Marco Giancotti , November 13, 2025 Cover image: Image by vackground.com, Unsplash
A part of me still hasn't recovered from learning that some people believe there is no such thing as an untranslatable word. I've written about why I disagree before, but that explanation didn't satisfy me completely. There was a stronger argument to be made, I thought, but I couldn't put it into words. Now I remember, though: you need to see language as (a little bit) like math. Call me crazy, but I think that language translation is like a change of basis in linear algebra.
Me making weird connections like this might simply be an occupational hazard. Both my PhD research and my first job had to do with controlling the position and orientation of spacecraft and rocks in space, which means that I spent years juggling vectors, matrix multiplications, and reference frames almost daily. Still, I think it is simple enough to be understood by anyone, so hear me out.
(You might remember linear algebra from high school. It's that subfield where you write about stuff like this:
M = [ 1 0 0 2 1 0 3 3 1 ] ( x y ) M = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 3 & 1 \end{bmatrix} \begin{pmatrix} x \\ y \end{pmatrix} M = 1 2 3 0 1 3 0 0 1 ( x y )
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.)
During those same mathematical days, I was also learning Japanese. The language fascinated me for many reasons, like its beautiful dissociation between written and spoken words and its many unique quirks, but I was also struck early on by something a bit more meta: how hard it is to translate things to and from it.
These two concurrent interests made it hard for me not to see a connection. For almost a decade now, I've held it in a corner of my mind without telling anyone, perhaps because I thought it would be seen as too outrageous, but hey! Now LLMs are popular and they literally handle words and concepts as vectors with linear algebra operations, so maybe my analogy isn't that out there. Let me give it a try.
The Case of Vectors
Contrary to popular belief, a vector is not "a list of numbers" but an abstract object with no predefined way to express it.
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