Grant Sanderson: 3Blue1Brown and the Beauty of Mathematics | Lex Fridman Podcast #64

Sanderson argues that writing the exponential function as e^x over-emphasizes repeated multiplication and obscures its true nature as the solution to a simple differential equation governing processes where rate of change is proportional to value, especially in the complex plane.

Physical observations (e.g., Pythagorean theorem in our space) ‘discover’ patterns that then guide which abstract structures we ‘invent’ (like R² with a specific metric), which in turn generate new ideas that feed back into further discoveries.

Effective teaching, Sanderson says, starts with specific, visual, low-level instances (like actual arrows for vectors) and only then introduces general definitions, letting the learner’s brain infer patterns rather than beginning with formalism.

Reading or watching lectures is not enough; working through exercises and trying to explain or teach concepts (even via code or videos) solidifies understanding far more than passive exposure.

They suggest that physicists focus on phenomena that admit compressible, elegant descriptions, and that minds like ours may only be capable of perceiving and modeling the ‘simple’ slice of a possibly more complex reality.

Sanderson frames infinity as the property of ‘always being able to add one more’ rather than an actually completed totality, and says higher-dimensional spaces are useful via state spaces, even if we cannot directly visualize them.

Sanderson finds the Riemann zeta function especially beautiful because it tightly links simple entities (natural numbers, primes) in non-arbitrary ways yet still contains vast unsolved mysteries, unlike Euler’s formula which he feels he now fully ‘sees through.’