DARPA to 'radically' rev up mathematics research. And yes, with AI
- Reference: 1745775487
- News link: https://www.theregister.co.uk/2025/04/27/darpa_expmath_ai/
- Source link:
So to accelerate – or "exponentiate" – the rate of mathematical research, DARPA this week held a Proposers Day event to engage with the technical community in the hope that attendees will prepare proposals to submit once the actual Broad Agency Announcement (BAA) solicitation goes out. Whoa, slow down there, Uncle Sam.
DARPA's project, dubbed [1]expMath , aims to jumpstart math innovation with the help of artificial intelligence, or machine learning for those who prefer a less loaded term.
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"The goal of Exponentiating Mathematics (expMath) is to radically accelerate the rate of progress in pure mathematics by developing an AI co-author capable of proposing and proving useful abstractions," the agency explains on its website.
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Speaking at the event, held at the DARPA Conference Center in Arlington, Virginia, DARPA program manager Patrick Shafto made the case for accelerating math research by showing just how slowly math progressed between 1878 and 2018.
These fields have experienced changes but mathematics hasn't, and what we want to do is bring that change to mathematics
During that period, math advancement – measured by the log of the annual number of scientific publications – grew at a rate of less than 1 percent.
This is based on [5]research conducted in 2021 by Lutz Bornmann, Robin Haunschild, and Rüdiger Mutz, who calculated the overall rate of scientific growth across different disciplines amounts to 4.10 percent.
Scientific research also brings surges of innovation. In life sciences, for example, the era of Jean-Baptiste Lamarck (1744-1829) and Charles Darwin (1809-1882), the period between 1806 and 1848 saw a publication growth rate of 8.18 percent. And in physical and technical sciences, 25.41 percent growth was recorded between 1793 and 1810, a period that coincided with important work by Joseph-Louis Lagrange (1736–1813).
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"So these fields have experienced changes but mathematics hasn't, and what we want to do is bring that change to mathematics," said Shafto [7]during his presentation .
DARPA's proposed innovation accelerant is artificial intelligence. But the problem is that AI just isn't very smart. It can do high school-level math but not high-level math.
As noted on one of Shafto's slides, "OpenAI o1 (Strawberry) continues to abjectly fail at basic math despite claims of reasoning capabilities."
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Nonetheless, expMath's goal is to make AI models capable of:
auto decomposition – automatically decompose natural language statements into reusable natural language lemmas (a proven statement used to prove other statements); and
auto(in)formalization – translate the natural language lemma into a formal proof and then translate the proof back to natural language.
Robin Rowe, founder and CEO of the AI research institute Fountain Abode, attended the event. As he explained to The Register , he majored in math in college but found it dull so he went into computer science.
Nonetheless, he said, he found it interesting that the goal appears to be creating an AI mathematician that can serve as a coworker, one that's equivalent to a graduate student capable of helping with proofs.
That is, he allowed, a higher level of competency than is currently exhibited in AI models.
"We have chain-of-thought now," said Rowe. "And so this is like chain-of-thought on steroids."
[9]As ChatGPT scores B- in engineering, professors scramble to update courses
[10]Google DeepMind's latest models kinda sorta take silver at Math Olympiad
[11]Alibaba pits people against AI in its annual mathematics competition
[12]Finally! A solution to 42 – the Answer to the Ultimate Question of Life, The Universe, and Everything
For Rowe, the question is how AI can be made better at advanced math.
"Patrick Shafto, who is the project manager for this, he wrote the [13]paper [PDF] on Bayesian induction, which is the idea that you can figure this out using a large language model," said Rowe.
"That's not the way I lean, but it's the way that a lot of the room lean because that's sort of the obvious next step if you're going to use existing technology.
What I think we need is mathematical reasoning
"For the people in the room, they're like, 'Oh, you know, LLMs have got a lot better in the last year. We'll just keep going.' It's an indication of DARPA's concern about how tough this may be that it's a three-year program. That's not normal for DARPA.
"But for myself, what I think we need is mathematical reasoning. The bids aren't in yet, but that's the direction that we plan to take. But there are other people there who also had a different take, such as doing geometric mathematical reasoning and things like that. There's probably a dozen different ways to do this."
In other words, Rowe isn't sure that focusing on natural language is the right path. He suggests models based on visual or audio input will be more adept at advanced math.
"Do we choose to go with the Bayesian induction on LLMs, which seems like kind of what you would first think of if this was your field," asked Rowe. "Or do we go with something more radical like geometric modeling and doing it visually, for instance, instead of using words at all.
"And it wasn't discussed in the room, but there are mathematicians who do audio calculating in their heads – they feel numbers as musical tones. And so there's a lot of wild stuff that people could propose if we model it on how mathematicians actually do proofs in real life, because there are many different methodologies. Most people just know about the common ones, because these other things require that you have some kind of genius ability that isn't normal."
That said, Rowe is optimistic. "I think we're going to kill it, honestly. I think it's not going to take three years. But I think it might take three years to do it with LLMs. So then the question becomes, how radical is everybody willing to be?" ®
Get our [14]Tech Resources
[1] https://www.darpa.mil/research/programs/expmath-exponential-mathematics
[2] https://pubads.g.doubleclick.net/gampad/jump?co=1&iu=/6978/reg_software/aiml&sz=300x50%7C300x100%7C300x250%7C300x251%7C300x252%7C300x600%7C300x601&tile=2&c=2aA6pC2pvd-6awguK-Fb3fAAAAkw&t=ct%3Dns%26unitnum%3D2%26raptor%3Dcondor%26pos%3Dtop%26test%3D0
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[5] https://www.nature.com/articles/s41599-021-00903-w
[6] https://pubads.g.doubleclick.net/gampad/jump?co=1&iu=/6978/reg_software/aiml&sz=300x50%7C300x100%7C300x250%7C300x251%7C300x252%7C300x600%7C300x601&tile=4&c=44aA6pC2pvd-6awguK-Fb3fAAAAkw&t=ct%3Dns%26unitnum%3D4%26raptor%3Dfalcon%26pos%3Dmid%26test%3D0
[7] https://youtu.be/xNWpp_w5wMU?feature=shared&t=989
[8] https://pubads.g.doubleclick.net/gampad/jump?co=1&iu=/6978/reg_software/aiml&sz=300x50%7C300x100%7C300x250%7C300x251%7C300x252%7C300x600%7C300x601&tile=3&c=33aA6pC2pvd-6awguK-Fb3fAAAAkw&t=ct%3Dns%26unitnum%3D3%26raptor%3Deagle%26pos%3Dmid%26test%3D0
[9] https://www.theregister.com/2025/04/23/whats_worth_teaching_when_ai/
[10] https://www.theregister.com/2024/07/26/google_deepmind_maths/
[11] https://www.theregister.com/2024/03/15/alibaba_ai_math_challenge/
[12] https://www.theregister.com/2019/09/07/three_cubes_problem/
[13] https://www.mit.edu/~jbt/temp/feeney-heit/tenenbaum-kemp-shafto-final.pdf
[14] https://whitepapers.theregister.com/
Isn't the promise of LLM for helping in this endeavor really applying advanced pattern recognition
on many different inputs. Not just text; not just existing mathematical equations.
But also applying these recognition analyses on some of the alternative sources mentioned such as the audio-visual one or geometric modeling seems like a logical extension.
I remember when we were using PCRE to try to decode genetic sequences via the protein alphabet. This technique kept improving but it was still mainly an initially hand-created algorithm. Same with the tensor models which have expanded to handle so much more for which it was constructed.
Definition of math advancement
>math advancement – measured by the log of the annual number of scientific publications<
Not being a mathematician i wonder if their reaction to this "measurement" is similar to mine, see icon
Re: Definition of math advancement
Goodhart's law in action
Re: Definition of math advancement
When I was in uni (longggg time ago) one of my maths profs told us there are errors in roughly half of the peer-reviewed, published mathematical papers.
Re: Definition of math advancement
and the statistics profs were trying to work out in which half.
Dear ChatGPT. How many times does the letter R appear in the word raspberry.
ChatGPT: The letter R appears twice in the word Raspberry.
And you want an AI to do maths?
Sure! We get many bigger numbers. Bigger is better, you know. We also get bigger papers. Bigger is better, you know. And we get bigger earnings. Bigger is better, you know. And finally we get bigger. Absolutely bigger because bigger is better, you know.
Now also hallucinating bigger small letters. Bigger is better, you know.
Maths is about confirmation of sparky ideas
Not average or consensus.
1000 people say this over the years. Along comes me with my P=NP so I can't possibly be right.
AI might well have a role in warning people where proofs have been scored to stop reinventing the wheel or disproving the wheel. That's only hints though and so many million Eulers miles from proof.
In the olden days we'd wake up in the middle of the night and realise the power of our knowledge or plod on, angry at our ignorance to furtle another day.
I wonder if these AI-masters can give examples of realms of mathematical knowledge that are 'ripe'?
So how long will it take
....before we get the LLM deciding to effectively do a Bergholt Stuttley Johnson and insist that pi is exactly 3 instead of 3.14?
Re: So how long will it take
Well, I tell you that I heard that 22/7 is a vulgar fraction. And they teach that to my kids in school? How dare they! I want good clean fractions.
AI Genius?
Einstein's proof of Pythagoras' theorem is supremely elegant (https://en.wikipedia.org/wiki/Pythagorean_theorem see section on proof by similar triangles). It appears to have been a stroke of genius. Similarly for Archytas' finding a construction for the cube root of 2, in 3 dimensions (https://www.joerg-enderlein.de/archytas) to solve the problem of doubling the cube.
It is not the number of papers published in pure mathematics that is important but their quality, and the usefulness of the theorems. We are actively using, as cutting edge cryptography, mathematics done hundreds of years ago.
I doubt that an LLM or AI could generate anything of the usefulness of the mathematics created / discovered* by Fermat, Newton, Gauss, Euler, Bernoullie, Poincaré, Archimedes, Apollonius, Cantor, Hilbert etc. Nor the physics discovered by Einstein, Heisenberg, Bohr, Dirac, etc.
We can 'exponentiate' mathematics, but will the extra papers be any use? The 'joke' is that physicists are expensive because they need billions spent on particle accelerators, chemists are less expensive because they only need vast complexes , Bunsen burners, fume cabinets etc., whereas all mathematicians need are paper, pencils and a waste paper basket, but philosophers are the cheapest, because they don't need the wastepaper basket.
Still, we can wait and see what DARPA's AI achieves, but it does seem to me to be somewhat inconsistent with the major de-funding of serious science by the current Trump administration. All of the mathematics that I know of was produced as an attempt to model reality, or is some extrapolation from that, even the most obscure mathematical logic, or multi-dimensional geometry is based at source in trying to understand reality, and some inspiration or insight into how to model it.
Genius icon 'cos Archytas' construction is hard.
* Depending on your philosophy, pure mathematics is either discovered or invented, take your pick, I have every intention of ignoring this argument.
The exponent will be 0.01
Converging bigly