How I Think About Math, Lecture 1: Relations

See the slides (PDF). (You may want to use your PDF viewer’s presentation mode; there are a lot of pseudo-animations that could get annoying to scroll through.)

Update: Today, I drew up the field axioms in this notation. I’m almost to the point where I can define linearity!

Last week at Hacker School, I floated the idea of giving a presentation about linear algebra. Over a decade after taking it in college, I finally feel like I understand linear algebra well enough to express clearly, to an audience of programmers, most of the concepts from linear algebra that they might find useful.

I figured the very first thing to present would be the concept of linearity itself. After all, a linear operator is just any operator that commutes with addition and scalar multiplication. But wait– what is “commuting”? Well, no problem, “A and B commute” just means that composing A with B yields the same operator as composing B with A. But wait– what is “composing”? I could start my presentation by defining a category, but that would be unnecessarily scary given category theory’s fearsome reputation. Besides, John Baez showed me last week that categorical diagram notation has its boxes and arrows counterintuitively swapped. But wait– I could just use Baez’s new notation, instead! Then my entire discussion of linear algebra will be based on concrete, non-fearsome relations, instead of “morphisms.”

So…I got about as far as defining “commuting.” (Linear algebra will have to wait.) * * *

Note: I’m skirting the edge of what Baez’s formalism actually allows; in his work so far, diagrams always depict morphisms, rather than logical assertions. I’m still working on the semantics of quantifiers in this notation, so it’s conceivable some of the examples in these slides will change as I learn more.