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Perhaps we should start admitting to ourselves that infinity is not really something that actually exists but instead it's just an imaginary concept that we cre

Perhaps we should start admitting to ourselves that infinity is not really something that actually exists but instead it's just an imaginary concept that we created but that has no correspondence to anything that makes sense in our universe? Or at least, some versions of infinity (I will leave which ones, exactly, to the experts). Hence all the paradoxes that derive from it. I mean, perhaps we can work with some version(s) of infinity, without creating a plethora of paradoxes. The natural numbers, for example, might be something that is infinite and makes sense. However, perhaps talking about a single set that contains an infinite amount of natural numbers might not make sense (or perhaps it still might?). Or perhaps we just need a new language for talking about infinite things without re-using the language we use for describing (finite) sets, so that we don't try to reuse concepts from finite objects with infinite objects. Or perhaps the only objects that exist (and can be meaningfully talked about) are the computable ones. And/or perhaps "computable function" is one that can use infinite amounts of time but only finite amounts of storage (rather than infinite amounts of storage). Would it make sense to go back to classical finitism and re-evaluate some of the choices that were made along the way to where we are now? I can't help but think about the many paradoxes that arise from the very (supposed) existence of infinity. For example, everybody thinks the halting problem is undecidable. The halting problem is clearly not undecidable on finite-state machines (i.e. real computers), it's only undecidable if you use imaginary Turing machines (which supposedly can have literally infinite state) as a model. But of course, Turing machines cannot exist in our universe. So it's rather ironic that people say the Halting problem is undecidable, when this is only true for (impossible to construct) imaginary machines but it's not true for the real machines that we actually have and that are possible to construct. Or think about Hilbert's paradox of the Grand Hotel. Or the Banach-Tarski paradox. Or Galileo's paradox. Or other such paradoxes that only exist if you believe that imaginary infinite objects could possibly exist. Most people will tell you "well, it's just that our intuition doesn't work for infinite things", but in this specific instance, it's really hard for me to believe that it's our intuitions that are mistaken rather than that we actually made some conceptual mistakes along the way. To me, this argument sounds more like gaslighting rather than a real convincing argument. I'm not saying our intuition is always correct, though. For example, the quantum properties of our universe are clearly not intuitive but it's hard to argue that our universe is not quantum, after all the scientific experiments that have clearly verified this to be true. I don't think that invalidates my argument, though, because the convincing argument in that case is not "sorry, you must believe my theory that the universe is quantum" despite not existing any evidence that this was true, but rather, "sorry, but here's all this real evidence that our universe truly is quantum". Am I crazy here?

tsimionescu 4y ago
The thing is, infinity is extremely useful for taking things to the extreme. The halting problem is a very good example: as you rightly point out, the halting problem is trivially decidable for any finite Turing machine. However, the core of the argument remains true, we just need to complicate it significantly if we want to remain within the realm of finite machines: instead of saying "there is no TM that can decide if any arbitrary TM halts", we need to say something like "there is no finite TM that can decide if any TM smaller than itself halts in an amount of time less than that TM". They are equivalent concepts on some level, but the second one is significantly more complex, and harder to prove (assuming you don't appeal to the infinite version, from which it easily follows). That is, even if you seek to do maths on finite but unbounded sets, you will find mostly the same properties that infinite sets have, but you'll have a much harder time working with the concepts. It's true that there are some problems that infinite sets have that unbounded finite sets don't, but those may well be a worthy price to pay. And either way, regardless of intuition, the logic is valid. The Banach-Tarski "paradox" is perfectly logically valid, though unintuitive. So, even if you could say that it's useless and a result that we should avoid having in our theories, you can't say that it's wrong, since it doesn't violate any rule of logic.
- wizeman 4y ago
  I can't really give a substantial response to your first argument, as I'm not certain about all the implications that you mention. Your second argument, though, doesn't sound very convincing to me. There are many possible consistent logical frameworks (perhaps an infinite amount of them? hah!) that are clearly not adequate for describing our universe. Or at least, they wouldn't make sense for us. For a trivial (and perhaps stupid) example, consider an alternate universe where we decided to use a logical framework which can represent natural numbers, but where the digit "2" is represented as "9" and the digit "9" is represented as "7", etc. Let's say you have a 1-to-1 mapping from one digit to another, compared to our usual representation. Or perhaps the mapping is a lot more complicated than that, and not just about digits, but about entire numbers. Let's say we decided to do that, for no good reason. Or maybe there was a good reason at the time, but since then we have figured out that it doesn't make sense anymore. Yes, you could do exactly the same math in this logical framework as we do in our usual ones, and no contradictions would arise. However, would that really be a wise idea? Wouldn't that just lead to making unnecessary mistakes, sometimes even conceptual ones, when considering that we would be prone to confusing numbers in this system with the numbers that we use normally? I think language is also important here, not just logical soundness. Especially if we want our intuitions to be helpful. Just to bring this back to something less abstract again, I will just mention the following: > However, the core of the argument remains true, we just need to complicate it significantly if we want to remain within the realm of finite machines: instead of saying "there is no TM that can decide if any arbitrary TM halts", we need to say something like "there is no finite TM that can decide if any TM smaller than itself halts in an amount of time less than that TM". Well, I don't believe the core of the Halting problem to be true, and that's exactly the language problem that I'm talking about: confusing language leads us to making conceptual mistakes. For example, time is not necessarily relevant here. I could argue that what the Halting problem really demonstrates is that to analyze whether a finite-state machine (FSM) halts, you need to use an FSM that can have more state than the machine being analyzed (which is why it stops working if you believe you can have infinite state). This FSM with more state would always be able to decide whether the other FSM halts in finite time, regardless of whether the other machine halts or whether it never does. In fact, that's exactly what cycle-detection algorithms do, such as the tortoise and hare algorithm.

tsimionescu 4y ago

The thing is, infinity is extremely useful for taking things to the extreme. The halting problem is a very good example: as you rightly point out, the halting problem is trivially decidable for any finite Turing machine. However, the core of the argument remains true, we just need to complicate it significantly if we want to remain within the realm of finite machines: instead of saying "there is no TM that can decide if any arbitrary TM halts", we need to say something like "there is no finite TM that can decide if any TM smaller than itself halts in an amount of time less than that TM". They are equivalent concepts on some level, but the second one is significantly more complex, and harder to prove (assuming you don't appeal to the infinite version, from which it easily follows). That is, even if you seek to do maths on finite but unbounded sets, you will find mostly the same properties that infinite sets have, but you'll have a much harder time working with the concepts. It's true that there are some problems that infinite sets have that unbounded finite sets don't, but those may well be a worthy price to pay. And either way, regardless of intuition, the logic is valid. The Banach-Tarski "paradox" is perfectly logically valid, though unintuitive. So, even if you could say that it's useless and a result that we should avoid having in our theories, you can't say that it's wrong, since it doesn't violate any rule of logic.

wizeman 4y ago
I can't really give a substantial response to your first argument, as I'm not certain about all the implications that you mention. Your second argument, though, doesn't sound very convincing to me. There are many possible consistent logical frameworks (perhaps an infinite amount of them? hah!) that are clearly not adequate for describing our universe. Or at least, they wouldn't make sense for us. For a trivial (and perhaps stupid) example, consider an alternate universe where we decided to use a logical framework which can represent natural numbers, but where the digit "2" is represented as "9" and the digit "9" is represented as "7", etc. Let's say you have a 1-to-1 mapping from one digit to another, compared to our usual representation. Or perhaps the mapping is a lot more complicated than that, and not just about digits, but about entire numbers. Let's say we decided to do that, for no good reason. Or maybe there was a good reason at the time, but since then we have figured out that it doesn't make sense anymore. Yes, you could do exactly the same math in this logical framework as we do in our usual ones, and no contradictions would arise. However, would that really be a wise idea? Wouldn't that just lead to making unnecessary mistakes, sometimes even conceptual ones, when considering that we would be prone to confusing numbers in this system with the numbers that we use normally? I think language is also important here, not just logical soundness. Especially if we want our intuitions to be helpful. Just to bring this back to something less abstract again, I will just mention the following: > However, the core of the argument remains true, we just need to complicate it significantly if we want to remain within the realm of finite machines: instead of saying "there is no TM that can decide if any arbitrary TM halts", we need to say something like "there is no finite TM that can decide if any TM smaller than itself halts in an amount of time less than that TM". Well, I don't believe the core of the Halting problem to be true, and that's exactly the language problem that I'm talking about: confusing language leads us to making conceptual mistakes. For example, time is not necessarily relevant here. I could argue that what the Halting problem really demonstrates is that to analyze whether a finite-state machine (FSM) halts, you need to use an FSM that can have more state than the machine being analyzed (which is why it stops working if you believe you can have infinite state). This FSM with more state would always be able to decide whether the other FSM halts in finite time, regardless of whether the other machine halts or whether it never does. In fact, that's exactly what cycle-detection algorithms do, such as the tortoise and hare algorithm.