- Maps and meaning
- Foundations of mathematical thinking
- One Ring to Rule Them All
- Abstraction for mortals
— a sequence introducing mathematical thinking to Verbals
Thesis: (verbally loaded) mathematics is just philosophy, but where everything is precisely what we mean
Maps and meaning
— lays down the foundations of modern Verbal mathematics, from basic category theory to homotopy type theory
Book review: Conceptual Mathematics
— what it says on the tin
The four intuitions
— In defense of B. Mazur's thesis that mathematical practice can be partitioned by "intuitions"
The Categories were Made For Math
— material set-theory vs structural set-theory (ETCS)
When is one thing equal to another?
— condenses this paper by B. Mazur
The Eightfold Path to the Calculus of Constructions
— a subsequence introducing type theory
A Fork in the Road
— an introduction to homotopy type theory
Not Math Made for the Categories
— cornerstone post ending with univalence and deriving first-order logic from homotopy type theory
Foundations of mathematical thinking
— a comprehensive review of mathematical problem solving (basically Schoenfeld + Zeitz + vignettes from the mathematical problem solving community)
Beyond How to Solve It
—Schoenfeld's research; pay lip service to Michalewicz
- Eventually, learning to code will supplant most of the mathematical drudgery you have to do. But it is known: what you cannot build with only sticks and stones, you do not understand. [link to EY]
Paper as the first REPL
—on not getting stuck; exercise vs problem distinction from Knuth (then explain the levels of problems in terms of complexity theory)
- But essentially, when you're solving a problem as opposed to doing an exercise, you're doing an interactive session with a REPL. A very primitive one, but still a loop.
- It's interesting to note how long it took us to start asking questions like "What are the kinds of problems we can solve?" and "How much resources would it take to solve such problems?" (i.e., on the computational complexity of different sorts of questions)
- And that's the thing. When you feel stuck it feels bad. You associate it with all sorts of bad feelings (probably because not being able to solve the problem was behaviorally tied to not getting good grades which was behaviorally tied to getting punished if not by overbearing parents then by fearmongering pronouncements of not amounting to anything in the real-world). But really, getting stuck just translates to not knowing the next action to take. And the main difference between seasoned problem-solvers and people like you and me [Footnote: aside from them overwhelmingly being composed of Maths] is that it takes them a while to exhaust their list-of-possible-next-actions. Hence why they keep telling you that mathematics is cumulative (though give them hell because they don't specify precisely how it accumulates).
- Like most feelings human, this next-action-exhaustion feeling is something whose effect on you you can modulate. You can do exposure therapy on this feeling until it doesn't hold power over you anymore. And once you're desensitised, we can start building mathematics.
- To help you get settled in the world of problem solving, I'll do exactly that: exposure therapy.
Problem solving in the world of wordsmiths
— on analogies and seeing at a glance vs working it out; introduces the concept of analogy tables
- on verbal people as a budget Scotts (+ a footnote on not idolising our community leaders)
- "I cannot claim to speak for all of us, but if you're anything like me, imprecise concepts crystallise within half a second (and if they don't, then I have to expend some effort into actually working out all the details).
- "I don't know if I have slow cognitive tempo [like Jonah Sinick] but I feel like I'm more of a Hilbert than a Gauss (really, anyone who feels like the latter probably does not need this sequence). It takes me a bit more time to digest objects from the Math universe, but once I do I can put it in my Hyperlinker and get a bunch of connections to vastly different fields for free."
Think aloud protocol 0: problems from category theory
— here is a list of heuristics, here are problems which may require the use of these heuristics, now record yourself narrating what you're doing as you're doing the problem (if you don't solve the problem within 15 minutes, it's okay; again we're not giving out grades here); introduces public commitment list
- I'm not gonna give out grades. But I do want you to do the think-aloud protocol. So for the next few weeks, whoever sends me their recording of doing the problem will be put on this public list. It's not a trophy or anything, but I hope acknowledging the superiority of people who actually tried over people who just skimmed this piece and reveled in their helpless state. [Footnote: Again, I don't mean to harp on people who have psychological blocks against doing mathematics. God knows it took me so long to get through mine. But I feel like we as a community don't value agency enough and I'm not about to contribute to that slide by tolerating it.]
Think aloud protocol I: a geometric problem
— opens with Grothendieck deriving Lebesgue integration by repeatedly asking what a 'volume' really is;
Think aloud protocol II: a combinatorial problem
— the Beast, the most Mathematical of all Maths objects; contrary to public perception, there's at least one book that treats combinatorics in the algebra intuition style: [insert book here]
One Ring to Rule Them All
— a digression to geometric algebra; i.e., go through vast swaths of the math/physics curriculum using Clifford algebra
Occam's Beautiful Chainsaw
- "If you wish to know mathematics as it is practiced, then check Wikipedia. Pick up standard textbooks. Trawl the Annals, because I will not teach you mathematics in the form known by most professional mathematicians. But if you want the shortest reverie from numbers; to linear algebra; to calculus: real, complex, and multidimensional; to geometry from Euclid to Riemann; and then periodically foray into Newton's Laws, electrodynamics, general relativity, and quantum mechanics; then let's begin."
Magnitude, Direction, Orientation
— a pseudohistory of numbers and geometry
- "What are we doing here? We have a map from the category Real to the category Euclid^3 […]"
- But geometry without algebra is dumb, and algebra without geometry is blind.
— what do you get when you combine esotericism with a mathematical bent? A proof of the Pythagorean theorem in terms of arrows, generalised to n–dimensions
Abstraction for mortals
— category-theoretic thinking for the rest of us
The Bayesis of Categories
—categorical view of Bayesian probabilities
—Bayesian causal networks in category theory
Of Cats and Ologs
—category theory applied to biological and linguistic systems; what category-theoretic thinking can bring to philosophy
You Could Have Invented Quantum Teleportation (If You Had Cats)
—quantum mechanics in terms of categories; see Coecke
I, For One, Welcome Our New Cat Overlords
—category theory applied to neural networks
[a pun on death and cats]