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> For anyone that spends enough time working in the given field, the notation is by far the easiest thing to figure out. I want to emphasize this because the w

> For anyone that spends enough time working in the given field, the notation is by far the easiest thing to figure out. I want to emphasize this because the weirdness of math notation comes up often. The difficulty of grasping mathematical concepts far outweighs notational concerns. Getting used to notation in a specific field is an absolute drop in the bucket compared to understanding the theorems and proofs, and often the reason notation is so idiosyncratic and inconsistent is because it's meant to try to reflect human intuition, which is idiosyncratic and inconsistent in comparison to the mathematics from which it is derived. Once you really get into mathematics, notation is so far down the list of things that make learning mathematics hard that I don't think it's worth the associated cost of trying to change it on a large scale.

t-3 5y ago
I have to vehemently disagree. Notation does more to keep mathematics the domain of nerds than anything else. It's unintuitive, uninformative, excessively complicated jargon which exists only to gatekeep and save paper. It's simply backwards to continue to use such an error-prone system in times of modern technology.
- gjulianm 5y ago
  > It's unintuitive, uninformative, excessively complicated jargon which exists only to gatekeep and save paper. As a mathematician, notation exists because it's easier to read things with good notation than without. I have some advanced mathematics books in my table and in most of the pages you'll find far more text than notation. Notation is usually explained when it starts being used, usually in the definition. For example, the problem with understanding the notation "Aut(L|k)" is not the notation, because if the author wrote "group of field automorphisms of L fixing k elementwise" I'd be equally lost if I don't understand those words. And once I understand those words, the notation makes it easier to read and understand things whenever the author is talking about that group, because your brain automatically identifies the concept associated with the notation instead of having to read the words and linking them with the concept.
- coliveira 5y ago
  People who don't understand math have this phantasy that they would understand everything only if mathematicians decided to use normal language instead of math symbols. This is just a phantasy, there is no simplification if you stop using math symbols, in fact it is just the opposite. Try to describe some intricate math without symbols and you'll see this.
  - t-3 5y ago
    I do complex math without using greek letters and other untypable characters all the time. It's called programming. The order of operations is explicit, the operations themselves are explicit, and there is very little room for ambiguity. The notation is only for saving space, not because it's the be-all end-all absolute best solution for representing mathematics.
    - BeetleB 5y ago
      > I do complex math without using greek letters and other untypable characters all the time. It's called programming. This is simple math. Start writing formal proofs for your programs. You will then be in the realm of something more complex (and still far lower than the level in a research paper). Even a proof of the complexity of your algorithm (let alone the correctness) will get ugly without this notation.
    - coliveira 5y ago
      Sorry to tell you, but most programming is just elementary school math and boolean operations. In many cases it doesn't even require high school math. I'm not saying it is easy, but it certainly doesn't require any sophisticated math.
  - brimble 5y ago
    My first step to trying to understand an unfamiliar equation or proof is, if I am to have any hope of figuring it out, always breaking it down by term and operation and figuring out WTF each one means in context. Usually on paper. In normal words. Typically I end up turning it into something more "algorithmic" (OK, what happens as the value "passes through" this operation, and then on to this next one...). I'm 1000% useless at understanding equationally- or identity-focused writing, and have to both painstakingly decipher every term and translate it to something more algorithm-like before I can do anything more than stare at the page and drool. So... yes, I do think it would help if at least some of that were already done for me. As it is it takes so damn long that I don't bother unless I have a good reason.
simias 5y ago
I'm not a computer scientist but as a programmer I enjoy dabbling in computer science papers and while I often do understand the underlying concepts (sometimes very well since I've been using these concepts daily for decades) I often find myself incapable of following a paper just because the notation is hopelessly obtuse. The immediate example for me would be lambda calculus which makes a lot of sense to me when expressed in code (be it lisp, python or Rust) but just looks like vomit in theoretical papers, for instance: (λy.M)[x := N] = λy.(M[x := N]), if x ≠ y and y ∉ FV(N) This looks like a perl one liner from an obfuscation contest. Why is it that software engineers in a few decades have come to the conclusion that proper variable naming is important and abuse of sigils a bad idea but mathematics don't feel the same way?
- abdullahkhalids 5y ago
  I would recommend two things: 1. Since you seem to someone who reads papers sometimes, I would encourage you define your "programming notation" and start proving theorems in that notation. Do it for sufficiently complex proofs: the ones that require a few different lemmas, span several pages, and use some non-trivial algebra. Then, attempt to present your proof to someone in your notation. Judge for understanding. 2. You might very well succeed. But if you don't, wonder why out of many many software engineers (or profs at software engineering oriented degrees) over the past few decades who read math/cs papers, no one attempted to use "programming notation" to write a book or notes to communicate mathematics to others?
  - SAI_Peregrinus 5y ago
    WRT 2, plenty of people use more "programming" style notation. Most cryptography papers these days define algorithms as sequences of steps that look quite a bit like code, just with Greek letters for many of the variable names (but italicized words for function names). EG this paper[1] picked from the last week of IACR preprints shows the style. That's just the first one I clicked, the title looked interesting. Most of the rest use a similar style. The big reason for single-letter variable names is that historically multiplication can be denoted by concatenation. It makes formulae shorter, so they don't need line breaks. Personally I don't think that's a huge benefit, particularly when there's more than one sort of product possible so you end up needing to explicitly denote multiplication anyway. [1] https://eprint.iacr.org/2022/106.pdf https://eprint.iacr.org/2022/106.pdf (Page 7-8)
abecedarius 5y ago
People are different. For the kind of person who's likely to get heavily into math, any infelicities of notation are lower-order problems, that's true. For other people, the function of math in the school system seems to be mainly to sort them by math ability -- and for that, artificial barriers aren't a problem. So there's little demand for better UX design for nonspecialists. Sucks if you think society would be better off if more people appreciated/understood/applied more math.

t-3 5y ago

I have to vehemently disagree. Notation does more to keep mathematics the domain of nerds than anything else. It's unintuitive, uninformative, excessively complicated jargon which exists only to gatekeep and save paper. It's simply backwards to continue to use such an error-prone system in times of modern technology.

gjulianm 5y ago
> It's unintuitive, uninformative, excessively complicated jargon which exists only to gatekeep and save paper. As a mathematician, notation exists because it's easier to read things with good notation than without. I have some advanced mathematics books in my table and in most of the pages you'll find far more text than notation. Notation is usually explained when it starts being used, usually in the definition. For example, the problem with understanding the notation "Aut(L|k)" is not the notation, because if the author wrote "group of field automorphisms of L fixing k elementwise" I'd be equally lost if I don't understand those words. And once I understand those words, the notation makes it easier to read and understand things whenever the author is talking about that group, because your brain automatically identifies the concept associated with the notation instead of having to read the words and linking them with the concept.
coliveira 5y ago
People who don't understand math have this phantasy that they would understand everything only if mathematicians decided to use normal language instead of math symbols. This is just a phantasy, there is no simplification if you stop using math symbols, in fact it is just the opposite. Try to describe some intricate math without symbols and you'll see this.
- t-3 5y ago
  I do complex math without using greek letters and other untypable characters all the time. It's called programming. The order of operations is explicit, the operations themselves are explicit, and there is very little room for ambiguity. The notation is only for saving space, not because it's the be-all end-all absolute best solution for representing mathematics.
  - BeetleB 5y ago
    > I do complex math without using greek letters and other untypable characters all the time. It's called programming. This is simple math. Start writing formal proofs for your programs. You will then be in the realm of something more complex (and still far lower than the level in a research paper). Even a proof of the complexity of your algorithm (let alone the correctness) will get ugly without this notation.
  - coliveira 5y ago
    Sorry to tell you, but most programming is just elementary school math and boolean operations. In many cases it doesn't even require high school math. I'm not saying it is easy, but it certainly doesn't require any sophisticated math.
- brimble 5y ago
  My first step to trying to understand an unfamiliar equation or proof is, if I am to have any hope of figuring it out, always breaking it down by term and operation and figuring out WTF each one means in context. Usually on paper. In normal words. Typically I end up turning it into something more "algorithmic" (OK, what happens as the value "passes through" this operation, and then on to this next one...). I'm 1000% useless at understanding equationally- or identity-focused writing, and have to both painstakingly decipher every term and translate it to something more algorithm-like before I can do anything more than stare at the page and drool. So... yes, I do think it would help if at least some of that were already done for me. As it is it takes so damn long that I don't bother unless I have a good reason.

simias 5y ago

I'm not a computer scientist but as a programmer I enjoy dabbling in computer science papers and while I often do understand the underlying concepts (sometimes very well since I've been using these concepts daily for decades) I often find myself incapable of following a paper just because the notation is hopelessly obtuse. The immediate example for me would be lambda calculus which makes a lot of sense to me when expressed in code (be it lisp, python or Rust) but just looks like vomit in theoretical papers, for instance: (λy.M)[x := N] = λy.(M[x := N]), if x ≠ y and y ∉ FV(N) This looks like a perl one liner from an obfuscation contest. Why is it that software engineers in a few decades have come to the conclusion that proper variable naming is important and abuse of sigils a bad idea but mathematics don't feel the same way?

abdullahkhalids 5y ago
I would recommend two things: 1. Since you seem to someone who reads papers sometimes, I would encourage you define your "programming notation" and start proving theorems in that notation. Do it for sufficiently complex proofs: the ones that require a few different lemmas, span several pages, and use some non-trivial algebra. Then, attempt to present your proof to someone in your notation. Judge for understanding. 2. You might very well succeed. But if you don't, wonder why out of many many software engineers (or profs at software engineering oriented degrees) over the past few decades who read math/cs papers, no one attempted to use "programming notation" to write a book or notes to communicate mathematics to others?
- SAI_Peregrinus 5y ago
  WRT 2, plenty of people use more "programming" style notation. Most cryptography papers these days define algorithms as sequences of steps that look quite a bit like code, just with Greek letters for many of the variable names (but italicized words for function names). EG this paper[1] picked from the last week of IACR preprints shows the style. That's just the first one I clicked, the title looked interesting. Most of the rest use a similar style. The big reason for single-letter variable names is that historically multiplication can be denoted by concatenation. It makes formulae shorter, so they don't need line breaks. Personally I don't think that's a huge benefit, particularly when there's more than one sort of product possible so you end up needing to explicitly denote multiplication anyway. [1] https://eprint.iacr.org/2022/106.pdf https://eprint.iacr.org/2022/106.pdf (Page 7-8)

abecedarius 5y ago

People are different. For the kind of person who's likely to get heavily into math, any infelicities of notation are lower-order problems, that's true. For other people, the function of math in the school system seems to be mainly to sort them by math ability -- and for that, artificial barriers aren't a problem. So there's little demand for better UX design for nonspecialists. Sucks if you think society would be better off if more people appreciated/understood/applied more math.