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Sunday, 1 December 2013

Maths and languages

I should be working on one of three things due this week, and I won't even tell you what those are because that's not what I'm writing about now. 

Maths and languages

I think the first time I was marvelled by languages I was 9 or so and learning about code languages. I'm not sure why it's taught to children at all, other than to stimulate their brain activity or something along those lines, but in the book where we were meant to be learning English we were taught about encoding information by switching letters with symbols. It was magical. Imagine being able to express things knowing only those who had your key would be able to know them. It's the cheap fun behind codes in cereal boxes but it goes a lot deeper than that. 

I've always been impressed by polyglots and dream of speaking every language ever (or, failing that, at least 10 languages, but I'll settle for being fluent in 4 or 5). More than the average person, I've been interested in understanding ancient/dead languages. Learning a language, to me, is like sharing a secret. The more books I can read in their original language, the closer I am to what their true forms were in the authors' minds. Understanding an ancient language is the key to the secrets that can no longer be spoken (see: The Voynich manuscript). Being able to speak a language that only you and a few people in a room are able to understand allows you to speak to them and only them. Creating a language of your own and choosing who you teach it to is just another way to do that. There have been attempts at creating universal (or at least "common") languages. You can see them in Middle Earth (common tongue) and you can see them in India (Hindi). Esperanto never really took on. English is trying to take over but will fail miserably. I may not know enough languages but I have yet to come across a complete one. There are always words you can't quite translate that only have a meaning in the specific context of the language they belong to, like "saudade," "lassitude," "Schadenfreude," and even your ever-so-simple "nice." Moreover, I don't think there should be a complete language. On the one hand, it's not practical (I don't know about possible), but on the other, there's a certain beauty in not being able to name every single thing. 

There's a good chance that we're not naming anything at all because communicating with words is ultimately an act of faith: you are counting on the other person using the same set of symbols and sounds to represent an abstract (or not so abstract) notion. Even when that notion is very unambiguous, there's no guarantee that both you and the other person think of the exact same thing when you think of it. That's part of the reason why we have dictionaries and that's the whole reason mathematics rely on definitions and axioms. 

Through minor variations in the words as they are translated into other languages, what you write in mathematical form is very unambiguous. In fact, what language is needed is to translate the maths into words, meaning that "$x = 5$" is a statement which can be translated into other languages as "ex equals five," "equis es igual a cinco," "X gleich fünf," or "x égal cinq." 

(Fun fact: it took me forever to write that last couple of sentences not so much because I didn't know how to say them, but because I didn't know how to write them and even now I'm not sure I did it right. How come I can't find the names for letters in German and French which are so handy en English and Spanish?)

You can agree beforehand what a mathematical object is and then use a word (in whatever language) to refer back to that maths definition. Take for example a circle as the set of all points at a fixed distance from an origin. Whether I say "circle" or "círculo," they both refer to a set of the form $\{x \in \mathbb{R}^2 | d(x, O) = r\}$ where $O$ is the origin and $r$ is the radius. As far as languages I know of go, maths is as unambiguous a language as there is. If there is no word for a particular structure in maths, you may have trouble finding one, but as long as you can describe it with mathematical symbols its meaning is clear to anyone. That's why it's beautiful knowing maths and being able to "speak" its language. Whether maths really are the language of the universe or just the language we've collectively come up with to describe it, they're so pure and precise you can't help but be extra aware of the fact that everything is a fabrication of your mind. And it's wonderful. 

I don't know if it's unfortunate, then, that abstract concepts like feelings can't be written down in the language of maths. It sound very deterministic, but if it's possible at all it would be horribly impractical: start by modelling the very tiniest of physical entities* (good luck deciding which) and then model their interactions on an ever greater level until you reach the complexity of life. Then tear down psychology and medicine and explain every neurochemical interaction through that. Now that you've mastered all of the above, using as many axioms, definitions, lemmas, corollaries and theorems as possible, I present to you a final challenge: use it in a sentence.

*There's an unmentioned debate there that does not elude me. Physics use maths to describe observations. Before we go to Inception levels of nonsense, you have to wonder whether physics uncover the underlying maths that make the universe work or if the maths are just our closest approximation to something purer.

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