Learning Methodology — Thurston Facilities

Source: [[On Proof and Progress in Mathematics]] — William Thurston (1994), Section 2

Understanding is not monolithic — it is a collection of separate mental facilities that work together loosely

Facilities listed by Thurston:

Human language / linguistic

Vision, spatial sense, kinesthetic (motion)

Logic and deduction

Intuition, association, metaphor

Stimulus-response

Process and time

The derivative example (p.3): 7 different ways of conceiving the same object — each activates a different facility

My learning process (applied [[April 26th, 2026]])

Isolate — take one concept out of the full system

Engage a facility — build a concrete artifact that activates one subprocess

Recast — put the concept back into the full model with multiple facilities now firing

Intermediate artifacts are not illustrations of understanding — they are the understanding at a tractable scale (also [[Dynamicland]] influence)

Premises

If I can't write the program, I don't actually understand the system

Rereading is a bad study strategy

Chunking is necessary — the full system is too large to hold in working memory at once

Examples from [[Poincaré Embeddings for Learning Hierarchical Embeddings - Nickel]]

"Walking downhill" → spatial/kinesthetic facility for gradient descent

A, B, C with actual coordinates → logical/stimulus-response facility for the loss function

Whiteboard score function sketch → visual facility for geometry-lives-in-the-score-function

Code (not yet written) → process/time facility — the final test of understanding