• 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