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Banach-Tarski and the Paradox of Infinite Cloning

quantamagazine.org

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Re: Banach-Tarski and the Paradox of Infinite Cloning

#101

Post by tsimionescu »

ithkuil wrote: post #90
tsimionescu wrote: post #79

Earlier quoted context omitted.

I'm not sure that it could be, actually. You can't use a finite amount of evidence to verify that something is infinite, so any infinity can always be replaced with a huge (or minuscule) number and the theory would make the same measurable predictions.

I'm not talking about proving that there exist infinite things. I'm talking about using the abstract concepts of infinity as a useful mathematical tool to produce predictions. Notable example: calculus

Sure, but calculus makes infinity sufficient, but not necessary for describing the physical world. Integers, rationals, and apparently complex numbers (presumably those with rational components) are actually necessary for describing the physical world, given our current understanding. Irrational numbers and infinities are extremely useful, but not strictly necessary.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#102

Post by Tomte »

inetsee wrote: post #94

Can someone explain the flaw in my reasoning here? Assume I have a sphere made of pure iron. I divide the sphere into individual iron atoms. I divide this group of atoms into two groups of atoms. I take each of those groups of atoms and form them into 2 spheres. How is it that these two new spheres are not either less dense or smaller that the original sphere?

> I divide the sphere into individual iron atoms.

You have highly restricted the act of choosing sets of points here. B-T doesn't say that any "division" results in that unintuitive outcome.

Note that points are infinitesimally small and infinitely many, and atoms in your iron sphere are neither.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#103

Post by snet0 »

tsimionescu wrote: post #100
rytill wrote: post #69

Earlier quoted context omitted.

As the other comment implied, infinity and exactitude are two sides of the same coin. Exactitude is infinite precision. No finite amount of empirical evidence can afford infinite precision, so you’re back in math-land.

That's true, though I have a hard time squaring that with observations of classical objects. Would it make sense to say that I have approximately 2 legs, or can I actually be confident in saying I have exactly two legs? Even with things like MWI, in any particular world I would still have an exact integer amount of legs, as far as I understand. Perhaps the problem here is one of mixing intuition (the idea of 'an obje…

We have a very precise (although one might struggle to describe it) idea of what qualifies as a human leg. The number 2 is basically defined (at least in common usage) as the number of things you have when you have one thing and then another thing. I'd point out that, as you mentioned, this is a different level of abstraction to physics and mathematics.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#105

Post by danparsonson »

hprotagonist wrote: post #51
danparsonson wrote: post #34

Earlier quoted context omitted.

OP didn't argue that finite numbers are physical objects, they said that infinities are not present in the universe. For example, I could in theory hand you 7 electrons but there are not infinity electrons for me to hand to you.

it's OK. electrons don't "exist" discretely, either. At best, when you "hand me 7 electrons", you're directing me towards the fat part of 7 probability distributions, so we're back to math again...

Well that's a bold assertion about a theory with whose implications we are still grappling; electrons may not be point-like entities but they are nonetheless quantifiable 'packets' of energy, are they not?

Re: Banach-Tarski and the Paradox of Infinite Cloning

#106

Post by tsimionescu »

snet0 wrote: post #99
tsimionescu wrote: post #96

Earlier quoted context omitted.

I was talking about exactitude in general, as a requirement for a physical interpretation of the integers. > I think it is not impossible for us to prove that the universe was created by a god, if we found some hidden message in subatomic particles or cosmic dust or something. That's actually a good point, there could be scientific proof of some intelligent creator in principle. The fact that there is no reason a pri…

Exactitude in general is impossible, though. We assume that the charge on an electron is a constant, but there are limits to the precision. An infinite value is theoretically testable. It simply implies that for however long you make your ruler, the value is larger. That is a prediction. You may not reach a conclusion, as you said, in finite time, but that is still a prediction. The problem with replacing an infinity…

> Exactitude in general is impossible, though. We assume that the charge on an electron is a constant, but there are limits to the precision.

Yes, others have pointed that out and I am in fact conflicted right now.

> The problem with replacing an infinity in a model with an arbitrarily large number is that, given enough time and a long enough ruler, you'll surpass that number, meaning your model is incorrect. In defence of "science", you're adding an arbitrary number into a model that you expect to be incorrect. That's not how it should work.

I'm not against using infinities in scientific practice at all. I'm just pointing out that, when it comes down to it, that infinity is never necessary in the logical sense.

> If the model says there's a singularity, we don't then say "okay but well that clearly doesn't make sense, so put a limit on the formula that clamps the values to uh 10^45". That is unscientific.

Sure, picking some random big number would be unscientific. But saying "the model predicts a singularity or growing to infinity, so we're probably missing some piece of the picture that sets an upper bar" is not unscientific. It is in fact the common practice - just like no one believes that black holes or the early universe had an actual singularity at the center, we normally just believe the models break somewhere at those levels, and more powerful models (quantum gravity) will actually put a cap. Or how we keep saying that we know an upper bound for the possible mass of a photon, but don't actually know that it really is 0, and we keep trying to measure it.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#107

Post by tsimionescu »

snet0 wrote: post #103
tsimionescu wrote: post #100

Earlier quoted context omitted.

That's true, though I have a hard time squaring that with observations of classical objects. Would it make sense to say that I have approximately 2 legs, or can I actually be confident in saying I have exactly two legs? Even with things like MWI, in any particular world I would still have an exact integer amount of legs, as far as I understand. Perhaps the problem here is one of mixing intuition (the idea of 'an obje…

We have a very precise (although one might struggle to describe it) idea of what qualifies as a human leg. The number 2 is basically defined (at least in common usage) as the number of things you have when you have one thing and then another thing. I'd point out that, as you mentioned, this is a different level of abstraction to physics and mathematics.

I would also need to know more QM. I don't know if the theory would actually allow a small fraction of an electron or an electron and a bit to actually exist - common descriptions suggest that it wouldn't. If it doesn't, then electrons could be counted just as much as mathematical objects and legs.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#108

Post by danparsonson »

layer8 wrote: post #84
danparsonson wrote: post #34

Earlier quoted context omitted.

OP didn't argue that finite numbers are physical objects, they said that infinities are not present in the universe. For example, I could in theory hand you 7 electrons but there are not infinity electrons for me to hand to you.

But the electron field has different values at different points in spacetime, and we have no evidence that either the number of points (locations) or the number of different possible values at those points is finite. Unless we posit that they are finite in number, infinity is quite present in the universe.

Would not an infinite electron field in a finite (observable) universe result in an infinite energy density and therefore the entire universe would collapse into a black hole?

Re: Banach-Tarski and the Paradox of Infinite Cloning

#109

Post by snet0 »

tsimionescu wrote: post #106
snet0 wrote: post #99

Earlier quoted context omitted.

Exactitude in general is impossible, though. We assume that the charge on an electron is a constant, but there are limits to the precision. An infinite value is theoretically testable. It simply implies that for however long you make your ruler, the value is larger. That is a prediction. You may not reach a conclusion, as you said, in finite time, but that is still a prediction. The problem with replacing an infinity…

> Exactitude in general is impossible, though. We assume that the charge on an electron is a constant, but there are limits to the precision. Yes, others have pointed that out and I am in fact conflicted right now. > The problem with replacing an infinity in a model with an arbitrarily large number is that, given enough time and a long enough ruler, you'll surpass that number, meaning your model is incorrect. In defe…

I think we're not disagreeing, if we were before. Adding limits to functions without rationale other than avoiding infinities is arbitrary and unscientific. Given that we assume that the universe doesn't actually have the capacity for infinities (is this a scientifically grounded assumption, given the impossibility in measuring them?), finding them points us towards holes or limitations in the model, e.g. GR vs quantum gravity.
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