10 ms·

Annotated equations for increased readability and understanding of papers

192 points by visviva 5y ago

muth02446 5y ago
Mathematics seems to be obsessed with one-liners Single letter variable names (often from different alphabets) and complex expressions that should be broken down would never make it through a code review ;-)
- Ar-Curunir 5y ago
  unlike programming, you're not mutating and updating mathematical variables. Having "intelligible" names doesn't actually help that much to understand the core mathematical concepts anyway.
  - nomel 5y ago
    > core mathematical concepts anyway But it can really help understand the practicalities of it, like any paper involving models, which are pretty numerous outside of pure mathematics. Unfortunately, I've seen many papers that "leave it to the reader" when it comes to equations used, with no translation between variable names and what they actually represents.
    - Ar-Curunir 5y ago
      I feel like that's an issue of the paper, and not the idea; instead of making the math verbose, the paper should provide a legend or key that documents each symbol, or at least the main ones.
- kergonath 5y ago
  Variable names with more than one letter (excluding superscripts and subscripts) are awful to read and difficult to parse. Kerning issues become confusing as you try to figure out where are the implied multiplications. There is nothing wrong with single letters as long as they are chosen appropriately and are clearly defined in the text.
  - deleted 5y ago
    [deleted]
  - DennisP 5y ago
    Maybe the problem is implied multiplications. Most programming languages don't have them, and are easier to read with longer variable names.
- omaranto 5y ago
  > Mathematics seems to be obsessed with one-liners Not at all. Formulas in a math paper always come with at least a paragraph of text explaining them. You aren't supposed to read the formula without the explanation. > Single letter variable names (often from different alphabets) and complex expressions that should be broken down would never make it through a code review ;-) That's done for readability. The text explains the terms of the formula and the formula concisely summarizes the precise relationship between them.
  - netizen-936824 5y ago
    Theoretically, we could do the same with comments no? Make the code extremely compact and explain it, perhaps even in a format like literate programming?
    - omaranto 5y ago
      Yes. But it would be tough to change the culture. Math papers are meant to be read, so people pay attention to the explanations. Computer programs (at least if they garner users!) are run much more often than read, so it's natural to focus on the code and neglect the comments.
      - nomel 5y ago
        > But it would be tough to change the culture. I think it would be tougher to find anyone willing to read programs written like this. Requiring an explanation to understand the variables is very similar to encoding the variables names and putting a lookup table below. Why force someones eyes to dart back and forth between the program and lookup table, just to get an idea of what the variable are, rather than also including a very rough explanation of what they variables are doing, in the program itself, by giving them meaningful names? We only have so much working memory. Giving variables names frees up a significant amount.
        netizen-936824 5y ago
        I've personally written code that was immensely difficult to read. Branchless stuff with single letter variables. Even though I commented it thoroughly, its ridiculously difficult to read and modify. This was a good lesson for me, I won't do that again and I'm far more verbose now. My thinking is that maybe math should learn from things like that as well, perhaps expanding equations and being more verbose within the equation instead of an explanation next to it would make maths more accessible to a broader audience.
- enaaem 5y ago
  It’s important to see patterns in equations and verbose variables makes it harder to see.
  - nitrogen 5y ago
    Patterns occur in code, too, especially when allowed to span multiple lines with indentation.
- joppy 5y ago
  It's very hard to see the patterns in mathematical expressions or equations without compressing the notation down to be very small, which is the main reason this is done. However, mathematicians break complex expressions down into simpler reusable parts all the time...
- jstx1 5y ago
  This is one of my gripes with the Julia programming language. The community really likes Greek letters and math notation mixed in with their code. And I just can’t.
  - enriquto 5y ago
    Ha! this is precisely the thing that I love about Julia! Only thing missing is to have the empty string as the product operator. I want to write "y = ax + b" and that it computes the product of "a" with "x".
    - jstx1 5y ago
      What happens when you have "a", "x" and "ax" as variables in scope?
      - enriquto 5y ago
        I'm not a language designer, but maybe there should be a warning or a compilation error or something, instead of a silent but arbitrary decision. But it gets ugly easily, with three-leter variable names that can be grouped in different ways. If I designed a new language from 0, I'd like to allow only single letter variable identifers (possibly with unicode sub-indices and modifiers).
        jstx1 5y ago
        > If I designed a new language from 0, I'd like to allow only single letter variable identifers (possibly with unicode sub-indices and modifiers). In the nicest possible way - this is absolute madness.
        enriquto 5y ago
        Do you think so? It may seem risky in itself, but I think this would capture many current practices in mathematical notation. The language could aid you to clarify your code. For example, by forcing each variable to be declared explicitly at the beginning of its scope, on a line all by itself. Then the compiler could put a warning if one of these declarations didn't have a comment.
- fault1 5y ago
  mathematical notation would be kind of unreadable without the terse and compact symbology, imho. it makes it much easier for the mind to parse. mathematics is partially a language (maybe one of the most universal ones), once you learn common conventions, it becomes easier. Obviously it's no problem for human minds to learn huge numbers of symbols, just like some CJK languages with a large number of ideographs that have to be understood in context. I do think auto-annotation, as well as other types of interactivity are great, though!
  - deleted 5y ago
    [deleted]
- dunham 5y ago
  The alphabets (and typefaces, like blackboard bold or cursive) are mathematicians' version of syntax highlighting. They are often used to represent different types of objects.
- snicker7 5y ago
  Mathematicans read very slowly relative to, say, computer programmers. A page can easily take an hour or more. Bad notation (e.g. variable names) can interrupt one's flow.
- red_trumpet 5y ago
  I think mathematical formulas are partly optimized to be easily manipulated with a pen on paper. Imagine trying to derive a slightly complex function where the variables are called `growth_rate`, `duration` and `distance`.
choeger 5y ago
Finally! This is super useful! I wanted to do stuff like this ten years ago for small code samples but didn't get very far with TikZ. Doing it for equations is even more complicated to get into a pleasent shape, I guess.
- ffhhj 5y ago
  > I wanted to do stuff like this ten years ago for small code samples but didn't get very far with TikZ. I'm a no-code tools developer. Yours sound like an interesting problem to solve, could you elaborate on your use case? What programming languages? What kind of information did you need to annotate? Which are your expectations on usability?
jesprenj 5y ago
It would be very good to include this in a future release of TeXLive.
- jimhefferon 5y ago
  Basically, anything on CTAN under a Free licence gets into TeX Live.
selwynii 5y ago
While I like the annotations, I have to say they make it difficult to make out the original equation. It parses and highlights variables and gets in the way of seeing a cohesive whole equation. I think an annotated version would be good as a supplement.
- sideshowb 5y ago
  I'd prefer definitions of terms to appear when you hover the mouse over them. And the proofing software to enforce that all terms are thus defined. Somehow in 2021 at least, undefined terms were still slipping through the review process :-/
rq1 5y ago
I was skeptical when I clicked but that’s actually amazing. It will eventually allow us to make even more complex constructs.
- mateo1 5y ago
  No it won't. It will allow a bunch of people to skim-read detailed research publications and miss all the nuances that matter. No equation, no matter how many colours and arrows you cram into it, has any meaning on it's own. If you want to know what the author means by each of the symbols, read the damn paper. Yes, in some fields certain symbols are known by tradition to mean something and older papers that were published with a limited audience in mind omitted some descriptions which is awfully annoying, but this is solved by having a section describing what your abbreviations and symbols refer to, not with highlighter spam.
  - rsfern 5y ago
    > It will allow a bunch of people to skim-read detailed research publications and miss all the nuances that matter. I have to disagree, there’s nothing stopping readers from doing this already, I think it’s quite common I do agree that one needs to actually read the paper to really understand the context of some equation, but honestly flipping between the abbreviation table or the dense prose of “where x is foo, y is bar, zzz” it’s just not readable. I don’t think it’s would make sense to use this style for every equation ever, but I think it has utility
  - rq1 5y ago
    I disagree. These annotations help to bind the variables to the context and the “detailed nuances”. Yes this is called a lookup table, and while it can seem obvious to you when you have spent 1y working on the same subject, it is not for the others. People will anyway skim-read it, as I do, and put it aside for detailed reading, if the overall idea is good or if I found an interesting technique. Anyway, the best papers I read were short and concise. Cf. historically Lebesgue paper about measure theory and his correspondance with Picard, compared to Borel’s theory.
yunruse 5y ago
Tangentially, one of the things I always enjoyed seeing in the wild was semantic colouring in equations. The educational YouTuber 3Blue1Brown [0] and to some extent the maths side of YouTube in general tend to make use of them, colouring like terms the same to give a subtle visual aid, similar to pretty printing. [0] https://youtube.com/3blue1brown https://youtube.com/3blue1brown
cinntaile 5y ago
I want the explanations but without the highlighting. I get why it's highlighted but personally I find it distracting.
contravariant 5y ago
I don't think I like this as a replacement for simply defining terms in the text. You could do both but something like "S a subset of Range(X)" when the equation is about the probability that X is in S doesn't seem important enough to take that prominent a position in the equation. I can see this style of annotation it being helpful when you need to add some context or want to explain a pre-existing formula. And I can definitely see this being used to highlight key ideas during a tricky derivation, stuff like partial integration could be a lot neater if you can point out the part of the equation that you're partially integrating.
afthonos 5y ago
Rendering the different symbols in different colors might be enough, as long as those colors are used in the text defining the symbols. Used sparingly (i.e. only for the symbols that are relevant to the argument), it can greatly enhance readability. I feel like these annotations go slightly overboard; the entire equation/annotation construct is visually pleasing, but hard to parse.
glorifying 5y ago
I’m not sure I understand what the point of this is. If I’m reading academic papers on a particular topic then I already know the notation the math is written in. If I’m reading a paper outside of my field then I’m not really interested in the details of the equation but more of its application and what it means.
- visviva 5y ago
  > If I’m reading academic papers on a particular topic then I already know the notation the math is written in. This is not true in my field - notation is all over the place, even for the same topic.
  - creata 5y ago
    A famous quote in Lee's "Introduction to Smooth Manifolds": > differential geometry is the study of properties that are invariant under change of notation
    - webmaven 5y ago
      > differential geometry is the study of properties that are invariant under change of notation That ought to go on a t-shirt.
- nomel 5y ago
  > If I’m reading a paper outside of my field then I’m not really interested in the details of the equation but more of its application and what it means. This is fine, until you find the application interesting, and try to apply it. I've had to contact authors just to figure out which one of several parameters a, b, c, or d actually represented. To be fair, these came with apologies.
- burnished 5y ago
  Welp, you might be a special case. When I'm reading papers outside of my field its because I'm looking for understanding. Annotated equations in this fashion seem real neat for that purpose.
mbokinala 5y ago
I definitely see value in this, but it's currently too much of a hassle to use. It requires lots of tedious code duplication and requires the user to manually set positions and lengths for the arrows and annotations, defeating the purpose of allowing writers to focus solely on the content of their work. However, this would be a great thing to try to encapsulate in a package.
nitrogen 5y ago
From comments here it sounds like people already deeply familiar with the mathematical notation used in a domain really want to stick with the compact, black-ink-only equation representation. However, as an engineer who occasionally wants to translate papers from a mathematical field I rarely encounter into code I will use all the time, annotations like this would greatly increase the utility I could derive from a paper in a given amount of time. Also, as yunruse mentioned in [0], and as I have done in my own videos that include math, syntax highlighting for math is very useful for absorbing new math. How acceptable would that be to practitioners already deep in their fields? So, could annotations of this form be acceptable to both groups if they were switchable on or off in an online version of a paper? [0] https://news.ycombinator.com/item?id=29899938 https://news.ycombinator.com/item?id=29899938
- quazar 5y ago
  How would this help you? Now these explanations are just in the paragraph next to the equation and are not limited to 3 words.
  - DonaldPShimoda 5y ago
    > Now these explanations are just in the paragraph next to the equation and are not limited to 3 words. Not necessarily. Will Crichton, a PhD student at Stanford University, has been working on a tool called Nota [0] that he's hoping might serve as a platform for online paper publication. I'd suggest checking out the initial presentation of Nota [1] to see some ideas of how syntax highlighting-type effects can be made as toggle-able capabilities that are not intrusive nor dislocated as you suggest. [0] https://nota-lang.org https://nota-lang.org [1] https://willcrichton.net/nota/ https://willcrichton.net/nota/
  - nitrogen 5y ago
    It activates more information channels into my brain. Prose is just one transfer channel, and for me, not a very efficient one (though I love reading!). Mathematical notation is one more. But that leaves out word association, color, spatial relationships, conceptual graphs, diagrams, etc. The more channels I can activate, the faster I can build intuition, and intuition facilitates practical application. Here's an example of how I like to use color from my own modest work: https://youtu.be/iyjIVSnrPSo?t=424 https://youtu.be/iyjIVSnrPSo?t=424 Another example: https://youtu.be/iyjIVSnrPSo?t=345 https://youtu.be/iyjIVSnrPSo?t=345 In this case, each independent variable in the equation gets its own color, and that color corresponds to a label on the graph above. The imaginary unit also gets a special color, which matches the color of its definition, for those of my audience who are unfamiliar with or out of practice with complex numbers. And the syntactic elements are muted to make the shape of the equation "pop" for those whose eyes are not routinely trained to read math notation. Elsewhere in the video I use different colors to represent positive, negative, and zero frequencies. So to restate, it's all about giving information as many paths into my brain as possible.
yummypaint 5y ago
Seeing this presented for use in papers is a very good thing. These annotations look alot like slides many people make (sometimes showing relations between multiple equations, etc). However, much of this readability is customarily stripped out for the sake of uniformity. I think most journals im familiar with enforce specific formatting for equations.
orforforof 5y ago
For many of the papers I read (engineering) I would love to be able to click on symbols in an equation and have it link me back to where that term was first defined in the paper, either in another eqn or in the text. Tracing back in this way is a major pain point e.g. when trying to reproduce calculations. Seems like something journals/latex could easily implement in an automated fashion (maybe already possible?)
- diarrhea 5y ago
  I use glossaries-extra. It allows the usual glossary stuff (acronyms), but can also be used to define symbols. All symbols then become hyperlinks, linking to the nomenclature at the beginning or end of the document. And in that nomenclature, glossaries- extra allows one to have a column of "first used", specifying the page of first occurrence.
riffraff 5y ago
I understand math is optimized for writing, but honestly I've always felt it would be better if we just used long variable names instead of single letters.
- xyzzyz 5y ago
  It’s not. The entire point of math papers is to get the idea across to other mathematicians. Unlike code, which can be useful even if others don’t understand it, a paper that can’t be understood is entirely useless. You are just not in the target audience, that’s all.
- deleted 5y ago
  [deleted]
wolfi1 5y ago
why not link directly to the github repo? but that aside, it's quite neat
- visviva 5y ago
  At the time I posted this, the README for the repo did not include the images that are there now.
tpoacher 5y ago
On one hand I like this, and have used similar visuals in presentations, on the other hand I feel it's a less a problem related to math literacy per se, and more needing a workaround relating to the artificial limits and style restrictions imposed by journals (as well as the limitations of a paper medium, to some extent). The whole "pay £1000 per extra page" by itself has been enough to reluctantly cause me to cram math inline and assume symbols are known to the field without too much explanation.
blippage 5y ago
TBH, I don't think the annotations add anything. You can simply write " Lap(x | mu, b) = ... where b>0, a scalar parameter. " What is needed is some better way to motivate the concepts behind any maths that is confusing for people who might be confused.
VladimirGolovin 5y ago
I was reading Russian political Telrgram earlier today, and one of the better channels ("Толкователь") had a post about a scientific study on why people prefer conspiracy theories over scientific information. The study's summary says that this is because conspiracy theories are communicated in a language that is understandable, while scientific communication uses language that is opaque for most people. Which leads me to the conclusion that any effort towards making scientific communication more understandable to general public -- including the annotated equations -- is a good thing.
- webmaven 5y ago
  > The study's summary says that this is because conspiracy theories are communicated in a language that is understandable, while scientific communication uses language that is opaque for most people. Well, when you remove the constraint to be truthful, you can do a lot of things to make statements that are more plausible and convincing to a layman. In other words, of course when the main requirement is just to sound good you end up with statements that are more convincing. Did you know that 74.13% of all statistics are completely made up? It's true! I've got a report from the Institute of Data & Information Oriented Technology Studies right here!
avnigo 5y ago
Here are the examples in long-form rather than a tweet: https://github.com/synercys/annotated_latex_equations https://github.com/synercys/annotated_latex_equations
_Nat_ 5y ago
Definitely something that could be nice in an active-document. By analogy, Wikipedia articles have a feature where, if you mouse-over a link, it'll display a preview of the link-target (https://en.wikipedia.org/wiki/Wikipedia:Tools/Navigation_popups https://en.wikipedia.org/wiki/Wikipedia:Tools/Navigation_pop... ). This can be an efficient way for a reader to conveniently glance at the definition of a term while reading. Mathematical notation varies significantly between fields, and many papers define custom symbols with their own paper-specific definitions. So this sort of feature isn't just something for math-newbies, but rather something that could significantly benefit experts too. The same sort of annotations might be more situational in non-active documents, where there'd be more of a trade-off between helpful-annotations and clutter.
blauditore 5y ago
I always wondered why no one is doing syntax (or, semantical) coloring/highlighting for stuff like mathematics or sheet music. It could even help with simple plain text. The reason is probably that "it's always been like that", just like keyboard layouts. Though one could argue that coloring would be not be very disruptive and backwards-compatible, so it wouldn't hurt traditional reading much.
- Wowfunhappy 5y ago
  > It could even help with simple plain text. I've always found it interesting how video games (e.g. Legend of Zelda) will make certain key words in text a different color. It works well—but why do we only do it in video games?

192 points by visviva 5y ago

Mathematics seems to be obsessed with one-liners Single letter variable names (often from different alphabets) and complex expressions that should be broken down would never make it through a code review ;-)

Ar-Curunir 5y ago
unlike programming, you're not mutating and updating mathematical variables. Having "intelligible" names doesn't actually help that much to understand the core mathematical concepts anyway.
- nomel 5y ago
  > core mathematical concepts anyway But it can really help understand the practicalities of it, like any paper involving models, which are pretty numerous outside of pure mathematics. Unfortunately, I've seen many papers that "leave it to the reader" when it comes to equations used, with no translation between variable names and what they actually represents.
  - Ar-Curunir 5y ago
    I feel like that's an issue of the paper, and not the idea; instead of making the math verbose, the paper should provide a legend or key that documents each symbol, or at least the main ones.
kergonath 5y ago
Variable names with more than one letter (excluding superscripts and subscripts) are awful to read and difficult to parse. Kerning issues become confusing as you try to figure out where are the implied multiplications. There is nothing wrong with single letters as long as they are chosen appropriately and are clearly defined in the text.
- deleted 5y ago
  [deleted]
- DennisP 5y ago
  Maybe the problem is implied multiplications. Most programming languages don't have them, and are easier to read with longer variable names.
omaranto 5y ago
> Mathematics seems to be obsessed with one-liners Not at all. Formulas in a math paper always come with at least a paragraph of text explaining them. You aren't supposed to read the formula without the explanation. > Single letter variable names (often from different alphabets) and complex expressions that should be broken down would never make it through a code review ;-) That's done for readability. The text explains the terms of the formula and the formula concisely summarizes the precise relationship between them.
- netizen-936824 5y ago
  Theoretically, we could do the same with comments no? Make the code extremely compact and explain it, perhaps even in a format like literate programming?
  - omaranto 5y ago
    Yes. But it would be tough to change the culture. Math papers are meant to be read, so people pay attention to the explanations. Computer programs (at least if they garner users!) are run much more often than read, so it's natural to focus on the code and neglect the comments.
    - nomel 5y ago
      > But it would be tough to change the culture. I think it would be tougher to find anyone willing to read programs written like this. Requiring an explanation to understand the variables is very similar to encoding the variables names and putting a lookup table below. Why force someones eyes to dart back and forth between the program and lookup table, just to get an idea of what the variable are, rather than also including a very rough explanation of what they variables are doing, in the program itself, by giving them meaningful names? We only have so much working memory. Giving variables names frees up a significant amount.
      - netizen-936824 5y ago
        I've personally written code that was immensely difficult to read. Branchless stuff with single letter variables. Even though I commented it thoroughly, its ridiculously difficult to read and modify. This was a good lesson for me, I won't do that again and I'm far more verbose now. My thinking is that maybe math should learn from things like that as well, perhaps expanding equations and being more verbose within the equation instead of an explanation next to it would make maths more accessible to a broader audience.
enaaem 5y ago
It’s important to see patterns in equations and verbose variables makes it harder to see.
- nitrogen 5y ago
  Patterns occur in code, too, especially when allowed to span multiple lines with indentation.
joppy 5y ago
It's very hard to see the patterns in mathematical expressions or equations without compressing the notation down to be very small, which is the main reason this is done. However, mathematicians break complex expressions down into simpler reusable parts all the time...
jstx1 5y ago
This is one of my gripes with the Julia programming language. The community really likes Greek letters and math notation mixed in with their code. And I just can’t.
- enriquto 5y ago
  Ha! this is precisely the thing that I love about Julia! Only thing missing is to have the empty string as the product operator. I want to write "y = ax + b" and that it computes the product of "a" with "x".
  - jstx1 5y ago
    What happens when you have "a", "x" and "ax" as variables in scope?
    - enriquto 5y ago
      I'm not a language designer, but maybe there should be a warning or a compilation error or something, instead of a silent but arbitrary decision. But it gets ugly easily, with three-leter variable names that can be grouped in different ways. If I designed a new language from 0, I'd like to allow only single letter variable identifers (possibly with unicode sub-indices and modifiers).
      - jstx1 5y ago
        > If I designed a new language from 0, I'd like to allow only single letter variable identifers (possibly with unicode sub-indices and modifiers). In the nicest possible way - this is absolute madness.
        enriquto 5y ago
        Do you think so? It may seem risky in itself, but I think this would capture many current practices in mathematical notation. The language could aid you to clarify your code. For example, by forcing each variable to be declared explicitly at the beginning of its scope, on a line all by itself. Then the compiler could put a warning if one of these declarations didn't have a comment.
fault1 5y ago
mathematical notation would be kind of unreadable without the terse and compact symbology, imho. it makes it much easier for the mind to parse. mathematics is partially a language (maybe one of the most universal ones), once you learn common conventions, it becomes easier. Obviously it's no problem for human minds to learn huge numbers of symbols, just like some CJK languages with a large number of ideographs that have to be understood in context. I do think auto-annotation, as well as other types of interactivity are great, though!
- deleted 5y ago
  [deleted]
dunham 5y ago
The alphabets (and typefaces, like blackboard bold or cursive) are mathematicians' version of syntax highlighting. They are often used to represent different types of objects.
snicker7 5y ago
Mathematicans read very slowly relative to, say, computer programmers. A page can easily take an hour or more. Bad notation (e.g. variable names) can interrupt one's flow.
red_trumpet 5y ago
I think mathematical formulas are partly optimized to be easily manipulated with a pen on paper. Imagine trying to derive a slightly complex function where the variables are called `growth_rate`, `duration` and `distance`.

choeger 5y ago

Finally! This is super useful! I wanted to do stuff like this ten years ago for small code samples but didn't get very far with TikZ. Doing it for equations is even more complicated to get into a pleasent shape, I guess.

ffhhj 5y ago
> I wanted to do stuff like this ten years ago for small code samples but didn't get very far with TikZ. I'm a no-code tools developer. Yours sound like an interesting problem to solve, could you elaborate on your use case? What programming languages? What kind of information did you need to annotate? Which are your expectations on usability?

jesprenj 5y ago

It would be very good to include this in a future release of TeXLive.

jimhefferon 5y ago
Basically, anything on CTAN under a Free licence gets into TeX Live.

selwynii 5y ago

While I like the annotations, I have to say they make it difficult to make out the original equation. It parses and highlights variables and gets in the way of seeing a cohesive whole equation. I think an annotated version would be good as a supplement.

sideshowb 5y ago
I'd prefer definitions of terms to appear when you hover the mouse over them. And the proofing software to enforce that all terms are thus defined. Somehow in 2021 at least, undefined terms were still slipping through the review process :-/

rq1 5y ago

I was skeptical when I clicked but that’s actually amazing. It will eventually allow us to make even more complex constructs.

mateo1 5y ago
No it won't. It will allow a bunch of people to skim-read detailed research publications and miss all the nuances that matter. No equation, no matter how many colours and arrows you cram into it, has any meaning on it's own. If you want to know what the author means by each of the symbols, read the damn paper. Yes, in some fields certain symbols are known by tradition to mean something and older papers that were published with a limited audience in mind omitted some descriptions which is awfully annoying, but this is solved by having a section describing what your abbreviations and symbols refer to, not with highlighter spam.
- rsfern 5y ago
  > It will allow a bunch of people to skim-read detailed research publications and miss all the nuances that matter. I have to disagree, there’s nothing stopping readers from doing this already, I think it’s quite common I do agree that one needs to actually read the paper to really understand the context of some equation, but honestly flipping between the abbreviation table or the dense prose of “where x is foo, y is bar, zzz” it’s just not readable. I don’t think it’s would make sense to use this style for every equation ever, but I think it has utility
- rq1 5y ago
  I disagree. These annotations help to bind the variables to the context and the “detailed nuances”. Yes this is called a lookup table, and while it can seem obvious to you when you have spent 1y working on the same subject, it is not for the others. People will anyway skim-read it, as I do, and put it aside for detailed reading, if the overall idea is good or if I found an interesting technique. Anyway, the best papers I read were short and concise. Cf. historically Lebesgue paper about measure theory and his correspondance with Picard, compared to Borel’s theory.

yunruse 5y ago

Tangentially, one of the things I always enjoyed seeing in the wild was semantic colouring in equations. The educational YouTuber 3Blue1Brown [0] and to some extent the maths side of YouTube in general tend to make use of them, colouring like terms the same to give a subtle visual aid, similar to pretty printing. [0] https://youtube.com/3blue1brown https://youtube.com/3blue1brown

cinntaile 5y ago

I want the explanations but without the highlighting. I get why it's highlighted but personally I find it distracting.

contravariant 5y ago

I don't think I like this as a replacement for simply defining terms in the text. You could do both but something like "S a subset of Range(X)" when the equation is about the probability that X is in S doesn't seem important enough to take that prominent a position in the equation. I can see this style of annotation it being helpful when you need to add some context or want to explain a pre-existing formula. And I can definitely see this being used to highlight key ideas during a tricky derivation, stuff like partial integration could be a lot neater if you can point out the part of the equation that you're partially integrating.

afthonos 5y ago

Rendering the different symbols in different colors might be enough, as long as those colors are used in the text defining the symbols. Used sparingly (i.e. only for the symbols that are relevant to the argument), it can greatly enhance readability. I feel like these annotations go slightly overboard; the entire equation/annotation construct is visually pleasing, but hard to parse.

glorifying 5y ago

I’m not sure I understand what the point of this is. If I’m reading academic papers on a particular topic then I already know the notation the math is written in. If I’m reading a paper outside of my field then I’m not really interested in the details of the equation but more of its application and what it means.

visviva 5y ago
> If I’m reading academic papers on a particular topic then I already know the notation the math is written in. This is not true in my field - notation is all over the place, even for the same topic.
- creata 5y ago
  A famous quote in Lee's "Introduction to Smooth Manifolds": > differential geometry is the study of properties that are invariant under change of notation
  - webmaven 5y ago
    > differential geometry is the study of properties that are invariant under change of notation That ought to go on a t-shirt.
nomel 5y ago
> If I’m reading a paper outside of my field then I’m not really interested in the details of the equation but more of its application and what it means. This is fine, until you find the application interesting, and try to apply it. I've had to contact authors just to figure out which one of several parameters a, b, c, or d actually represented. To be fair, these came with apologies.
burnished 5y ago
Welp, you might be a special case. When I'm reading papers outside of my field its because I'm looking for understanding. Annotated equations in this fashion seem real neat for that purpose.

mbokinala 5y ago

I definitely see value in this, but it's currently too much of a hassle to use. It requires lots of tedious code duplication and requires the user to manually set positions and lengths for the arrows and annotations, defeating the purpose of allowing writers to focus solely on the content of their work. However, this would be a great thing to try to encapsulate in a package.

nitrogen 5y ago

From comments here it sounds like people already deeply familiar with the mathematical notation used in a domain really want to stick with the compact, black-ink-only equation representation. However, as an engineer who occasionally wants to translate papers from a mathematical field I rarely encounter into code I will use all the time, annotations like this would greatly increase the utility I could derive from a paper in a given amount of time. Also, as yunruse mentioned in [0], and as I have done in my own videos that include math, syntax highlighting for math is very useful for absorbing new math. How acceptable would that be to practitioners already deep in their fields? So, could annotations of this form be acceptable to both groups if they were switchable on or off in an online version of a paper? [0] https://news.ycombinator.com/item?id=29899938 https://news.ycombinator.com/item?id=29899938

quazar 5y ago
How would this help you? Now these explanations are just in the paragraph next to the equation and are not limited to 3 words.
- DonaldPShimoda 5y ago
  > Now these explanations are just in the paragraph next to the equation and are not limited to 3 words. Not necessarily. Will Crichton, a PhD student at Stanford University, has been working on a tool called Nota [0] that he's hoping might serve as a platform for online paper publication. I'd suggest checking out the initial presentation of Nota [1] to see some ideas of how syntax highlighting-type effects can be made as toggle-able capabilities that are not intrusive nor dislocated as you suggest. [0] https://nota-lang.org https://nota-lang.org [1] https://willcrichton.net/nota/ https://willcrichton.net/nota/
- nitrogen 5y ago
  It activates more information channels into my brain. Prose is just one transfer channel, and for me, not a very efficient one (though I love reading!). Mathematical notation is one more. But that leaves out word association, color, spatial relationships, conceptual graphs, diagrams, etc. The more channels I can activate, the faster I can build intuition, and intuition facilitates practical application. Here's an example of how I like to use color from my own modest work: https://youtu.be/iyjIVSnrPSo?t=424 https://youtu.be/iyjIVSnrPSo?t=424 Another example: https://youtu.be/iyjIVSnrPSo?t=345 https://youtu.be/iyjIVSnrPSo?t=345 In this case, each independent variable in the equation gets its own color, and that color corresponds to a label on the graph above. The imaginary unit also gets a special color, which matches the color of its definition, for those of my audience who are unfamiliar with or out of practice with complex numbers. And the syntactic elements are muted to make the shape of the equation "pop" for those whose eyes are not routinely trained to read math notation. Elsewhere in the video I use different colors to represent positive, negative, and zero frequencies. So to restate, it's all about giving information as many paths into my brain as possible.

yummypaint 5y ago

Seeing this presented for use in papers is a very good thing. These annotations look alot like slides many people make (sometimes showing relations between multiple equations, etc). However, much of this readability is customarily stripped out for the sake of uniformity. I think most journals im familiar with enforce specific formatting for equations.

orforforof 5y ago

For many of the papers I read (engineering) I would love to be able to click on symbols in an equation and have it link me back to where that term was first defined in the paper, either in another eqn or in the text. Tracing back in this way is a major pain point e.g. when trying to reproduce calculations. Seems like something journals/latex could easily implement in an automated fashion (maybe already possible?)

diarrhea 5y ago
I use glossaries-extra. It allows the usual glossary stuff (acronyms), but can also be used to define symbols. All symbols then become hyperlinks, linking to the nomenclature at the beginning or end of the document. And in that nomenclature, glossaries- extra allows one to have a column of "first used", specifying the page of first occurrence.

riffraff 5y ago

I understand math is optimized for writing, but honestly I've always felt it would be better if we just used long variable names instead of single letters.

xyzzyz 5y ago
It’s not. The entire point of math papers is to get the idea across to other mathematicians. Unlike code, which can be useful even if others don’t understand it, a paper that can’t be understood is entirely useless. You are just not in the target audience, that’s all.
deleted 5y ago
[deleted]

wolfi1 5y ago

why not link directly to the github repo? but that aside, it's quite neat

visviva 5y ago
At the time I posted this, the README for the repo did not include the images that are there now.

tpoacher 5y ago

On one hand I like this, and have used similar visuals in presentations, on the other hand I feel it's a less a problem related to math literacy per se, and more needing a workaround relating to the artificial limits and style restrictions imposed by journals (as well as the limitations of a paper medium, to some extent). The whole "pay £1000 per extra page" by itself has been enough to reluctantly cause me to cram math inline and assume symbols are known to the field without too much explanation.

blippage 5y ago

TBH, I don't think the annotations add anything. You can simply write " Lap(x | mu, b) = ... where b>0, a scalar parameter. " What is needed is some better way to motivate the concepts behind any maths that is confusing for people who might be confused.

VladimirGolovin 5y ago

I was reading Russian political Telrgram earlier today, and one of the better channels ("Толкователь") had a post about a scientific study on why people prefer conspiracy theories over scientific information. The study's summary says that this is because conspiracy theories are communicated in a language that is understandable, while scientific communication uses language that is opaque for most people. Which leads me to the conclusion that any effort towards making scientific communication more understandable to general public -- including the annotated equations -- is a good thing.

webmaven 5y ago
> The study's summary says that this is because conspiracy theories are communicated in a language that is understandable, while scientific communication uses language that is opaque for most people. Well, when you remove the constraint to be truthful, you can do a lot of things to make statements that are more plausible and convincing to a layman. In other words, of course when the main requirement is just to sound good you end up with statements that are more convincing. Did you know that 74.13% of all statistics are completely made up? It's true! I've got a report from the Institute of Data & Information Oriented Technology Studies right here!

avnigo 5y ago

Here are the examples in long-form rather than a tweet: https://github.com/synercys/annotated_latex_equations https://github.com/synercys/annotated_latex_equations

_Nat_ 5y ago

Definitely something that could be nice in an active-document. By analogy, Wikipedia articles have a feature where, if you mouse-over a link, it'll display a preview of the link-target (https://en.wikipedia.org/wiki/Wikipedia:Tools/Navigation_popups https://en.wikipedia.org/wiki/Wikipedia:Tools/Navigation_pop... ). This can be an efficient way for a reader to conveniently glance at the definition of a term while reading. Mathematical notation varies significantly between fields, and many papers define custom symbols with their own paper-specific definitions. So this sort of feature isn't just something for math-newbies, but rather something that could significantly benefit experts too. The same sort of annotations might be more situational in non-active documents, where there'd be more of a trade-off between helpful-annotations and clutter.

blauditore 5y ago

I always wondered why no one is doing syntax (or, semantical) coloring/highlighting for stuff like mathematics or sheet music. It could even help with simple plain text. The reason is probably that "it's always been like that", just like keyboard layouts. Though one could argue that coloring would be not be very disruptive and backwards-compatible, so it wouldn't hurt traditional reading much.

Wowfunhappy 5y ago
> It could even help with simple plain text. I've always found it interesting how video games (e.g. Legend of Zelda) will make certain key words in text a different color. It works well—but why do we only do it in video games?