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I want to portray a culture with a significantly different method of perceiving their own mathematical concepts, yet not so as to say something like just "they used a kind of number very often, that is almost never used by [insert name of relevant other culture(s)]." I could bore the audience to death by going over random abstract nonsense from set theory or category theory or whatever, and probably will at least injure my audience by this means, but my question here is: are there systems of numerals (not numbers) where the difference is not in the symbols but the colors of the symbols?

E.g., let 0 = X, then 1 = the same shape but in red, 2 = that in blue, etc. Let's say then that 8 = the same symbol but in light gray, so there are nine numerals. I'm not sure if this is feasible, but I'm guessing we might then have XX, in dark gray/black, for 9, then the double-X in red for 10, etc.

So, are there real-world languages that have this feature? Or, are there conlangs, maybe, that involve this construction? In Conway and Guy's Book of Numbers, for example, they introduce some quirky things called "nimbers" (for the game of Nim or something) and mark them out by Arabic numerals except colored red specifically, and I don't know if that was purely for the sake of convenience or if it was meant to signal a kind of difference between nimbers and non-nimbers that, even if not color-based in abstracto, is obscurely similar enough to a difference of coloration so as to explain the coloration decision in the text.

I guess my question more specifically is: have color-coordinated numeral systems been designed, and then shown to be either workable or unworkable? I'm not sure why the related culture would have even come up with this system; a placeholder idea could be that they associated darkness with nothingness = zero, though this suggests that they would have chosen white for infinity instead of 8. I mean, I did say they'd have 8 as light gray, not white, at least because it'd be impossible to read white text against a white backdrop. Maybe clear white would be for infinity because it's impossible to read (against the expected backdrop). At any rate, if this idea has already been explored IRL and shown to be relatively impossible as an effective form of numerical communication, I would have to scrap the idea of some culture emphasizing such a notation system (presumably, if some culture tried the system out, they would have either failed more broadly because of their use of this system, or they would have abandoned it).

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    $\begingroup$ If it works, it will bring us a step closer to finally being able to smell the color nine. $\endgroup$
    – Qami
    Mar 28, 2023 at 1:33
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    $\begingroup$ I'd expect this only from a significantly more advanced race, who somehow standardised their colour perception. For comparison, human colour perception shows a great deal of genetic (and possibly cultural) variability. They also need to be significantly far away in time from not just the idea of runiform/cuneiform writing but also from ink-based writing, as using a multitude of colours to write a number would not be practical even with the coolest multi-colour pens of the 80s. :) $\endgroup$
    – biziclop
    Mar 28, 2023 at 9:43
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    $\begingroup$ I'm not sure if merely replacing glyphs with colors is enough to make a significantly different method of perceiving mathematical concepts. $\endgroup$ Mar 28, 2023 at 13:21
  • $\begingroup$ @Qami Underrated comment. Mildly considering creating a sock puppet just to upvote your comment again. $\endgroup$
    – automaton
    Mar 28, 2023 at 16:31
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    $\begingroup$ Do females in your culture can see 0 through 9 while males can only see 0, 3, 6 and 9? $\endgroup$
    – Josh Part
    Mar 28, 2023 at 22:31

13 Answers 13

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If the colour distinction of the species is large enough

Communication works or fails by distinction clarity. We use speach and writing, because we can produce sounds and visuals that can be understood with clarity if learned. Colours should be no different, as long as the distinction is clear and we can reproduce it.

It might already be possible with our eyes. Many artists or designers have a great range of colour distinction. This is gained mostly from experience. Most people will be able to learn that a certain shade or shade range is a certain number.

You might think that some variables will make it impossible. Light can change the colour drastically. Morning-, day- or evening light all produce different colours. Not to mention that candles or light bulbs have a wide range as well. However, with experience this can be overcome. Just like you can understand someone talking to you in several circumstances. Crowded room, with a vacuum cleaner on or a silent room all influence the sound quality. It isn't foolproof, but it is certainly workable.

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    $\begingroup$ I'm tending towards this as the accepted answer. All the answers are helpful, and in light of the many mentions of the "ink availability" problem, also then mod what this answer mentions in its title, it occurred to me that cuttlefish, or bioluminescent animals generally, would be a kind of species where this kind of color-coding might be relatively practical, at least during "speech" (flashing colors at each other). One time I accepted an answer here, I was advised to delay, so I'm going to wait some more this time, but again, out of many good answers, this gets at a solution most sharply. $\endgroup$ Mar 28, 2023 at 11:59
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    $\begingroup$ This could also be in combination to a species disability to accurately recognize small shapes - if their brains are naturally built like some forms of dyslexia, colors may in fact be the only sensible solution for a numeral/writing system. $\endgroup$
    – Falco
    Mar 28, 2023 at 12:47
  • $\begingroup$ @KristianBerry The problem is that your question mutates. The title is about aliens, but the body is about human languages. And the body is more words and often a better indicator of your real question. $\endgroup$ Mar 29, 2023 at 2:55
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In fact, we do use color-coded numbers a lot.

The electronic color code is a well-known (but by no means exhaustive) example, widely used for indicating the values of resistors and capacitors. All electronicists know how to read it:

resistor color code

(image credit)

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    $\begingroup$ [Adjusts Monocle] * Electronicist * - en.wiktionary.org/wiki/electronicist $\endgroup$ Mar 28, 2023 at 12:56
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    $\begingroup$ @MindwinRememberMonica: Thank you. Very useful monocle. $\endgroup$
    – AlexP
    Mar 28, 2023 at 14:27
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    $\begingroup$ @MindwinRememberMonica As an electronicist, I've been missing that word from my vocabulary for so long! $\endgroup$
    – pipe
    Mar 29, 2023 at 19:59
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Usage adapts to the allowances of the writing system, not the other way around. That is, if we have a conflict between the "right way" that we want to write things, and the practical way that we can write things, practical wins.

That's how we get shorthand and abbreviations. That's how Þ became the Y in "Ye Olde Shoppe" (because Belgian printers didn't have typeset Þs) and how German oe became oͤ became ö (because they were successively easier to write) and then back to oe (because it was easier to print on English-language computers).

Unfortunately, as others have pointed out, colored symbols are really annoying to write in ink, because in pre-industrial civilizations, inks and dyes are really expensive (especially blues and purples; reds and browns are usually easier to get ahold of) and tend to fade relatively quickly, and it's hard to get consistent lighting when you're largely at the mercy of the sun and firelight.

So in situations where precision is important - like your tax records, for instance, or your census - scribes will be tempted to add little annotations. Not numerals, of course, the numerals are still there in living color, but reminders. Say, reminding you that a particular number is written in red, and not orange. Even if it starts to go a bit orange in the sun. Or apologizing to the reader for saying "green + yellow" everywhere instead of "purple" but, have you priced purple ink these days? Things like that.

The problem is - because this shorthand is far more convenient to read and write than the colored numerals, and because it's capable of expressing all the ideas that colored numerals can - it will generally tend to displace them over time. People will get so used to reading and writing the annotations instead that the numerals themselves become vestigial and are eventually forgotten.

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    $\begingroup$ This should be the accepted answer. $\endgroup$
    – RonJohn
    Mar 29, 2023 at 4:59
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    $\begingroup$ It's worth remembering that there is good evidence that all of mathematics and all of literature grew out of the need for accurate tax and census records. $\endgroup$ Mar 29, 2023 at 8:01
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The main problem with this is that it's more complex for no reason.

Sound writing systems can distinguish between different components easily. On paper, colors are easily distinguishable enough, but...

What do they do if they don't have the ink to write down number X? What about individual perception of colors, period? Colorblindness would make reading said system impossible, and some people perceive shades of red differently!

That isn't to say the system doesn't have its merits, as by using colors could achieve a much higher information density.

Essentially you'd be going from base 10, like we have, to base 16 million (roughly the amount of colors in the RGB color space). Impractical, but could have merit in some edge cases once your civilization goes digital.

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    $\begingroup$ Why "base 16 million"? There are also millions of possible shapes, but our numbering system is base 10. Similarly, there are millions of different colors, but the numbering system proposed by OP is base 9. It would be enough if the chosen colors were distinguishable by almost all members of the species. $\endgroup$
    – wimi
    Mar 29, 2023 at 20:32
  • $\begingroup$ @wimi because that's around the number of colors the RGB color space can produce $\endgroup$ Mar 31, 2023 at 3:40
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There are some Terran natural languages (or better, cultures) that use some rudimentary colo(u)r coding of numerals.

One is the use of counting rods in (ancient) China, where the positive numbers were marked by red, and the negative numbers by black.

The other is the "modern" accounting practice in the "western" world, where the negative numbers ("debt") are sometimes written in red, and the positive numbers in black or blue.

IMHO this is historically more limited by the colo(u)rs of available ink - dark blue or black is the default, and you can get red one cheaply (by pissing on lead plates - yes, really), but having any other bright, outstanding colo(u)r is rather difficult.

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Human numeral systems are all quite old, and pre-date the availability of a full range of coloured inks at low cost. That's a fairly recent development, starting with the rise of chemical synthesis in the late nineteenth century.

So no widely-used human system of numerals uses colours. I recall an educational toy that was used in the UK in the late 1960s and early 1970s, which had wooden blocks of different lengths, in colours related to their length. The idea was to give children something less abstract than numbers to count and do basic arithmetic with. It was not a success, and did not continue.

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    $\begingroup$ "It was not a success" - citation needed. en.wikipedia.org/wiki/Cuisenaire_rods. It became less popular, but you can still buy them. It was how I was introduced to mathematics, and it was how I eventually managed to convince myself that multiplication was commutative, even when using the identity element for addition . (It took some time to realize that in general x·0 ≠ x). $\endgroup$ Mar 29, 2023 at 8:10
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Any culture that values (or thrives from) precise numeric/mathematics will always have multiple way to express their numbers.

For example in English, we have:

And in non-English language, they may have their own numeral system which may be used interchangeably with Arabic numeral like Chinese or Japanese; which if the number is small enough they use 一 二 三 四 五 六 七 八 九 十 ..., and if it's very long, 1234567890 is used instead (e.g.: representing money).

So, regarding your question if there can be culture that represent their number in color instead of in symbol. Then the answer is yes, our culture already done it, not exclusively though.

As I claim in the opening (reworded): any culture that values (or thrives from) precise numeric/mathematics will NOT USE ONLY ONE WAY to express their numbers.

Our own culture does not represent number EXCLUSIVELY in symbols, but in multiple form.

Quoting your question:

are there systems of numerals (not numbers) where the difference is not in the symbols but the colors of the symbols?

Yes there are: it's called charts

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  • $\begingroup$ It took millennia of industrial progress to develop colors ubiquitous enough for charts. $\endgroup$
    – RonJohn
    Mar 29, 2023 at 5:02
  • $\begingroup$ Not really a millennia (1000 years): industrial revolution is only from 1700-1800 (about 100 years). Compared with how long it takes from only having spoken numbers to having written numbers (I don't know how long, googling for how long is prehistoric age said that it's 2.5 million years ago to 1,200 B.C.), its not too long $\endgroup$
    – Kristian
    Mar 29, 2023 at 15:45
  • $\begingroup$ So I think the progress of representing numbers should go like this: spoken > written (symbol) > visual (color). Spoken to written stage is very long cultural progress, that the amount of industrial progress needed from written to visual color seems trivially short. $\endgroup$
    – Kristian
    Mar 29, 2023 at 15:48
  • $\begingroup$ "industrial revolution is only from 1700-1800 (about 100 years", but of course that 100 years didn't spring out of nothingness. All of the earlier ages had to happen before the Industrial Age. $\endgroup$
    – RonJohn
    Mar 29, 2023 at 17:41
  • $\begingroup$ Also, using color as part of numbers requires that each person carry more pencils/pens. Not very practical. $\endgroup$
    – RonJohn
    Mar 29, 2023 at 17:43
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... in a way ...

When we talk about numbers, mostly we mean natural numbers ($\mathbb{N}$) or integers ($\mathbb{Z}$). Those are 1, 2, 3, ... and ... -1, 0, 1, 2, ..., respectively. Sometimes, we mean decimals or fractions ($\mathbb{Q}$). Those have been known since antiquity, in some societies with decimal notation and zero and in some without it (complicating arithmetic). It takes some pretty sophisticated reasoning to come up with numbers in between the fractions, which leads to real numbers ($\mathbb{R}$).

The writing symbols for one ("1", "I", and others) and two ("2", "II", and others) can be easily told apart. You can write them in the sand with a stick, on a wax tablet with a stylus, or on a stone with a chisel. If you want to write 1.5, you do not usually write one-and-a-half "I", you use a special notation like $1\frac{1}{2}$.

Now imagine a society which rejects the concept of integers as an illusion, and insists that any actual numbers are non-fraction real numbers to start with ($\mathbb{R/Q}$). If one thinks to have two distinct appearances of the same number, that merely means one did not look finely enough, there will be a tiny difference between them. Too small to measure, but it is there because the mortal world is imperfect or some moral reasoning like that.

Say you and I are merchants and strike a deal. "Light blue" handfuls of hack silver for "orange" wagonloads of olive oil. I send my teamster to your warehouse foreman, and in the torchlight they exchange "violet" handfuls of hack silver for "brown" wagonloads of olive oil. We go to the magistrate to complain, and the magistrate says "you swapped somewhat blue silver for somewhat red olive oil, the contract was fulfilled, how can you expect divine perfection from your hirelings?"

Doing that will of course cripple their science and economy. If a ledger or money order fades in the sun, does that change the amount? How much of a difference is too much of a difference when it comes to selling grain by the pound?

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Not sure if its the answer you are looking for but here is a rabbit hole: Not just a numerical system, but even a Turing-complete esoteric programming language that is really colorful: piet

Piet program that prints 'Piet':

enter image description here

Based on that, and with some poetic liberties, we can assume a set of nice abstract paintings that would in fact be "programs" with various "functions" including creating fractal numbers, pseudo-random number series etc...

In reality, there is such a think as information density and Shannon's theorem, which limits the size of a program to the complexity, size, and number/hue of colors of a painting, but the number of colors can be rather large...

More piet samples

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I think depending on the biology of your alien species it could work. There are many animals that can change the colour of their skin so completely that they become near invisible. Think the octopus or cuttlefish, as @Kristen Berry mentioned, but there are many others on land and in the sea.

To me, this indicates that they can sense colour to a finer detail than humans can. For centuries their lives have depended on their ability to hide. There has also been research on the hogfish from Duke University that indicates they have some ability to sense light through their skin.

Perhaps your species could use an ability like the ones in the animals above to sense wavelengths of light (different colours) through their skin. Dealing with numbers could be a hybrid of vision and touch. I envision them doing math kind of like reading in braille while also using their eyes.

They could have evolved from a species that had these colour-changing/sensing properties in their distant ancestry.

As to the ink problem, perhaps they have used technology to synthesize ink that can respond to signals from their bodies much like an octopus uses a variety of signals to change its colour and texture. That way they could subconsciously direct the colour of their numbers as they work.

A system like this seems feasible if you're able to step waaaaaay back from how humans evolved and sense the world.

This is a world I would be keen to read about or experience if you are working on something to publish.

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Colored tokens as an abstraction layer for counting

Colored tokens are extremely useful counting tools specifically because they are not married to a specific numeric system. I could take a set of poker chips and play a game with my kids where white = 1 point, red = 2 points, and blue = 3 points just to keep it simple, and then take the same set up chips to a high stakes poker game and call them \$100, \$500, and \$2500 respectively. The countable chips work the same way no matter how I use them; so, I can use a pocket full of colored tokens to work out all sorts of different kinds of problems.

Now imagine you have a highly multi-cultural civilization where you have some people running around using base-6, some base-10, some base-12, etc. Numeric math is largely married to your numeric base and involves a lot of tricks that stop working if using a different numeral system... so the only way to do complex base-16 math for the average person is to count, and tokens make counting super easy.

In this way, colored tokens could function a lot like an abacus, but without a fixed number of numerals per row. All I have to do when talking to a person from another background is establish what base system to use, and then we could easily work together using the same base system.

source : https://www.wallpaperflare.com/poker-chips-on-white-paper-casino-map-play-cube-ace-gambling-chip-wallpaper-wpdzv

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No human examples, as color blindness is relatively common in humans. Red Green in men is apparently 14%. So they'd be innumerate and forced to work at Starbox or deliver processed carbohydrates to morons for money.

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    $\begingroup$ Just a note that the 14% number refers to men who have less-than-perfect color vision. Most of them are not color blind and can pass a driving license test. $\endgroup$
    – AlexP
    Mar 28, 2023 at 8:16
  • $\begingroup$ Thanks Alex. A better test is probably Ishihara, as some countries barely test colour vision. I say that as someone who is hopeless at Ishihara but can easily distinguish traffic and marine navigation lights. Paler shades, however, are a real mystery. I deservedly fail. $\endgroup$
    – wingnut
    Mar 29, 2023 at 11:41
  • $\begingroup$ I was firmly convinced that Ishihara cards were the only test used for the purpose of assessing the color vision of would be drivers. I never encountered any other test. $\endgroup$
    – AlexP
    Mar 29, 2023 at 12:24
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Have you read Can Fish Count, by Brian Butterworth? There is an interesting discussion of counting practices, including the Yupno counting system from New Guinea: the names of the numbers are names of body parts, starting with left little finger is one; after the fingers the toes are used for counting; after that the sequence goes 21 left ear, 22 right ear, 23 left eye, 24 right eye, 25 nose, 26 left nostril, 27 right nostril, 28 left breast, 29 right breast, 30 navel, 31 left testicle, 32 right testicle, 33 penis. The Yupno speakers have a gift exchange culture; when it is my turn to give gifts, I need to be able to remember how many pigs you have me last time, so I can reciprocate (Presumably nobody ever gives more than "penis" pigs). This bears on an interesting point: every number system is a solution to a problem; the Yupno system solves the problem of making their gift exchange culture work.

Maybe you could consider the problem that your culture is trying to solve.

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