The editorial argues the story isn't 'AI does math' but how Tao works — treating the model as a tireless but unreliable postdoc, dense prompts, catching algebraic slips, and naming hallucinations openly. This posture sits deliberately between AI-maximalist and AI-skeptic camps and offers a template for expert-level AI-assisted work.
By submitting the shared ChatGPT conversation URL directly (rather than a write-up or paper), the submitter frames the transcript as the primary object of interest. The 785-point response validates that the community values seeing the unedited back-and-forth of a top mathematician probing a hard problem with an LLM.
The editorial emphasizes Tao explicitly does not claim the counterexample works; he uses ChatGPT to iteratively narrow down where the construction actually breaks. Framing it as a lab notebook rather than a result guards against the retracted-preprint history that surrounds the Jacobian Conjecture.
Terence Tao — Fields medalist, MacArthur fellow, and possibly the most cited living mathematician — posted a shared ChatGPT conversation on July 22 walking through a proposed counterexample to the Jacobian Conjecture. The post hit the top of Hacker News within hours (785 points and climbing) not because Tao claims a breakthrough, but because the raw transcript is one of the most instructive artifacts of AI-assisted expert work published to date.
The Jacobian Conjecture, posed by Ott-Heinrich Keller in 1939, is deceptively simple to state: if `F: k^n → k^n` is a polynomial map whose Jacobian determinant is a nonzero constant, then `F` has a polynomial inverse. It has resisted proof for 87 years, generated a graveyard of retracted preprints, and earned a spot on Smale's list of problems for the 21st century. Any serious counterexample would be front-page news in mathematics.
Tao doesn't claim the counterexample works — he uses ChatGPT to help him figure out whether it does, in the open, showing every prompt and every correction. The conversation reads like a lab notebook: he pastes the candidate map, asks the model to compute the Jacobian symbolically, catches an algebraic slip, feeds the correction back, asks for the inverse via formal power series, and iteratively narrows down where the construction actually breaks. When ChatGPT hallucinates a citation or overreaches on a claim, Tao names it and moves on.
The interesting story here isn't 'AI does math.' It's the working style. Tao treats the model as a tireless but unreliable postdoc — infinite patience for symbolic grunt work, zero credibility on anything he can't verify himself. That's a very different posture from either the AI maximalist ('the model will do it') or the AI skeptic ('the model is useless') camps that dominate developer discourse.
A few things jump out of the transcript. First, the prompts are dense. Tao doesn't write 'help me with this conjecture.' He states the exact map, the exact question, the exact expected form of the answer, and often the exact technique he wants applied. This is the same pattern you see when senior engineers get useful work out of Copilot or Claude Code: the prompt encodes half the answer, and the model fills in the mechanical middle. Vague prompts get vague slop; specific prompts get useful drafts.
Second, he never trusts a computation he hasn't checked. When ChatGPT produces a series expansion, Tao verifies terms by hand or asks the model to derive the same result a different way and compares. When it produces a definition, he restates it in his own words and asks the model to confirm. This is the discipline that separates productive AI use from the cargo-cult 'looks right, ship it' that's currently generating a wave of hallucinated npm packages and phantom API calls.
Third, the model is genuinely useful even when it's wrong. Several of the most productive moments in the transcript come from ChatGPT proposing an approach that turns out to be flawed — but the flaw exposes something Tao wants to understand. The model is functioning as a rubber duck with a symbolic algebra system bolted on. That's a real form of leverage, and it's not the same thing as correctness.
Community reaction on Hacker News broke along predictable lines. One camp reads the transcript as vindication: 'even Tao uses it, therefore it's real.' Another reads it as damning: 'even Tao has to correct it constantly, therefore it's not real.' Both are missing the point. The transcript is neither a proof of AI's power nor of its uselessness — it's a demonstration of what the tool is actually good for in the hands of someone who already knows the answer shape.
The direct translation to software engineering is uncomfortable but clear. Most developers using LLMs today are using them the way a first-year grad student would use ChatGPT on a Jacobian Conjecture problem: paste the question, accept the answer, hope for the best. Tao's transcript is what it looks like when a domain expert with strong ground-truth intuition drives the same tool. The output is night-and-day different, and the difference is entirely on the operator's side.
Three concrete practices from the transcript port directly to code:
Front-load context in the prompt. Tao's prompts include the exact objects under discussion, the notational conventions, and the constraint of the problem. In code, that's the file, the function signature, the invariants, and what 'done' looks like. If you find yourself in a five-turn back-and-forth clarifying what you meant, the first prompt was too thin.
Verify outside the loop. Tao doesn't ask ChatGPT to check its own work — he checks it against a hand computation or a second derivation. In code, that's your test suite, your type checker, and your ability to actually run the thing. Asking the model 'are you sure?' is theater; running the tests is not.
Throw away confidently. A meaningful fraction of the conversation is Tao noting a proposed step is wrong and moving on. He doesn't argue with the model, doesn't sunk-cost the previous 20 messages, doesn't try to salvage. The equivalent in a coding session is `git checkout .` without ceremony when a generated patch heads in the wrong direction. If your instinct is to keep debugging the model's output rather than restart from a better prompt, you're paying interest on a bad loan.
The Jacobian Conjecture will still be open next month, and this particular counterexample almost certainly won't be the one that cracks it — Tao's transcript ends without a verdict, and history suggests the null result. But the artifact itself is going to be cited for years, because it's the clearest public example we have of an unambiguous domain expert working with a general-purpose LLM at the frontier of their field. If you want to know what 'AI-augmented' actually looks like in five years, it looks a lot more like Tao's transcript than like anything currently shipping in an IDE. Read it — not for the math, for the method.
This is the second ChatGPT shared conversation I've seen today that is truly fascinating.The first one was someone proving another conjecture false by just repeatedly saying "keep going" to ChatGPT: https://x.com/DmitryRybin1/status/2079904005652893709What a w
Terrance Tao's chatgpt conversation is really interesting for a variety of reasons:1. The counter example wasn't just a brute force selection, the polynomial is structured in a very specific way that ends up getting the result.2. Terry Tao's questions are very specific and prompts the
It's crazy how he suggests simplifications over and over and gets led through the finding. Absolutely bonkers how you can use AI to understand something and map it to your own mental map so efficiently, and of course he's most interested in generalizing or finding a simpler sub-result that
Math has some of the most insanely dense and impenetrable nomenclature. I can generally keep my head mostly above water or at least near the surface reading from most STEM fields, perhaps leaning on google/wikipedia a bit, but man, mathematics just so quickly decouples from all common tractable
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It’s endlessly fascinating to read the AI transcript of an expert who _really_ knows how to cut to the chase. It just shows how much you can potentially squeeze out of these models. I’m also surprised to see that even Terrence Tao seems to use it in a way that resembles, in progression, how I use ll