Proving theorems with computers - Kevin Buzzard скачать в хорошем качестве

Proving theorems with computers - Kevin Buzzard

Stony Brook Mathematics Colloquium Kevin Buzzard, Imperial College March 18, 2021 It is well-known that Russell and Whitehead's proof of 1 + 1 = 2 from first principles ran to hundreds of pages. But it turns out that things have moved on since then. Computer proof verification systems are computer programs which can in theory understand modern pure mathematics, and can check human proofs, at least once humans have learnt how to speak their language. Modern systems have inbuilt AI's which can fill in certain kinds of gaps automatically, enabling humans (who have learnt their language) to write proofs in a way which is sometimes close to how they would present them on a blackboard. Right now a big problem in the area is that not enough mathematicians can speak the language these computers use, and the computer scientists who can are, for the most part, not particularly interested in formalising mathematics at MSc level mathematics or beyond. This has led to an impasse, which a small but growing group of mathematicians, myself included, are now actively trying to break down. How? By doing everything from teaching mathematics undergraduates how to use Lean (one of these systems) to formalising recent work of Peter Scholze in Lean, and many other things in between. I will give an overview of the area, show examples of what mathematics at various levels looks like in these systems, explain how working in these systems has changed how I think about mathematics, and explain why I believe all this stuff to be important. And if you don't think it's important, that's fine, give me access to your undergraduates instead, because they get it. We're turning math proofs into a computer game, and some of us believe that, like chess, one day the computers will play it better than us.