More Language Musings (ft. Geometry)
I was watching some math videos in French, as you do, and I had this realization that was fun to think about, which spiraled into something quite grandiose, as you might see if you read on. You might not even need to know much math to understand my point.
By the way, if you’re at all interested in math and can speak French, I highly recommend Scientia Egregia, he’s got an extremely thoughtful and high-level take on the whole thing that is the closest thing you can find on YouTube to a knowledgeable person explaining a concept to you one-on-one in its broader context, which grad school taught me is still the best way to learn something. His videos are, of course, only missing interactivity, but you can always rewind.
Anyway, back to the interesting thought, and something I had never noticed before in dozens of his videos. When he refers to things taking place at a point in space (for example evaluating a function at point p on a surface X) he says en ce point. This more roughly translates to in this point, not at this point. And of course, my brain is so used to thinking about the English way of saying it that it had actively ignored it.
I recently read The Origin of Consciousness in the Breakdown of the Bicameral Mind, and one idea that it represents in the very first chapter (and which I find convincing) is that metaphors were originally spatial. The point he’s building towards is that consciousness itself is a metaphor, and you don’t necessarily have to agree with that, but I find it pretty reasonable to believe that the metaphors closest to the origins of language generally have to do with navigating the real world:
And the adjectives to describe physical behavior in real space are analogically taken over to describe mental behavior in mind-space when we speak of our minds as being ‘quick,’ ‘slow’, ‘agitated’ (as when we cogitate or co-agitate), ‘nimble-witted’, ‘strong-’ or ‘weak-minded.’ The mind-space in which these metaphorical activities go on has its own group of adjectives; we can be ‘broad-minded’, ‘deep’, ‘open’, or ‘narrow-minded’; we can be ‘occupied’; we can ‘get something off our minds’, ‘put something out of mind’, or we can ‘get it’, let something ‘penetrate’, or ‘bear’, ‘have’, ‘keep’, or ‘hold’ it in mind.
As with a real space, something can be at the ‘back’ of our mind, in its ‘inner recesses’, or ‘beyond’ our mind, or ‘out’ of our mind. In argument we try to ‘get things through’ to someone, to ‘reach’ their ‘understanding’ or find a ‘common ground’, or ‘point out’, etc., all actions in real space taken over analogically into the space of the mind.
This struck me as a particularly interesting example because in the languages I’ve studied, the metaphors for (mathematical) points are slightly different!
- In English, we say at a point.
- In French, we say en ce point, or, in a point, even perhaps inside: “La tour Eiffel est en France”)
- In Japanese, we say に於ける (I use the kanji here, though it is never written that way, to say something about my guess regarding the etymology). The term is originally 置く, which means to place or put. The preposition (? I don’t know grammar terms) に (ni) means something like “toward”, and can be used more generally to describe movement: 駅に行きます, I go to the train station. Altogether, this bit of grammar reads like “regarding X” or “in terms of X”, but it stems from a metaphor for a concrete action: placing upon or towards a point.
Now, allow me to get lost in the Sapir-Whorf sauce and freestyle a bit about what it means that this metaphor is slightly different in these different languages, and how that might reflect on how each linguistic culture sees geometry and the world. This is not to be taken 100% seriously, just for fun.
In English, we think of a point as being infinitesimally small, and when we say things are happening at a point of interest (or even at a point in time), we are maybe evoking a metaphor of an all-encompassing, external, birds’ eye view. When viewing a map, it makes sense that the points, tiny dots, are abstract representations as opposed to inhabitable spaces. In this view of geometry, you are perhaps rotating a globe.
In French, and this is what struck me as strange, points are inhabitable. You can imagine living inside them, and looking around at all the mathematical stuff happening in your environment. In this view of geometry you are not looking at a map, you are living in the territory.
And in Japanese, you are placing points in a diagram. For example: 円の中心於て, which we might say as “at the center of the circle”, in Japanese reads perhaps closer to “place in the center of the circle”. I’m imaging placing building blocks or constructing a diagram or sculpture from scratch here, like the sangaku or geometrical puzzles placed as offerings at temples.
As I alluded to before, you can talk about traversing time as if it were physical space, and the same language mostly applies, in both French and English: “at this moment in time”, “timeline”, “go back in time”, “in the span of a minute”.
Again, don’t take this too seriously. Nothing is untranslatable here. All three languages can communicate and grasp the same mathematical concepts. But it’s just fun to consider how such a fundamental concept, not just of geometry but of space and time, can be viewed differently. If you really wanted to stretch things (and really don’t take me seriously here, I’m generalizing and exoticizing): the mathematical cultures reflect these linguistic differences too. Anglophone math is objective and external, classifying, observing. French math is always about exploring spaces — even such an abstract thing as functions, in the hands of Dirichlet, Fourier, Borel, Lebesgue, etc, live in spaces that can be mapped out. And Japanese math is all about relations and context, like the Yoneda Lemma. Again, not really! You can describe mathematicians from all three cultures in all three ways. Notably, I always thought it was funny that in the 20th century there were such fruitful collaborations between French and Japanese mathematicians in algebraic geometry and the beginnings of category theory, my two linguistic obsessions. But maybe there’s some tiny grain of truth in the stereotype.
To return to the point about metaphors all being originally concrete, I would have loved if Julian Jaynes (the author of that bit I cited above) had written about geometry. To me, geometry is math, and it is one of the two most fundamental human activities out there. The Mayans, Egyptians, Greeks, Indians, etc all independently pondered the subject in their deep histories. It is one of the first examples of the true power of metaphor. We begin by crawling around in the real world, building temples and farmland and cities, and we eventually have the idea to traverse an imaginary space of shapes instead, and once that metaphor becomes decoupled from practical concerns, anything is possible. It begins to take on a life of its own, free of referents. The other most fundamental human activity, I think, is language, which had a similar story of breaking free into the world of forms. There’s some other dimension where this stuff lives. “Where”? There I go again with the spatial metaphors…
I had a similar thought today when I read this excerpt about art inhabiting some alternate reality from Pale Fire by Vladimir Nabokov which I’m reading.
But in some of these portraits Eystein had also resorted to a weird form of trickery: among his decorations of wood or wool, gold or velvet, he would insert one which was really made of the material elsewhere imitated by paint. This device which was apparently meant to enhance the effect of his tactile and tonal values had, however, something ignoble about it and disclosed not only an essential flaw in Eystein's talent, but the basic fact that "reality" is neither the subject nor the object of true art which creates its own special reality having nothing to do with the average "reality" perceived by the communal eye.