Comment by TheOtherHobbes
4 years ago
If spacetime is continuous you effectively get infinite precision -> information density - at every point (of which there are an infinite number).
This seems unlikely for a number of reasons.
This doesn't mean spacetime is a nice even grid, but it does suggest it comes in discrete lumps of something, even if that something is actually some kind of substrate that holds the information which defines relationships between lumps.
> If spacetime is continuous you effectively get infinite precision -> information density - at every point (of which there are an infinite number).
This would be true if objects existed at perfectly local points. However we know that a perfectly localised wavefunction has spatial frequency components that add up to infinite energy. Any wavefunction with finite energy is band-limited. At non-zero temperature the Shannon-Hartley theorem will give a finite bit rate density over frequencies, and since the wavefunction is band limited it will therefore only have the ability to carry a finite amount of information.
This and the comment under it are making my point for me. Relativity assumes spacetime is continuous. Quantum theory implies that quantum phenomena are bandlimited and therefore the information spacetime can hold is limited.
The difference is we know what the quantised components of field theory are. We don't have any idea what the quantised components of spacetime are supposed to be, or how they operate.
The various causal propagation theories (like causal dynamical triangulation) may be the first attempts at this, but it's going to be hard to get further without experiments that can probe that level - which is very difficult given the energies involved. Without that, we're just guessing.
People complain about infinite information, but the theory of real fields is complete and consistent, while arithmetic is not:
https://en.wikipedia.org/wiki/Decidability_of_first-order_th...
I often wonder if we simply constructed math the wrong way around.
People tend to mentally construct the natural numbers from set theory, wholes from naturals, rationals from wholes, reals from rationals and so on.
But what if there is some universe (in the math sense), which is actually complete and decidable, it's just that the moment you take discrete subsets of it, you also remove the connections that make it consistent or complete.
The very act of formalising mathematical concepts into words and paper is a quantisation step after all, because both are symbols. Maybe there are proofs that can be inuitioned about (assuming brains are continuuous in some sense) but neither verbalised nor formalised.
> But what if there is some universe (in the math sense), which is actually complete and decidable, it's just that the moment you take discrete subsets of it, you also remove the connections that make it consistent or complete.
Presumably this hypothetical universe implements arithmetic, so it’s not complete.
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How do you know arithmetic is not complete or consistent? In ZFC arithmetic is complete and arithmetic is consistent. This assumes that ZFC is consistent. The second order Peano axioms are categorical so I assume you mean only the first order theory.
At any rate, what does any of this have to do with information capacity in the universe? Is the information capacity of the universe related to the consistency/completeness of arithmetic?
> How do you know arithmetic is not complete or consistent? In ZFC arithmetic is complete and arithmetic is consistent. This assumes that ZFC is consistent.
Aren’t you just begging the question by assuming ZFC is consistent to demonstrate that arithmetic is consistent?
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> If spacetime is continuous you effectively get infinite precision -> information density - at every point (of which there are an infinite number).
Even if space is continuous, that doesn't mean we can get information in and out of it in infinite precision.
Look at quantum physics. Maxwell's equations don't suggest existence of photons (quantized information). But atoms being atoms, they can only emit and absorb in quanta.
> If spacetime is continuous you effectively get infinite precision -> information density - at every point (of which there are an infinite number).
Without some smallest resolution, we'd need infinite amounts of information to track point particle displacement in one dimension.
Obviously that "something" is the float type used to calculate the simulation. Probably some ultra dimensional IEEE style spec that some CPU vendor intern booked anyways. ;)
Your mistake is that you didn't pony up for the "double" feature in your personal universe.
Im pretty sure my math is right here, so… It would be more appropriate to say they failed to shell out for the 256 bit processor. Because at 256 bits per int a vector of 3 ints can easily encodes any location in the observable universe as Planck length coordinates.
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Care to to enlighten us these number of reasons?