从陶哲轩与AI的一段长对话谈起：顶级数学家面对陌生公式时，究竟在想什么？

TL;DR - A commentary essay analyzing a public Terence Tao–ChatGPT conversation about the Jacobian Conjecture, using it to illustrate how a top mathematician converts brute computation into structural understanding — and how AI fits into that workflow. It matters as a grounded counter-narrative to "AI solves math," showing the human role stays in question-selection and modeling.

• The worked example is an explicit 3-variable polynomial map with constant Jacobian (locally invertible everywhere) yet three distinct preimages of one point; Tao's drive is not verifying these facts but explaining why the cancellations occur.
• Key moves traced: substituting recurring expression blocks as new coordinates (revealing the constant Jacobian as two coordinate-change volume factors cancelling), assigning scaling weights to expose weighted homogeneity, and reinterpreting the 3-to-1 fiber as "choose one of a cubic's three simple roots," normalized via a resultant condition on the linear×quadratic factorization.
• Non-properness is localized at infinity: when two roots collide, the derivative at the root vanishes, sending two sheets of the normalized factorization to infinity while one stays finite — explaining why everywhere-nonzero local Jacobian doesn't force global invertibility.
• On AI usage: ChatGPT handled algebraic expansion, substitutions, bookkeeping, and rapid recomputation, while Tao retained verification, direction-setting, and distinguishing verified identities from structural conjecture — the article explicitly declines to frame it as "AI cracked a famous conjecture."