Cantor showed that there are multiple sizes of infinity by use of a diagonal argument, which was a clever way to show that no matter how you might try, you can’t list out every real number. The proof method turns out to be very powerful, and it shows up frequently in different guises: the Entscheidungsproblem, Gödel’s Incompleteness Theorems, Russell’s Liar Paradox, among others. There is a shared structure of these arguments, and category theory, the “linguistics” of mathematics as Thricery (the maker of this video) puts it, shows how these different contexts rely on the same fundamental structure for their proofs.

One of the core concerns of my life has been connecting the dots across domains - looking for ways to insistently be interdisciplinary. I have a goal for myself to at once develop technical mastery in many domains, while retaining the humility and broad gaze of a generalist. Specialization when lacking in context is impoverished. That same specialization though provides profound insights when it’s coupled with appropriate ways to abstract, recontextualize, and synthesize - insights that reveal that a subtle characteristic of this corner of the world actually has the same structure and even a similar import somewhere else.

It’s because I’m striving for this that I have turned back to mathematics and am becoming a mathematician. Fields of math like set theory, graph theory, theoretical computer science, and category theory help me see more clearly each day how deep the structures can be that things share. Seeing how things are isomorphic reveals further lines of inquiry across their domains and in the details of each of them. It’s in situations like that that specialization and generalization cross-pollinate and let us direct our gaze both into details and out into broad structures at once.

To boot, as Galileo said, “La filosofia è scritta in questo grandissimo libro che continuamente ci sta aperto innanzi a gli occhi (io dico l’universo), ma non si può intendere se prima non s’impara a intender la lingua, e conoscer i caratteri, ne’ quali è scritto. Egli è scritto in lingua matematica, e i caratteri son triangoli, cerchi, ed altre figure geometriche, senza i quali mezi è impossibile a intenderne umanamente parola; senza questi è un aggirarsi vanamente per un oscuro laberinto.”