most general statement, the proof is almost always immediate ± some small tricks. While I do think that math requires a large amount of definitions to work and to put everything on rigorous footing, I also think there's this kind of weird way of writing where definitions and proofs are presented as "magical" and then shown to be right or useful later, rather than the other way around. This is fine for a few things, but I've found that most presentations that work don't follow this template (yet most mathematical papers I've read sadly do).
chongli 7y ago
The problem with math is that the only strategy we’ve discovered (with any success) for making a concept easier to learn is to connect it with a concrete, real-world analogy. This works because people find it easier to learn something if they can connect it to something else they already know. The Achilles heel of this approach is that it discards the abstract form of the concept such that the full generality is lost on the student. One example I see frequently around here is the much-maligned monad tutorial. Writers keep trying to connect it by analogy to concepts with which imperative programmers are already familiar. As a math student, I would much prefer to be given a basic statement of the definition of a monad, a list of properties, and maybe a theorem to prove that some structure is a monad instead of having to fall back to the definition. From there, I would just work through a few exercises and start to understand what it is and how it works. Most people don’t work this way, however. People seem to be extremely resistant to learning by experimentation. Instead, they prefer to be given all the answers. I suppose this isn’t really a problem that’s unique to math. We’ve also discussed how a lot of computer users fail to truly understand how the software works and instead prefer to follow steps by rote in order to carry out one particular task.
- lolc 7y ago
  People struggle with monads because it's a two-step process to learn them: Learn their mathematical properties, then learn how to apply them as a pattern. That's not a quick process you can do in half an hour. It doesn't help that the concept is alien to everyday processes. So yeah, analogy doesn't get you far with monads.
  - chongli 7y ago
    I think most monad tutorials skip over the math and try to teach people the applications and I think this is why people don’t get very far. But then people are probably not ready for the math if they haven’t gone beyond high school algebra. Very few high school students learn about abstract algebra topics such as group theory unless they’re studying on their own. Aside: as with monads, group theory has many disparate applications, from solving Rubik’s cubes [1] to particle physics [2] to crystallography [3]. Trying to teach someone group theory using only a Rubik’s cube may not be the best idea, in light of the abstract nature of the structure. [1] https://en.wikipedia.org/wiki/Rubik%27s_Cube_group https://en.wikipedia.org/wiki/Rubik%27s_Cube_group [2] https://en.wikipedia.org/wiki/Particle_physics_and_representation_theory https://en.wikipedia.org/wiki/Particle_physics_and_represent... [3] https://en.wikipedia.org/wiki/Crystallographic_point_group https://en.wikipedia.org/wiki/Crystallographic_point_group
knzhou 7y ago
No, there is no magic formula to make hard things easy. This is the same as the classic "monads are burritos" thing. A programmer spends a week trying to understand monads, and finally it clicks when he makes an analogy with wrapping burritos. So that must mean that burritos are the real way to explain monads, and everybody who didn't use the burrito analogy was being obfuscatory on purpose, right? Of course not! If that programmer than goes out and tells somebody else that "monads are burritos", they'll have no idea what that vague, nearly nonsensical statement means. You need the context of grappling with the idea over time for an analogy to actually help. Good textbooks are already extremely optimized for ease of learning; the authors of these books are universally people who started in your position, thinking that everybody would be so much easier to learn if it were just written in sensible language, usually the language they were happening to use when the idea finally clicked for them. And the result ends up hard to understand, because the concept genuinely is hard to understand.
CBVanderhuge 7y ago
The thing is that actually doing mathematics is a process where you take your intuition and find a way to formalize it into definitions and proofs and computations. A toddler could understand the difference between a smooth curve and a spiky curve, but to actually get a handle on them computationally, the best tool that we have is Calculus -- and even after decades of people iterating on how to teach it, Calculus is still a mystery to most undergrads who muddle through it. Similarly I could give you some intuition about manifolds by telling you that they're kinda like surfaces that can have higher dimensionality, but until you're able to actually understand and work with the definition of "locally homeomorphic to R^n", you're not going to be able to do anything with them. I think it's bollocks, this idea that mathematicians are obfuscating easy concepts with complex language. The language doesn't exist to get research grants and gatekeep, it's there because you need some succinct way to describe objects and concepts without running through a full definition every time. Obviously you can always improve communication, and from my experience most mathematicians try pretty hard to do that, especially in the classroom, but it's not easy.

by hannob 7y ago

mathgenius 7y ago

Absolutely. It's a problem with academia in general. Academics don't get rewarded for explaining things in an easy to understand way. Moreover, it is often a huge amount of work to distill results down like this. And if you succeed the response you may get is "well that is obvious". Anyway, this is my counter argument to the "formalization of mathematics": why don't we incentivize people to bring some clarity to the exposition? If mathematics is suffering from faulty proofs, then I have little sympathy. Make it simpler, organize the concepts. Muscular calculations only go so far.

ncmncm 7y ago
In other fields, writing survey articles is (or was, recently) an appreciated activity. Survey articles are explicitly supposed to bring people up on the important results and current state of a subject.
skela224 7y ago
There's the idea that pure math papers should not have a "Conclusion" section, where the author summarizes what has been done and says a few informal words. This doesn't seem to help the exposition but is nonetheless prevalent: if you haven't paid attention to the proofs and derivation, they aren't going to retell what happened in plain plain language. e.g. look at this advice from the academia stackexchange, to a young aspiring mathematician: "If you're relatively young and inexperienced and hoping for best results on the rapid publication of your work in strong journals, I would stick pretty mercilessly to the format: (i) strong introduction motivating your work and explaining clearly the value added both in the results themselves and the techniques of proof and (ii) the rest of the paper contains careful proofs of all the results, in a very clear, linear, easy to follow fashion, e.g. "Section A.B: Proof of Lemma C".

I_AM_A_SMURF 7y ago

> My impression often was that a lot of this stuff isn't so hard if you "get to it", but it's clouded in a lot of complicated language that makes it sound harder than it really is. I never dared to postulate that this is a more general problem of math, but your words sound like that's exactly what is happening. I "only" have a masters degree in Math, but this sounds really wrong to me. Most advanced math concepts are complex and can take months to understand for someone with an unrelated math background (e.g., say, Lebesgue measure theory or Galois theory). I wouldn't even know how to start teaching something like that to someone who doesn't have a formal mathematical background.

LolWolf 7y ago
I think the GP is trying to say that, once you get past all of the definitions and proofs, the underlying idea is almost always relatively simple. I do agree that it's pretty hard to be rigorous without setting up all of this edifice, but I've also found that it's very easy (at least, as an applied mathematician) to take an idea and continue generalizing it until the original application becomes a very small corollary of the new statement—one that appears usually unmotivated and "magical." In some sense, this is a wonderful thing: you can usually prove a much stronger statement using the same tools you used to prove the weaker one, but in some sense it makes many things unbelievably difficult to understand as a first or even second-time reader. To give an example, I was reading a book on dissipative PDE-solving as an optimization problem. The book's main chapter starts with a theorem that is so general and so far removed from any one specific application that, even as someone who works on several related fields, I struggled to understand its significance. The book then spends the next several chapters showing how existence/uniqueness of solutions to a bunch of PDEs are all corollaries (with some work) of this theorem. The problem I have is that the presentation would've been much clearer going the other way. Start with the easy-to-prove existence theorems for the PDEs you care about (which mostly relied on simple tools and could've been proven in a few lines) and build up the generalization to the "main theorem." There, the reader doesn't have to guess the meaning (and/or marvel at the "ingenuity") of the original statement since the motivation is perfectly natural and clear, and they certainly don't have to struggle through a proof without a clear ending. Plus, you usually get the nice side-benefit that, by the time you present the most general statement, the proof is almost always immediate ± some small tricks. While I do think that math requires a large amount of definitions to work and to put everything on rigorous footing, I also think there's this kind of weird way of writing where definitions and proofs are presented as "magical" and then shown to be right or useful later, rather than the other way around. This is fine for a few things, but I've found that most presentations that work don't follow this template (yet most mathematical papers I've read sadly do).

chongli 7y ago

The problem with math is that the only strategy we’ve discovered (with any success) for making a concept easier to learn is to connect it with a concrete, real-world analogy. This works because people find it easier to learn something if they can connect it to something else they already know. The Achilles heel of this approach is that it discards the abstract form of the concept such that the full generality is lost on the student. One example I see frequently around here is the much-maligned monad tutorial. Writers keep trying to connect it by analogy to concepts with which imperative programmers are already familiar. As a math student, I would much prefer to be given a basic statement of the definition of a monad, a list of properties, and maybe a theorem to prove that some structure is a monad instead of having to fall back to the definition. From there, I would just work through a few exercises and start to understand what it is and how it works. Most people don’t work this way, however. People seem to be extremely resistant to learning by experimentation. Instead, they prefer to be given all the answers. I suppose this isn’t really a problem that’s unique to math. We’ve also discussed how a lot of computer users fail to truly understand how the software works and instead prefer to follow steps by rote in order to carry out one particular task.

lolc 7y ago
People struggle with monads because it's a two-step process to learn them: Learn their mathematical properties, then learn how to apply them as a pattern. That's not a quick process you can do in half an hour. It doesn't help that the concept is alien to everyday processes. So yeah, analogy doesn't get you far with monads.
- chongli 7y ago
  I think most monad tutorials skip over the math and try to teach people the applications and I think this is why people don’t get very far. But then people are probably not ready for the math if they haven’t gone beyond high school algebra. Very few high school students learn about abstract algebra topics such as group theory unless they’re studying on their own. Aside: as with monads, group theory has many disparate applications, from solving Rubik’s cubes [1] to particle physics [2] to crystallography [3]. Trying to teach someone group theory using only a Rubik’s cube may not be the best idea, in light of the abstract nature of the structure. [1] https://en.wikipedia.org/wiki/Rubik%27s_Cube_group https://en.wikipedia.org/wiki/Rubik%27s_Cube_group [2] https://en.wikipedia.org/wiki/Particle_physics_and_representation_theory https://en.wikipedia.org/wiki/Particle_physics_and_represent... [3] https://en.wikipedia.org/wiki/Crystallographic_point_group https://en.wikipedia.org/wiki/Crystallographic_point_group

knzhou 7y ago

No, there is no magic formula to make hard things easy. This is the same as the classic "monads are burritos" thing. A programmer spends a week trying to understand monads, and finally it clicks when he makes an analogy with wrapping burritos. So that must mean that burritos are the real way to explain monads, and everybody who didn't use the burrito analogy was being obfuscatory on purpose, right? Of course not! If that programmer than goes out and tells somebody else that "monads are burritos", they'll have no idea what that vague, nearly nonsensical statement means. You need the context of grappling with the idea over time for an analogy to actually help. Good textbooks are already extremely optimized for ease of learning; the authors of these books are universally people who started in your position, thinking that everybody would be so much easier to learn if it were just written in sensible language, usually the language they were happening to use when the idea finally clicked for them. And the result ends up hard to understand, because the concept genuinely is hard to understand.

CBVanderhuge 7y ago

The thing is that actually doing mathematics is a process where you take your intuition and find a way to formalize it into definitions and proofs and computations. A toddler could understand the difference between a smooth curve and a spiky curve, but to actually get a handle on them computationally, the best tool that we have is Calculus -- and even after decades of people iterating on how to teach it, Calculus is still a mystery to most undergrads who muddle through it. Similarly I could give you some intuition about manifolds by telling you that they're kinda like surfaces that can have higher dimensionality, but until you're able to actually understand and work with the definition of "locally homeomorphic to R^n", you're not going to be able to do anything with them. I think it's bollocks, this idea that mathematicians are obfuscating easy concepts with complex language. The language doesn't exist to get research grants and gatekeep, it's there because you need some succinct way to describe objects and concepts without running through a full definition every time. Obviously you can always improve communication, and from my experience most mathematicians try pretty hard to do that, especially in the classroom, but it's not easy.