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In math, rigor is vital, but are digitized proofs taking it too far?

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Re: In math, rigor is vital, but are digitized proofs taking it too far?

#71
post #35

The problem with this ambition is that it turns mathematics into software development. There’s absolutely nothing wrong with this per se, however what happens is that, as in software, certain ideas get ossified. That’s why, for example, every OS has a POSIX layer even though technically the process/namespace/security model could be radically reimagined possibly to create more easily engineered, correct software. Math…

> what happens is that, as in software, certain ideas get ossified. That’s why, for example, every OS has a POSIX layer even though technically the process/namespace/security model could be radically reimagined possibly to create more easily engineered, correct software. Total amateur here, but it strikes me that one important difference is that performance matters in software in a way that it doesn’t in mathematics—…

Some proofs have become extremely long, and the raw size has created worries about correctness. It's easy to make a mistake in hundreds of pages.

Ultimately, a proof is an argument that something is true. The simpler "more elegant" proof is generally going to be more convincing.

Re: In math, rigor is vital, but are digitized proofs taking it too far?

#72

Great quote from Hilbert, I think it’s also a useful thought for software development. “The edifice of science is not raised like a dwelling, in which the foundations are first firmly laid and only then one proceeds to construct and to enlarge the rooms,” the great mathematician David Hilbert wrote in 1905 (opens a new tab). Rather, scientists should first find “comfortable spaces to wander around and only subsequent…

Yeah, I see a lot of people ( especially on HN) bemoaning any science that isn't a controlled double blind experiment with a large sample size. But exploratory science is just as important as the science that proves things. Otherwise we wouldn't know which hypotheses are useful/interesting to test.

Are they bemoaning that science is being done, or are they bemoaning that the experimental results have not yet reached high enough confidence to justify the conclusions being suggested?

Re: In math, rigor is vital, but are digitized proofs taking it too far?

#73

Earlier quoted context omitted.

Exactly this. LLMs really aren't built for discovering new mathematics, especially _interesting_ new mathematics. They're built to try the most obvious patterns. When that works, it's pretty much by definition not interesting. What LLMs are good at is organizing concepts, filling in detail, and remembering to check corner cases. So their use should help mathematicians to get a better handle on what's terra firma and…

LLMs and interactive theorem provers are vastly different. There are AI models that come up with workable formal proofs for ITPs but these aren't your usual frontier models, they're specifically trained for this task.

ITPs are far older than LLMs in general, sure, but that's a pedantic distraction. What everyone is talking about here (both the comments, and the article) are ITPs enriched with LLMs to make the "smart" proof assistants. The LLMs used in ITPs are not vastly different from the usual chatbots and coding assistants. Just a different reinforcement learning problem, no fundamental change in their architecture.

Re: In math, rigor is vital, but are digitized proofs taking it too far?

#74
post #22

Imagine a future where proofs are discovered autonomously and proved rigorously by machines, and the work of the human mathematician becomes to articulate the most compelling motivations, the clearest explanations, and the most useful maps between intuitions, theorems, and applications. Mathematicians as illuminators and bards of their craft.

Proofs of what?

Proofs tend to get generated upstream of people trying to investigate something concrete about our models.

A computer might be able to autonomously prove that some function might have some property, and this prove is entirely useless when nobody cares about that function!

Imagine if you had an autonomous SaaS generator. You end up with “flipping these pixels from red to blue as a servis” , “adding 14 to numbers as a service”, “writing the word ‘dog’ into a database as a service”.

That is what autonomous proof discovery might end up being. A bunch of things that might be true but not many people around to care.

I do think there’s a loooot of value in the more restricted “testing the truthfulness of an idea with automation as a step 1”, and this is something that is happening a lot already by my understanding.

Re: In math, rigor is vital, but are digitized proofs taking it too far?

#75
post #22

Imagine a future where proofs are discovered autonomously and proved rigorously by machines, and the work of the human mathematician becomes to articulate the most compelling motivations, the clearest explanations, and the most useful maps between intuitions, theorems, and applications. Mathematicians as illuminators and bards of their craft.

> Imagine a future where proofs are discovered autonomously and proved rigorously by machines, and the work of the human mathematician becomes to articulate the most compelling motivations

You've got the wrong idea of what mathematicians do now. There's not a proof shortage! We've had autonomously discovered proofs since at least Automated Mathematician, and we can have more whenever we want them - a basic result in logic is that you can enumerate valid proofs mechanically.

But we don't want them, because most proofs have no value. The work of a mathematician today is to determine what proofs would be interesting to have ("compelling motivations"), and try to prove them.

Re: In math, rigor is vital, but are digitized proofs taking it too far?

#76
post #35

The problem with this ambition is that it turns mathematics into software development. There’s absolutely nothing wrong with this per se, however what happens is that, as in software, certain ideas get ossified. That’s why, for example, every OS has a POSIX layer even though technically the process/namespace/security model could be radically reimagined possibly to create more easily engineered, correct software. Math…

> certain ideas get ossified.

That's fine in math. Math is true or it is not. People who overturn popular conjectures in math get fame, not approbation.

Being able to prove things in something like Lean means that stuff like Mochizuki's work on the abc conjecture could be verified or disproven in spite of its impenetrability. Or, at the very least, it could be tackled piecemeal by legions of students tackling a couple of pages every semester.

Re: In math, rigor is vital, but are digitized proofs taking it too far?

#77
post #58

Earlier quoted context omitted.

> mathematics is a social construct If you believe Wittgenstein then all of math is more and more complicated stories amounting to 1=1. Like a ribbon that we figure out how to tie in ever more beautiful knots. These stories are extremely valuable and useful, because we find equivalents of these knots in nature—but boiled down that is what we do when we do math

In my view mathematics builds tools that help solve problems in science.

This is known as “applied mathematics”.

Re: In math, rigor is vital, but are digitized proofs taking it too far?

#78
post #35

The problem with this ambition is that it turns mathematics into software development. There’s absolutely nothing wrong with this per se, however what happens is that, as in software, certain ideas get ossified. That’s why, for example, every OS has a POSIX layer even though technically the process/namespace/security model could be radically reimagined possibly to create more easily engineered, correct software. Math…

> That’s why, for example, every OS has a POSIX layer even though technically the process/namespace/security model could be radically reimagined possibly to create more easily engineered, correct software.

But that is because everyone has to switch to the new system. There are no shortage of experimental OSs that do things in different ways. They fail because of switching costs not because making them is hard.

A machine checked proof is valid if it happens once. You dont need the whole world to switch.

Re: In math, rigor is vital, but are digitized proofs taking it too far?

#79
post #18

Earlier quoted context omitted.

There are still many major oversimplifications in the core of math, making it weirdly corresponding with the real world. For example, if you want to model human reasoning you need to step away from binary logic that uses "weird" material implication that is a neat shortcut for math to allow its formalization but doesn't map well to reasoning. Then you might find out that e.g. medicine uses counterfactuals instead of…

People keep getting hung up on material implication but it can not understand why. It's more than an encoding hack--falsity (i.e. the atomic logical statement equivalent to 0=1) indicates that a particular case is unreachable and falsity elimination (aka "from falsity follows everything") expresses that you have reached such a case as part of the case distinctions happening in every proof. Or more poetically, "if my…

Material implication was not the default implication historically; it came as a useful hack by people who hoped that by enforcing it they could formalize the whole math and knowledge and have a sort of a "single source of truth" for any statement, and evaluate all statements purely syntactically. This proved to be futile as incompleteness theorem showed, and which material implication directly enabled by allowing self-referential non-sense as valid statements. There were many attempts to reconcile this with different logics but they all ended up weaker and unable to formalize all statements. We are now entering the next phase of this attempt, by using hugely complex reasoning function approximators as our "single source of truth" in the form of AI/LLMs.

I used to do a lot of proofs coming all the way from Peano arithmetics, successor operators and first-order tableaux method.

Re: In math, rigor is vital, but are digitized proofs taking it too far?

#80
It's easy to forget, as we all use digital tools in our day-to-day lives, that the world is fundamentally analog, and there's no way to escape that. Everyone trying to tell you otherwise is just selling snake oil, with one notable exception, which is mathematical rigor in proofs. It's understood now that a rigorous proof in math is exactly one that, in principle, can be digitized and checked automatically. Those are simply the same concept, so introducing a computer there is really a perfect fit of tool and purpose. If we can't use computers to automate the checking of mathematical proofs, then why have computers at all? It's the only serious thing people do that a computer can be literally perfect at!

To be clear, there's much more to math than writing down and checking proofs. Some of the most important contributions to math have been simply figuring out the right questions to ask, and also figuring out the useful abstractions. Those are both firmly on the "analog" side of math, and they are every bit as important as writing the proofs. But to say that we have this huge body of rigorous argumentation in math, and then to finally do the work of checking it formally is "taking it too far," is a really bewildering take to me.

No, I don't think formalizing proofs in Lean or other proof systems should dominate the practice of math, and no, I don't think every mathematician should have to write formal proofs. Is that really where we're heading, though? I highly doubt it. The article worries about monoculture. It's a legitimate concern, but probably less of one in math than in many other places, since in my experience math people are pretty independent thinkers, and I don't see that changing any time soon.

Anyway, the conclusion from all this is that the improved ability for mathematicians to rely on automated tools to verify mathematical reasoning would be a great asset. In my opinion the outcomes of that eventuality would be overwhelmingly good.

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