The mathematician and the oracle

On August 24, Levent Alpöge (@__alpoge__) posted on X.com that they had solved a maths problem with Claude’s help in a way that was contrary to a human proof of it, albeit a reputedly unsatisfactory one.

Let us set aside the validity of the new proof. In one reply to the post, @CircumjovialLLC asked @__alpoge__ to stop using Claude models that were not available to everyone to solve advanced maths problems.

Another account, @techlarper_, then replied implying that this argument does not make sense because, since the human brain is like a ‘closed model’ to which each person has access, @CircumjovialLLC’s argument is akin to asking smarter people to not solve tough maths problems, or to at least refrain from doing so unless they could explain their thinking.

Maybe @techlarper_ is a bot, maybe @CircumjovialLLC is too. I do not care. I think @techlarper_’s argument is mistaken and that @CircumjovialLLC is right to ask for what they did.

Say Selvi is an exceptionally good mathematician who can solve a problem that Shankar cannot. So Shankar says to Selvi, “Do not use your powerful mathematical abilities to solve this problem unless you can make the solution accessible to everyone.”

Sure, Selvi’s brain is a resource that she possesses individually. However, anyone can also inspect Selvi’s proof once she publishes it. And the fact that Selvi’s cognitive abilities cannot be reproduced by everyone is not considered to be a problem of mathematical practice. (Mathematicians agree that that is one of the ground rules of doing mathematics.)

Now, suppose Selvi has a secret oracle that instantly tells her whether a given mathematical statement is true. She uses it to announce that a conjecture is true, but now, she cannot reveal the oracle nor can she explain how it arrived at the answer. So Shankar objects: “Do not use that oracle to settle advanced mathematical questions in a forum where everyone is supposed to be able to scrutinise the work.”

If Selvi responds saying “your brain is also a closed model”, she will obviously be beside the point. Mathematicians do not treat Selvi’s brain as an oracle nor do they ever expect to be able to access it. (Another ground rule.) In effect, @CircumjovialLLC’s objection is not that human intelligence is unevenly distributed but that the epistemic process is inaccessible to or not reproducible by the relevant community of experts.

That said, what if mathematicians expect mathematicians to be restricted to solving maths problems using epistemic resources that they are all able to access?

@techlarper_ mistook unequal access to cognitive ability for unequal access to an epistemic instrument. One’s cognitive ability is an inalienable feature of human intelligence whereas one’s epistemic instruments refer to the means by which a mathematical proof has been obtained, and can be inspected.

Now, in the pre-AI epoch, so to speak, @__alpoge__ providing a complete and checkable proof would have sufficed for mathematicians; that an oracle helped @__alpoge__ would not have mattered. In this regard, @CircumjovialLLC’s objection may be misguided considering @__alpoge__ has also provided a paper detailing the proof.

Now, in an August 17 preprint on arXiv, mathematician Terence Tao wrote that the advent of AI in mathematics is stress-testing…

not our foundational framework for mathematical truth, but rather the largely implicit framework of mathematical values and practices: what we consider a contribution to be, what we reward, what we regard as understood, and who — or what — we regard as having done the work. [Emphases in the original] I argue that it will become necessary to make these unwritten goals of mathematics much more explicit; but once we have thoroughly examined and codified them, our community will emerge stronger and more resilient than before.

With this in mind, say the following statement is also a ground rule of doing mathematics, just one that has been rarely acknowledged in the open: A mathematician should not use an intellectual resource to produce a result if other mathematicians cannot access that resource yet are expected to verify and/or reproduce that result.

What would this statement say about mathematics as a whole? For starters, in this universe, mathematics would be partly procedural and partly constitutive rather than being concerned wholly with the truth of propositions.

Euclid’s algorithm to find the greatest common divisor is a good example of mathematics as procedure. Given two integers (30, 50), repeatedly divide the larger one by the smaller one (50/30), then replace the pair by the divisor and remainder (30/20) until the remainder is zero (20/10, 10/10, thus the answer is 10). Another mathematician can use this algorithm without also requiring some private insight that was available only to Euclid.

Another good example is the development, in the 17th century, of logarithms. Once natural philosophers drafted the logarithmic tables, any other philosopher or engineer could break down, say, a difficult multiplication problem, into a series of addition problems using the tables. Here, the tables were the intellectual resource, and they were public and reproducible, and the community agreed they were admissible as such a resource because other mathematicians could access and inspect them.

In effect, mathematics as procedure is a statement that, given the intellectual resources that mathematicians agree are permitted, a procedure that takes one from result X to result Y using that procedure is also permitted.

Now, in the new universe, there is a new condition: the permitted intellectual resources must themselves also be accessible to the people who are expected to verify the result.

Say Simran, a mathematician, possesses an oracle that can always correctly say whether a mathematical proposition is true or false. Simran asks it whether the Riemann hypothesis — a great unsolved problem in mathematics, explained here — is true. It says ‘yes’. So Simran claims, “The Riemann hypothesis is true.”

Now, suppose Simran’s colleagues do not have access to the oracle. In our ‘current’ universe, the proposition is true: Simran has in some sense discovered a truth.

But if we include the new rule, there arises a problem: Simran has not used any acceptable mathematical procedure at all. The oracle is excluded from the community’s permitted means of producing mathematical results since it is inaccessible to the mathematicians expected to verify or reproduce its result.

In effect, a true proposition obtained in this way will not have the status of being a mathematical result. Yet it can still be true.

Put another way, mathematics in the new universe would regard the means by which mathematicians arrive at a result to be subject to a norm of accessibility.

If you stuck with the post this far, dear reader, there are two questions for you. (i) Do you think an AI model is an extension of a mathematician’s cognition or is it a separate epistemic resource? And if the model is not accessible to mathematicians, (ii) do you think you inhabit the ‘old’ universe or the new universe?

I believe I inhabit the new universe, and in this universe @CircumjovialLLC’s comment is valid.

Say Selvi has an extraordinary mathematical intuition (like that of Srinivasa Ramanujan, say). She sees immediately that a particular proof is valid — but no other mathematician can see it. Here, if @CircumjovialLLC’s objection is that “Selvi should not use intellectual resources that her peers do not possess”, then Selvi’s intuition becomes insufficient.

But even here, there is a difference between Selvi and Claude: Selvi’s brain is not an external resource she has elected to use instead of some other resource that is also collectively available. Her superior ability is still only a difference between participants. Claude, on the other hand, represents a separate epistemic resource that has been introduced to the mathematical process.

In short, in the new universe, ‘unequal’ human abilities are still compatible with the universe’s conditions. Selvi can be cleverer than Shankar and Shankar does not need Selvi’s brain in order to be able to check Selvi’s proof.

However, if Selvi has a private machine that generates mathematical knowledge, and the community cannot access the machine, she has effectively introduced something novel, something different in kind rather than in degree: a non-shared, exclusive participant in the mathematical process. And this is why @techlarper_’s argument would not apply.

Let us plumb a step deeper. The new universe also makes room for a distinction between cognition as a private entity and method as a public one. That is, you are allowed to think whatever you like in your head: it will lie within the opaque machinery of your private cognition. But if you wish to bring that into mathematics, you have to convert that private cognition into a publicly accessible mathematical object — and mathematics already possesses the means in the form of proofs, constructions, arguments, counterexamples, etc.

Effectively, while @CircumjovialLLC was objecting to the distribution of cognitive resources within the practice of mathematics, @techlarper_ responded by focusing on the distribution of cognitive abilities among human mathematicians. Ergo @techlarper_ missed the point.

Addendum: Since I inhabit the new universe, I believe an intellectual resource cannot be permitted if it is not accessible to all mathematicians. But can some resource ever be accessible to truly all mathematicians?

Even if Anthropic avails the Claude model that @__alpoge__ used for free, using it requires a computer with sufficient hardware capabilities, an internet connection, and the technical literacy to use the model efficiently.

In other words, in this universe, the mathematical status of a result — rather than whether or not it is true — will benefit from improving public access to mathematics.