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Formalize it with Isabelle / Coq / Lean. Then it counts.
by pacala 9y ago
Formalize it with Isabelle / Coq / Lean. Then it counts.
- tobhahn 9y agoMath is more about figuring out why things are true than what is true. Mathematicians want to understand Mochizuki. They couldn't care less if hn thinks it counts.
- SomeStupidPoint 9y agoI don't know why this got flagged. At the point that we no longer understand the relationship between the formal proof witnesses (and really, the class of possible witnesses) and the axioms we choose, we can no longer do mathematics, because we can no longer meaningfully explore axioms -- our ability to make guided changes is destroyed by our inability to understand their effect. It's important for the community to understand something of why a thing, not just that it's true, because that's why drives the development of mathematics forward. (And indeed, particularly so in the ABC conjecture, which sits at a node between the nature of multiplication and addition, which don't usually have much to do with each other.) I actually wonder if US (and perhaps other) math education is harmful here: the focus on rote learning and just knowing that a thing is true (to mechanistically apply it) has conditioned people to not understand why the hesitance over proofs that humans don't understand -- for most of those people, they never understood the proofs anyway.
- pacala 9y agoThat is an excellent point. The future is mechanical proofs. Which will bring tools for proof search, proof refactoring, proof minimization, proof navigation, theorem generation. We'll be able to tackle _much_ harder problems, while still being able to get the gist of it. Sometimes I'm saddened that this future may come slower than we'd like due to imperfect funding structures. But I've grown a lot of patience over the years :)
- SomeStupidPoint 9y agoI'm actually very pro-formalization and mechanical verification -- both for mathematics and computer science. $HOBBYPROJECT involves automated theorem proving, while I'm trying to convince $DAYJOB to adopt some formal methods. I was just pointing out that the person got flagged for commenting that "witness and dump" isn't actually very useful for mathematics as a field, except as a signal that we should investigate a topic further. But in the case of the ABC conjecture, we already have plenty of incentive to investigate. I think mathematics and science have a lot of learn from computer science in terms of managing large models, proofs, etc -- and that we'll get a lot of automatic tools. That will all be really great. But there are proofs that are basically just brute-forcing a solution for which we have no higher-level understanding, and those don't really add much by way of knowledge to mathematics. At the point that those are all we can generate for "big" problems, we may be in trouble.
- pacala 9y agoAnother excellent point. Right now it's a "winner takes all" competition. It matters to prove a result, and much less to provide an "elegant" proof. I can only hope for a future where we measure the algorithmic entropy of a proof [log proof length][0], and results like "ABC theorem proof using half the bits as best known proof" become notable. [0] https://en.wikipedia.org/wiki/Kolmogorov_complexity https://en.wikipedia.org/wiki/Kolmogorov_complexity
- haskellandchill 9y agoI'm interested in your $HOBBYPROJECT. I'm hoping to develop some software similar to edukera.com, ie using proof assistants as educational tools for mathematics. Would love to swap some cool links and references. Thanks!
- SomeStupidPoint 9y agoI'm actually in the process of (poorly) implementing my own proofs engine to try and gain a deeper appreciation for the notions behind type theory (and how eg Coq operates): That a function is the same thing as a proof, with the theorem it proves being the type it constructs and its assumptions being the types it takes in. (Functions are proofs (which in this case, are things that take witnesses of their assumptions and produce witnesses of their theorem), values are witnesses that something is true, types are theorems, etc.) I have an interest in topology, and want to understand HoTT, but my intuition around type theory wasn't up to par -- so I'm trying to tackle it in a constrained setting (ie, not proving theorems about mathematics as such, but a narrower problem space). Figured there was nothing to do besides get into the messy bits of it. Can you tell me a bit more about what you're trying to do?
- skybrian 9y agoI don't see these approaches as necessarily opposed. It seems like in principle, formalizing a proof and verifying that it doesn't have any errors could be a preliminary step towards understanding it? (Though in practice, perhaps this is overly roundabout.)
- chuckdries 9y agoPeople have a lot of very valid complaints about common core, but at its core (hehe) it does seem to attempt to start to remedy your complaint about US math education at least at the high school level. Hopefully that can propagate upward.
- SolarNet 9y agoAlso, once enough people figure it out, one (or some) of them can make better explanations.
- sanxiyn 9y agoIndeed. We now actually have some math results of interest for which we have only formal proofs and no human proofs. https://arxiv.org/abs/1509.05468 https://arxiv.org/abs/1509.05468 is an example.
- mikebenfield 9y agoWe are very far from the point where that is feasible for many new results in math.