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Definitely not! The article you're referring to was about theoretical physics having surprising application to the real world, not pure math. The rabbit hole of pure math goes ridiculously deep, and only the surface layers are in any danger of accidentally becoming useful. Even most of number theory is safe - the Riemann Hypothesis might matter to cryptography (which is partly why it's a Millennium Problem), but to pick some accessible examples, the Goldbach Conjecture, Twin Primes conjecture, Collatz conjecture, etc. are never going to affect anyone's life in the tiniest way.
My career never went that way, so I've only dipped my head into the rabbit hole, but even I can rattle off many examples of fascinating yet utterly useless math results. Angels dancing on the head of a pin are more relevant to the real world than the Banach-Tarski paradox. The existence of the Monster group is amazing, but nobody who's not explicitly studying it will ever encounter it. Is there any conceivable use of the fact that the set of real numbers is uncountable? If and when BB(6) is found, will the world shake on its axis? Does the President need to be notified that Peano arithmetic is not a strong enough formal system to prove Goodstein's theorem?
These questions are all meaningful to me. I'm weird, though. I'm not even particularly good at math.
I hate dynamic programming, but it seems that you can't "jump ahead" when calculating prime numbers. This feels like computational irreducibility. The world in which this property exists, and the one in which it doesn't, are meaningfully different.
The Collatz conjecture, and BB, relate to the ability to generate large things from small ones. It seems relevant for this question: Can you design a society which is both novel and stable over infinite time? Would it have to loop, repeating the same chain of events forever, or is there an infinite sequence of events which never terminates, but still stays within a certain set of bounds? If we became all-powerful and created an utopia, we might necessarily trap ourselves in it forever (because you cannot break out of a loop. If you loop once, you loop forever). It may also be that any utopia must necessarily be finite because it reaches a state which is not utopian in finite time.
Some other questions are about the limitations of math. It's relevant whether a system of everything is possible or not (if truth is relative or absolute). If trade-offs are inherent to everything, then "optimization" is simply dangerous, it means were destroying something every time we "improve" a system. It would imply that you cannot really improve anything, that you can only prioritize different things at the cost of others. For instance, a universal paperclip AI might necessarily have to destroy the world, not because it's not aligned, but because "increase one value at the cost of every other value" is optimization.
I also have a theory that self-fulfilling prophecies are real because reality has a certain mathematical property. In short, we're part of the thing we're trying to model, so the model depends on us, and we depend on the model. This imples that magic is real for some definitions of real, but it also means that some ideas are dangerous, and that Egregores and such might be real.
None of these are meaningful in the way you mean. I am not that good at math, but I am good at mathematical model building and interpretation.
These are not meaningful because we can easily write different examples with different results, so the key question for, say, society is whether society satisfies another given property that is not the ones you mentioned.
Economics has models where agents who are part of the model and know or learn the model. Yet, self-fulfilling prophecies are not guaranteed or fully ruled out.
Economics would also have models that imply tradeoffs. Yet, in general not every improvement leads to a tradeoff because there are always dumb actions. Stop being dumb and you get an improvement without losing anything.
We can also come up with processes that generate large numbers from small and make that process loop or collapse or anything we want. The question is not whether such processes exist, but whether we can identify which kind better represents society, if any of them do.
I do think that some math is useful to recognize whether a kind of argument is plausible or ruled out. But most math is not even useful for that.
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You can, actually. Testing whether a specific number is prime is actually pretty easy (disclaimer: there are subtleties here I won't go into), and doesn't require computing the numbers earlier than it. It's factoring a number which is apparently hard (although there are still much faster methods than iterating over the numbers before it). This is why RSA is practical: it's computationally very easy to search for 1000-digit prime numbers, but very hard to recover two of them after they've been multiplied together.
I think the rest of your questions veer more into spirituality, philosophy, and ethics than math, so I'm not sure I'm the right person to ask. I have all the spirituality of a wet fart. But I can tell you that the Collatz conjecture is not relevant when discussing the future of civilization. :)
That doesn't seem like a way to generate prime numbers directly, but to sort of chip at the problem by creating a scaffolding around it and then getting close and closer. It doesn't feel elegant like some math does. And yeah, I think that pure maths is largely useless (because its scope is wider, i.e. less restricted than our reality). We can find interesting properties in math which hints at properties in reality, though. At high levels of abstraction, these things overlap. "The dao of which can be spoken is not the real dao" is a logical conclusion, since you can judge the limits of a system from within said system. Gödel did the same with math. You can use a similar line of thinking to derive that everything is relative (there's nothing outside of everything, so there can be no external point of reference).
Maybe this is "abstract reasoning" rather than math? I'm not sure what it is, but this ability is useful in general. I don't suffer from the philosophical problem of "meaning in life" because I recognized that the question was formulated wrong (which is why there's no answer!). I also figured out enlightenment, which you usually cannot reach by thinking because it requires not thinking. But you can sort of use thinking to show that thinking is the cause of the issue, and then "break free" like that.
Edit: Nietzsche came up with his "Eternal recurrence" through logic, showing that if time goes back infinitely, the world would already have been looping forever. Same with his "Perspectivism", that there's no facts, only perspectives. He wasn't a mathematician, he was just highly intelligent.
But I'm sort of weird, most subjects I think about don't fit any common categories
If your primary issue is that the algorithm is probabilistic, then good news: there’s also a polynomial-time deterministic algorithm for testing primality. (Just don’t pay attention to the constant factors.)
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If the phase space of events requires a continuum to describe, then this sounds like a classic chaotic system: it never reaches the same state twice, but it also converges to and moves within the "chaotic attractor" subset of that space.
If the events come from a finite set, that's a problem. Even if you make the system stochastic or otherwise somehow set up an infinite sequence with no repeats, does it matter? At some point you'll have reached every point that you're ever going to reach. Personally, if the best utopia we can ever come up with is "you get to experience every bit of goodness possible before you're done, but there's only a googleplex or whatever of those", I'll be happy with that. Others' opinions may differ. When I first read the idea (in Stephen Baxter's Manifold: Time) it was presented as existential horror; that dude is really good at introducing interesting ideas in depressing ways.
That's my whole answer; feel free to ignore the following digression.
The problem of coming up with an infinite utopia also reminds me of the biggest flaw in the excellent television series (spoilers)The Good Place.
(seriously, spoilers)
What if, instead,
Or maybe I'm just too much of a nerd, because
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https://people.seas.harvard.edu/~salil/am106/fall18/A Mathematician%27s Apology - selections.pdf
https://mathoverflow.net/questions/116627/useless-math-that-became-useful
That thread, in general, seems to have a great many examples. Other quotes from it:
I hope this shores up my claim that even branches of maths that their creators (!) or famous contemporary mathematicians called useless have a annoying tendency to end up with practical applications. It's not just in the natural sciences, I've certainly never heard cryptography called a "natural science".
Also, see walruz's claim below , that even what you personally think is useless maths is already paying dividends!
Maths is quite cheap, has enormous positive externalities, and thus deserves investment even if no particular branch can be reliably predicted to be profitable. It just seems to happen nonetheless.
No, I'm sorry, but you really don't know what you're talking about here. The field of pure mathematics is much larger and stranger than you know, and it takes years of intensive study to even reach the frontier, let alone contribute to it. Conic sections and integral transforms are high-school or early university math, and knowing them makes you as much of a pure mathematician as knowing how to change your car's oil filter makes you a CERN engineer. (And, for the record, conic sections were certainly never useless - even other people in that thread you linked called out that ridiculous claim. And non-Euclidean geometry is useful in many other realms than special relativity, like, oh, say, navigating the Earth!)
While there is zero chance of any of the math I linked above being useful, I admit that cryptography isn't the only example of surprising post-hoc utility showing up. As theoretical physics has gotten more abstract (way way beyond relativity), some previously existing high-powered math has become relevant to it. (The Yang-Mills problem, another Millennium Problem, unites some advanced math and physics.) But I absolutely defy the claim that there is a "tendency" for practical applications to show up. Another way to frame the fact that 0.01% of pure math has surprised us by being useful over the last 2,000 years is... that we're right that it's useless 99.99% of the time. I wish I had that much certainty about the other topics we discuss here!
BTW, did you not realize that @walruz was joking? What he linked is a fun Magic: The Gathering construction. If the Twin Primes conjecture is true, then the loop never ends. If it's not true, it does end, after 10^10^10^10^whatever years. It may be slightly optimistic to describe that as "paying dividends"... (Also, the construction only exists because of a card that specifically refers to primes in its rules. You can't claim that math has practical application because it's used to answer trivia questions involving that same math!)
I was leaning into the joke. MTG nerds are a different breed.
On the topic of conic sections, the poster claimed:
They very much didn't start out that way.
That depends on how strict you want to be on the definition of non-Euclidean, spherical geometry, a limited subset, was used in celestial and terrestrial navigation as early as the first century CE, though real usage only boomed in the Age of Sail.
But that was for a very specific purpose, the idea that space itself was non-Euclidean came about much later. That is a lag of about 21 centuries.
I am happy to acknowledge availability and recall bias here. If there are topics in maths that have remained utterly useless and purely theoretical to this day, I am unlikely to have heard of them.
My overall point is that:
Maths is incredibly productive on net.
Even if we do have "99.9%" certainty that a particular field is unlikely to have practical applications, the benefits in the unlikely case that it does are usually substantial. If I came across a normal lottery and saw that my ticket had a 0.1% chance of winning billions, then I'd be spending quite a lot of money on lottery tickets.
Ergo, it is immensely sensible to subsidize or invest in maths as a whole. The expected value from doing so is positive. Our entire society and civilization runs on mathematical advancements.
I have no quibbles with these points! I think what you should take away is that the distribution of potential practicality is far from uniform. There are fields that we can be very, very, very sure aren't practical. If we were horribly utilitarian about things, we could easily, um, "optimize" academic math without losing out on any future scientific progress.
Also, lest my motivations be misunderstood, I'm happy that we fund pure math for its own sake. I took a degree in it. I love it. I just don't want it to be funded under false pretenses.
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I think it's still open to debate whether, in the absence of subsidized pure math research, we'd get the same mathematical advancements "never", "much later", "as soon as we need them", or "practically just as soon".
The fact that everybody thinks of (even their own!) pure math as "useless", right up until it turns out to be the foundation for quantum physics or something, is perhaps the best evidence for "never". I got my PhD in Applied Math (unspoken motto: do you want respect, or do you want job offers?), and it feels almost criminal when you hear about a mathematician coming up with an abstract toy only for someone more focused on science and engineering to come along generations later and say, "whoa, that solves my problem; yoink!"
As evidence for "as soon as we need them": the applied mathematicians haven't been just swiping everything; if you don't find something that solves your problem off-the-shelf, you take what you have and you expand it and tweak it and invent more of it until you do, and in the end you're still proving new theorems, just motivated by "this is how I can guarantee when my new algorithm will converge" rather than "theorems are fun!"
As evidence for "much later": the trouble with "do you want job offers" is that some job offers let you publish more than others, and if you're not getting subsidized via something like academic grants or civilian national lab research, it's downhill from there. Math is in part a cooperative team sport, and it doesn't work as well when you want to score in the "free advancement of human knowledge" basket but you're lucky if you get to shoot for "patent" rather than "trade secret" or "national security" instead.
And as evidence for "practically just as soon", I refer back to "theorems are fun!" There are some people who you can shunt off to a job as a patent clerk and it still won't stop them from playing with tensor calculus; if these are the sort who make the critical-path advances then we still get the advances.
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What about things like quaternions, which suddenly became relevant when we needed to interpolate 3D transforms and do rotations without Gimbal lock? The current best process for calibrating cameras is to use dual quats, which also means needing dual number theory. Were those areas originally expected to be useful for engineering? My understanding is no, but I'm not a mathematician.
That's a good question. I'm not sure of the exact reason quaternions were invented - you can indeed stumble on them just by trying to extend the complex numbers in an abstract way - but the Wikipedia article suggests they were already being used for 3D mechanics within a couple of years of invention. (BTW, "number theory" involves integers, primes, that kind of thing, not quaternions. Complex numbers do show up though.)
You could ask the same question about complex numbers too, but they originally arose from the search for an algorithm to solve cubic equations, which is a fairly practical question. That they later turned out to be essential for electronics and quantum mechanics is a case of some new applications of an already useful math concept.
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The twin primes conjecture actually has some applications: https://old.reddit.com/r/BadMtgCombos/comments/1feps3y/deal_infinite_damage_for_4gru_as_long_as_the_twin/
Hehe, I stand corrected!
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