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
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