OpenAI says an internal version of its new model has solved ten previously open mathematical problems, by having several agents collaborate over a long period. In July, a fifty-year-old conjecture was tackled with sixty-four sub-agents. These demonstrations exist. They are long, technical, and the number of humans capable of fully verifying them can sometimes be counted on one hand. Is a proof that almost no one can read a proof?
We have looked at these systems' capacity to have ideas. Here is the next question, and it is older than one might think.
What a proof is, and for whom
One readily imagines that a proof is an object: a sequence of logical steps, valid or not, regardless of who reads it. That is true in principle, and it is not how real mathematics works.
A proof is accepted when a community has read it, understood it, and found no flaw in it. It is not just a logical object, it is a social act: someone convinces other people, who stake their credibility by validating it.
This point was theoretical as long as proofs remained readable. It no longer is when they exceed what a human can absorb.
The precedent, and what it teaches
The problem is not new. In 1976, the four-colour theorem was proved with the help of a computer that checked thousands of cases. Part of the community long refused to regard this as a proof, precisely because no human could go through it.
The solution that has since prevailed is instructive: rather than reading the proof, one checks the program that checks it. Proof assistants — software whose logic is simple enough to be audited itself — mechanically verify each step. Trust shifts from the demonstration to the verification tool.
This is the same shift we described in our article on trust: you do not understand the object, you understand the device that checked it.
Proof assistants existed before language models, and their combination is promising: a model proposes, a formal verifier validates. In this setup, you do not need to trust the model, only the verifier. The problem is that this is not always how results are announced. A proof produced in natural language by a model, not translated into verifiable form, rests on the same thing as a human proof that is too long: the word of whoever produced it. And we have documented how poorly calibrated that word is here.
The problem of abundance
There is a difficulty that the 1976 precedent does not raise: volume.
An exceptional proof mobilises the attention of an entire community. A hundred proofs a month will not. If systems produce results faster than humans can examine them, a growing share of what is announced will never be verified — and will not be contradicted either.
This creates an unprecedented grey zone: results that are neither established nor refuted, that pile up, and on which other results come to be built. This is exactly the mechanism we described regarding modernised scientific code: what looks right is taken up before it is verified.
What seems to hold
Two requirements, and they are more modest than full human verification.
Verifiable form. A result produced by a model should be expressed in a way that a formal verifier can check. Without that, it is not a proof, it is a claim.
Public distinction. It should be possible to distinguish, in what is announced, what has been formally verified from what has not. Today, announcements mix the two.
What to take away
The question in the title has an answer, and it is an old one: a proof that no one can verify is not yet a proof, but it can become one if you can verify what verifies it.
What the models add is not a new philosophical problem, it is a problem of pace. Mathematics took thirty years to accept a computer-assisted proof, by building the tools that made it acceptable. They will not have thirty years this time.
One thing remains unchanged, and it holds beyond mathematics. A result is established only when someone else has been able to challenge it and has failed. As long as challenge remains possible, knowledge advances. The day we accept results because they are too numerous to be challenged, we will have stopped knowing and started believing.