Advice to a Beginning Graduate Student (2001)

Practical guidance on graduate research emphasizes four activities: reading, studying, thinking, and writing. Read nonlinearly and aggressively - open books anywhere, mark margins, and write as you read to internalize difficult material. Treat writing as a cognitive tool: with paper and pencil your mind gains the power of a Turing machine rather than a finite automaton. When stuck, invent small hints for yourself or search for clever shortcuts; study published solutions not as defeats but as opportunities to learn how you should have thought (the author estimates roughly half the time new work differs substantially from your attempt, a quarter matches but is weaker, and a quarter improves it). Contradictions and paradoxes are highlighted as powerful vehicles for discovery, illustrated by examples from logic, computability, physics, and even a brief paradox-of-consciousness aside.

Advice on starting and surviving a PhD focuses on picking problems, dealing with advisors, and deep sustained focus. Advisors vary: some already know answers, some propose answerable problems they haven't solved, others steer into unfamiliar areas that force independent learning; both models can succeed but require knowing which you have. Begin by reading broadly, talking to others, and showing your engagement. In the middle years concentrate narrowly until you understand a topic completely while keeping a sense of broader context. Use small examples, experiment, and view failures and published overlaps as learning chances; examples include an unsolved optics problem and a surprising O(n) median algorithm. Above all, enjoy and habitually exercise the intellectual muscle of research and writing.