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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

#111

Post by bigbillheck »

Ericson2314 wrote: post #29
mjw1007 wrote: post #20

Earlier quoted context omitted.

I remember sitting in maths lectures and wishing that when they did thing like prove the intermediate value theorem they'd make it clearer that what was going on wasn't so much "We're rigorously proving that this thing that seems obvious is true" as "We're checking that the formalisation we introduced earlier is fit for purpose". I think things like the Banach-Tarski theorem are the other side of that coin: they're s…

Ultimately this crowd wants to change the practice of mathematics in the real world, so they are very accomiadating. See https://golem.ph.utexas.edu/category/2021/06/large_sets_1.ht... for tackling the "large cardinal pissing contest" that is much of modern set theory. Your very statement is a good retreat from platonism with blinders, acknowledging the inherit "moral relativism" that there are many possible foundati…

> Ultimately this crowd wants to change the practice of mathematics in the real world, so they are very accomiadating

PhD mathematician in industry here. The way I see it, foundations is to the rest of mathematics the way music theory is to music: it needs to be a describer, not a prescriber. (If I were less charitable I'd have said "ornithology is to birds").

> the mainstream formalizations have clearly failed in that mathematicians that aren't logicians or set theorists would rather engage with them as little as possible

On the contrary, ZFC has been a tremendous success in that most mathematicians don't need to worry about it at all.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#112

Post by layer8 »

danparsonson wrote: post #108
layer8 wrote: post #84

Earlier quoted context omitted.

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?

Integrals over a finite interval can have (and often do have) a finite size even though the interval contains an infinite number of points, with an infinite number of different values at those point.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#113

Post by throwawaygal7 »

bobthechef wrote: post #19
robot_no_419 wrote: post #8

Earlier quoted context omitted.

Mathematics doesn't really "exist" in the first place. It's more of a language that's rich enough and with enough logic to describe/approximate the laws of physics that actually do exist.

So quantity, structure, formal necessity don't exist? Mathematics has nothing to do with laws of physics. Even if the laws of physics[0] were different, these mathematical[1] truths would remain the same. [0] Laws of physics don't actually exist. They're shorthand generalizations about features of particulars. The notion of some kind of abstract disembodied "laws" that somehow "govern" everything is absurd. [1] For c…

I have never understood the opposition to the law of physics you seem to hold.

What is the alternative? That we merely observe rigid patterns that are baked into physical reality? Isn't whatever is 'baked in' more or less a 'law of physics'?

If these are just 'brute facts' are they not then 'laws'? Maybe governance is too strong an word for the correspondence but what is the alternative?

Re: Banach-Tarski and the Paradox of Infinite Cloning

#114

Post by danparsonson »

jhgb wrote: post #49
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.

That sounds like a weird interpretation of "to be present in the universe" to me. Also I was under the impression that it's unknown whether the universe contains an infinite number of electrons or not.

Weirder than 'numbers are physical objects'? Can you suggest a more suitable interpretation of the OPs intent?

Re: Banach-Tarski and the Paradox of Infinite Cloning

#115

Post by throwaway81523 »

bigbillheck wrote: post #111
Ericson2314 wrote: post #29

Earlier quoted context omitted.

Ultimately this crowd wants to change the practice of mathematics in the real world, so they are very accomiadating. See https://golem.ph.utexas.edu/category/2021/06/large_sets_1.ht... for tackling the "large cardinal pissing contest" that is much of modern set theory. Your very statement is a good retreat from platonism with blinders, acknowledging the inherit "moral relativism" that there are many possible foundati…

> Ultimately this crowd wants to change the practice of mathematics in the real world, so they are very accomiadating PhD mathematician in industry here. The way I see it, foundations is to the rest of mathematics the way music theory is to music: it needs to be a describer, not a prescriber. (If I were less charitable I'd have said "ornithology is to birds"). > the mainstream formalizations have clearly failed in th…

Imho, software people should study foundations (particularly proof theory) much more than they do. That is how you ensure correct code, after all. The people deeply involved in software verification are basically logicians. Also, the test suite for the HOL Light proof assistant (used in software verification) uses a large cardinal axiom, sort of. It uses a version of itself with the large cardinal added, to prove the consistency of the normal version without the large cardinal. Neither version can prove itself consistent, because of Gödel's theorem, but the one with the large cardinal can prove the consistency of the one without it.

One can say that if either is inconsistent then they can prove everything, but that makes it even sharper: the large cardinal is used purely to give engineering assurance of software correctness and not real mathematical rigor. So it's a pure engineering use of one of the most "out there" mathematical objects. It doesn't seem worse than using IEEE floating point arithmetic to design airplanes....

Re: Banach-Tarski and the Paradox of Infinite Cloning

#116

Post by hodgesrm »

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?

Your sets are finite. B-T depends on properties of infinite sets.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#117

Post by danparsonson »

layer8 wrote: post #112
danparsonson wrote: post #108

Earlier quoted context omitted.

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?

Integrals over a finite interval can have (and often do have) a finite size even though the interval contains an infinite number of points, with an infinite number of different values at those point.

Right, because the integral of a function is not a straight sum of values of that function evaluated for every number in the interval; the integral of y=x dx for 0<=x<=1 is not 0+0.1+0.11+0.111+0.1111+...+1. Electrons have a fixed energy, so cramming an infinite number of them into a finite space necessarily requires infinite energy.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#118

Post by ncmncm »

I was initially appalled by Banach-Tarski.

Looking into it more closely, it turned out to be both trivial and not notably meaningful, like most surprising results involving uncountable infinity. Nothing that affects us involves actual infinities, so infinities are just a convenient approximation that often produces correct-enough answers. Anything infinities imply that seems crazy trivially is.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#119

Post by perl4ever »

Until recently I never questioned the idea that, say, the positive integers and the odd positive integers are equivalent because they can be paired, but this cloning thing seems like something that falls out of that. And it seems like that view of infinity isn't actually necessary if Cantor style cardinality is not the last word.

In the paragraph on nonstandard analysis in the Wikipedia page on infinity, it says:

"The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers"

https://en.wikipedia.org/wiki/Infinity

I can't say anything precise or mathematical, but after I read the above, I have an "obvious in hindsight" feeling. If H=inf is different from H + 1, how much different is it? 1/inf or an infinitesimal amount! And an infinitesimal is not nothing.

The quanta article says "You can add or subtract any finite number to infinity and the result is still the same infinity you started with" but this seems like just a dogma for non mathematicians?

Re: Banach-Tarski and the Paradox of Infinite Cloning

#120

Post by bigbillheck »

ncallaway wrote: post #64
thaumasiotes wrote: post #57

Earlier quoted context omitted.

The whole idea of a proof system is that there are some things you can't have without also having other things. The Banach-Tarski theorem is a consequence of things we want. You don't get to pick and choose everything at once.

> The Banach-Tarski theorem is a consequence of things we want Is it? I think the parent comment is saying: “maybe we shouldn’t want things that result in Banach-Tarski” Maybe it’s a hint that the underlying axioms we’ve selected aren’t exactly what we want. You’re right that we can’t pick and choose the results of our axioms, but we do explicitly get to pick and choose the axioms we start with. If we choose bad axio…

> Is it? I think the parent comment is saying: “maybe we shouldn’t want things that result in Banach-Tarski”

OK, which axiom do you want to replace?

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