Full argument is elaborated here "Indeterminism in Physics, Classical Chaos and Bohmian Mechanics. Are Real Numbers Really Real? by Nicolas Gisin": https://arxiv.org/abs/1803.06824
"Information" is not a physical quantity, and there cant be a "volume" of it. Nor does this have anything to do with real numbers.
It is impossible for there to be any system extended in space and time to "zoom infinitely" into a continuous range and hence record an infinite amount of information. No one claims this, and the formulation of physics (entirely on real numbers) does not require it.
Rather to say, eg., space is continuous, is to say its unbroken. There is no physical quantity which is becoming infinite.
For example in E = 1/2mv^2, a particle has kinetic energy in virtue of being matter in motion -- it is motion and matter which are basic. Energy is just a system of accounting which tracks motion in the aggregate over time (with kinetic/potential just being the future/past in the accounting) hence why energy conservation is just a temporal invarience.
When making arguments about the physical properties reality has (eg., whether aspects are continuous) you need to be exceptionally clear what your terms mean, and terminology in physics isnt designed for this.
There are no "information saturated volumes", this is a series of abstractions piled on top of each other.
All the words in this area have quite complex formal definitions that are have quite difficult to unpack semantics, you cannot just go around saying "saturated volumes" -- it is this sort of language which breeds cranks, and pop sci does it with abandon.
This entire discussion is a matter of several PhDs, and to be done only well by people with PhDs in the matter (philosophy of physics), or equivalent research. It's not possible to scrap fragaments of what compusci bloggers say and derive much that's likely to be actually correct.
He's also written papers that are basically philosophy of physics. It would be interesting to go over what he has actually said on this topic.
Various Hawking-Bekenstein results about black holes relate to information density, especially, shockingly, that information is proportional to surface area, not volume. This makes perfect sense because a black hole has all its incoming matter and energy sprawled, flattened and red-shifted on its horizon (to a distant observer). It can never export its internal state to the outside world, so you might never expect a volume's-worth of states to be exposed.
The idea was generalized by 't Hooft to the Holographic Principle, for 2D screens encoding the state of 3D volumes on the other side.
However, the full AdS/CFT Correspondence only applies to a certain type of AdS space, not our actual dS space. At the moment, it seems half of theoretical physics doctoral students are trying to extend AdS/CFT to dS space (obviously - strings :) and half of observational astrophysics doctoral students are desperately hoping to show we live in AdS space - LOL
At first I was inclined to agree with you that this is an appeal to authority, with the caveat that such appeals do not always constitute a fallacy. For example, if we both agreed that such a Gisin is an expert whose opinion on this topic can be trusted, then his statements are valid evidence for one way or another.
But then I realised that the very claim being challenged is whether Gisin knows what he's talking about. Floating his credentials and experience feels like a valid contribution. For what it's worth, back in my PhD days I read several of his papers and saw a couple of his talks at conferences, and can confirm he's one of the leading researchers in the field and is particularly thoughtful and careful in his work.
There's a problem for people who think reality can be modelled by computable functions of finite inputs: this makes classical physics non-deterministic, because chaos requires infinite precision for determinism.
So either you go for "reality is deterministic and continuous, and not computable" or "reality is non-deterministic, and discrete, and not computable"
either option in this fork includes properties that offend the minds of the people who want everything to be discrete.
I lean towards a preference for determinism & continuity (via, in QM, superdeterminism) since that's trivial to justify on our best physics
> I argue that there is another theory, similar but different from classical mechanics, with precisely the same set of predictions, though this alternative theory is indeterministic
and in the footnote he describes indeterminism to mean:
> given the present and the laws of nature, there is more than one possible future
Out of curiosity, why do you lean towards superdeterminism and not other deterministic interpretations of QM such as Many-Worlds or Bohmian mechanics?
Yeah. As they said, it’s computer science leaking out.
It can be misleading to reason about entropy, which is the relevant physical concept, as if it were strictly equivalent to information as computer scientists understand it. Entropy works perfectly fine with continuous densities of states and real numbers.
The correct setting is a rigged Hilbert space: given an algebra of operators A on a Hilbert space H, let S be the maximal subspace of H such that |sa| is finite for any s in S, a in A. These are your states. Operators in A don't necessarily have eigenvectors in H, but they do have eigenvectors in the space S* of all continuous linear functionals on S. So <x|, for instance, is just the map `psi -> delta_x(psi)`.
This has the advantage of not having any funky "rigged" states suddenly appearing in your calculations and is also exactly how we deal with non-projective measurements in finite dimensional quantum mechanics.
See here, for example
The spectral theorem, rather than decomposing X in terms of a sum of eigenvectors & eigenvalues instead decomposes it as an integral over the spectrum with respect to the (spectral) projection-valued measure.
Now it is fair to question whether this "observable" is really observable, but it certainly works out mathematically consistently in the normal way we do things in quantum mechanics.
That being said, Gisin's approach is still interesting and his results can still be valid. But he starts with the assumption that real (irrational) numbers are unphysical, i.e. that – in a sense – our observable from above can actually only take on certain (rational) values, and then he derives certain predictions from that.
¹) Putting "information" in quotation marks here because no one really knows what it is.
There are many interesting ways to probe this problem.... here's one:
Say I tell you to imagine a circle, an ideal Platonic circle in a Cartesian coordinate system (real coordinates, first uneasiness). Let's ignore translation, so it is centered at (0,0). I tell you the radius. Can you imagine the circle with Plato? Model the circle? Reproduce the circle? Do you need pi? Does the circle include or encode pi? But pi is has infinite information.
Perhaps all you need is the square root function? But that's also an infinite Taylor series expansion. You can plot and recreate the circle to any precision if you have a square root function. The series will only need to run to the required precision. The circle will always be granular, depending on the number of terms you use in pi, or the square root function. Yeah, right, obvious, so why is that a problem?
What if I tell you the circle is the physical manifestation of equipotentials of a stationary charge (say, nucleus), or mass (say Earth), with inverse square law - so basically a geometric fall-off with range determined by spatial (circular, spherical) considerations. What is the force at some distant point? Do you need pi? Do you need square root function? Or reciprocals? How does the other charge or mass feel the 56,323rd decimal place of the force due to the potential?
Maybe it doesn't, because by the time it has felt the second decimal place, time has moved on, the charges/masses have moved on, and the nuance of what would've/should've been felt in a never changing universe are never experienced. There is a modified differential equation that relates various time derivatives to precision of experienced forces (this almost sounds like relativity :)
The discrete explanation with photons goes like this: the force is produced by radiating photons. They automatically encode the geometric expansion as inverse square law, because of their pathways, no need for pi, or sqrt functions. But that is statistical, the accuracy is only as good as the number of photons that can arrive from the source. The circular/spherical nature of the force only emerges over time, as photons arrive and act. The accuracy of smooth circularity and inverse square only establishes itself over time...
Elapsed time affects experienced precision - hmmm, interesting.
How would you quantify such a thing, where time changes the precision of what you feel? Well, the other obvious example is the Heisenberg Uncertainty Principle. This is just a simple and obvious example of Fourier Analysis for any theory based on a linear wave equation. It almost doesn't need stating, and if it must have a name, it is certainly Fourier, not Heisenberg. Anyway, any math/physics/engineering student knows Fourier to their core, and it gives a nice solution to the information problem: coordinates may be real-valued degrees of freedom, but there is no way to mathematically or physically resolve all coordinates and their derivatives to infinite precision. It's just not possible, even if the underlying equations/reality maintain the fiction of real-valuedness.
Fourier combines time, waves, amplitude, velocity (momentum, etc.) with a specific expression for possible information. A picture is worth a thousand words at this point, just look at a wave-packet, it's obvious. Fourier is a masterwork, and vastly underappreciated as a fundamental limit on knowledge, in a real world sitting on smooth continuous waves.
So Fourier sets limits on knowledge, even in the wavy world of the smooth continuum. Of course, I do not believe in the smooth continuum anyway, but Fourier is my wingman to fight the real-infinitists on their own smooth turf.
It does not, according to any sane way of defining its information content. For example the Kolomogrov complexity of pi is clearly finite - I can write down a program for a Turing machine which will run (forever) and keep writing down digits of pi as it does so.
That's also a different point than the parent's. Seems they're saying if you were to specify pi as the limit of some expansion that describes the physical process of photons arriving in some area, then that specification's information increases with more terms added. Pi, being almost random by every statistical measure, has as much information as a random string, in fact, in any normal conception of information. You cannot wave that away by machine manipulation tricks or by defining a new constant, and this is borne out also by the parent's physical argument that in reality there are no low-complexity universal constants, but that there may be limits to information density (in space and time).
Continuous physics can be a manipulation of limiting quantities without being literal.
I wrote "Kolmogorov complexity" not "original Kolmogorov complexity" so this isn't particularly relevant. The application of the concept to the infinite string which represents pi is essentially trivial.
> Plus, the program would need to store that for whichever special strings you choose you will use this function but for other ones not. That will be a giant table that is part of your program.
I honestly can't parse this.
> Pi, being almost random by every statistical measure, has as much information as a random string
This is wrong. You can consider something like a simple communication task. Alice and Bob share a phone line and she is attempting to tell him a number. Every second the line allows her to send a bit to Bob. For a truly random number she has to use the line infinitely many times to tell him the number. To send pi she can send a finite number of bits which amount to a program to compute pi and he can do the computation on his end.
You're assuming Alice and Bob have already pre-synchronized what kind of computing machine is going to be used, one in which pi is the output of a relatively short program, as opposed to another type of machine where some other random-looking number has that property (random to you, pseudo-randomly generated via some machinery for all you know). You are assuming many things away.
Also it is absolutely not trivial to extend Kolmogorov complexity to infinite strings. There are multiple formulations and they are a lot more difficult than for finite strings. Not the computation part but the complexity assignment part.
Theres a bunch of fine detail in getting it down to defining an actual number measuring complexity which I don't care about at all, all I care about (in the context of this discussion) is that the number is finite.