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> There are so many _symbols_ used in mathematics. It's all as if they're spells written by wizened greybeard wizards and passed down to apprentices throughout

> There are so many _symbols_ used in mathematics. It's all as if they're spells written by wizened greybeard wizards and passed down to apprentices throughout the centuries. Think of it as jumping into a new codebase you have never worked with before. It is full of strange functions calling each other for unknown reasons. It takes a while to get a good intuition for why. M/L could be written Field-extension(M, L). That might make it easier to Google what's going on, rather than having to pick up a group theory book. And indeed in this case it would be extra helpful, because M/L could also mean "quotient group" or even "division" depending on the type of the objects M and L. I personally prefer M:L for field extensions. But Googling individual concepts is hardly a good way to learn group theory. So I don't think it matters too much. For anyone that spends enough time working in the given field, the notation is by far the easiest thing to figure out. In general it is nice that mathematics - still being largely a handwritten endeavor - has the freedom to introduce new notation where it makes sense. See https://mathoverflow.net/questions/42929/suggestions-for-good-notation https://mathoverflow.net/questions/42929/suggestions-for-goo... for cool examples of new notation improving the lives of everyone.

dwohnitmok 5y ago
> 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.
- 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.
jcelerier 5y ago
> Think of it as jumping into a new codebase you have never worked with before. It is full of strange functions calling each other for unknown reasons. It takes a while to get a good intuition for why. Having done consulting, I frankly disagree. Except pathological cases most of the codebases I've been involved in it's possible to open a file at random at quickly make sense of what happens even without prior exposure. With math papers my experience is that you have to go 3/4 references deep before startit to understand where the notation comes from which wastes hours and hours. They should provide normalized names that could be searched for in e.g. a coq-based database of math concepts or something like that.
- gjulianm 5y ago
  > With math papers my experience is that you have to go 3/4 references deep before startit to understand where the notation comes from which wastes hours and hours. Most of the time you'd need even more hours to understand not the notation but the concept themselves. In this list there isn't that much weird notation compared to weird words. For example, 'The Krull dimension of a noetherian integral domain is finite'. Even without notation, I'd expect a non-math major to have to go read 8-10 wikipedia pages just to understand "Krull dimension".
- benibela 5y ago
  And in codebases you can always search/grep for function names, so it easy to find the definition of everything. In math it is a strange symbol in a PDF and you cannot search for at all
  - BlueTemplar 5y ago
    This is more a limitation of computers and search engines. The move to using more Unicode like in Julia (and hopefully away from PDF to (m)HTML) should make this issue better.
    - benibela 5y ago
      It is also because the notation with single symbols is too short In code someone would write "lightspeed" and that can be searched. In a paper, they write "c" and when you search for it you find thousands of unrelated things.
bitcurious 5y ago
> Think of it as jumping into a new codebase you have never worked with before. It is full of strange functions calling each other for unknown reasons. It takes a while to get a good intuition for why. I think about this analogy from time to time, when I get frustrated by maths. Myself not being mathematician but on occasion needing to read a research paper, my frustration is typically begins with the raw syntax of maths. It’s so obviously optimized for the combined limitations of the human brain’s limited short term memory and the human hand’s speed of writing/typing. Which probably is the correct optimization for high level practitioners. For communication with non experts, it’s dreadful, but I think fixable post-facto with a auto-notation tool. Think black.py for maths.
- samatman 5y ago
  Here's the lens I use, hopefully it will help. A terse equation or statement in mathematics represents what you will understand, the rest of the paper is to get you there. Wanting to read the main equations and understand them, so as to use that knowledge reading the paper, is backward. If the main equation was ax^2+bx+c=0, you wouldn't need to read the paper, right? It's about the quadratic equation, it's aimed at primary students. But that's how it works, we keep pounding away at the paper: taking notes, following references, thinking a lot about each of the definitions, then eventually, you look at the main equation and say "oh. ok. I know what this says now".
jimbob45 5y ago
>Think of it as jumping into a new codebase you have never worked with before. It is full of strange functions calling each other for unknown reasons. It takes a while to get a good intuition for why. Software engineers have built careers off of making code more readable, both in terms of the language itself and the codebase it applies to. Mathematicians seem to go out of their way to make their work as obtuse as possible. Can you imagine if software engineers had stopped at C? If we were all still trying to apply the spiral rule [0] to read everything? That's what reading math papers feels like. It feels like you stopped at C and are unwilling to even consider that there could be a better way to express things. [0]http://c-faq.com/decl/spiral.anderson.html http://c-faq.com/decl/spiral.anderson.html
- gjulianm 5y ago
  > Mathematicians seem to go out of their way to make their work as obtuse as possible. Mathematicians go out of their way to make their work as understandable as possible. If they wanted to make it as obtuse as possible, you'd just see a picture of chaotic scribbles on a blackboard. > That's what reading math papers feels like. It feels like you stopped at C and are unwilling to even consider that there could be a better way to express things. Do you have any suggestions? Mathematics is a hard subject because it deals with concepts in several levels deep of abstraction. That's a complexity that you can't escape. You can't expect to read a math paper (aimed at researchers in the same field) and understand everything, the same way it would be unreasonable to ask you to write your code in a way that someone who isn't familiar with either programming or the concepts your code works with to understand it if they went to read it.
  - jimbob45 5y ago
    If I give you suggestions, you're just going to say, "well that's the way we've always done it and people will be confused if we change now", which is exactly what they said about C until someone decided enough was enough. Even something as simple as the set of integers being the letter Z requires you to understand German to know why we chose Z (and not I) in the first place. Or d/dx being totally unintuitive for anyone not well-versed in differential calc. I'm quite certain that every student learning calc thinks "Well the d's just cancel out there". Granted, I've heard the intuition behind the notation and I understand why it's tolerated. Still, it seems less helpful than it could be if mathematicians weren't married to tradition.
brimble 5y ago
> Think of it as jumping into a new codebase you have never worked with before. It is full of strange functions calling each other for unknown reasons. It takes a while to get a good intuition for why. I've basically abandoned all languages/ecosystems that aren't, at the very least, extremely grep-friendly. Static types preferred so I can get that sweet, sweet auto-refactoring, jump-to-references/definition, docs-on-hover, et c. If we expected people to read code like they're expected to read math, I'd have taken the huge pay hit and left the profession by now. It'd be a choice between that or a spiral into depression ending in tragedy. I don't know how math-loving folks can stand it.
- mannerheim 5y ago
  For textbooks, it's a non-concern because the meaning of each equation is explained in the surrounding text. The amount of new notation (by which I mean symbols such as / for group division or notation for differentiation and integration, rather than variable names) introduced by any given textbook is not that much, and as others have pointed out, by the time you finish reading the associated theorems, examples, and have done some of the exercises (very much underrated part of learning mathematics), if you understand the concept represented by the notation, the notation itself is trivial to remember. Variables, as opposed to notation, tend to be local in scope, so 1) it isn't vital to have a memorable name and 2) it wouldn't really help to have a 'proper' name, either. As an example, if we're talking about two elements from a group, we usually talk about two elements, a and b, in a group, G. Could you call them elements 1 and 2 from an ExampleGroup? Maybe, but there's no clarity provided since these variables are only scoped for a given section. Additionally, longer equations means that it takes longer to read through and check steps like algebraic manipulation in a proof, or application of a lemma or theorem. What is globally scoped are things like theorems, lemmas, definitions, etc., and those are easy to track down since they are by convention numbered after chapter and section.

dwohnitmok 5y ago

> 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.
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.

jcelerier 5y ago

> Think of it as jumping into a new codebase you have never worked with before. It is full of strange functions calling each other for unknown reasons. It takes a while to get a good intuition for why. Having done consulting, I frankly disagree. Except pathological cases most of the codebases I've been involved in it's possible to open a file at random at quickly make sense of what happens even without prior exposure. With math papers my experience is that you have to go 3/4 references deep before startit to understand where the notation comes from which wastes hours and hours. They should provide normalized names that could be searched for in e.g. a coq-based database of math concepts or something like that.

gjulianm 5y ago
> With math papers my experience is that you have to go 3/4 references deep before startit to understand where the notation comes from which wastes hours and hours. Most of the time you'd need even more hours to understand not the notation but the concept themselves. In this list there isn't that much weird notation compared to weird words. For example, 'The Krull dimension of a noetherian integral domain is finite'. Even without notation, I'd expect a non-math major to have to go read 8-10 wikipedia pages just to understand "Krull dimension".
benibela 5y ago
And in codebases you can always search/grep for function names, so it easy to find the definition of everything. In math it is a strange symbol in a PDF and you cannot search for at all
- BlueTemplar 5y ago
  This is more a limitation of computers and search engines. The move to using more Unicode like in Julia (and hopefully away from PDF to (m)HTML) should make this issue better.
  - benibela 5y ago
    It is also because the notation with single symbols is too short In code someone would write "lightspeed" and that can be searched. In a paper, they write "c" and when you search for it you find thousands of unrelated things.

bitcurious 5y ago

> Think of it as jumping into a new codebase you have never worked with before. It is full of strange functions calling each other for unknown reasons. It takes a while to get a good intuition for why. I think about this analogy from time to time, when I get frustrated by maths. Myself not being mathematician but on occasion needing to read a research paper, my frustration is typically begins with the raw syntax of maths. It’s so obviously optimized for the combined limitations of the human brain’s limited short term memory and the human hand’s speed of writing/typing. Which probably is the correct optimization for high level practitioners. For communication with non experts, it’s dreadful, but I think fixable post-facto with a auto-notation tool. Think black.py for maths.

samatman 5y ago
Here's the lens I use, hopefully it will help. A terse equation or statement in mathematics represents what you will understand, the rest of the paper is to get you there. Wanting to read the main equations and understand them, so as to use that knowledge reading the paper, is backward. If the main equation was ax^2+bx+c=0, you wouldn't need to read the paper, right? It's about the quadratic equation, it's aimed at primary students. But that's how it works, we keep pounding away at the paper: taking notes, following references, thinking a lot about each of the definitions, then eventually, you look at the main equation and say "oh. ok. I know what this says now".

jimbob45 5y ago

>Think of it as jumping into a new codebase you have never worked with before. It is full of strange functions calling each other for unknown reasons. It takes a while to get a good intuition for why. Software engineers have built careers off of making code more readable, both in terms of the language itself and the codebase it applies to. Mathematicians seem to go out of their way to make their work as obtuse as possible. Can you imagine if software engineers had stopped at C? If we were all still trying to apply the spiral rule [0] to read everything? That's what reading math papers feels like. It feels like you stopped at C and are unwilling to even consider that there could be a better way to express things. [0]http://c-faq.com/decl/spiral.anderson.html http://c-faq.com/decl/spiral.anderson.html

gjulianm 5y ago
> Mathematicians seem to go out of their way to make their work as obtuse as possible. Mathematicians go out of their way to make their work as understandable as possible. If they wanted to make it as obtuse as possible, you'd just see a picture of chaotic scribbles on a blackboard. > That's what reading math papers feels like. It feels like you stopped at C and are unwilling to even consider that there could be a better way to express things. Do you have any suggestions? Mathematics is a hard subject because it deals with concepts in several levels deep of abstraction. That's a complexity that you can't escape. You can't expect to read a math paper (aimed at researchers in the same field) and understand everything, the same way it would be unreasonable to ask you to write your code in a way that someone who isn't familiar with either programming or the concepts your code works with to understand it if they went to read it.
- jimbob45 5y ago
  If I give you suggestions, you're just going to say, "well that's the way we've always done it and people will be confused if we change now", which is exactly what they said about C until someone decided enough was enough. Even something as simple as the set of integers being the letter Z requires you to understand German to know why we chose Z (and not I) in the first place. Or d/dx being totally unintuitive for anyone not well-versed in differential calc. I'm quite certain that every student learning calc thinks "Well the d's just cancel out there". Granted, I've heard the intuition behind the notation and I understand why it's tolerated. Still, it seems less helpful than it could be if mathematicians weren't married to tradition.

brimble 5y ago

> Think of it as jumping into a new codebase you have never worked with before. It is full of strange functions calling each other for unknown reasons. It takes a while to get a good intuition for why. I've basically abandoned all languages/ecosystems that aren't, at the very least, extremely grep-friendly. Static types preferred so I can get that sweet, sweet auto-refactoring, jump-to-references/definition, docs-on-hover, et c. If we expected people to read code like they're expected to read math, I'd have taken the huge pay hit and left the profession by now. It'd be a choice between that or a spiral into depression ending in tragedy. I don't know how math-loving folks can stand it.

mannerheim 5y ago
For textbooks, it's a non-concern because the meaning of each equation is explained in the surrounding text. The amount of new notation (by which I mean symbols such as / for group division or notation for differentiation and integration, rather than variable names) introduced by any given textbook is not that much, and as others have pointed out, by the time you finish reading the associated theorems, examples, and have done some of the exercises (very much underrated part of learning mathematics), if you understand the concept represented by the notation, the notation itself is trivial to remember. Variables, as opposed to notation, tend to be local in scope, so 1) it isn't vital to have a memorable name and 2) it wouldn't really help to have a 'proper' name, either. As an example, if we're talking about two elements from a group, we usually talk about two elements, a and b, in a group, G. Could you call them elements 1 and 2 from an ExampleGroup? Maybe, but there's no clarity provided since these variables are only scoped for a given section. Additionally, longer equations means that it takes longer to read through and check steps like algebraic manipulation in a proof, or application of a lemma or theorem. What is globally scoped are things like theorems, lemmas, definitions, etc., and those are easy to track down since they are by convention numbered after chapter and section.