So a good fraction of famous old math puzzles with no practical use are famous because they are in some way similar to problems people actually care about. If you can solve the toy problem that is in some way simpler, maybe you can use the same methods to solve the "real" problem.

One open question is whether these machine solutions to these problems will act as springboards to future research, either when given to human mathematicians, or when used to train future machine models.

One of the most famous mathematicians of all time studied number theory. He wrote an apology to humanity for all of the very smart and capable people wasting time studying number theory, since these things are clearly unless old puzzles with no practical use whatsoever. Now, 100 years later, these results underpin all of modern public key cryptography.

And then there's G. H. Hardy, famous number theorist, who wrote an "Apology" for mathematics in the sense of defending the study of number theory for its own sake because it's beautiful.

Which one were you referring to?

It is one in the same! The apology is, of course, defending the beauty of the pursuit but at the same time the writing acknowledges the (alleged) utter lack of practicality.

It's just really hard to affirm that something has "no practical use whatsoever". Maybe it doesn't have use "now", maybe it doesn't have "direct" use but can be used for another finding that is useful, Math has a long story of finding out stuff that turns useful later on.

True. Though ~99% is still useless in the end.

We are not at the end, how could you possibly be confident in this claim?

As they say: 80% of marketing is useless, you just don't know in advance which 80%. Same with math

The problem is more that the puzzles had an original context and reasons why they arose from "real" problems directly or mathematical problems trying to solve "real" problems etc. but the presentation and abstraction hides this pretty well for people not "near" the problems.

The meaning of "practical use" is all about context - who/when/where/why/what - and so it would be kind of hard to definitely claim, in an intellectually honest way, that a piece of math is an old puzzle with "no practical use whatsoever".

Unless you want to be the guy in the 19th century making fun of Boole algebra for having no practical use. You might be right, but not for long.

Isn't most of math this way? Occasionally, we find a practical use for it, but that's usually not the point.

Yes, that's correct. I've done research in pure math, but left after the postdoc level. I enjoyed it for its artistic sake but it's not something that gives anything practical. Most pure mathematicians don't ever work on practical problems either. They just teach and in return get to do their hyperspecialized research. It's done for its own sake. (Some pure math eventually ends up having applications but it's rarer and rarer and it wasn't very common to begin with. Like applications to crypto or whatever, but that was always a tiny slice to begin with.)

Most modern pure math has no practical use whatsoever. It's amazing how few people understand this. Mathematics is a service department and maybe one in a hundred thousand results have any practical use. (Speaking as someone with a PhD and over a decade in pure math with several published papers...)

Thanks to Martin Gardner's books back in the day we had genious hackers doing just that, recreative programming... until it guided our lifes: GNU/Linux, GNU itself, GNU Emacs, the BSD's, Rogue with the Curses library (and tons of Unix serious stuff), Unix itself to play SpaceWar in a PDP10...

even the ZMachine it's an astounding example of running virtual machines on 8 bit machines 20 years earlier than Java.