Linear Algebra Explains Why Some Words Are Effectively Untranslatable
27 comments
·November 14, 2025pjsg
seanhunter
Not sure this approach really accounts for the difference between a language like German where you have one compound word for a concept that would require multiple words in English. For one good example, the German "Nomenkompositum" is "compound noun" in English.
z500
If you ignore the spaces, the only real difference between German and English compound nouns are the infixes between elements to show bracketing. Case in point: Nomenkompositum
suddenlybananas
That's just a difference in orthography. English could easily have had an orthographic standard where we write "compoundnoun" for compounds. This is in contrast with a language like French, where compound nouns are relatively rare. Compare English "Olive oil" and German "Olivenöl" with French "huile d'olive". In French you need to have a preposition to combine the two nouns, whereas English and German do noun-noun composition.
adrian_b
You are right but neither yours nor those of the previous posters are good examples of compound nouns.
These examples have just the meanings of a noun + adjective or of a noun + noun in genitive case, where some languages are lazier than others and omit the markers of case or of adjectival derivation from noun, which are needed in more strict languages.
There are also other kinds of compound nouns, where the compound noun does not have the meaning of its component words, but only some related meaning (usually either a pars pro toto meaning or a metaphorical meaning). Those are true compound nouns, not just abbreviated sequences of words from which the grammatical markers have been omitted.
Such compound words were very frequent in Ancient Greek, from where they have been inherited in the scientific and technical language, where they have been used to create names for new things and concepts, e.g. arthropod, television, phonograph, basketball, "bullet train" and so on.
This kind of compound words are almost never translatable, but they are frequently borrowed from one language to another and during the borrowing process sometimes the component words are translated, but the result is not a translated word, it is a new word that is added to the destination language.
drivebyhooting
That isn’t a proof. Synonyms can bolster the enumeration sans augmenting novelty.
manwe150
That is the crux of the article premise: each synonym conveys similar denotations (principle component is I think what the article called it), but usually with some difference in connotations (the off axis contributions). You can nudge the languages vectors towards each other by adding enough synonyms and modifiers together, but they are always a little bit off even still
mannykannot
True, but many languages now have words that were absent from their earlier vocabularies. Shakespeare did not have the option to use 'telephone', 'semiconductor' or 'entropy'.
James_K
I think the reasonable reader will conclude it's unlikely for any two languages to share exactly the same vocabulary, accounting for synonyms.
cortesoft
Yeah, I was interpreting 'untranslatable' to mean what it says, but they meant 'untranslatable with only a couple words', which is a very different claim.
epistasis
> If the mere sight of the above is like a punch in the face for you, don't worry. I'm not going to math you to death in what follows. I will only remind you of a tiny basic part of it that I think relates to languages.
Yes, that mathematical expression is like a punch in my face, but not for the reason you think. I am offended that the rank of the matrix does not match the dimension of the matrix, not that I'm seeing a matrix.
tptacek
It's a 3x3 matrix with 3 independent rows. The rank matches the dimension.
nph278
Just read Wittgenstein (The Blue+Brown Books / Philosophical investigations), and this confusion will go away. The difference between translation, definition, and explanation needs to be understood.
proteal
I think a succinct way to describe my thoughts on linear algebra/language is that language has high dimensionality (ie many different basis vectors that may not necessarily be orthogonal) and that individual languages use a unique coordinate system to express thought. Each language is a lossy approximation of all conceivable thought and some languages can more efficiently represent the “all thoughts” vector space because they have basis vectors that point in more uncommon directions (like the go to japan example). So while you can more or less point to any thought in any language, some thoughts are easier to express in certain languages, which the post (and me) agree to be untranslatable words.
I tried to find the really interesting article about language and color that describes how some cultures use different naming schemes for colors but couldn’t find it. It talked about how back in the day we don’t know orange as a color, we just thought it was red-yellow and only after the fruit was distributed did the word for the color catch on. Here’s the best article I can find that talks about this phenomena https://burnaway.org/magazine/blue-language-visual-perceptio...
MangoToupe
> Each language is a lossy approximation of all conceivable thought
I'm not quite sure I understand this—I do have mental sensations/processes sans language, but I would not characterize them as "thoughts". To me, a thought is inherently linguistic, even if they relate to non-linguistic mental processes. So to me, learning a new language is very literally learning how to think differently.
rich_sasha
It's a tenuous analogy, but if you along with it, you can take it further.
You could consider the "cost" of expressing a word as some kind of metric or norm on the vector. What in one language/basis is a simple Kronecker delta, in another is a very complex vector (of course if it were the same vector in two bases, it would have the same length, but we could rather think of translation as an affine transformation, say).
And finally, with two bases, they need not span the same vector space. You can have a three-coordinate vector space all you like, if you have only two basis vectors you ain't spanning it. At best you can hope for an orthogonal projection from one to the other, and lose some nuance.
Eventually, with bilinguality, you learn not to translate words. Concepts live in different languages and describe a reality. Usually you can describe that reality in two different languages, but sometimes not.
raincom
There is a better thesis coming from the late philosopher W.V.O Quine: indeterminacy of translation [1]
d-lisp
Are they multiplying a 3x3 matrix by a 2 component vector ?
tptacek
In that one case, yeah; I don't think they're going for anything more than general illustration here.
magicalhippo
The text that follows does take on a new meaning though, for those that know linear algebra:
If the mere sight of the above is like a punch in the face for you, don't worry.
Almost makes me wonder if it was intentional.
seanhunter
Yeah that made me twitch also.
wjholden
It also has mixed square brackets and curved parentheses. I stopped reading the article when I saw this.
zvmaz
> I hope these rather unorthodox leaps between linguistics and mathematics helped make it almost obvious that some words and ideas are untranslatable in practice. I also hope you don't take the analogy too seriously, because it won't go much further than this.
Phew! Thanks for clarifying.
mannyv
Communication/language depends on shared context. The more context you share the shorter the trigger for evoking that thing and that context. And if you share no context communication becomes very difficult.
I wasn't aware that that idea was in dispute.
pixl97
And honestly, without a lot more communication even with a person that speaks your language you have no idea if you actually have a shared context. While an American from NYC and one from some backwater town in Kansas share a lot of context but there is a lot of context they don't, so as communication becomes more detailed between them it's very likely that 'translation' between each other will be somewhat incorrect.
This is also why lawyer speak is so particular. Language is fuzzy in most cases. Only language that relates to discrete physical objects gets closer to the binary state of exactness described in the article.
behnamoh
Tangent: I really like vornoi diagrams and part of me thinks there's a hidden, precious concept they represent. I didn't get their relation to the article but was wondering if they have applications in engineering/sciences.
triclops200
This article assumes that concepts are somehow precise coordinates within a single language; that's not the case, at best, speakers of a language mutually approximate a relatively consistent representation, but like, look at a word like yeet or whatever: we decided as a society on its meaning while it was being developed, as it were. Furthermore, it never rigorously defines what it means by translation. It claims 上京 is a single basis meaning moving to Tokyo, for example, but that isn't even an accurate translation: the individual components represent superior/greater/above and Tokyo and as an idiomatic phrase it represents the concept of moving to the capital for a better life. Something like "moving on up" or the like in some vernaculars of English, and idioms translating to idioms is a form of translation. It's disingenuous to represent the first concept as a single basis but not the second. Similarly, it claims mono no aware (物の哀れ) is unable to be translated, but, again, more literally "translated" is saying "the sorrow within things" character by character, and, only as an idiom has the full contextual understanding. It's not really a single point even if it's rather accurately located in a hypothetical embedding space by Japanese speakers. Imo, an English translation of the concept is "everything is dust in the wind", only 2 more individual conceptual units than the original Japanese phrase, and 3 of them are mainly just connecting words, but it's understood as a similar idiom/concept, here.
Concepts are only usefully distinguished by context and use.
By the author's own argumentation: nothing is translatable (or, generally, even communicatable) unless it has a fixed relative configuration to all other concepts that is precisely equivalent. In practice, we handle the fuzziness as part of communication and its useless to try and define a concept as untranslatable unless you're also of the camp that nothing is ever communicated (in which case, this response to the author's post is completely useless as nobody could possibly understand it enough internally for it to be useful. If you've read this far, congrats on squaring the circle somehow)
The article seems to think that a word is untranslateable if there is no single word in the target language. If I'm not misreading the article, then this is completely obvious -- just consider the number of words in English and the number of words in almost any other language, and you will find that there are more English words than the other language. It is now clear that there exist English words that don't correspond to a single word in the other language.