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I'm a sympathetic mathematician but I found this whole passage bizarrely off-base: > mathematics is in a weird and, I’d say, unique position of always having t

I'm a sympathetic mathematician but I found this whole passage bizarrely off-base: > mathematics is in a weird and, I’d say, unique position of always having to defend what they do with respect to it’s value for other disciplines. I again stress that this is something that would be considered absurd when it comes to any other discipline. > People don’t expect any return on investment from physical theories - noone bashed string theory for failing to make useful predictions. Huh? Tons of people bash string theory for failing to make useful predictions! There was a whole book criticizing it called "Not Even Wrong", meaning it didn't even reach the point of being testable. 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. I also support the argument that math is intrinsically and aesthetically valuable, but for those who don't agree, the argument above may make some headway.

tylerhannan 4y ago
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.
canjobear 4y ago
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".
- civilized 4y ago
  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-physics-that-only-math-can-solve-20210610/ https://www.quantamagazine.org/the-mystery-at-the-heart-of-p...
  - hermitcrab 4y ago
    Not sure that is much of a return on investment for the millions (?) of person-years of research that was spent on string theory.
    - civilized 4y ago
      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.
      - hermitcrab 4y ago
        >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!
        alexvoda 4y ago
        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.
    - FridgeSeal 4y ago
      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.
- daxfohl 4y ago
  Dead?
- layer8 4y ago
  String theory continues to be the leading candidate for quantum gravity. Very far from dead.
hermitcrab 4y ago
>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.
analog31 4y ago
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.
cubefox 4y ago
> 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.
- civilized 4y ago
  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.
  - cubefox 4y ago
    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.
boris_m 4y ago
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.
- civilized 4y ago
  Are you sure about that? How much have you researched string theory? See for example https://profmattstrassler.com/2022/03/20/a-prediction-from-string-theory/ 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!

tylerhannan 4y ago

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.

canjobear 4y ago

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".

civilized 4y ago
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-physics-that-only-math-can-solve-20210610/ https://www.quantamagazine.org/the-mystery-at-the-heart-of-p...
- hermitcrab 4y ago
  Not sure that is much of a return on investment for the millions (?) of person-years of research that was spent on string theory.
  - civilized 4y ago
    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.
    - hermitcrab 4y ago
      >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!
      - alexvoda 4y ago
        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.
  - FridgeSeal 4y ago
    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.
daxfohl 4y ago
Dead?
layer8 4y ago
String theory continues to be the leading candidate for quantum gravity. Very far from dead.

hermitcrab 4y ago

>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.

analog31 4y ago

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.

cubefox 4y ago

> 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.

civilized 4y ago
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.
- cubefox 4y ago
  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.

boris_m 4y ago

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.

civilized 4y ago
Are you sure about that? How much have you researched string theory? See for example https://profmattstrassler.com/2022/03/20/a-prediction-from-string-theory/ 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!