> the world is continuous
Is it though? Does it matter one way or the other? Do we think reality is the math in some way, or is the math a really darn good model of the reality?
Is it though? Does it matter one way or the other? Do we think reality is the math in some way, or is the math a really darn good model of the reality?
This happens the same way in which steel demonstrates isotropic behavior although its microscopic structure is anisotropic.
So there is no easy way to prove or disprove continuity of space.
However, what's missed here is that discrete is a necessary but not sufficient condition.
Once you give any sort of plausible account of how reality could be discrete, as you've done here, you end up with non-computable aspects (eg., typically randomness). So the metagame is lost regardless: reality isnt a computer (/ no complete physical theories of reality are computable).
Though the meta-meta-game around "simulation" is probably internally incoherent in itself -- whether reality is a computer or not would really have nothing to do with whether any properties had by it (eg., mass) are simulated.
Since either you take reality to have this property and hence "simulation" doesn't make sense, or you take it to be faked. If it's faked, being computable or not is irrelevant. There's an infinite number of conceivable ways that, globally, all properties could be faked (eg., by a demon that is dreaming).
Also continuous doesn't mean uncomputable either, because in many cases the infinite amount of computation for continuum does not add anything interesting and finite approximation works good enough.
> So the metagame is lost regardless: reality isnt a computer (/ no complete physical theories of reality are computable).
I don't see any evidence for this. For now we do not have a proof for one way or another. If for instance it turns out that quantum computers really can run Shor's algorithm factoring very large numbers, it would be a good evidence for continuum, but we are not there yet.
But even that would not be an evidence for reality not being a computer, since it will still allow the possibility of reality being a computer that can perform operations on real numbers.
The "Nondeterministic" in NFA means its transition function goes from states to sets of states, instead of from states to states. Informally, it can explore multiple paths in parallel for the cost of one. They're not probabilistic.
Thus, this semantics implicitly encodes the notion that the machine is nondeterministically choosing the next state in each execution.
The decision problem of whether an NFA accepts a string w is what allows for the informal parallel interpretation, that it accepts iff you imagine the computation is forking off a new thread at each nondeterministic branch. But to say that this not nondeterministic or not probabilistic is like saying the Many-Worlds Interpretation means there is no real superposition, or something like that. It's like saying a throw of a dice does not really involve probability because of a symmetry argument that a dice has six equal sides. Mainly, I don't understand that, because I see probability as a way to implement nondeterminism: a system is probabilistic only because it is making nondeterministic choices according to some probability distribution. And checking Sipser 2nd ed. p.368: "A probabilistic Turing machine is a type of nondeterministic Turing machine in which each step is a coin-flip step".
Anyhow, my main issue was that the original commenter casually claimed that probability makes things (physics) uncomputable. But Turing computability has nothing to do with probability, since as I recall the closest concept is the Non-deterministic Turing Machine (NDTM) and with that it is a basic proof to show that NFAs vs. DFAs, as well as NDTMs vs. DTMs, are computationally equivalent and there are theorems for that.
Meaning either they are using an idiosyncratic definition of computability or are ignorant of an introductory course on theory of computing which explains formally what Turing/Church's theories were about when clarifying the concept of computability. Okay or maybe they have a deeper philosophical disagreement with computability and complexity theorists - maybe they reject Sipser's definition above - but these are standard undergraduate curricula in CS by now and it could be argued that perhaps it is the non-CS experts who haven't thought deeply enough about what computability really is and would benefit from actually learning from these subdisciplines. I don't know, as they did not reply.
This characteristic is observable for metals as well. Steel becomes less flexible as it's worked because it's grains become smaller and more chaotic - A microscopic property with a macroscopic effect.
Everything we can see move on the grid is at least 20 orders of magnitude bigger than the grid spacing. Any macroscopic objects we can experiment with are more like 30+ orders of magnitude bigger than the grid spacing and consist of numerous atoms that will all be moving within the object due to thermal jiggling over distances orders of magnitude bigger than the grid spacing.
I kind of suspect "is the universe continuous versus discrete" will come down to that. I don't know what a hybrid of such things looks like. With our current conceptions it seems impossible. But it always does, before the breakthrough comes and then in hindsight all the people of the future will get to look back at us going "How could they not see this obvious thing?", to which my only defense is that you, dear future reader, only think it's obvious because it was handed to you on a silver platter and you'd be as confused as we are if you were back here with us.
These are two very different questions. As for the former, I don't believe there is consensus at this point with good arguments for and against. As for the latter, if we can reliably prove that the reality is not analog but digital, it has consequences at various levels, and we might make better choices when using math to describe it/make approximations.
What matter is a subjective topic. What we all have in common is logistics constraints. So if some people set as a goal something that requires to settle if reality is more easily handled when modeled in continuous or discrete manner for logistical reasons, then it this scope it matters. But whatever you settle on, human brain is thus built that it can always assume that the perfectly fitting model is only valid in its scope which is built on top of an other more subtle level of reality which is on it’s part better modelized with an antithetic approach.
Now, on a very personal out of blue opinion, I fail to see how any causal series might happen without an underlying continuous flow of event. I mean, supposing causal discontinuity is to my mind as relevant as supposing that universe as it is right now, actually just appeared, without anything we can think about it being relevant, and in the next instant could be completely different or nonexistent since universe is not bound in any remote way to what we might expect on our delusional just created sense of causality.
You say you can't comprehend how something can move from 1 to 2 discretely. But the paradoxical notion of infinite continuous change has been known since antiquity. It's faith either way.
Discrete doesn't mean state changes are wholly globally arbitrary. Imagine a graph with nodes and edges, a state machine as computer sciences call it. I think it's easy to agree that the universe could be parsed by a regex ;-) Heck, imagine an integer on the number line that can go up or down.
Worlfram has written a ton about this. Despite all his issues, his math is solid. (Which is not to say his physics is true.)
I didn’t mean that, sorry if my words that induced you to believe so.
What I want to point out is that, to my mind, if I assume a discrete foundation of universe, on meta-cognitive level I must recognize it implies everything I experiment through my current attention might possibly be a just made up state without any compelling ontological relation to anything I can recall or think of. So, as far as I’m concerned, believing in continuity is just a lazy way to relax on a metaphysical Gordian knot.
> It's faith either way.
Yes and no. It’s probably easier to change scientific perspective to whatever model apply best for some purpose than to adhere to some philosophy about Nature.
Zeno didn't believe that the latter was possible. But he wasn't stupid, he obviously knew that motion was happening all the time in real life. His paradox really only makes sense in the context of Eleatic philosophy which assumes that reality is an illusion because change is fundamentally impossible (how can something come from nothing?).
If you want to reframe it in more modern terms, Zeno's paradox shows a contradiction in axioms. If you want to get rid of the contradiction, you have to change some of the axioms.
In real analysis, loosely speaking, we remove the axiom that an infinite process cannot result in a finite outcome - this way we are allowed to sum (some) infinite series, for example. But we don't "know" if reality behaves that way.
The atomists found a different solution: they argued that reality was fundamentally discrete. This way, Zeno's paradox also doesn't arise.
my mistake - that should have read "are both continuous".