Formulas Are Compressed Reasoning
Across eleven tracks you met formulas for vector geometry, transformations, probability, optimization, and differentiation. Memorizing them without a model of what they assert is fragile. But treating formulas as disposable syntax is also a mistake: notation states assumptions, domains, units, and exact relationships that intuition alone can blur.
Use Intuition and Formalism Together
Start with a conceptual question—what changes, what is conditioned on, which axes interact—then use the formula to make that question precise. Test it on small cases, check units and limits, and let a numerical library handle arithmetic after the mathematical problem is specified.
Computers can differentiate recorded operations and solve linear systems, but they do not guarantee that the chosen objective, model, or data answers the intended question. People must inspect the assumptions and interpret the result.