Hornbein said:
Water is made of molecules. The molecules are countable. So the ocean has a countable infinity of atoms. (Real numbers don't really exist, so I doubt that uncountable infinities physically exist.)
The following is just a bunch of thoughts I've had about it. I hope it doesn't sound too out there.
I think it's important to be clear in how we assign existence to numbers in general. For example, does the number 9 exist, and if so, in what form? Well, it exists as an abstract concept. And any instance where that concept exists physically is presumably through some physically embedded information that encodes it. But encodings are observer/system dependent also. There are no atomic 9's floating around in space-time. What we know of and see is that some physical system acts on our instruments and or our senses and then our minds through some chain of events, and the final result is a mental cognition of something classified a 9 with some context attached. This is the case even when simply doing pure mathematics I think. In all cases, 9 is not what it seems like to us (at least me), and existence, even of abstract mathematical objects might not be what it seems at the metaphysical level.
Even though 9's aren't floating around as raw atomic things, if you hypothetically had some kind of system for unambiguously identifying some underlying configuration of something, through the chain of events that eventually reaches your mind, as a one to one representation of a natural number, then maybe you could say some microscopic physical representation of the number 9 exists, with respect to that system.
When it comes to any computable number, including numbers like ##\pi, \sqrt{2}##, those numbers could be represented by a finite discrete configuration interpreted through some such system. As an example, we can take the set of all Turing machines, and definitely one of them prints the digits of ##\pi##, and there is one that prints every other computable number too (although there will be multiple different ones that print the same numbers), hmm?
But it would seem that such a system to identify an uncomputable number would not be accessible to beings like us. At least that is what the Church-Turing thesis seems to imply (although that is not proven) depending on how we define computable and what consciousness is. Even our models of quantum and analog computing are no exception. But we cannot rule out some form of hyper-computation made possible through something we don't know about. We can say that if the universe is doing hyper-computing, nobody seems to know how that might be exactly. If the universe isn't doing hyper-computing then I suppose we can say that it is not computing uncomputable numbers.
But what would that mean for the universe to compute an uncomputable number anyways? All such numbers are infinite. So whatever sense you can think of it being computed, maybe it is through some kind of 1d signal through some kind of temporal fluctuation, or maybe it's somehow producing some infinite configuration that exists all at once and represents the number. In the temporal case, it's hard to fathom. The number surely has a beginning. So do you consider infinite extension of time in both dimensions? Or maybe just infinite extension of time in one direction. If that is the case, you could say some finite time has passed at any moment, if you haven't produced infinitely many fluctuations in that finite amount of time, then the number isn't finished, and the number that has been computed thus far is actually a prefix also of a computable number.
If there were infinitely many fluctuations over finite time or space, then you've basically got infinitely deep physical systems in terms of time and length scales occurring.
Now suppose we think of spacetime as a grid that is infinitely divisible, and fluctuations occur at the smallest scales, and all of the way up. We are now defining some fluctuation of an infinitely small point, and imagining whether those fluctuations across time or space somehow are printing uncomputable numbers. Now we want to figure out how they could print all real numbers. Well, first we want some hypothetical system to be able for an oracle to break down these local processes in ways that uniquely represent all real numbers at once. So what are the rules? Can you have a combination of spatial and temporal encoders? Can they overlap (e.g. share fluctuations)? If you allowed non-local patterns and overlap, then you could trivial say all real numbers are covered (like a crossword puzzle where the letters spelling words can be shared and don't need to be connected, all you need is all the letters and every word is on there). So maybe that's not allowed. Maybe overlap is allowed, but only locally (like a crossword puzzle). I don't know.
Maybe we can start with a model of something basic and study that. So we could suppose a scalar field on a grid with infinitely many points (like ##\mathbb{R}^2##). Then imagine each point in that space has a scalar value that can fluctuate infinitely many times in a finite amount of time. But somehow those fluctuations are all part of a single system, with local causality, like a cellular automata. And now we can ask, could each of those points (which there are uncountabley many of) be a generator for a unique real number? If not, how about ##\mathbb{R}^n## for some ##n##? Or do we need ##\mathbb{R}^{\infty}##? Anyways, I don't know the answer, but maybe someone does?
Or maybe in such a space, without temporal evolution, could we have a crossword puzzle including all real numbers?
Or maybe the space can be a countably infinite grid with infinite dimensions instead and the numbers can be in some crossword puzzle in spatial form?
Anyways, maybe instead of thinking of the physical embedding of the numbers themselves, we want to consider a number conceptually as a potential property. For example, just having the points in some space like ##\mathbb{R}^n##, so that we have an uncountable number of points. But an uncountable infinity of nothing isn't very satisfying.
That a single uncomputable number could exist in some physical form less trivial than that is hard to imagine, let alone an uncountable infinity of them.
I guess all of these questions are really just questions about math (some of them probably have answers I don't know about), since I've just tried to figure out some formal concepts that could be used to map our notion of existence to our concept of numbers. But I'm not sure how it could be any other way that we talk about anything and be precise without it ending up being an exercise of some sort of logic and mathematics.
