Is AI Replacing Mathematicians? Discussing Google’s AlphaGeometry

Hey, I used Macsyma, a MIT based algebra system, in the late 1970s (I was an undergrad at Hopkins and then grad at MIT). I used that for complex PDEs. It was not based in neural net but symbolic manipulation. It was grant from ARPA and three Professors developed, Joel Moses, Carl Engelman and William Martin, all at MIT.

For someone like me with limited math vocabulary and some math is one of many necessary skills, watching the steps is a gold mine. Thank you for that concise and insightful introduction.

I’ve worked with mathematics almost my whole life, mostly programming statistical models into business processes, and just asking chatgpt or AlphaGeo is just crazy. It show step by step so you can check, and o’my its fast. It’s not doing anything unique, just advanced mathematics 10-100 times faster than me, and I looove it!

Not sure how this ended up on my feed but good video. No hype or hysteria just straight info.

Yeah but an AI doesn't get a buzz from smashing a tough maths problem.

The thing to consider is that this is just two models working in concert together. Imagine what happens when they start training a model for each specific math domain. Then you mixture of experts them. With a really really creative (playing with the attention values) LMM as the head model over seeing the agent work flow. You can already do this with coding agents to get them to build software on a local machine. I'd be very curious to see what a MOEMM (mixture of experts maths model) would be capable of solving if it had the knowledge domains of geometry, calculus, trig, and probably more.

I have a concept relating to anomalous galaxy rotation velocity. I can work the concept as a geometric consideration, and Claude was able to conceptualize my work and develop the mathematical formulation. What a gift

Intuition and inspiration are what separates intelligence from algorithms.

Computers don't get tired so when at some point AI is close enough to human performance, scaling it is trivial by just adding more nodes to the cluster. When the AI is able to teach itself and create better versions of itself, we don't even need to do much to make it better anymore. We are not there yet, but I could easily see how AI might replace humans almost completely in this type of domain in say 50 years. Same for tasks like programming, any kind of planning and optimisation, probably engineering generally.

Wasn't a new faster matrix dot product algorithm discovered by AI?

Very interesting, if the Ai is to be extended into other areas of maths I can foresee a couple of snags 1) Could it be that proof becomes so long and tedious that it would be impossible for a human to check it , and if we could not check it how would we know if the machine is lying? 2) Mathematics is essentially a creative process and humans are great at thinking outside the box. For example take the invention of complex numbers using the letter "i" to stand for a number that does not exist is an idea that appears to have no purpose but we know how useful it is when it's possibilities are explored. The idea of exploring such barren areas is a human trait and difficult to mimic in machines.

Surely the Wolfram system is the bulk of the work? The AI is giving it a user-friendly front end to it. Regarding lack of training data, the Wolfram system got a lot of stuff uploaded by hand. This is because you don't want it to learn mistakes. Another good AI I saw the other day designed a CPU chip from scratch. They built it up and got it running and it was about as powerful as a 486.

I mean, we all know the answer. I'm 52, and here's what I remember about the past several decades' prevailing wisdom regarding AI. Everyone, and I mean everyone, thought that human creativity would be the very last thing AI could learn to do, if ever. And it turned out that the creative fields were among the first to be surpassed by AI, as it works just like human creativity. It mathematically deconstructs all of the samples that preceded it and then kit-bashes them back together in new syntheses. Maths were expected to be among the first things to go the way of the dodo in terms of job opportunities. I am a housepainter. I enjoy some life of the mind, as a history and philosophy buff and a self-taught programmer, and I do those things because I find them personally rewarding. Programming is not unlike woodworking, it's just that in the end, your product exists in abstraction and the materials (electrons) are dirt cheap. My suggestion to anyone in a professional field that is going to be overtaken by AI sooner or later is to not define yourself with one interest or one role, and in particular not with a professional role. Of any kind. Or you'll be left with nothing.

There's only really one question one needs to ask to know whether AlphaGeometry is a step forward: If you were to make a list of all of the geometrical constructions that have ever been made in high school geometry (drop a perpendicular, add a midpoint etc), how long would that list be, and what would the algorithmic complexity be of brute forcing that list? Because if a computer can run that algorithm, then alphageometry is at best a speedup. But I personally think that list of constructions wouldn't be more than 100 things, and on diagrams with fewer than 100 lines/curves, I cannot imagine that training and running a billion+ parameter LLM network is more efficient than brute forcing it. And if that's the case then automated theorom provers are still state of the art and we've just added a really expensive "speedup". For context, we know that just adding midpoints together with the deduction engine would already get you to above the average student's performance and very close to alphageometry. And you can do that without any AI. I suspect that just a handful of constructions would probably already beat it.

Great video as always. As a video idea could you try doing a STEP past paper for the exams coming up in about a month and a half

Excellent summary Ellie.

I agree. It's kinda a tool which can manipulate already formalized maths. Some kind of improved brute force (but it's not brute force but hope you get what I mean). But unsolved maths is not formalized yet so AI can't solve it while has no human-like intuition...

An AI solving maths is just another case of maths solving maths. It's like using pythogoras to get the magnitude of a vector. You're using linear algebra and calculus to solve a geometric problem.

Good presentation - Thank you - I cannot but wonder how to define the limitations of the approach - obviously it's more powerful that just pure sequential algorithms problems but it seems intuitively that it's just a subset of family sequentially solvable problems - if that makes sense

Does it solve problems from other mathematical areas except geometry?