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> A better argument for advanced math: it's not always clear in advance what math will be useful. A popular example is Riemannian geometry, which had no clear real-world value when developed, but ended up being used for General Relativity a few decades later.

This is the classical argument, but I don't think it is plausible. It seems even less likely that, say, transfinite set theory will ever become remotely useful than string theory ever becoming useful.

The argument "it might be useful in the future" can justify research in any theory whatsoever, no matter how esoteric. It's like defending an outlandish conspiracy theory by pointing out that it is possible that it is true. That's technicality correct, but what matters here is the probability that it is true, which might well be close to zero. Similarly, while we can't rule out that transfinite set theory might have a useful application someday, this probability is so small as to not being worth discussing.

I think people should admit that they are interested in theoretical math (as opposed to applied math) for its own sake, like people who are interested in cosmology or string theory or theoretical philosophy or whatnot. Pointing to the technical possibility of future usefulness seems to be a dishonest fig leaf.

Actually, no, there are results from transfinite set theory which are foundational to so much mathematics that there would be extensive damage to the theoretical foundations of very practical math if it were unavailable.

For example, the distinction between countable and uncountable is important in analysis and measure theory. Countable subsets of the real line have Lebesgue measure zero, and this result is used in many theorems in probability and stochastic process theory with practical implications.

Now that we've seen a statement which was rated as almost certain is incorrect, it seems to reinforce my original point: it is not so clear what kind of math has applications, so it is good to develop math broadly.

I agree that it shouldn't be our only argument, but it does pack a significant punch.

Most mathematicians never use ZFC or any similar "foundational" system. That something can serve as such a foundation doesn't remotely mean that it is useful. Math is not like a house which breaks down without a foundation.

The distinction between countable and unaccountable infinity is precisely something that finitists point out as being useless. For example, analysis existed well before Cantor, and it's notion of limits and convergence ironically provides a potential notion of infinity which doesn't treat it as a mathematical object, contrary to the set theoretical notion of actual infinity, where we have a zoo of transfinite numbers.

And the fact that the countable/uncountable distinction can be integrated in some practical theories doesn't show that it has any practical implication. That would only be the case if those practical theories wouldn't exist otherwise. But analysis existed before Cantor, and other practical theories, like probability theory, could have existed before him. They are perfectly compatible with being a finitist.

If we are talking about "most" mathematicians, then you should also admit that most mathematicians are not a member of the finitist church, and that most mathematicians appreciate the distinction between countable and uncountable.

Yeah, I'm not sure I follow the logic of "transfinite set theory isn't useful because any applications it has could be replaced with finitist math", when this is something that no one wants to do.

Reminds me of that famous old HN comment about how Dropbox is pointless because an equivalent service could be set up with Linux utilities. Sure, but there are reasons people don't want to do that. Most people prefer the ease of use of mainstream foundations.

To be fair to the original commenter, I would be willing to bet some money that, say, large cardinal theory won't inspire any applications in the next 20 years. But next 100 years or 1000 years, I wouldn't. Maybe there will be some weird cross-fertilization with other more applied fields that leads to something.

"transfinite set theory isn't useful because any applications it has could be replaced with finitist math" would be a misleading way to phrase it. Like, analysis is not an "application" of transfinite set theory. It doesn't presuppose the existence of any transfinite numbers. No piece of applied math does. There is nothing which needs to be replaced with "finitist math".

To repeat my earlier comment: analysis on mainstream foundations uses infinite sets of multiple transfinite cardinalities, and there are results like "countable sets have Lebesgue measure zero", and the difference between finite, countable, and uncountable sums/unions/intersections, that are relevant to proving results that have practical significance, as well as (crucially) to avoiding erroneous calculations.

If you want to prove the same results in a finitist framework, there is nontrivial work to do, and few mathematicians are interested in doing it.

If your contention is that only calculations matter, not proofs, I would agree with you that transfinite set theory may not be relevant. You can do calculations without any rigor at all. But I think the position that proofs are of no practical value is untenable. It is historically simply not the case that engineer or physicist intuition is a sufficient guide to deriving results reliably.

I do believe we are getting sidetracked by the example. The original argument was that some fields are studied for their own sake. And that is a good thing.

Pretending that the reason to study them is that they may prove useful in some undetermined future is just a distraction used to convince others, usually in order to provide funding or as a means to gain status. We should simply embrace that things are worth studying for the sake of knowledge itself instead of letting the pursuit of knowledge be corrupted by all devouring capitalism which reduces everything to a monetary value by denying the existence of any other kind of value.

I'm actually interested in what these results of practical significance would be. Practical, as in actually useful for technology or such.

I'm also not sure what you mean with "finitist framework". As I said, analysis existed before set theory, it doesn't require any special framework. In fact, the system of natural numbers, real numbers, and complex numbers can be axiomatized just with second-order logic. Without set theory, let alone a transfinite set theory like ZFC. And normal mathematicians wouldn't even use a formal logic here, they would just write down those axioms in plain English.

Hmm. Measure theory without sets? The entire thing is about mapping sets to numbers?

Most mathematicians would include measure theory in analysis since it is needed for Lebesgue integration, and what is analysis about if not integrals?

If you're wondering about applications of measure theory you'll see plenty here: https://en.m.wikipedia.org/wiki/Measure_(mathematics)

That was not the issue here though. Whether one is finitist or a Cantorian or agnostic doesn't play a role in applied math precisely because applied math doesn't require any transfinite set theory.

If your issue is that applied math doesn't need infinite sets, then you are just plain wrong. Applied math uses infinite sets, and real numbers, and measure theory, etc. all the time. Granted, what set theorists are interested in most of the time, isn't particularly relevant for applied math. Except of course in the sense that if you one day have a question about your (applied) math which has to do with sets, it is nice that there are people who might have an answer for your question. I've had various such questions when working on Practal.

Researchers at universities only have to justify their work in very broad strokes when it comes to mathematics, if at all.

It's true that most researchers are motivated like you say, because they like it, because if you do it due to utility, you'll have a hard time. But, the research money that was released to the researchers was justified that it might (and probably will) become useful one day. It has paid off many times in the past.

I'm not sure work in theoretical math has really often turned out to be useful. There are some examples, but it seems likely that Einstein & Co would otherwise have simply come up with concepts from Riemannian geometry themselves, ad hoc, as the need arose. There are in fact several cases where useful math (such as integration) has been reinvented multiple times by scientists who were unaware that it already existed. I think the people who do theoretical mathematics don't even believe themselves in practical applications of their research, they just mention this possibility because it sounds good in their grant applications.

This reminds me of a logician who is interested in non-classical logic, and then writes in his research proposal that it might have applications for AI. Of course this would be GOFAI, which doesn't work, but the guys reviewing the grant application wouldn't know. Or a historian who is interested in neolithic culture in India, and now has to justify how this research could be useful. He probably could write something far-fetched, but the truth is that it very probably won't be useful. Which doesn't mean that it isn't of intrinsic interest.

> The argument "it might be useful in the future" can justify research in any theory whatsoever, no matter how esoteric. It's like defending an outlandish conspiracy theory by pointing out that it is possible that it is true.

Have you read Against Method? It goes even further.

https://en.wikipedia.org/wiki/Against_Method

>noone bashed string theory for failing to make useful predictions

That stuck out for me. Lots of people have criticized string theory for exactly that reason. A theory that makes not testable predictions is not a scientific theory.

Mathematics doesn't have to have real word applications. But science has to make predictions that can be tested experimentally.

Just to quibble a little bit, but the entire effort of string theory (et al) was to find a testable theory. It's not that they weren't interested in useful predictions, but just that they didn't find any. It wasn't that they weren't making the effort, but that the effort has so far failed.

But the day is young, as they say. It took humanity 1000 years to figure out how to solve quadratic equations. Physics theory has had a lot of quick successes, but you still never know when the next one will arrive. We might figure it out in a year, or a decade, or a century, or we might never figure it out.

Will probably edit it, as many people complained. The focus here is useful predictions, in the sense of predictions that have utilitarian value, but string theory doesn't make any predictions at all, which is another issue altogether.

Are you sure about that? How much have you researched string theory?

See for example https://profmattstrassler.com/2022/03/20/a-prediction-from-s...

Search for SUEP to cut to the chase.

That aside, thank you for putting this out there. You're explaining advanced math and that's awesome. This is just a quibble about the intro. I'm checking out the rest now (I was only ever a brief and superficial user of category theory). I think I have already learned some things and been refreshed on others!

Even in the late 90s when I was studying Topology there was a bit of furor over the lack of application...

Until, suddenly, it was quite applicable in evolutionary biology and knot theory in context of D?NA enzymes.

When I think of string theory, the first thing that pops into my head is "mathematically sophisticated theory that sucked in a generation of physicists but is dead now because it doesn't make any testable predictions".

But some of the fancy math in this area is turning out to be helpful for QFT calculations, and it's becoming clear that we need more powerful math to even formulate proper quantum field theories. Check this article out for a briefing: https://www.quantamagazine.org/the-mystery-at-the-heart-of-p...

Not sure that is much of a return on investment for the millions (?) of person-years of research that was spent on string theory.

Compared to what? Civilization spends on a lot of speculative stuff. Venture capital is the art of investing in failures 999 times out of 1000. And personally I'd rather have tens of billions spent on speculative theory research than on, say, Meta's VR boondoggle.

>I'd rather have tens of billions spent on speculative theory research than on, say, Meta's VR boondoggle.

We can certainly agree on that!

I believe the lesson here is not that we shouldn't invest in speculative theory research. It's that we need more breadth, we need to not get stuck researching only one option.

I for one would much rather seen millions of person-years spent on plumbing the depths of various bits of theory, than have that same energy directed towards making ads/social-media/consumerism better. Not exactly the same “circles”, but the point still stands.

Dead?

String theory continues to be the leading candidate for quantum gravity. Very far from dead.