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Comment by cubefox | Hacker News Reader

This illustrates how unimportant this problem is. A prior solution did exist, but apparently nobody knew because people didn't really care about it. If progress can be had by simply searching for old solutions in the literature, then that's good evidence the supposed progress is imaginary. And this is not the first time this has happened with an Erdős problem.

A lot of pure mathematics seems to consist in solving neat logic puzzles without any intrinsic importance. Recreational puzzles for very intelligent people. Or LLMs.

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It shows that a 'llm' can now work on issues like this today and tomorrow it can do even more.

Don't be so ignorant. A few years ago NO ONE could have come up with something so generic as an LLM which will help you to solve this kind of problems and also create text adventures and java code.

The goal posts are strapped to skateboards these days, and the WD40 is applied to the wheels generously.

Regular WD40 should not be used as bearing lubricant!

Exactly!

I don't get your pessimism...

Nothing of it was even imaginable and yes the progress is crazy fast.

How can you be so dismissive?

You misread my comment.

You mean like a small rocket build? Okay :)

You can just wait and verify instead of the publishing, redacting cycles of the last year. It's embarrassing.

It's hard to predict which maths result from 100 years ago surfaces in say quantum mechanics or cryptography.

The likelihood for that is vanishingly low, though, for any given math result.

> "intrinsic importance"

"Intrinsic" in contexts like this is a word for people who are projecting what they consider important onto the world. You can't define it in any meaningful way that's not entirely subjective.

Mathematical theorems at least have objectively lower information content, because they merely rule out the impossible, while scientific knowledge also rules out the possible but non-actual.

You have it backwards. Mathematical theorems have objectively higher information content, because they rule out the impossible and model possibilities in all possible worlds that satisfy their preconditions. Scientific knowledge can never do more than inductive projections from observations in the single world we have physical access to.

The only thing that saves science from being nothing more than “huh, will you look at that,” is when it can make use of a mathematical model to provide insight into relationships between phenomena.

There is still enormous value in cleaning up the long tail of somewhat important stuff. One of the great benefits of Claude Code to me is that smaller issues no longer rot in backlogs, but can be at least attempted immediately.

The difference is that Claude Code actually solves practical problems, but pure (as opposed to applied) mathematics doesn't. Moreover, a lot of pure mathematics seems to be not just useless, but also without intrinsic epistemic value, unlike science. See https://news.ycombinator.com/item?id=46510353

I’m an engineer, not a mathematician, so I definitely appreciate applied math more than I do abstract math. That said, that’s my personal preference and one of the reasons that I became an engineer and not a mathematician. Working on nothing but theory would bore me to tears. But I appreciate that other people really love that and can approach pure math and see the beauty. And thank God that those people exist because they sometimes find amazing things that we engineers can use during the next turn of the technological crank. Instead of seeing pure math as useless, perhaps shift to seeing it as something wonderful for which we have not YET found a practical use.

Even if pure math is useless, that’s still okay. We do plenty of things that are useless. Not everything has to have a use.

I’m not sure I agree. Pure math is not useless because a lot of math is very useful. But we don’t know ahead of time what is going to be useless vs. useful. We need to do all of it and then sort it out later.

If we knew that it was all going to be useless, however, then it’s a hobby for someone, not something we should be paying people to do. Sure, if you enjoy doing something useless, knock yourself out… but on your own dime.

Applications for pure mathematics can't necessarily be known until the underlying mathematics is solved.

Just because we can't imagine applications today doesn't mean there won't be applications in the future which depend on discoveries that are made today.

Well, read the linked comment. The possible future applications of useless science can't be known either. I still argue that it has intrinsic value apart from that, unlike pure mathematics.

There are many cases where pure mathematics became useful later.

https://www.reddit.com/r/math/comments/dfw3by/is_there_any_e...

So what? There are probably also many cases where seemingly useless science became useful later.

Exactly, you're almost getting it. Hence the value of "pure" research in both science and math.

You are not yet getting it I'm afraid. The point of the linked post was that, even assuming an equal degree of expected uselessness, scientific explanations have intrinsic epistemic value, while proving pure math theorems hasn't.

I think you lost track of what I was replying to. Thorrez noted that "There are many cases where pure mathematics became useful later." You replied by saying "So what? There are probably also many cases where seemingly useless science became useful later." You seemed to be treating the latter as if it negated the former which doesn't follow. The utility of pure math research isn't negated by noting there's also value in pure science research, any more than "hot dogs are tasty" is negated by replying "so what? hamburgers are also tasty". That's the point you made, and that's what I was responding to, and I'm not confused on this point despite your insistence to the contrary.

Instead of addressing any of that you're insisting I'm misunderstanding and pointing me back to a linked comment of yours drawing a distinction between epistemic value of science research vs math research. Epistemic value counts for many things, but one thing it can't do is negate the significance of pure math turning into applied research on account of pure science doing the same.

"You replied by saying "So what? There are probably also many cases where seemingly useless science became useful later." You seemed to be treating the latter as if it negated the former"

No, "so what" doesn't indicate disagreement, just that something isn't relevant.

Anyway, assume hot dogs taste not good at all, except in rare circumstances. It would then be wrong to say "hot dogs taste good", but it would be right to say "hot dogs don't taste good". Now substitute pure math for hot dogs. Pure math can be generally useless even if it isn't always useless. Men are taller than women. That's the difference between applied and pure math. The difference between math and science is something else: Even useless science has value, while most useless math (which consists of pure math) doesn't. (I would say the axiomatization of new theories, like probability theory, can also have inherent value, independent of any uselessness, insofar as it is conceptual progress, but that's different from proving pure math conjectures.)

It really speaks to the weakness of your original claim that you're applying this level of sophistry to your backpedaling.

There are 1135 Erdős problems. The solution to how many of them do you expect to be practically useless? 99%? More? 100%? Calling something useful merely because it might be in rare exceptions is the real sophistry.

So when you said "so what, hamburgers (science) taste good (is useful)", you were implicitly making a point about how bad (mostly not useful) the hot dogs (math research) was? And that's the thing that supposedly wasn't being followed on the first pass?

That brings us full circle, because you're now saying you were using one to negate the other, yet you were claiming that interpretation was a "failure to follow" what you were saying the first time around.

It's hard to know beforehand. Like with most foundational research.

My favorite example is number theory. Before cyptography came along it was pure math, an esoteric branch for just number nerds. defund Turns out, super applicable later on.

You’re confusing immediately useful with eventually useful. Pure maths has found very practical applications over the millennia - unless you don’t consider it pure anymore, at which point you’re just moving goalposts.

No, I'm not confusing that. Read the linked comment if you're interested.

You are confusing that. The biggest advancements in science are the result of the application of leading-edge pure math concepts to physical problems. Netwonian physics, relativistic physics, quantum field theory, Boolean computing, Turing notions of devices for computability, elliptic-curve cryptography, and electromagnetic theory all derived from the practical application of what was originally abstract math play.

Among others.

Of course you never know which math concept will turn out to be physically useful, but clearly enough do that it's worth buying conceptual lottery tickets with the rest.

Just to throw in another one, string theory was practically nothing but a basic research/pure research program unearthing new mathematical objects which drove physics research and vice versa. And unfortunately for the haters, string theory has borne real fruit with holography, producing tools for important predictions in plasma physics and black hole physics among other things. I feel like culture hasn't caught up to the fact that holography is now the gold rush frontier that has everyone excited that it might be our next big conceptual revolution in physics.

There is a difference between inventing/axiomatizing new mathematical theories and proving conjectures. Take the Riemann hypothesis (the big daddy among the pure math conjectures), and assume we (or an LLM) prove it tomorrow. How high do you estimate the expected practical usefulness of that proof?

That's an odd choice, because prime numbers routinely show up in important applications in cryptography. To actually solve RH would likely involve developing new mathematical tools which would then be brought to bear on deployment of more sophisticated cryptography. And solving it would be valuable in its own right, a kind of mathematical equivalent to discovering a fundamental law in physics which permanently changes what is known to be true about the structure of numbers.

Ironically this example turns out to be a great object lesson in not underestimating the utility of research based on an eyeball test. But it shouldn't even have to have any intuitively plausible payoff whatsoever in order to justify it. The whole point is that even if a given research paradigm completely failed the eyeball test, our attitude should still be that it very well could have practical utility, and there are so many historical examples to this effect (the other commenter already gave several examples, and the right thing to do would have been acknowledge them), and besides I would argue they still have the same intrinsic value that any and all knowledge has.

> To actually solve RH would likely involve developing new mathematical tools which would then be brought to bear on deployment of more sophisticated cryptography.

I doubt that this is true.

It already has! The progress that's been made thus far, involved the development of new ways to probabilistically estimate density of primes, which in turn have already been used in cryptography for secure key based on deeper understanding of how to quickly and efficiently find large prime numbers.

It's unclear to me what point you are making.

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