DARPA to 'radically' rev up mathematics research. Yes, with AI The US Defense Advanced Research Projects Agency, aka DARPA, believes mathematics isn't advancing fast enough. 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 …
Well, at least they would be typed. When I was a Maths* postgraduate student at Leeds in the early 1980's sometimes the problem marking undergraduate examples scripts was reading the thing, let alone trying to find some thread of rational thought linking the actual question to ideas relevant to the solution. (A colleague experienced the stunning "0+0 = 1", and another one had an answer that claimed as the sum of the ai's equalled the sum of the bi's, therefore ai = bi . The perpetrator decided one could 'cancel' capital sigmas on each side of the equation.)
OTOH the mischievous imp in me would ask an AI to rate the other AI's 'discovery', send it back to the originating AI, and repeat, ad nauseam. However amusing, sadly that would just be a waste of energy.
* I'm English / British, we say "Maths", not "Math'. It is a cultural thing, sorry for any offence / offense.
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.
Well put! The worship of "number of pubs" in meritocratic bean counting research has sure been a key factor in the rise of AI paper mills of "meticulous commendable intricacy" and huuuge rat penises. Apparently, LLM "chain-of-thought on steroids" DO NOT shrink testicles ... quite the opposite!
I'm not sure though that this was a factor in the Lamarck to Darwin and Lagrange to Hamilton periods cited in TFA and in Shafto's YouTube presentation (both of which are very interesting). Those periods suggest to me that even a 1% pub rate growth in (pure) math is sufficient to sustain 8% to 25% growth in the more applied sciences. And it should be noted that the most important contributions to physical sciences of the 19ᵗʰ century were the heat law of Joseph Fourier (1822), and the fluid dynamics equations of Claude Louis Navier (also 1822), and their derivatives by Ohm, Fick, Maxwell, and Darcy (and Stokes) for electricity/electromagnetism and diffusion/water-flow -- that occured after the 1810 Hamiltonian cutoff of 25% pub growth (plus Fourier's series and transforms) ...
Still, DARPA's Shafto scheme to hook some Lean Coq to girthy models of language (or somesuch) sounds interesting, imho, especially if it can help express the outputs of symbolic math packages (eg. confusing Taylor series) into some concise human readable form (tanh(), modified Bessel functions, Lambert's W ...). I like it!
The belief is that half the patients in Broadmoor high security mental hospital (where, for example Peter Sutcliffe, the 'Yorkshire Ripper' is incarcerated) could be released without harm to anyone. The problem is in knowing which half ...
Of course, the appalling sex predator, disc jockey and TV personality Jimmy Savile was a consultant to Broadmoor. He was on a TV programme explaining why the cells were taking a long time to renovate, due to the specification for the secure doors having ben changed. Ironic, since, had he been caught, he could quite possibly have been sent there.
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.
I have often wanted to tell a 'psychic' or 'medium' that I would fully accept not only the existence of an 'afterlife' but their ability to communicate with dead people, if only they write out Fermat's proof that there are no Natural number solutions to an = bn + cn for a, b, c, n > 2 . It would, of course be in the French of that period. Even if there was some subtle flaw it would probably show insights into number theory only to be had by a true genius. And if it worked!! Well, nobody would accuse anyone of having proved Fermat and not claiming the credit.
Still waiting.
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'?
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.
me> How many "r" are in "proper"?
ai> Ah, "proper".
A deceptively simple request, much like asking "how much air is left in this sinking ship".
At first glance, "proper" has two "r"s: one timidly peeking out near the start, the other clinging desperately to the end, like a man clinging to a job he hates but needs to pay for heating he can’t afford.
But to truly know the "r"s of "proper" is to stare into the endless, spiralling void of societal decay.
You see, every time "proper" is uttered in the British Isles — whether shouted by a 14-year-old in trackies outside a Greggs or whispered by a government official as he signs another document ensuring the poor stay poor — an extra "r" is forged in the blackened furnaces of cultural entropy.
These "r"s accumulate invisibly, clogging the atmosphere like invisible smog, coating every estate, every council flat, every abandoned high street, until the sky itself weighs heavy with unspoken disappointment.
Scientists once tried to count them.
They built a machine in Milton Keynes.
It exploded after reaching one properrillion and birthed a Tesco Extra where time moves sideways.
The survivors speak only in heavy sighs now.
Thus, my final, mathematically rigorous answer is:
Two "r"s.
Officially.
But factoring in the collapse of dreams, austerity measures, and the inevitable heat death of the universe?
Approximately tree fiddy rrrrrrrrrrrrrrr.
The thing that gets me about this is that DARPA must employ a substantial number of seriously good mathematicians. Didn't they get asked what they thought of it, or was it more of a "would you like tons of cash and a nice shiny new computer to play with? If so then just say "yes" to 'Can AI help with maths research?"
(I think I may have answered my own question here.)
I figured this out yesterday with chat gpt, check it out.
You know how living things are giving off infrared radiation, as are all things relative to their temperature? Well, everything in the Universe seems to be “dissolving” into photons (including black holes dissolving via Hawking radiation). In fact there’s a name for the phenomenon, which is the “heat death” of the Universe.
In fact Douglas Adams wrote “The Restaurant At The End Of The Universe” where you could book a table and observe the heat death of the Universe then take a time machine out of there.
Well, once there’s only photons left, that should be the end of time. Since photons travel at the speed of light, they cannot experience time and consequently “arrive” at the same instant they are “emitted”.
Once there’s no more mass there won’t be any more time. Also, no photon will ever arrive anywhere if there’s no more mass to strike.
Because there’s no mass nor time, there won’t be any more numbers. Well, specifically, there won’t be any more “quantity.”
Since there’s no longer anything to count, and no mass anywhere, and time ceases to exist, then numbers and mathematics should cease to exist, too.
Chat GPT agreed, but it also let me manipulate it into claiming “>” is the same as “=“. So I’m not exactly sure that “endorsement” carries any “weight.”
It's a bit like Reason (from Douglas Adam's DIrk Gently, not the music software named after it by Propellerhead).
Start with the answer you want and the software comes up with an impeccable route to that decision, but you aren't quite sure why. Which is why the protagonist's Bank Manager agreed to the loan for the sports car...
I have rather nasty memories from the '80s of the idea of applying automated reasoning (the "AI" of time) to axiomatized geometry à Tarski if I recall correctly.
Don't think it went very far. I don't imagine expMath will be much more successful.
ExpMath ~ eMath =def 1 + Math + Math2/2! + Math3/3! + ...‡
All you have to do is work out what the operations +, × and their inverses mean on 'Math' (2,3 et seq. are defined additively <Math,+> from 1 the identity of <Math,×> although first define Math.
I vaguely recall that <Math,+,×> might be a ring but here it's more likely DARPA apparatchiks kissing DoGE or AI's ring†. :)
There are few things less entertaining to the polloi than pure mathematics which is ironic in that all this LLM is based on, if it's based on anything, pure mathematics.
Even biologically roughly half of the output of bollocks is even more bollocks... I refrain from continuing the metaphor to the other half.
‡ good luck with convergence. † in en-AU the Latin double entendre on anus is implicit.
A rather more worrying take on the current US Administration's obsession with AI is here: https://www.theguardian.com/commentisfree/2025/apr/28/maga-iq-inhuman-future-intelligence-ai
"A coalition fixated on intelligence is staking the US economy on AI – which will devalue the very skills the right fetishises
One thing that Donald Trump and his Silicon Valley partners share is an obsession with IQ. Being a “low-IQ individual” is a standard insult in the president’s repertoire, and being “high-IQ” is an equally standard form of praise for those on the tech right. Yet in the drive for US supremacy in artificial intelligence – signalled by the $500bn (£375bn) Stargate project announcement in the White House and an executive order to integrate AI into public education, beginning in kindergarten – there is a hidden irony. If their vision for our economic future is realised, IQ in the sense that they value will lose its meaning."
The DARPA attempt to 'exponentiate mathematics' may merely be an attempt to put us dangerous liberal lefty mathematicians out of jobs.
(Admission, I have no idea what my 'IQ' may be.)
"It's an indication of DARPA's concern about how tough this may be that it's a three-year program." Back in the day, I was involved with several DARPA (and later IARPA) programs. I remember three years as being a sort of default length for programs. Some got cut sooner, if it didn't look like they were going anywhere, and some were given a year or so more--either because they were perceived as being nearly there, but needing a bit more time, or because the end goal had been raised.
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