Harmonic

Interesting update: a few days ago, Nathanson presented a talk at the New York Number Theory Seminar explaining how Aristotle solved some of his problems.

Interesting update: a few days ago, Nathanson presented a talk at the New York Number Theory Seminar explaining how Aristotle solved some of his problems.

I think a radical new viewpoint is emerging on the many activities that mathematicians do. Perhaps a novel profession of mathematical engineering is emerging from the early chaos of AI for mathematics. I can see very clearly that the coordination, setup, technical pursuit, and orchestration of AI systems scaled for massive mathematical efforts and projects will require a special engineering mindset that is currently lacking, or almost completely absent, in mathematical projects. The existence of such a profession is not in opposition to mathematical tinkerers who use their artisanal craft to produce genuinely novel content. As with any kind of content, someone needs to adapt it to the grand scheme of things. This is why these roles are starting to appear complementary rather than competitive. Maybe this is a temporary activity, soon to be replaced by computers, but I think the major role of mathematical engineers will be to stay in touch with the tinkerers and provide a human cushion around their internal activities. I am enjoying this kind of activity (*), where you orchestrate with models and see how the project itself becomes a challenge in design and scale. This might well mean more jobs for mathematicians. In the long run, I suspect we may become secondary cognitive powers in parts of the mathematical information chain. But I do not think this will happen very soon across the whole system. And I hope it never happens at the most human layer: the joy people feel when a new idea is born. (*) This project is essentially a lambda-calculus lab, fully integrated with classical topics such as Church’s lambda calculus, the Aristotle formal proofs system, and extensions over particular papers. I presented it to students at the workshop in Warszawa-Falenty and was very pleased with the result. I am now using this framework for proof development. What strikes me most is that this is primarily an engineering challenge: the mathematics entering the pipeline is being handled, structured, and formalized, but not radically developed inside the pipeline itself.

To my mind, what seems most important here is not so much the results themselves, nor even the particular methods used in the proof. What really matters is the workflow: the closed loop from autonomous discovery of the right constructions, to formal verification of the proof in Lean, to informalised exposition back into a manuscript that mathematicians can read, understand, rewrite, and improve. It suggests a framework in which formal proof is not merely a static final certificate, but an active part of mathematical research: a medium through which ideas are found, organised, tested, explained and made available for human judgment. That dynamic loop is the real story for me. The natural next question is how far it can be pushed.

What will mathematics look like 7 years from now? I’m really curious to hear your brief opinion.

🚨SHOCKING: Apple just proved that AI models cannot do math. Not advanced math. Grade school math. The kind a 10-year-old solves. And the way they proved it is devastating. Apple researchers took the most popular math benchmark in AI — GSM8K, a set of grade-school math problems — and made one change. They swapped the numbers. Same problem. Same logic. Same steps. Different numbers. Every model's performance dropped. Every single one. 25 state-of-the-art models tested. But that wasn't the real experiment. The real experiment broke everything. They added one sentence to a math problem. One sentence that is completely irrelevant to the answer. It has nothing to do with the math. A human would read it and ignore it instantly. Here's the actual example from the paper: "Oliver picks 44 kiwis on Friday. Then he picks 58 kiwis on Saturday. On Sunday, he picks double the number of kiwis he did on Friday, but five of them were a bit smaller than average. How many kiwis does Oliver have?" The correct answer is 190. The size of the kiwis has nothing to do with the count. A 10-year-old would ignore "five of them were a bit smaller" because it's obviously irrelevant. It doesn't change how many kiwis there are. But o1-mini, OpenAI's reasoning model, subtracted 5. It got 185. Llama did the same thing. Subtracted 5. Got 185. They didn't reason through the problem. They saw the number 5, saw a sentence that sounded like it mattered, and blindly turned it into a subtraction. The models do not understand what subtraction means. They see a pattern that looks like subtraction and apply it. That is all. Apple tested this across all models. They call the dataset "GSM-NoOp" — as in, the added clause is a no-operation. It does nothing. It changes nothing. The results are catastrophic. Phi-3-mini dropped over 65%. More than half of its "math ability" vanished from one irrelevant sentence. GPT-4o dropped from 94.9% to 63.1%. o1-mini dropped from 94.5% to 66.0%. o1-preview, OpenAI's most advanced reasoning model at the time, dropped from 92.7% to 77.4%. Even giving the models 8 examples of the exact same question beforehand, with the correct solution shown each time, barely helped. The models still fell for the irrelevant clause. This means it's not a prompting problem. It's not a context problem. It's structural. The Apple researchers also found that models convert words into math operations without understanding what those words mean. They see the word "discount" and multiply. They see a number near the word "smaller" and subtract. Regardless of whether it makes any sense. The paper's exact words: "current LLMs are not capable of genuine logical reasoning; instead, they attempt to replicate the reasoning steps observed in their training data." And: "LLMs likely perform a form of probabilistic pattern-matching and searching to find closest seen data during training without proper understanding of concepts." They also tested what happens when you increase the number of steps in a problem. Performance didn't just decrease. The rate of decrease accelerated. Adding two extra clauses to a problem dropped Gemma2-9b from 84.4% to 41.8%. Phi-3.5-mini from 87.6% to 44.8%. The more thinking required, the more the models collapse. A real reasoner would slow down and work through it. These models don't slow down. They pattern-match. And when the pattern becomes complex enough, they crash. This paper was published at ICLR 2025, one of the most prestigious AI conferences in the world. You are using AI to help you make financial decisions. To check legal documents. To solve problems at work. To help your children with homework. And Apple just proved that the AI is not thinking about any of it. It is pattern matching. And the moment something unexpected shows up in your question, it breaks. It does not tell you it broke. It just quietly gives you the wrong answer with full confidence.