Terence Tao, 'Machine Assisted Proof'

Terence Tao, UCLA, gives the first of three AMS Colloquium Lectures at the 2024 Joint Mathematics Meetings in San Francisco. This lecture is entittled, "Machine Assisted Proof."

[mydwchannel]

Lean should quote Tao on their website: "It took me a month to learn"

[kellymoses]

It is refreshing to hear a mathmaticion of Tao's reputation saying "This is very far from my area of expertise".

[lukaleko]

I find it fascinating how this modern trend of proof formalization is moving mathematics more and more towards something resembling programming: similar to how a compiler checks for validity in your syntax and then executes your code, the proof system checks for validity in your proofs, and gives you a quasi-guarantee of correctness.

[ehfik]

thank you. mr. tao speaks perfectly well!
the haters havent watched a single minute of this video.

[pranaygodha]

🎯 Key Takeaways for quick navigation:

03:30 🧠 *Introduction to Machine-Assisted Mathematics*
- Historical overview of machine-assisted mathematics and traditional methods.
05:51 🖥️ *New Modalities: Machine Learning Algorithms*
- Machine learning for pattern generation using large databases.
09:00 🧩 *Satisfiability Solvers (SAT Solvers) and Applications*
- SAT solvers explained with applications, complexity, and limitations.
11:17 🤖 *Large Language Models (LLMs) in Mathematics*
- Potential of large language models, including limitations.
13:20 🔄 *Formal Proof Assistants*
- Evolution of proofs using formal proof assistants.
19:42 🧩 *The Complexity of Hales' Proof*
- Challenges and controversies in Hales' proof.
20:38 🖥️ *Challenges in Refereeing and Formalization*
- Refereeing difficulties and controversies during formalization.
23:35 📉 *Peter Schulzer's Liquid Tensor Experiment*
- Schulzer's collaborative effort in formalizing mathematics.
26:09 📚 *Benefits and Discoveries in Formalization*
- Schulzer's formalization revealing errors and improving collaboration.
27:15 🔄 *Blueprint Approach in Formalization*
- Use of blueprints aiding in formalization progress.
29:17 🌐 *Collaboration and Scalability in Formalization*
- Formalization projects enabling large-scale collaborations.
31:56 🕰️ *Time and Effort in Formalization*
- Current formalization time decreasing, aiming for widespread adoption.
37:05 🤖 *Machine Learning in PDE and Mathematics*
- Increasing use of machine learning in solving mathematical problems.
38:02 🧶 *Machine Learning in Knot Theory*
- Machine learning predicting knot signatures with high accuracy.
42:23 🤖 *GPT-4 Solving Mathematical Problems*
- GPT-4 solving IMO problems, integrating with tools for reliability.
46:31 🤯 *GitHub Co-Pilot for Code Generation*
- GitHub Co-Pilot automating code generation tasks.
50:08 🚀 *Future of AI in Mathematics*
- Improving AI formalization times for future collaboration.

Made with HARPA AI

[TL-fesi]

49:13 "The Geometry of the theories themselves" this is mind-blowing

[zhw]

Feels cool that I witnessed him learning the tool when i was also learning lean from the community 😅

[JesseBusman]

Interesting. I tried to learn Lean a while ago and gave up.
I think I'll give it another shot :)

[kellymoses]

I'm just a programmer with the highest math I ever took was Calc 2 but I found this fascinating.

[jtpmath]

I was interested in artificial intelligence and AI-generated proofs a few years ago but I dropped the idea. Back then I really thought category theory would help with this in a way that will reduce the amount of redundancy in mathematics. I can't help but feel that some famous theorems are redundant or have already been proved in some other logically equivalent statement. However, I don't know how to go about checking this, and categories made sense. I was learning about Haskell at the time for some applied category work, and could probably talk to Professor Riehl (sp) about this, but I haven't spoken with her in a while and I don't really know where to begin.

I'd say start with theorems that we already know to be true, obviously, and ones that seem different but are categorically equivalent.

For instance, when I was reading Introduction to Measure Theory, I asserted that the category of measurable spaces forms an Abelian category. I'm not even sure if that's true or if it makes sense, but could I create some sort of AI to answer this question for me?

First I would need to tell the computer what an Abelian category is, but I can't let it use the definition from Wikipedia, because that has too much ambiguity, and we're trying to define a category independent of the notion of "set" so most programming languages cannot do this. I figured this is where "types" come from in programming but still, I haven't figured this out yet.

I'm now interested in using Julia because it sounds prettier. There are some theorems that could possibly be proven if we were to create something in Julia (and maybe Haskell) but I would need to take an algorithms course, a course in Julia and a course in category theory before I can even prove/disprove what I'm saying.

I really think algebraic geometry will benefit a great deal from these assisted proofs, because so much of it in higher dimensions is combinatorial, that we can check things with much better precision. For this reason, I began learning a bit of complex visualization and Galois theory in Julia, but nothing is conclusive yet, since I usually don't have the discipline to even sit down and create anything.

But anyway, I will stop here and continue watching the lecture. I'm at 16:40 before my attention went elsewhere and I apologize for that. Let's keep going!

[zerosumgame]

Historical talk. This will go down as a turning point in mathematics

[HighKingTurgon]

Terry Tao, the only public speaker who speaks fast enough

[rishovmondal]

What if we train an LLM to turn human-readable blueprints to technical formalisation into lean? Like a custom GPT

[phillipgonzalez]

Is the volume on this video very low for anyone else?

[juliand.]

It seems to me like the people taking on individual Blueprint tasks are the modern equivalent to the human computers from the last century.

[humbertoospino]

maths is difficult field for most students specially in university, personally maths were really hard for me but even thought most of my teachers knew how bad my math skills were they still got the time to tutor me even thought other students were more proficient on the field and could keep up with better scores. when a professor see you pay so much attention you kind of gain the symphaty of them really fast. so i manage to pass with a c+ the statistics class and trigonometry as well.

[PRIMARYATIAS]

As a Software Dev who likes advanced math it is nice to see practices used in Open Source Software Development cross pollinating into Mathematics. Collaboration with computer validation is the key to tackle the ever bigger challenges and searching for interconnections between the ever bigger vast subfields of Mathematics. No human can understand everything so the computer puts the ideas to test (There are only two options on the computer: either the program does what it is supposed to do or not), So large scale collaboration is necessary. In fact all of what we know and have is based on a ton of collaboration and building/expanding over what other people done (like we have in software dev: pull requests for bugs, forking projects that take original projects into a whole different trajectory etc.). Nice to see Mathematics moving forward.

[mauricekalevra]

He uses the Boadilla latex theme, my favorite.. :)

[The_Conspiracy_Analyst]

Ah yes the OEIS. You know you're getting somewhere when your line of investigation produces relevant sequences that aren't found on it.

[mooncop]

let's goooo! san francisco swimming!