Inside Every Calculus Is A Little Algebra Waiting To Get Out
summary
Claim: calculus reduces to algebra by treating derivatives as algebraic structure, implemented with dual numbers (a + bε, ε²=0) and proven via microcancellation and Skolemization. The talk shows a direct Kotlin/Haskell-style encoding that yields forward automatic differentiation (and backward AD via CPS), derives product/chain rules and sin′=cos by pure algebra, and notes limits: vanilla dual numbers give only first‑order univariate derivatives, higher‑order or multivariate needs nested/ tree duals and laziness. An axiomatic infinitesimal formulation is elegant but not directly executable.
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chapters (16)
- 0:00Start
- 2:04Differentiable Programming
- 2:47Chain Rule
- 3:12Derivative of a Function
- 4:10Why Is Differentiable Probabilistic Programming about Integration
- 8:41Real Numbers
- 15:38Dual Numbers
- 18:11Taylor Expansion
- 19:37Structural Equality
- 20:09Micro Cancellation
- 24:45The Derivative of Sine Equals Cosine
- 26:23Forward Automatic Differentiation
- 27:39Scotlandization
- 27:58Examples of Using Scholarsation
- 35:16Axiom for Differentiation
- 45:19Differentiation
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