Inside Every Calculus Is A Little Algebra Waiting To Get Out
summary
Claims that synthetic differential geometry and dual numbers give a purely algebraic foundation for differentiation, replacing limits with an axiom that skolemizes into a point‑free alpha/beta/gamma isomorphism and yields product, chain and trig rules by symbol‑pushing alone. Shows a direct implementation: transliterate the dual‑number algebra into Kotlin for forward automatic differentiation and explains backward AD as CPS. Notes practical limits: need multiple infinitesimals for multivariate or higher derivatives and warns of perturbation confusion; gives an integration axiom but no easy numeric recipe.
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chapters (10)
- 0:00Introduction to Software 2.0
- 3:49Algebraic foundations
- 9:09Dual numbers and derivatives
- 12:30Mechanical differentiation
- 17:32Implementation and point-free
- 20:15Isomorphisms and axioms
- 30:07Universal properties
- 37:15Limits of dual numbers
- 41:07Algebraic integration
- 44:00Questions and answers
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