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> Peripheria > Is there an infinity in nature? 
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Tomasz Grysztar
In my recent discussion of 2adic numbers I made a remark about their infiniteness, explaining that I prefer to think about it in terms of just a potential infinity. We never deal with actually infinite sequences of digits, we always have a limit, but the language of 2adic numbers allow to considers various rules without having to worry what the particular limit we may have. We can think of it as always being sufficiently large to not be exceeded by anything we do. My assemblers in particular demonstrate this idea, as I defined them to give results of computations as if done on 2adic numbers, even though they actually have limits on the number of possible digits.
Despite my background in theoretical mathematics, I am generally inclined to think about everything in our world as finite, as I don't think we have a way of knowing about anything that is not. If we try to keep splitting anything in half, we are going to reach Planck length after a few dozens of steps and we do not know if it would be possible to continue the process in any sense. But even if we could continue it indefinitely, how would we know if there is truly no limit there, even if always out of our reach? In the modern science we see more and more how building blocks of the nature can be explained in terms of information. Maybe this is simply because we use the thinking tools and language that feels most natural to us nowadays, in the information age. But I also think it is related to the fact that everything that we can experience and explore is actually finite, and everything finite can be deconstructed into sequences of bits. At this point someone is certainly going to bring quantum computing into discussion, as it seems to directly contradict everything I just said. But I do not think it does. After all, if quantum states are not actually infinitely precise, but just with high enough precision for us to not be able to notice the difference, we would never know. I should note, however, that I find it plausible that there may be limits of quantum precision that are in our reach, and I think that it could be a great opportunity to expand our knowledge. After all, if we made quantum computers powerful enough to be hit by such limitations, it would be a discovery that could pave the way for some new theories of physics. What brought this idea back to my attention was a guest post on Backreaction blog, which references a recent publication, Discretisation of the Bloch Sphere, Fractal Invariant Sets and Bell's Theorem. It is a fascinating example of a theory that can give explanations similar to quantum mechanics while being finite in all aspects (down to states of the system being simply a finite strings of bits). It even shows how this finiteness would place upper limits on qubit entanglement, which could indeed be a testable constraint on the abilities of quantum computers. This confirms that my suspicion was not entirely ungrounded, and if there really are such limits in nature, I sincerely hope that they are in our reach to test someday. That article also gives an excellent quotation of David Hilbert, which I am going to reproduce here, as I think it greatly summarizes what I attempted to say. David Hilbert wrote: The infinite is nowhere to be found in reality, no matter what experiences, observations, and knowledge are appealed to. 

12 Mar 2020, 11:31 

Tomasz Grysztar
revolution wrote: Just because we can't ever experience anything outside of our universe* doesn't automatically mean that there is just nothing. Moreover, even if we clarify that the thing we talk about exists in some other world (like "Gandalf exists in the world of MiddleEarth", or "number 2 exists in the set of natural numbers"), it always is an abstraction that is in some way rooted in our reality nonetheless. There are many structures in our universe that work like natural numbers (well, there is a reason why they are called "natural") and number 2 is present in each of them. MiddleEarth, although a fictional universe, exists in our universe as a collection of concepts passed from mind of the author to the minds of the readers (and then evolving further). However, for things that are completely outside of our universe we lack any anchor, so we are not able to say anything factual about them. We could assume that absolutely everything that we could think of exists there, but it would not make any difference for us (as you noted, by the very definition of "our universe")  therefore I consider such speculations pointless. 

12 Mar 2020, 13:25 

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