> 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.

Are they multiplying a 3x3 matrix by a 2 component vector ?

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.

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.

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.

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.

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...

> 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.

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)

There is a better thesis coming from the late philosopher W.V.O Quine: indeterminacy of translation [1]

[1] https://plato.stanford.edu/entries/quine/#IndeTran

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.