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Comment by dannykwells | Hacker News Reader

I think articles like this are hilarious because they act like written down proofs are a gold standard of unassailable logic. Of course they aren't - what constitutes proof to one person (or in one age) may or may not constitute proof to others. Look at Mochizuki's work - someone so convinced they are right, and yet no one else can understand it:

https://www.nature.com/news/the-biggest-mystery-in-mathemati...

Another good example: Euler's proofs in his day would be failed in even a basic real analysis class now.

Having known many many pure mathematicians, I believe that they don't want to use computers for the simple reason that many of them don't know how to, and thus denying the legitimacy of computer-generated (or assisted) proofs protects their livelihood/careers.

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Most mathematicians would love for a machine to check their proofs. The problem, as Buzzard says, is that is much too expensive. Another way to put is, at the moment the math you can do with machine-checkable proofs is a bit boring, compared to math on the cutting edge outside formal methods.

>denying the legitimacy of computer-generated proofs protects their livelihood/careers.

While this might be an issue, the bigger problem is that there are no interesting new theorems that were obtained in this way yet.

What is true is that there is still serious prejudice against theorems verified by reducing to a finite (yet large) number of concrete cases, which are then verified by an algorithm. This is the case even after the whole brouhaha over and resolution of the 4 colors theorem by Hales.

Good catch on the generated vs. assisted - I edited my comment.

Hales didn't resolve the 4 colors theorem, Gonthier did that. Hales proved the Kepler Conjecture and afterwards facilitated its formal verification via the Flyspeck project.

> Hales didn't resolve the 4 colors theorem, Gonthier did that.

Gonthier didn't resolve the 4 colors theorem, Appel and Haken did that. Gonthier just created the first computer-checked proof for the Four-color theorem.

Yes, thank you. Mixed up machine assisted proofs.

>> denying the legitimacy of computer-generated proofs protects their livelihood/careers.

> While this might be an issue, the bigger problem is that there are no interesting new theorems that were obtained in this way yet.

I think the issue is that there is a need for computer verified proofs, not necessarily computer generated proofs. It's much easier to create a proof verifier than a proof generator, and verifiers can give us much greater confidence that the proofs are correct (regardless of whether a human or computer generated them).

Why do you insist mathematicians become programmers? They have different skills, have different ways of thinking, and solve different problems. As an analogy, plenty of famous musicians were obviously very musically talented but didn’t know how to read or write musical notation. Proofs are just a medium to convey mathematical ideas to other mathematicians. I bet if Erdos were required to publish his ideas in a theorem prover language his mathematical output would have at least halved.

> Why do you insist mathematicians become programmers?

No one is insisting that. We're insisting that proofs actually be correct, something that demonstrably cannot be done (to an adequate level) when we depend only on humans. Humans get tired and make mistakes such as failing to notice missing or invalid steps. Humans are still really good at coming up with proofs - but not so good at verifying them.

> Proofs are just a medium to convey mathematical ideas to other mathematicians.

I disagree. For example, claiming that a^n + b^n = c^n for any integer value of n greater than 2 when all values are integer is clearly a mathematical idea. But that idea is not a proof of Fermat's Last Theorem; it is simply an idea. A proof should establish, beyond doubt, that a claim is actually true.

> I bet if Erdos were required to publish his ideas in a theorem prover language his mathematical output would have at least halved.

I'm not sure about that, Erdos was pretty smart. There's relatively little investment in theorem prover languages; a little investment would go a long way. For example, today you have to prove lots of little things that really should be in libraries and tools, but the lack of investment holds things back.

This also assumes that "mathematical output" is acceptable if it is not computer-verified. At some point in the future I think a "proof" that is not computer-verified will be viewed as an unproven claim. Humans simply aren't very good at verifying proofs - and computers are extremely good at it.

Well, I guess what I meant is that proofs are for convincing other mathematicians. If someone is unconvinced by the proof they’ll try to fill in the gaps or further specify certain parts of the proof themselves. If that doesn’t work, they might ask for clarification. Eventually a consensus on the validity of the proof will be reached. I agree if the proof were presented in a machine verifiable format to begin with we wouldn’t need this process. I think though, that the reason it isn’t done this way is partly because the effort required isn’t worth it yet. My professor would actually get annoyed if I am too specific in my proofs because it overall makes the proof more difficult for a human to understand, which as I said is ultimately the purpose of a proof.

Additionally the proof itself sometimes conveys an interesting idea that isn’t captured in the theorem. Some theorems have many different proofs that use different methods to prove the same theorem. These different proofs provide different perspectives that lead to a better understanding of the topic.

I have a background in both computer science and pure math and am very interested in highly formal methods, I just don’t think we should disparage mathemicians for doing more human oriented math.

> Some theorems have many different proofs that use different methods to prove the same theorem. These different proofs provide different perspectives that lead to a better understanding of the topic.

Agreed, but there's no conflict. You can create multiple formal proofs, each of which provide different perspective that can lead to a better understanding.

> Well, I guess what I meant is that proofs are for convincing other mathematicians.

Not everyone agrees with this claim, and this disagreement is the essence of the "crisis". Is the purpose of a proof (1) to show that something is actually true, or (2) to convince another mathematician that it's true (whether or not it's actually true)? Historically the two were conflated, but that is no longer the case. Now we must ask if #1 or #2 is the real goal of a proof, because we now can choose between those alternatives.

In short, I disagree with you. I believe that the purpose of a proof is to show that something is actually true (#1). Whether or not you can convince a human - any human - is historically how we determined #1, but increasingly this alternative (#2) is no longer an adequate measure of anything.

Mathematicians are humans. They can be fooled, fail to see errors due to fatigue, or simply be overwhelmed because they have limited time. When humans were the only way to verify proofs, we had to live with those limitations - because there were no alternatives.

But computers today can verify proofs. They can do it far more accurately than humans, they don't get tired, and they can do it 24x7. Computers aren't as good (today) at creating proofs, but they don't need to create proofs to verify proofs. We can take a number of actions that make computer-verified proofs far far more reliable than proofs merely checked by humans. As I stated earlier, it's odd that we make humans do the verification job, because a computer is much better at doing it.

The question now is: should we continue to accept proofs that are only human-verified? Why? Why is it acceptable if "other mathematicians" can be convinced, if the proofs are invalid? Do we really believe mathematicians can't make mistakes?

It's certainly useful to show "summary proofs" so that humans can see the big picture. But once you have the details, they can be summarized. If only "summaries of proofs" are available that aren't computer-verified, that means that errors may lurk that would have been countered by a formal proof. Large proofs are easily shared via the Internet, so size is no longer a serious concern.

I don't disparage mathematicians for doing more "human oriented math". Indeed, we should honor the work of past math giants, who had no alternatives available. But we need to move beyond "human oriented math" because it is not trustworthy enough and we have a better alternative available.

I'm not saying we can instantly do that today, by the way. There are tools that make it possible, but it's hard. It will take persistent work over many years to transition to a world where proofs are routinely computer-verified. There will need to be decisions on what to prioritize on formalization. Different approaches will need to tried, and I doubt there will be a single approach everyone uses. People with different skills will need to work together. Libraries and tools will need creation and refinement. But the first step in that journey is to have a general agreement that it's a journey worth taking.

Do mathematicians and those who fund them care about truth, or do they care about merely convincing other mathematicians? We must now choose between those two options. I choose truth.

> Look at Mochizuki's work - someone so convinced they are right, and yet no one else can understand it:

I don't think this is a good characterization of Mochizuki or his Inter-universal Teichmuller theory. The vast majority of new results in both pure and applied mathematics are accepted by the mathematics community without incident - this applies to extremely sophisticated and groundbreaking work as well. For example, if you look at the work of almost any Fields medalist in the last century, you'll see that it was contemporaneously well-received.

What you're citing is one example of 1) an extremely complicated proof, 2) delivered to the mathematics community in a very unorthodox manner, 3) by an author who has demonstrated significant reluctance to refine their work to make it more comprehensible to others. This is very uncommon and doesn't actually support your point (not that your point doesn't have merit, but this just isn't really applicable).

I feel your framing of Mochizuki's IUT could be interpreted as saying that he somehow developed a robust proof which is, for a variety of reasons, not accepted by the community at large. This is not the case - his proof is not robust! There is significant evidence that it doesn't actually work. Furthermore, if your attempt at a proof does not agree with the general consensus of the mathematics community, it makes more sense to conclude that your proof is likely in error than it does to use it as an example of how proofs are subjective, and different people could require different levels of rigor. That's almost never the case in real world peer review - you're more likely to have your math paper dinged for a subjective assessment of how interesting or useful it is than because the reviewer actually finds it to in error.

I'm also going to have to strongly disagree with you in regards to Euler's proofs. Analysis as a discipline is far more mature than it was in Euler's time, but Euler still clearly proved the things we attribute to him. I'm not really sure what your expectation is here - we have more sophisticated means of articulating proofs, and likewise much more mathematical scaffolding to support new theories. But Euler's work is rigorous.

> Having known many many pure mathematicians, I believe that they don't want to use computers for the simple reason that many of them don't know how to, and thus denying the legitimacy of computer-generated (or assisted) proofs protects their livelihood/careers.

Trading anecdote for anecdote: I've never known a single mathematician who felt that computer-generated proofs posed a threat to their careers. I don't mean to call your experience into question, but frankly I'm a little shocked that this is your interpretation of their reluctance to use computers. In my experience mathematicians are loathe to use computers not because they feel threatened by them, but because they just don't need to and never learned to. That's a very different attitude!

As for my own opinion: the legitimacy of computer-generated proofs and the work of professional mathematicians are independent of each other. It would be naive to think that computer-assisted proofs would actually threaten a mathematician's livelihood. From the discussions I've had with mathematicians both in person and online (such as on /r/math), I'm fairly sure most of them would consider this idea incoherent and not actually be concerned about it. In fact, most mathematicians I know (myself included) would be thrilled to have more computer-assisted proofs available.

> I'm also going to have to strongly disagree with you in regards to Euler's proofs. Analysis as a discipline is far more mature than it was in Euler's time, but Euler still clearly proved the things we attribute to him. I'm not really sure what your expectation is here - we have more sophisticated means of articulating proofs, and likewise much more mathematical scaffolding to support new theories. But Euler's work is rigorous.

What about the Euler product formula and using it to prove the infinitude of primes? At best it has gaps that can be easily filled, but it wouldn't meet modern standards of rigor. See http://www.math.harvard.edu/~elkies/M259.06/euler.pdf (the best source I can find on my phone)

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