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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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