Doug Hofstadter is an inveterate observer of his own mind. His wonderful satirical essay on sexist language builds on a multi-year compilation of sexist words: every time he heard a word that privileges males (like “chairman”, “clergyman”, “foreman”, “manpower”) he’d write it down. After years (maybe decades?) of acutely observing language for this phenomenon, he wrote that killer essay. Only someone like that could explain the personal-ness of ideas and metaphors so well.
Around minute 33, he cuenta la cuenta of how an image/feeling from his past was called up years later, mogrified, contrasted, or intertwined with something else that he was experiencing at the time.
He also talks about what I would call “personal slang”. For example Doug says he thinks of the words “sourgrapes” as just one word. I would take this farther and say that, for me, entire stories can sometimes take up “just one mental unit”. For instance ∃ an economic argument that harsh factory labour in the Philippines is good because it’s better than picking trash out of a garbage heap. I could rattle this argument off so quickly in my head that I would be still spitting it out in person for five to ten minutes after I finished the thought to myself. What I mean is that since the argument is encoded as a single unit in my mind, it just takes me a millisecond to conceive the whole thing. So if I’m reading something on-topic that misses the point of that argument, I can quickly call up the entire speech and dismiss what I’m reading in the next millisecond.
I have many other personal metaphors (some of them are more original than repeating a popular argument). Some of them come from a mathematics textbook. And from these, a blog is born.
Do you ever watch a really awesome musical and wish, when you’re leaving the theatre, that people in regular life would burst out in choreographed song-and-dance? Or that no-one would look askance at you if you belted a song while you’re walking down the street, feeling intensely cheerful or melancholic? …Well, I’ve wished that before…and I’ve also wished that I could use mathematics in my speech (it’s sometimes possible with the right crowd, if there’s a chalkboard handy). And that’s what i’m trying to accomplish here.
A few months before I started this blog, I was auditing a class on the applications of differential geometry to psychology. I realised that due to X years of mathematical modelling and reading books about median voter theorems, hyperfunctions or Fourier transforms, my vision of the world had become totally abnormal.
Over the past several years I’ve felt increasingly like I need to share these shapes. You know, because it’s extremely weird to want other human beings to understand what you’re thinking / feeling on a deeper level, and stuff. And I feel like there are certain ideas that I have where some abstract idea from a maths book interpenetrate with whatever else is going on in my life, such that I think these unshareable thoughts like:
- My sister’s utility function must be dominated by interaction terms.
- When Jesus said “Let he who is without sin cast the first stone”, is kind of like the Dirac norm (trivial norm). What if we applied that to auto insurance claims?
- Trying to figure out what I need to improve in my business—it’s a bit like sensitivity analysis, isn’t it?
- Seeing university as an optimisation problem—balancing between various levels of classes, and a composite variable for “other” (non-class) activities.
- A line bundle varies from point to point—so an ideal anything could vary from instance to instance!
- What does the space of all possible bands look like? What would the distance function(s) be?
- Reading a book, everybody brings some part of themselves with it. It’s like the meaning you take away is a convolution of yourself with the author’s intent.
- The word “X is like Y” — it’s basically just saying ∃ a functor between X and Y, isn’t it?
- And phase transitions, superposition, posets, abstract geometries, fuzzy set membership, noncommutativity, convex hulls, metric and quasimetric spaces, curves/paths, state space, graphs, probability distributions — there are a lot of other metaphors that have just become part of my personal language. And now they can show up whenever I think about anything.
The difference between this blog and what you’d read on Wikipedia or in a paper or textbook is, I’m trying really hard to not be didactic. Sometimes I do define something or share an “aha!” moment in case it’s also the “aha!” for someone else. But when I do that it’s as preparation for a story I want to tell later. Like my wish that people would spontaneously break out in song-and-dance, I wish that I could just talk normally using these metaphors—like at a party or something, and people would know what i mean, and it wouldn’t seem either braggy or academic.
In Plato’s vision of an ideal government, The Republic, the “gold” people—the philosopher-kings—would train during their youth in corporalità and geometry, among other things. Geometry was supposed to teach the philosopher-kings abstract reasoning and about the true forms of the world.
Well, mathematics came a long way in the 19th & 20th centuries, such that I find the Platonic ideal much more believable now. For me, I want to understand quasimetrics, homotopy, cohomology, CW-complexes—those are the imagination-tools I want to have to see the world. If a deep understanding of surgery theory is more valuable to Gregory Perelman than a million bucks—that’s pretty convincing proof, to me, that this is personal language worth acquiring.