Philosophy and Math/Stat

← Reading

Geometry from iteration

The volume of an $n$-dimensional ball becomes manageable when I slice it into lower-dimensional balls. Integrating two coordinates gives the recurrence $V_n(1)=\frac{2\pi}{n}V_{n-2}(1)$; with $V_0(1)=1$ and $V_1(1)=2$, it leads to $V_n(R)=\frac{\pi^{n/2}R^n}{\Gamma(n/2+1)}$. A high-dimensional integral can emerge from repeated ordinary integrals. This is why the derivation stays with me. See Calculus.

Examples before abstraction

A diagram of an approximating sequence may suggest a limit before I can prove it. A counterexample tests the exact reach of a theorem. The proof then explains why the intuition works and where it stops. See Analysis 1.

Models are approximations

Differential equations describe change with explicit assumptions; statistical models describe uncertainty with theirs. Neither is nature itself. A useful model has to be checked against the phenomenon and the data. See Differential Equations and Statistical Learning.

Projection and inference

A least-squares projection separates a fitted component from a residual. That simple geometric picture connects linear algebra with regression, while reminding me that unexplained variation remains. See Advanced Linear Algebra and Generalized Linear Models.

从例子走向证明 · From examples to proof

例子和反例帮助我看清定理的适用范围；证明则解释其中的逻辑。Examples and counterexamples show where a theorem applies; proof explains why.