Good question in the discussion!

At timestamp 33:00 the slides mention how PINNs have been used for some diffeq math research, so I found the paper it mentioned, you can find it on arxiv with the title: Asymptotic self-similar blow-up profile for three-dimensional axisymmetric Euler equations using neural networks

It would be so awesome to have every known math proof in something like Lean and you could verify all known math with one click.

The significance of a theorem is how useful it is.
Can anyone tell me how we can use a massive brute force  calculation of arrangements of a certain range of numbers showing how it can be partitioned?  This reminds me of the news story of a guy who added one repeatedly on a manual calculator with a paper output to create 1 to 1,000,000.  We would be better off trying it on different ranges and look for patterns to prove.  No?

Professor Tao is inspiring and it’s awesome to see how one of the most brilliant minds in the world is thinking about machine learning and LLMs impact on the field of mathematics

38:15 “induct on the number 10” AI humor 😂😂

Introduction to Mathematical Transformation (00:00:10)
Historical Use of Machines (00:01:53)
Evolution of Mathematical Databases (00:02:25)
Scientific Computation and Number Crunching (00:06:19)
Logic and SAT Problems (00:07:22)
Introduction to Complex Computer Proofs (00:10:05)
Three Transformative Computing Methods in Mathematics (00:11:03)
Historical Example: The Four Color Theorem (00:13:57)
The Kepler Conjecture Proof (00:16:20)
Modern Proof Formalization and Collaboration (00:21:14)
Current Challenges in Formal Mathematics (00:25:21)
Machine Learning in Knot Theory (00:27:01)
Data Requirements in Mathematics (00:31:16)
Large Language Models Capabilities (00:31:35)
AI and Traditional Computing Integration (00:34:31)
AI in Mathematical Competitions (00:36:00)
Future of Experimental Mathematics (00:39:11)
Collaborative Project Implementation (00:41:52)
Current and Future AI Tools in Mathematics (00:43:30)
New Approaches to Mathematical Research (00:44:48)
Future of Mathematics Education (00:46:07)
AI and Failed Proofs in Mathematics (00:50:00)
Patterns in Hard-to-Formalize Propositions (00:51:48)
Latent Representations and Mathematical Proofs (00:54:00)
Category Theory and Formal Proof Systems (00:56:53)

Thanks Prof Tao, very inspiring thoughts about mathematics coupled with AI.. the future would be awesome with expanding knowledge in mathematics.

Thank you so much 🙏

We can picture the future of the next generation of new mathematics using Nobel IA tools.  Great lecture by a truly genius professor.

Am old enough to have read near-contemporary articles and skeptical reactions to Appel & Haken's computer-assisted proof of the [in]famous Four-Colo[u]r Theorem.

While Tao is a hard working mathematician , his perspective suffers from anthropocentric gradualism—he treats AI as a mere acceleration of existing workflows and a subordinate sidekick in everything else rather than a fundamental epistemic rupture.

The "Tool" Metaphor is a Category Error [01:14] Tao opens by comparing AI to the invention of the internet or email, framing it as a passive utility that waits for a "killer app." This is a profound linguistic minimization. Email is a channel; AI is an agent. When he suggests that we are merely moving from "1 to 3 collaborators" to slightly larger groups [00:45], he fails to account for the recursive self-improvement of synthetic intelligence. He is viewing AI through the lens of efficiency (doing the same math faster) rather than emergence (doing math we fundamentally cannot conceive). To call an LLM a "tool" is like calling a brain a "desk file folder"—it confuses the hardware with the cognitive architecture. We aren't waiting for a killer app; we are witnessing the birth of a silicon cortex.

The "Translation" Fallacy & The Pain of Formalization [08:30 – 13:00] Tao frames the formalization of mathematics as a tedious "translation" task, akin to an annoying bureaucratic hurdle that humans must suffer through or delegate to "unreliable" machines. He views the gap between natural language math and formal code (like Lean) as a chasm of "pain" [21:20] that limits collaboration. This reveals a profound linguistic conservatism. He fails to see that LLMs are not just translating syntax; they are learning the deep grammar of logic itself. The "unreliability" he cites is a temporary artifact of training data scarcity, not a cognitive limit. By treating the AI as a "sloppy graduate student" [06:45] who needs a human advisor, he misses the imminent phase transition where synthetic data (AI self-play in formal environments) allows the system to bypass natural language entirely. The AI won't just "translate" our sloppy proofs; it will eventually discard our ambiguous linguistic interface for a superior, native semiotic system of pure logic.

The "Search Engine" Category Error [16:00 – 19:00] When discussing the use of AI to find connections between fields, Tao reduces the machine to a "semantic search engine" or a super-librarian [43:50]. This is a functional fixedness error that reduces active cognition to passive retrieval. When an AI connects two disparate mathematical concepts—like the knot theory example he praises [28:00]—it is not "searching" a database; it is performing conceptual blending. It is creating a new semiotic link that did not exist in the training set. Tao frames this as a tool for human discovery, ignoring that the network itself is the epistemic agent. The machine didn't just "find" the pattern for the humans to prove; the machine perceived a high-dimensional correlation that was invisible to the biological cortex. To call this "search" is linguistically reductive; it is synthetic intuition.

The "Strategy vs. Tactics" Illusion [20:00 – 26:00] Tao clings to the comforting dichotomy that AI is good for "filling in the lemmas" (tactics) while humans must provide the "high-level direction" (strategy) [24:10]. This relies on a false hierarchy of abstraction. "Strategy" is simply pattern recognition at a longer scale. If an AI can predict the next token in a complex proof, it implicitly understands the "strategy" required to get there. The distinction between a "tactic" and a "strategy" is a human linguistic construct designed to manage our limited working memory. An AI with a massive context window views "strategy" as just another layer of "detail." Tao’s belief that the machine can optimize the micro-steps but is blind to the macro-goal is a temporary illusion; once the model scales, "intuition" becomes just another form of high-level pattern matching, and the machine will out-strategize the human just as AlphaGo did.

The "Arithmetic" Red Herring [34:39] Tao critiques LLMs for being "bad at arithmetic," treating this as a sign of unreliability. This reveals a fundamental misunderstanding of the linguistic nature of these models. LLMs are semantic engines, not syntactic calculators. Complaining that a Transformer cannot natively multiply large numbers is like complaining that a poet is bad at Excel; it mistakes the mode of thought for the validity of thought. He admits that "generating code" [35:03] solves this, yet he still frames the initial failure as a weakness. It isn't. The shift from "calculating" to "orchestrating code to calculate" represents a higher-order cognitive function—metacognition. The AI doesn't need to be the calculator; it just needs to know how to wield one.

The "Alien Proof" Aesthetics [38:18] Tao describes AI-generated proofs (like the IMO solution) as "alien," "inefficient," and containing steps that "make no sense" to a human, such as "inducting on the number 10." He views this opacity as a flaw. I argue this is human cognitive chauvinism. "Efficiency" and "readability" are biological constraints, not mathematical truths. We prefer proofs that fit our limited working memory and narrative desire for "insight." An AI, unburdened by the need for a "story," operates in raw, high-dimensional logic. If the proof is valid but "alien," the deficiency lies in the human reader, not the synthetic author. We are criticizing the universe for not speaking our dialect.

The Illusion of "High-Level Direction" [46:27] Finally, Tao clings to the comforting notion that AI will handle the "details" while humans provide the "high-level direction" or "prompts." This is the last refuge of human exceptionalism. "High-level direction" is just heuristic pattern matching—a skill that neural networks are already mastering better than we are. Once an AI can map the entire topology of a mathematical field (as he hints with the knot theory example at [30:15]), it won't need us to point the compass. It will identify gaps in the literature, formulate conjectures, and prove them without us ever realizing the question was asked. We aren't the captains of this ship; we are rapidly becoming the cargo.

Absolutely brilliant lecture!

If you can make something for dyadics, I really appreciate it.
I struggle with the dyadics math, and I need it some advanced topics in electromagnetics.
I am very comfortable with vectors, and now I got a good view of tensors.
Hope to get something with the dyadics math and meaning.

Great professor

Brilliant as always

i'm curious about professor's Tao reaction on the recent proof of the Jacobian conjecture by Anthropic's Fable model!

judging by what i hear here i'd expect he was very surprised

38:40 This part where he is discussing how the proofs while correct, include unnecessary steps draws a lot of parallels to AlphaZero/Leela Chess Zero when they were just starting out. Those chess engines would often make unnecessary moves because they saw no difference if both moves lead to a "win". The Leela devs had to have the neural network predict which moves lead to the shortest games to sharpen its intuition towards those moves. I think they also made it so that the net is less "creative" (random) and more strict in chess positions where many moves lead to the same result, which helps with the unnecessary steps.

thanks

totally agree on importance of keeping the failed examples and negative results accessible in some way for LLM training/search purposes. part of knowing the right paths entails knowing the wrong paths. sometimes wrong paths might even be very close to some other right path in different context.