A lesson I picked up from language learning is showing up in mathematical explorations. Relationships among things are perhaps even more fundamental than things themselves.

When you are learning a new language, a naive approach is to try to learn new words individually, flashcard style. The intuition is that because you know what the thing is in a reference language like your native language, you can just learn the word as a symbol pointing to that same thing, all on its own. You write your new vocab word on a flashcard and then what it means in English or whatever on the back of the flashcard. You flip the card over and over, but the language doesn’t seem to materialize. When you do this, each word is a lonely atom, maybe mapped to a word or two in another language, but not linked with words and phrases in its own language - let alone linked to the world of experience!

A much richer and easier way to go about learning a new language is to start your learning through phrases and interactions. Embed the language in contexts. Relate your nouns with verbs, use particles to flip and turn the meanings of your phrases, and have adjectives and adverbs ornament your sentences. The vocabulary starts to become a language when you think about and inhabit thinking in relationships. How can these words be chained together? What’s the difference between two thoughts when I switch these out?

Learning those relationships reveals deeper structures and opens the way to analogy. To boot, relationships are things that are tractable to think about and concretize - consider how language can be represented computationally as vectors and the famous word2vec example of the vector difference between the pair king and queen and the pair man and woman’s being approximately the same. In a different domain, consider one of the simple models of computation, deterministic finite automata. They are directed graphs, where the nodes are states your computation can be in, and the edges are transitions between them based on the input to your automaton. If you were to have just the states with no transitions, you would have a poor automaton, hardly distinguishable from a set. When there are transition functions however, it can come alive with that structure.

In this video, Math Infinitum walks through a ladder from concrete everyday arithmetic up through layers of mathematical abstraction up to category theory. He then makes an homage to Kant and Hegel, and the questions we face in philosophy. Can we know a thing in itself? I am not sure whether we can or not, but for anything taken alone I would be pessimistic that we could. Truly alone, I think it would be like pure being in Hegel’s Science of Logic - indistinguishable from nothing. Mathematics that deals in relationships opens up the world, and I hope that it will help me grapple with my own questions about being, becoming, and passing away.