Using Emojis in Mathematical Notation | Hacker News Reader

Using Emojis in Mathematical Notation

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24 pointsrmeertens72 comments

Yeah, no. Please don't do this. This has been discussed previously w.r.t commit messages, and I'd argue that the reasons for not doing so with maths are largely the same: https://news.ycombinator.com/item?id=21760021

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They're bad for accessibility. Don't work with screen readers . Hard to make out for people without perfect vision. Harder to type out.

They don't render well on many systems. They can't be handwritten. How are they do be pronounced? "croissant emoji squared plus girl wearing hat"?

Conventions exist for what symbols to use where in science and math. Don't mess with this. Kids won't magically find math easier if you use emoji in place of symbols.

Instead, please focus your efforts of improving teaching methods.

It is kinda ironic, in that one of my favorite math papers is Polya's On Picture Writing. Which seems to imply that math with pictures should work fine.

I can't disagree with you, though. Curious what makes it fine there, but bad here.

Both have the same problem and Polya mentions it in his paper. Both are to various degrees incompatible with technology (printing press, keyboards, teletypes, text mode screens, CLI, screen readers, etc.).

The reason the printing press became so revolutionary in latin alphabet using countries even though paper, printing and movable type were invented in China, is because it is very easy to make a machine that uses it. The English alphabet is more or less a common denominator of all other latin alphabet-based languages. Any non-latin/greek/cyrillic script is very hard to adapt to the various forms of technology throughout history.

But both papers illustrate a very real problem. Mathematical notation is horribly, infuriatingly, idiotically overloaded. And, no, context is not enough to divine the meaning. There are plenty of examples where a symbol is used for multiple meanings in the same paper or the same lesson.

My opinion is that the cause of all these problems is a very brain dead decision in math to allow adjacency of symbols to signify multiplication. This results in no longer being able to use multiple character names for things (variables, constants, etc.). And this is crippling. It results in the use of modified characters as symbols, use of characters from other languages as symbols and now the proposal of use of emoji as symbols. Because why not, emoji are plentiful and are now easier to input, store and display than ever.

The same issue (diversity of names) was solved by almost all by having mandatory separator characters(space, tab, comma, semicolon, etc) that are not allowed in names. Imagine a programing language where it is impossible to tell at first glance the meaning of something like "TotalWeight". Is it one variable? Is it Total * Weight? Is it TotalW * eight? Is it Total * W * eight? Is it T * o * t * a * l * W * e * i * g * h * t? We can rewrite this last one as aeghilo(t^2)TW . This is mathematical notation. And it will not change because it is entrenched.

This argument falls by simple counter example: cos, sin, log are all multi character names that are routinely used with no significant syntax confusion without having to use explicit multiplication indicators. So longer names are routinely used despite apparent impossibility due to multiplication confusion.

No, notation is terse for other reasons. Largely driven by the very common practice of writing it by hand. If you give your new object a long name and use that very name as a symbol you'll quickly discover that others who are interested will just abbreviate it to a letter to save effort writing it.

And in programming we generally frown uppon global use single letter identifiers. There are of course exceptions (jQuery comes to mind).

That is why we have aliasing in most shell languages or the using directive in C# and C++, or try-with-resources in Java. Aliasing is there when you need it but it is only used locally.

In math it kind of is the default. Trigonomerty is the exception.

Nobody really programs by hand, so I'm not sure what that remark aims towards? The practice of maths and programming are so different I don't know where you think there is any clarity gained from direct comparison.

Nobody programs by hand ANY MORE. Everybody did in the age of punchcards.

I think there is clarity to be gained from the history of programming. Programming went from very very hard (punch cards) to write to very easy to write (intelisense). And we also went from code being mostly written to code being mostly read.

With each step there was a move to either make best practices more verbose or terser. Verboseness has been a solution to defects. Better languages have been a solution to verboseness. For example: when typing bugs became common, hungarian notation became a best practice. The noise was an acceptable tradeoff. When better type checking became available, hungarian notation was abandoned because it produced noise. The compiler already did al the work for you. Improvements in the tools happened at the same time as improvements in the language.

All of this was motivated by growing code complexity resulting in increasing team sizes.

Math is also growing in complexity and collaboration is increasingly digital. But so far, I have only seen tool improvements, the language has mostly been stuck. If the language does not evolve tools will evolve much much slower.

I'm going to be honest, there is a reason mathematicans drop multiplication (or in general, the most important operator in whatever you are working on), because it would double the size of basically everything you write.

In group and semi-groups, where there is only one operator, it would just produce "visual noise", but introducing a * between every pair of things you write down. I have done this in beginner classes, but it rapidly gets boring.

While I understand how it makes it harder to read for non-experts, when you are writing maths, and maths papers, as a job you don't want to double the size of everything and scatter *s around for no reason.

The best parallel I can think of (this isn't good, sorry) would be like attaching the type of every variable to every place you use it -- this would technically make it easier to follow a snippet of code, but create lots of visual noise.

> The best parallel I can think of (this isn't good, sorry) would be like attaching the type of every variable to every place you use it -- this would technically make it easier to follow a snippet of code, but create lots of visual noise.

At some point we did this very thing and it was the recommended way to write code. And it did indeed create a lot of noise. My understanding of how we moved on from that is that namespacing, aliasing, and in general better code structure started being used. Things are no longer global. How do you make parts of mathematical proofs non-global to the entire proof?

https://en.wikipedia.org/wiki/Hungarian_notation

Honestly, I think we could, and should, do much better.

If people generally moved on from LaTeX (which is a whole other argument, and one I've had, and lost, on several occasions) to something with more semantic content, it would be much easier to make papers where it was easy to change the style of formatting, make it clear which symbols are "the same", etc.

> which is a whole other argument, and one I've had, and lost, on several occasions

Care to link to it if it's available? I am curious to read that debate.

Nothing concrete, I occasionally moan about it on Twitter and here, and in person, for example here:

https://news.ycombinator.com/item?id=17856184

Ironically, given some of the discussion we've been having, the biggest problem with LaTeX is that it is, once compiled into a PDF, extremely non-accessible and that's very hard to fix (to really understand why it's hard to fix, you need to look at how LaTeX works inside, it's a rewriting system and at no point has anything like a semantic description of the maths).

Also, emojis are hard to write by hand, which is an important feature to some people.

I find it very hard to internalize math without going through the derivations with pen and paper. Though I must admit that I have slowly gotten to the point of preferring to proof-read calculations for publication by typing out the intermediate steps in latex and then commenting them out.

Let the person who can write a decent \xi make the first claim that all greek letters are easy to write...

Huh, aren't \xi and \zeta just "squiggle" and "squiggle with a loop"? :P

I disagree. In maths, the main competition is Greek letters. Kids today arent going to know their eta from their gamma. I find many undergrads can't remember the Greek alphabet.

Sure, girl wearing hat might be a bad choice, but I think "simple emoji like tree, fire, snowflake (for example). Do a good job of replacing Greek letters.

Another place I use emoji, is for a set of Labella which should have no order. Mathematica a often say they are naming some things 1,2,3, but please forget the integers have arthmetic and ordering. If you use fire, tree, snowflake as the names, no-one assumes an ordering, or arithmetic.

> Kids today arent going to know their eta from their gamma.

I was inclined to agree with you, but then I remembered the zodiac sign emoji. Capricorn is basically η with a looped tail and taurus is gamma ɣ with a bigger loop. I'm pretty sure kids wouldn't be able to write them by hand (neither would I), but they have the advantage that if you type their names on a phone keyboard, you just need to recognize the symbol and tap it. Maybe all that's needed for Greek letters to compete with emoji is equal treatment by keyboards.

> Kids today arent going to know their eta from their gamma. I find many undergrads can't remember the Greek alphabet.

Not sure what is the problem here. Even without knowing Greek nor the Greek alphabet, its letters are sufficiently diverse and can be easily told apart.

I can understand emojis can make math more enciting for young children, but if high school+ people cannot get past foreign-looking Greek letters I doubt they would find the motivation to progress much further in understanding the concepts involved.

In undergrad lectures I constantly have students refering to "fork" or "curly n", or just misnaming different Greek letters (and, I find myself sometimes doing the same). The Greek is (to me) unnessary block to discussion.

And greek letters and emoji have the same issue of dificult type-ability, non-ASCII-ness, dificult printability and dificult displayability. Emoji only add color.

The real solution is normal multicharacter names.

I am happy for people to hate emoji, but I don't think non-ASCII-ness is a problem (lots of languages aren't in ASCII anywhere), and in general printing and displaying is easy -- it seems I can type emojis basically anywhere I can type anything else.

True, they don't display well in terminals, but I wouldn't not be willing to limit my maths to an ASCII monospaced terminal anyway, and that isn't how anyone writes or creates maths (unless they are typing LaTeX, in which case it will be nicely formatted anyway). I want super/sub scripts.

The reason I mentioned non-ASCII-ness is because top-parent mentioned them:

> Yeah, no. Please don't do this. This has been discussed previously w.r.t commit messages, and I'd argue that the reasons for not doing so with maths are largely the same: https://news.ycombinator.com/item?id=21760021

I'm sorry, I don't buy any of those reasons.

emoji are perfectly usable by screen readers -- if we really wanted to help people with screen readers we would move on from unstructured terminal output and programs which just dump screens of data. You know that thing that sometimes happens when a program running in the background outputs to the terminal -- that REALLY messes up screen readers.

Fair enough. We should make sure software including screen readers properly support Unicode in a useful way. Lack of support in the present is not a reason to avoid emoji. And math notation is already very non-ASCII so arguments related to plain text mode debugging are not applicable. There is probably an argument to be made about color blindness.

But overall my argument is that emoji are not a solution to the problem of very limited naming options for things in math, lack of significance of the symbol used and that of homonymity . The solution is using words. And those words may contain Unicode characters. And alias longer names to symbols on case by case basis.

> Kids today arent going to know their eta from their gamma.

Are you saying kids will not be able to distinguish between the two symbols? I highly doubt that. They may not remember the names of the symbols, but emojis have the same problem.

Sorry, yes I mean they won't know how to say them out loud, which makes discussion hard, whereas they will know snowflake or fire.

Heck, I teach graph theory and I keep forgetting the damn names mid-lecture.

Yes. You have to memorize the mapping between the meaning and arbitrary Greek letters whereas the meaning is more obvious with emoji. But both (greek letters, emoji) are bad solutions to a very stupid problem math has. See my comment here https://news.ycombinator.com/item?id=25207527

> How are they to be pronounced? "croissant emoji squared plus girl wearing hat"?

I knew a girl named 胡伊人 Hú Yīrén, who for a long time had her wechat display name as three emoji, a tiger face, a hand holding up one finger, and a human girl's face. (The intent would have been to read these as "tiger", "one", and "person", 虎一人 hǔ yìrén.)

But emoji didn't appear in notifications, and my phone was set to English rather than Chinese, so whenever I got a message from her the notification would read "[Tiger][No. 1][Girl] ...". I started thinking of her as "tiger number one girl", a bit.

On the other hand, how much sense does it make to keep overloading the same finite set of ~50 symbols? And while the admonition to stick to convention is a good general rule, there are both places where there exists conflicting conventions as well as areas where no convention exist at all.

As for the criticism that emoji can't be handwritten or pronounced: show me anyone doing a proper xi, and ask the greek what they think of how foreigners pronounce their letters (or indeed how one pronounces bold face letters). Clearly these are problems that have been solved before, and so can be solved again to the same level of quality.

Further, when handwriting you really have far more freedom than when typesetting, I had a friend that taught me the useful shorthand of just drawing a sphere to indicate that the surface integral was over a sphere. I tend to name partial results things like boxes or triangles when I work out because lugging around a second or even third "a" is just not as clear. Importantly, it's also much more fun to call your partial integral "small house" than "iii".

All that being said, the example choices in the article seem like straight up bad choices: naming the sides of a triangle a specific direction seems unhelpful in the fairly common case of several triangles with opposite orientation for example.

I think emoji would make it challenging for students. Students already struggle to decipher the handwriting and foreign notation of their professors. Adding emoji to that mix would make it even harder to keep up, and many emoji (and mathematical symbols, to be fair) are difficult to write down.

I would have struggled during my math and stats degrees if I had to distinguish between emoji and mathematical notation.

Depends on the medium people will use to do math in the future, there may be a material reason why handwriting is the main method, but I would say its just as likely that it goes the way of other, less structured texts as the tools get better.

It doesn't really matter if you can't write down an emoji if you don't write anything down.

>Math is hard to understand.

Because of the concepts used.

>I have to translate every character back to the concept meant by this character

Using emojis for variables won't help with that.

>Every scientific domain has its usual notation for specific concepts

Does that imply each emoji be used for a single concept across all domains? Not only that won't be possible (considering people don't even always use same variable for something, e.g. Pythagorean theorem being a^2+b^2=c^2, α^2+β^2=γ^2, x^2+y^2=c^2, ...) but is also a bad idea even if it could work as it will imply you've to remember every single emoji used.

Again, math is about concepts not what symbol you use for a variable. Using emojis won't make anything easier to teach or to understand.

I find it ironic that your examples were single-character roman & greek letters. Why not Cyrillic? Phoenician?

I wouldn't use pictograms everywhere, but they could be elegant in certain contexts. E.g. having a dedicated pictogram for various base physics quantities (e.g. for mass, length, time, etc).

Because the latin and greek alphabets are widely known and understood, at least an order of magnitude more than Cyrillic and at least seven orders of magnitude more than Phoenician.

This is like asking why I'm writing in English even though it's not my native language, I do that because almost every educated person understands English, while almost nobody would understand if I wrote in Italian.

Pictograms would be difficult to standardize and hard to reproduce, unless you commit to a small set of them, which would just become an alphabet in disguise.

I appreciate the Western convention, but a significant fraction of the world uses pictogram-like written languages (e.g. Chinese). Using pictograms to represent words or concepts isn't that far-fetched.

Yes, and there is a reason why Chinese is so hard to learn: There are so many characters that learning them takes years.

Chinese being difficult to learn has nothing to do with the current discussion. And the wealth of symbols of chinese script is beneficial to mathematical notation where names are forced to be single character. They are descriptive and diverse which cannot be said about latin or greek letters. But they are foreign to westerners.

The real issue is forcing names to be of single character lenght.

> Chinese being difficult to learn has nothing to do with the current discussion.

It has everything to do with the current discussion. GP was suggesting that a "pictogram-like written language […] isn't that far-fetched". And I was countering that argument by saying that a large alphabet (a large set of pictograms) is hard to learn or at least takes a lot time, as can be seen from trying to learn Chinese.

That time is much better spent wrapping one's head around the math itself.

> And I was countering that argument by saying that a large alphabet (a large set of pictograms) is hard to learn or at least takes a lot time, as can be seen from trying to learn Chinese or oher asian scripts.

And my counter to that is that the same way you do not learn greek in order to use some greek characters in math, the same way you do not learn chinese in order to use some chinese characters in math.

There are cases where "Δ" read as "delta" makes perfect sense for the delta between two values. Whereas "η" read as "eta" is not a obvious synonym for efficiency, or for the coefficient of viscosity, or for the metric tensor in Quantum Field Theory. And "c" read as "c" is not an obvious synonym for speed of light, or for 100, or for the space of convergent sequences. But "光" read as "Guāng"(chinese) or "hikari"(japanese) or "light"(english) might be obvious for something related to light. You do not need to learn the original pronunciation of the character to use it. Just have a 1 to 1 mapping between notation and notion. Same way Chinese and Japanese have completely incompatible pronunciations for the same characters.

And even better, just use words instead of characters.

Precisely. As an analogy, the Julia programming language allows Unicode variable names.

https://docs.julialang.org/en/v1/manual/variables/#man-varia...

I'll willing to bet a significant chunk of the "emoji" alphabet is better known and understood than the Greek alphabet nowadays.

It is true you would have to stay in some reasonable-sized subset, and I can't begin to imagine the arguments if someone tried to define "common emoji" or something..

> E.g. having a dedicated pictogram for various base physics quantities (e.g. for mass, length, time, etc).

What if you have two different lengths, one being the distance between two objects and the other being the position of the center of mass over time? What about their two masses? And, in relativity, their two proper times?

Now you're back to using subscripts and the Latin alphabet because what better way to describe the precise object you're talking about in a given language (here: English) than the language itself?

>>I have to translate every character back to the concept meant by this character

>Using emojis for variables won't help with that.

Yes, they will, because there is an amount of meaning embedded in the graphical shape of the emoji used whereas there is none for a greek/chirillic/phoenician/etc. character.

But the real solution is to not limit names to a single character.

See my comment here: https://news.ycombinator.com/item?id=25207527

This paragraph convinced me:

https://i.ibb.co/WsgQYHq/PNG-image.png

In my experience, math is 99% handwriting. You only put it into Latex once you solved the problem. Exploration and experiments are done on with pen and paper.

This is also the reason why notation is often terse, and highly domain/author specific. You get tired of writing long_variable_names very quickly if you do it OVER AND OVER by hand. Now think about how much work and confusion it would be to replace those symbols with little paintings...

Math exploration and experiments are done on pen and paper because the notation is incompatible with keyboards.

I assure you there is plenty of exploration done in tools like Jupyter.

I think this can just as well be an argument that this is all safe as it implies the only emoji used will be those that can easily be handwritten.

I need to see way more than Pythagoras' theorem to be convinced. Show me a couple of pages of "complex" concepts, with something like ten equations involving summations, triple indices and whatnot. Something like this [1] (no affiliation, first random thing I found). I bet emojis will make it even worse.

[1] https://www.jefkine.com/general/2016/09/05/backpropagation-i...

I feel this links quite closely to the 'Tau' debate around notation [0]. The idea behind using Tau is that it makes equations more easily relatable to Circles and highlights rotation where the concept exists. One of the cases for these emojis is to more easily highlight function of variables and to highlight the equation. Both of these are also along the same lines as coloured equations [1].

Something I strongly disagree with in both the Tau manifesto and this Emoji manifesto, is the notion that Tau and Emoji should be encorporated into the official literature. Pi is 'wrong' - always use Tau. Einstein's papers would be easier to understand if you used the fire emoji instead of E.

Of course not. But in the other hand, the responses to these arguments attack that aspect of the claim rather than the intent behind them. "You can't use Tau because all textbooks use Pi. You would need to reprint all textbooks in the world". "You can't use emoji because the support is bad, you can't draw them and you can't pronounce them."

What we're missing, and where all of these belong, is in a formal explanation format. Most attempts to break down and make concepts more palatable tend to be blogs, YouTube or even broadcast media. We don't see anything in between 'formal paper or textbook' and 'colourful diagram aimed at beginners'. Where are the colourful diagrams for your latest paper on Flat Chains in Banach spaces?

[0] https://tauday.com

[1] https://betterexplained.com/articles/colorized-math-equation...

Never knew about colorized equations. That looks brilliant.

Thank you for connecting all these concepts together. And I agree that this is one of the hurdles preventing people from becoming more than beginners. We need a gentler progression in abstraction. For many people, learning math is like learning Vim.

I don’t know, my initial thought is that it’s fairly distracting and makes me pause longer to think. Anyone sensing the opposite?

emoji is both the singular and plural form of the word.

And, English grammar is rightly descriptive rather than proscriptive. Today's nonstandard usage is tomorrow's standard English.

This is definitely not universally readable (even for people with average eyesight) and looks like it won’t be for a long time. I skimmed through it and saw several lines filled with a series of [?][?] on a device with the latest available OS. I checked on another device (with a different OS) and could read the emoji replacements. The publisher may be excluding a large number of people with this method.

Emoji in math has a useful niche, illustrating basic mathematics for young schoolchildren, and for simple recreational puzzles. Beyond this, the accessibility issues raised by others seem to be a greater concern to me.

My physics teacher in high school would sometimes replace variables with smilies to make a point that the symbol representing the variable is not the thing itself and I think it helped some students who had problems solving the exact same equation with different variable names. Of course naming conventions are important and were mostly used, but the smilies always stuck with me for some reason.

I've done this, too, when teaching remedial algebra in graduate school. I'm pretty sure it helped some people, while not confusing those who were already confused anymore.

Reading this in Firefox all I see is lots of [?] boxes.

I almost thought the entire article was sarcasm until I looked through Chrome.

Many math papers do use weird LaTeX characters [1], for example Halloweenmath [2]. And even though it does not look like there is a shortage of symbols for writing math, emoji can be used too, just be tasteful about it.

  [1] http://tug.ctan.org/info/symbols/comprehensive/symbols-letter.pdf
  [2] http://www.ctan.org/pkg/halloweenmath

If anyone writes a (sci-fi) novel and needs a fictional currency, some of those characters could be useful as an unit symbol.

Or go back in history I suppose (𐆖, 𐆗, ₶):

* https://en.wikipedia.org/wiki/Currency_symbol#List_of_histor...

Yeah, there is a lot of interesting symbols to use.

Why is math restricted to single-character identifiers in the first place?

energy = mass · lightspeed²

↑ As you would do in programming.

Proofs can be very long, writing out full pages with single letters let alone full words would be difficult to parse

Source code for software is much, much, much longer and structured in directories and version controlled and automatically syntax checked and automatically tested and even edited simultaneously remotely across the globe.

And in case of some languages we can even use the full Unicode for variable names (Julia). The only restriction is we have mandatory separators and adjacency does not automatically mean multiplication.

See for example VS Code Live Share for the last one if you are not aware of it.

That's perfectly fine, and sometimes done in expository writing.

https://www.amazon.com/Algebraic-Geometry-Projective-Varieti...

This get's very tedious to write if you use this over and over in proofs.

If you had to tediously write it over and over in proofs, why can't just you give it a name or an abstraction?

↑ As you would do in programming.

The reason I found is because in math adjacency is interpreted as multiplication. There is no mandatory separator character.

See my comment here: https://news.ycombinator.com/item?id=25207527

This is just terrible to look at.

kill me now yeet me to the astral plane

Where all the emojises exist?

This is great; emojis to entice younger generations into the world of math. I love it!

Reply on news.ycombinator.com