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Q&A

How expressive is a color-based language?

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In the answers to this question I learned that some cephalopods communicate via color:

Some cephalopods are capable of rapid changes in skin color and pattern through nervous control of chromatophores.[8] This ability almost certainly evolved primarily for camouflage, but squid use color, patterns, and flashing to communicate with one another in various courtship rituals.[7] Caribbean reef squid can send one message via color patterns to a squid on their right, while they send another message to a squid on their left.[9][10]

That is super-cool.

I'm developing an alien, intelligent, water-dwelling race that needs to reach a reasonably advanced level, by which I mean they understand science, technology, and abstract thought like philosophy. I'm leaning toward deriving them from cephalopods. As advanced creatures they'll need to communicate ideas that are linguistically more complex than mating proposals, "follow me to the food", and "extreme danger that way, 200 meters". Because they're water-dwelling, it seems unlikely that they would develop a written language to accompany this language. Therefore the idea of words composed of letters might not make sense (maybe they "chunk" at the word/concept level), and that's fine.

Absent an accompanying written language, is it plausible that the cephalopod language could develop into something that can be used to discuss complex or abstract ideas? If not, what changes do I need to make to enable this level of communication? (Am I going to need a written language?)

Part of me thinks that a language based on color, patterns, and flashing could easily be as advanced as a written language -- what are words but symbols on a page, after all, and is that so different from splotches on skin? Another part of me thinks that this sort of system might have more inherent limitations than, say, people typing at each other in real time in Slack.

Perhaps an advanced cephalopod language is more like sign language than text. I don't know how people using sign language who've never used a written language communicate about philosophy or math.

Having written this question, I only now realize the vague similarity to The Story of Your Life, in which a linguist cracks a very unusual "written" language by the "heptapods". I don't know how realistic that is, though.

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written celaphopodic language My thoughts are relying to the concept, that one single image displayed on the skin (acco …

7y ago

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The complexity of the language based on colour change should not be, ceteris paribus, too different from our audio-oral …

8y ago

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Absent an accompanying written language, is it plausible that the cephalopod language could develop into something th …

8y ago

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You need to look at sign languages, which have non-linear elements and and have a somewhat tighter relationship between …

8y ago

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Abstract ideas such as philosophy Think about logographic writing systems where each character has its own meaning, but …

8y ago

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We actually use images to capture our speech for analysis To make a spectrogram like one of these, we split out the h …

8y ago

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There is no reason why a color-based language could not be as expressive as any human language. Building blocks Every …

8y ago

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I will approach this from a linguistic angle. First, I expect that you want a language capable of abstract expressions. …

8y ago

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While the quoted example is extremely primitive, you are overlooking the possibility of shapes (glyphs) enhancing the la …

8y ago

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Given the complexity of colouration that cephalopods are capable of, one can easily imagine their language being vastly …

8y ago

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Binary Consider how complex a computer is and what it is capable of, yet all its code is composed of almost the most ru …

8y ago

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I will approach this from a linguistic angle.

First, I expect that you want a language capable of abstract expressions. (words like "truth" "beauty" "disappointment" and the like) Abstract concepts are often derived from object-specific concepts, for example "adversity" from "opponent", and I assume you wouldn't want the same symbol to express both the objective and the abstract concept. This implies that whatever form of language that is used must have a near-infinite potential for new vocabulary.

In the case of human spoken language, combinations of the couple dozens or so syllables are sufficient for the tens of thousands of vocabulary in an average human language. In the case of a colour language, one may replicate the structure by having a similar number of basic symbols, similar to the syllable of human spoken languages.

Assuming you are basing your creature design on cephalopods, a basic symbol could be a certain arrangement of colours on the creatures' tentacles. For instance, a creature may have four arms specialized for displaying coloured messages, if there are 6 distinctly recognizable colours, this makes a total of 1296 different permutations, more than enough to construct an analogue of human language. (I believe that the colours used for any variation of the colour language will not span through the whole spectrum, rather, only a few regions, e.g. red, yellow, green, cyan, blue, magenta, will be used, because colour usage spanning across the whole spectrum may cause ambiguity for small variation in hue, while discrete intervals allows for better error-tolerance)

In the particular case of colour languages, features of human language such as tone of voice, loudness and the like can be expressed by the saturation of the colours or small variation in the hue of the colour. For example, a creature may display a slightly yellowish shade of red to indicate the lack of commitment to what is being said. (similar to this: "A was friend with B, apparently", except without showing doubt by adding extra words.)

In summary:
permutations of colours displayed on different regions of the body form the equivalent of syllables, the combination of these "syllables" form words; discrete intervals in hue are used to avoid ambiguity; saturation, slight variation in hue etc. are analogous to human languages' tones of voice, accent and implicit messages.

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Given the complexity of colouration that cephalopods are capable of, one can easily imagine their language being vastly more complex than a spoken or written language.

You'd begin with the obvious, basic ideas being expressed by broad tones or colours, as present-day cephalopods do. Red means I'm angry, white means I'm scared, etc. Simple nouns could be expressed with approximate representations of the object in question, and coupled with colour. White circle = I'm afraid of the ball.

More complex ideas can be conveyed by symbolic language, just like our writing. Unlike our writing, the interactive nature of the medium gives you enormous flexibility.

Motion gives you a whole new set of possibilities. You can indicate the direction of the noun, you can indicate tense with movement forward or backward, indicate urgency with speed of flashes...the possibilities are endless.

A similar idea was explored by sci-fi author Ben Bova in Jupiter, and further developed in Leviathans of Jupiter. The novels posit a sentient species of massive organisms living in an ocean deep inside Jupiter, communicating with patterns of colour on their skins, and human attempts to communicate with them.

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We actually use images to capture our speech for analysis

Spectrogram

To make a spectrogram like one of these, we split out the higher frequencies from the lower frequencies. With this, we can see the way our vocal tract changes the sounds. On these images, it is clear how the different formats (peaks) shift from one frequency to another as part of speech. Thus, at a bare minimum, your race could simply have a streaming spectrogram running across their surface.

Of course, one would expect the race to be more efficient with their space. If one looks at these, it is immediately clear that we're really only using one dimension. Your species would likely be able to adjust their colorings in two dimensions, similar to that of a Cuttlefish. (and honestly, the question could end with the first 2 minutes of that video alone! Cuttlefish are amazing!)

If I were designing a language based around such color schemes, I would mostly focus on three things: size, spatial frequency, and temporal frequency. Size would be a major factor in long distance communication. If the signal is too small, it simply cannot be seen from a distance. "Whispers" might be done using only a small part of the body, while "shouts" might involve vibrant colors across their entire body. Spatial frequency would also be very important. The viewers eyes have limited angular resolution. If your signal is too high frequency, the eyes fail to distinguish it. A message may contain several "layers" to it, some of which are easy to see low spatial frequencies while nuances appear in the higher frequencies.

We can see this sort of thinking in JPEG compression, where images are broken up using the same spatial frequency system, and more bandwidth is given to correctly rendering the lower frequencies. JPEG actually breaks up an image into segments using the following patterns of 8 pixels:

Different spatial frequencies

As you can see, the cells near the bottom right contain more detailed information than those on the top left. "Speech" done with high frequencies clearly requires you to be closer to the individual to "hear." Most of the content, however, is found in the low frequencies. You can see this clearly in this clever picture, which is more compressed on the left, and less compressed on the right. The effect in this picture would be similar to what would happen if one viewed a "speaker" from far away (left) vs closer (right)

Cat image, with varying compression ratios

The final question is temporal frequency, which is how fast you change your colors. Obviously the faster you change, the more information you can convey per second. However, it also calls for the listener to be paying more attention. If you were speaking to a crowd, you might rely on slower "speech" to ensure everyone has time to see what you are "saying." If you are one on one, you might speak quickly.

Also, consider feedback. We tend to think of speech as a one way process. This is because of our biology. When we begin speaking, our vocal chords vibrate so loudly that we actually disengage our eardrum slightly, using 3 bones specially constructed for the purpose. As a result, other people are actually quieter in our ears while we are speaking. Also, it is hard to distinguish the sources of sounds, so our sounds overlap with others. An alien speaking with color would have no such issues (unless they speak using luminescent colors like glow sticks!). I would expect their language to naturally develop a sort of two way communication, where the listener is constantly responding to the speaker to let the speaker know that they are getting the full message. It might even evolve to be a full blown duplex communication!

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While the quoted example is extremely primitive, you are overlooking the possibility of shapes (glyphs) enhancing the language considerably. Tribes and extended family units could develop their own glyphs and grammar... creating complex diversity just like humans.

For example, you could have a glyph for "individual", and the color shade gives more meaning including "male", "female", "relative", "mother", "father", etc. Including the possibility of representing something very complex with a single color: "my mother's father's oldest brother who is now deceased".

As far as learning the symbols themselves, its probably not much different than a human toddler learning "mamma" and "dadda", as well as "nose" etc... its just that these aliens skip the auditory learning and go straight to literacy.

While written language (outside of their bodies) would be more difficult especially in the aquatic environment, it would certainly be possible... starting with "crude" sculptures and eventually creating technology like our computer screens. Poetry would be both a "verbal" expression as well as fine art, and in fact their depth of visual artistry would probably exceed nearly all other alien species.

Meanings behind the colors might be instinctual across the entire species, such as flashing red equals danger (fear), black equals anger, and a pinky glow is embarrassment, light blue equals agreement, etc. These emotions responses would affect the background of any glyphs and allow primitive communication across the entire species (trade language). If the color of the background is entirely subconscious, it could also prevent their culture from ever developing a concept of dishonesty.

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Absent an accompanying written language, is it plausible that the cephalopod language could develop into something that can be used to discuss complex or abstract ideas?

You don't need written language to discuss complex or abstract ideas. You need written language to store the complex or abstract ideas so that others can access them and build more complex ideas on top of them.

Color is as complex if not more complex than sound. Our written language is not more complex than our spoken language, merely more persistent. Consider the difficulties if someone had to read to you every book that you want to read. That's an oral tradition. Not having a storable language is more of an issue than color versus sound.

Note that braille and Morse code are fully capable of expressing anything in our language. Yet both are essentially two state languages. Raised/smooth for braille and long/short for Morse (perhaps silence is a third state).

It's not clear to me why they wouldn't be able to store language. Can't they just make little cephalopod statues as their equivalent of cave paintings? I'm struggling to understand why they wouldn't have a stored language. They already have the concept of a visual language. They just need to put the appearance on something other than themselves. A written language would seem more natural under those circumstances than under ours.

I wonder if there is a cephalopod somewhere wondering if sapience could develop in creatures with purely sound-based language. How would such things learn to store sounds rather than a simpler visual language?

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Binary

Consider how complex a computer is and what it is capable of, yet all its code is composed of almost the most rudimentary elements possible: zeros and ones (binary code). So, the computer software you and I are using right now to communicate is being transmitted to each other via binary and all languages the human race has ever developed have also been translated into binary. In addition, all philosophical texts, all mathematics of superstring theory, general relativity, all sciences, all humanities, all arts have been translated into binary formats that we access all day long while using digital devices. So, even just sequences of black and white blocks of colors on chromatophores are fully sufficient to display binary code.

Think of the cephalopod skin as an LCD with a specific resolution, a set number of colors it can represent and also a size of the display. So their communication devices have a huge range of possibilities. I haven't found an easy way to convert chromatophore size into human DPI or screen resolution, but the cells are the size of single cells, so that is a pretty small unit and so I feel sure their skin is capable of 1080 resolution and a refresh rate certainly quick enough to change as quickly as the other entities read the whole message.

So, they have a medium, their skin, that is capable of displaying complex patterns, characters or signs, as quickly as they can be read and the ability, technically speaking, of making as complex of signs as their intelligence and nervous system are capable of producing.

So, I don't see any problems at all with extremely complex of data being transmitted via skin.

Recording

However, how do they remember it all? Do they record it in written form, or do they have savant level photographic memory skills?

Squid at least have razor-sharp 'teeth' that ring the suckers found on some squid tentacles:

enter image description here

These could be used to inscribe very complex symbols on appropriate surfaces. In addition, cephalopods are known for their tool use, so they could use implements to carve or write on surfaces not appropriate for their tentacle rasps. This could be similar to how dragons record their language.

Opposable Thumbs vs. Tentacles

I think a good case could be made for a bunch of long tentacles being comparable in nimbleness and manual dexderity to opposable thumbs. Since opposable thumbs have been hailed as one of the main reasons humans managed to outperform so many other species in tool use, perhaps if given enough incentive, cephalopods could begin to use their tentacles as effectively as we use our thumbs. This mimic octopus has already mimicked legs even though it's not obvious why--is it just for fun? Regardless, they seem quite capable of complex dexterity.

Memorized Content in an Oral/Visual Only Language

In ancient India, when Sanskrit was an oral language only, pundits were still required to memorize the entire text ascribed to their family lineage. There were four major texts of the Vedas and each was quite long. The Rig Veda, for example, is a collection of 1,028 hymns and 10,600 verses, organized into ten books. All of that had to be memorized verbatim, as any deviation was considered a sin. There were even pundits who memorized all four vedas, earning them the title "Chaturvedi", "Knower of the Four Vedas".

So, humans can memorize entire sets of books verbatim and so it is not unthinkable that cephalopods could do something similar if their language was visual only (instead of oral only, as it was in India.)

Sign Language

Also, some human sign language systems have two sets of symbols: one for the letters of the alphabet, and one for whole words. In addition, some sign languages convey much of their content through non-manual signs. Postures or movements of the body, head, eyebrows, eyes, cheeks, and mouth are used in various combinations to show several categories of information, including lexical distinction, grammatical structure, adjectival or adverbial content, and discourse functions. In ASL (American Sign Language), some signs have required facial components that distinguish them from other signs. An example of this sort of lexical distinction is the sign translated 'not yet', which requires that the tongue touch the lower lip and that the head rotate from side to side, in addition to the manual part of the sign. Without these features it would be interpreted as 'late'.

So, I assume similar analogs of symbolic communication could take place in cephalopods. In fact, they have even more appendages to work with than humans do, so perhaps their sign language could be even more robust than human sign language.

Imagine combining tentacle positions, body posture and color changes in skin all coupled together into one very elaborate method of communication.

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The complexity of the language based on colour change should not be, ceteris paribus, too different from our audio-oral form of communication. If there is going to be anything that makes is more or less complex, it is the intellect of the species.

Consider your chromatic language - as others indicated in their answers, there are millions upon millions of shades that can be discerned by human eyes (let's forget now that their eye may be much more limited or much more evolved and that the underwater conditions will probably limit some colours) so it would allow for huge variety in possible colouring etc.

BUT

Human audio-oral communication uses similarly vast field of sound frequencies for communication and yet our languages are what they are and have typically around 30-40 phonemes (minimal distinctive units of sound). Thus the problem is of the language forming is not the richness available but to the contrary, on its limiting and articulation.

The principle of the language is to use finite means (which our brains can encode and comprehend) to express infinite meanings. Thus we use frequency bands between some 20 to 20kHz, which we mix heavily into those 30-40 phonemes, from these we create several thousand morphemes, from which we create tens of thousands of words and infinite number of sentences. It would be very easy to expand on this and to think up of a system that would enable much richer form of language, one that is not so much confined by the parametres of linearity etc. but it is our brains that do not allow this.

Probably also the biological limitations would not either because we still need to have these 30-40 phonemes whose realisations are unbelievably different if you take them only as individual tokens, i.e. the [a] sound pronounced by you in one sentence will look wildly different from the same sound in a different sentence pronounced by a person of different gender, age category and in different mood. Yet we are able to recognise this token from very minimal context and map it to a particular phoneme.

Thus, if we ever hope to consider the colour changes communication to be a language, it would have to follow these same fundamental principles and if the neurology of the organism in question were not principally dissimilar from our, it would probably be subject to very similar type of limitations.

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There is no reason why a color-based language could not be as expressive as any human language.

Building blocks

Every spoken language consists of a set of phonemes, sounds that can be combined into words and sentences. English has about 40 phonemes, depending on how you count them, but some languages have as few as 11. From these few sounds we make with our mouth, nose, and throat, we can construct hundreds of thousands of words to talk about everything from astrophysics to zoology.

Phonemes themselves do not have any meaning; they only carry meaning once they are combined into words. So the expressive power of a language does not depend on the number of phonemes it has. At worst, having fewer phonemes just means that words will be longer.

So, the colors of your cephalopods can be analogous to the mouth sounds of humans. Call them chromemes instead of phonemes. You can generate words by stringing several chromemes together through time, just like human languages string several phonemes together through time. Once you have a way of building words, add grammar and abstract thinking and you've got yourself a language.

How important is written language?

Absent an accompanying written language, is it plausible that the cephalopod language could develop into something that can be used to discuss complex or abstract ideas?

Your question seems to assume that written language is necessary for developing complex or abstract ideas. I don't think that's true. Ancient societies came up with plenty of complex and abstract ideas, passed on through spoken language, before they ever invented writing. Stonehenge and other ancient monuments show detailed understanding of engineering and astronomy. Oral tradition, as Thom Blair III mentions, can convey extremely lengthy stories, law codes, or sets of proverbs. Philosophical questions have been pondered by humans long before the advent of writing.

Written language is certainly helpful. It allows the preservation and communication of concepts across long distances of space or time, increasing the cross-pollination of ideas. It's possible that we wouldn't have modern mathematics or science without writing. Or maybe it just would have taken longer.

So if your cephalopods can develop a form of writing--a way to capture their speech in a more permanent medium--they will benefit. But whether they do or not, their color language will certainly have the power to express any ideas they come up with, no matter how complex.

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You need to look at sign languages, which have non-linear elements and and have a somewhat tighter relationship between morphemes and phonemes than is found in spoken languages.

If you were signing the bus went past me, you would make the sign for bus and then hold your hand in the “vehicle classifier” handshape, then with the other hand make the sign for me (probably pointing at yourself), then hold that hand in the “person classifier” handshape, then bring the vehicle past the person. (Example drawn from Irish Sign Language, of which I have slight knowledge.) The grammatical structure of the language is, as you can see, fundamentally non-linear: goes past me is all one action, made with two classifiers (handshapes which represent categories of objects (vehicles, people)), with the specific meanings of those classifiers (“bus”, “me”) established beforehand. (There is no indication of either tense or aspect in this little example, because I don’t know enough about ISL grammar to describe how they should be indicated; however, they are not necessary for my point.)

Sign languages are perfectly capable of abstract expression, including philosophy and mathematics (inasmuch as any non-written language is capable of mathematics*).

In English, phonemes can have inherent meaning: gl- usually means something to do with sight, as in glare, glint, gleam, glitter, glossy, glaze, glance, glimmer, glimpse, and glisten (or not, as in gladiator, glucose, glory, glycerine, and globe).** This sort of thing is usually obscure and not very obvious: most phonemes (sound units) in the language have no inherent meaning; morphemes (meaning units) tend to be multi-phonemic. In sign languages, the two tend to be closer: many morphemes are a single phoneme (hand shape, movement, touch, etc.).

Iconicity (onomatopoeia) is to some extent a feature of all languages, sometimes quite abstract (see the bouba/kikki effect), and sometimes obvious (Wikipedia gives hiccup, zoom, bang, beep, moo, and splash as examples of onomatopoeia in English). There is some evidence that sign languages have a higher degree of iconicity than spoken languages, but by no means is everything iconic. Signs that were once iconic can be simplified down to basic elements such that iconicity is no longer apparent. Or the sign for light may be based on holding a candle. Or the sign may not be iconic at all (in ISL, the sign for day involves putting one hand on the chest and moving the other hand (in the shape of a clenched fist) down past it).

Plenty of discourse markers are given on the face: raised or furrowed eyebrows, for example, or puffed cheeks, or direction of eye-gaze. And these may be held over the full length of a phrase (providing further example that sign languages are less linear in nature than are spoken languages).


So how can you build your cephalopod language? Celaphapods have tentacles that can move. They don't have hands that can form complex handshapes. That, then, is where the colour comes in. Colour may have a direct morphological meaning (red means anger, for example), but no doubt would also have more abstract meanings (perhaps red as an intensifier, or brighter hues as an intensifier), and direct meanings (red means red). Tentacles can move and gesture; colour can change during the course of a sign, or be held over the whole thing. Different tentacles can sign different concepts at the same time, and bring them together to show a relationship (as in my the bus went past me example from earlier).

Colours on the body of the animal (either the whole body or in patches of colour, maybe even moving patches of colour), meanwhile, function in a manner roughly equivalent to movement of the face, head, and shoulders in sign languages, functioning mainly as discourse markers, intensifiers, and indicators of tone and emotion, though there is no reason why other grammatical features, such as tense, aspect, or evidentiality, could not also be indicated there.


* Sign languages can be written, of course, but they often aren’t.

** Examples drawn from An Introduction to Language (6th Edition), by Victoria Fromkin & Robert Rodman.

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written celaphopodic language

My thoughts are relying to the concept, that one single image displayed on the skin (accompanied by movements) contains many information.

  • Therefore it doesn't need to be fast flashing of images like in Morse-Code.
  • Second their image and information procession skills respective their pattern recognition would be quite high evolved.
  • As sort of writing/storing these complex patterns plus their accompanied movements seems to be quite difficult (see the post about writing sign language) this might occur quite late in their development
  • therefore the memory skills will be quite elaborate maybe even be eidetic
  • if they have eidetic memory the use cases of stored language reduces

usecases of stored language

  • information which is regarded to have high value for later generations in time scales where the conservation of information could not be guaranteed even by eidetic memory and high emotional/spiritual/religious affection
  • fast transport of information over vast distances
  • Instructions/manuals and calculations which enable later generations to still support complex structures or understand their basical calculations although the visual information chain breaks. This might apply to science about the "up there" (above water level) and the "down there" (deep sea) like deep sea industry using underwater volcanos, underwater mining facilities or complicated medicinal facilities (surgery, complex breeding techniques, genetic engeneering), etc.
  • minor important instructions for others which will arrive at your place while you are on your farm ("Welcome - feel yourself at home. Please don't forget to xy. I'll be around by tomorrow.")

Information transport

For information transport under water they won't necessarily need written/stored information. They would rather need to use another medium than light unless they can produce lasers. But it seems to be easier to use what is already existing in nature around them.

  • Whales and dolphins have elaborate sound capabilities which enable them to communicate complex information over large distances underwater.

So it seems to be quite likely, that they either develop means to a) code their visual language into the sound capabilities of whales and b) alter the physiology or train those animals to function as a kind of "radio transmittors".

Only problem will be that they need to develop a visual/sound translating device of some kind.

storing/writing visual information of higher value

  • As they will likely alter their surroundings to fit their needs and develop some kind of architecture (which is definitely needed for under water mining facilities, deep sea volcano industry or medicinal facilities) plus thinking visually the thought of mosaics might be a near one to them.
  • They could use pearls, mother of pearls, shells, gems, etc.
  • Problem might be underwater growth, algae, shells, etc. which might cover these images after time making it hard to read, so it would be likely to put them under a cover like in a box, like a shrine which is only opened at certain occasions.
  • the mosaics could depict the actual visuals on their skin surrounded by signs for the movements
  • as this won't be everyday communication this might have some ritual (emotional, spiritual, religious) and/or artistic value

storing/writing visual information of everyday value

While I think that developing this would be quite unlikely there might be some ways of doing this.

  • s.o. mentioned the possibility of carving circles or half-circles into softer material using the "teeth" of their tentacles
  • so it should be possible to develop a complex alphabet similar to cuneiform script on this basis
  • this could likely be regarded being inferior to the mosaic approach, as it would need more space but done faster on the other hand
  • this would need a material which is soft enough to be carved without ruining the teeth on the long run, this might be a problem
  • maybe a more likely possibility especially for everyday notices while you are not around to tell directly would be a script build of knots on a series of ropes (which would be certainly developed for making nets to farm/control sea creatures) I think some first nation Americans used this style of writing if I am not mistaken.
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Abstract ideas such as philosophy

Think about logographic writing systems where each character has its own meaning, but you can combine them to create more advanced meanings. In such writing systems, the meaning of a word may be the sum of its characters - for example, you combine the characters for 'small' and 'man' to make the word 'boy'; you combine the characters for 'sword' and 'man' to make 'soldier'.

You can apply this principle to your color-based language. Say that red means 'danger', green means 'food', blue means 'home', yellow means 'person'. Red-green might mean 'poison' or 'don't eat this'. Blue-green means the food you have stashed at home. Red-blue means something bad is happening at your house. Blue-yellow means 'family', but yellow-blue means 'friend'. Red-blue-yellow means 'invader'. Red-yellow-blue means 'enemy'. Green-yellow means 'glutton'. Red-green-yellow means 'wants to eat you'.

As others have noted, you can take this much further by using shades, frequency, or other characteristics of how you display the colors in order to express greater meaning. Maybe a quick flash of yellow-blue means friend, but yellow that fades slowly into blue means 'lover'. You can see how a language built on a foundation as simple as a handful of colors representing different meanings could evolve into a mechanism for expressing complex or abstract ideas.

Complex scientific ideas

This topic is moot based on the criteria you gave in your question. Knowledge of advanced science requires the use of tools and instruments. For example, for two of your creatures to have a conversation about hydrogen, they must have some method of examining or manipulating the elements that allowed them to discover hydrogen in the first place. If they are sufficiently advanced to create a working underwater chemistry set, creating an underwater writing system should be trivial.

Humans' ability to communicate math with spoken language isn't limited by our vocabulary, but by our working memory and other limitations of the brain. Very few humans can do advanced mathematics in their heads; big breakthroughs in mathematics have always been made on paper or with tools. Unless their brains are far more advanced or otherwise work far differently than human minds, it is unreasonable to expect that these creatures would need to communicate advanced math using only colors. If their brains are different enough that they can do advanced math in their heads, they'd probably have a much easier time than you would coming up with a complex language for expressing math using only colors.

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