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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

#21

Post by Ericson2314 »

boxfire wrote: post #13
Ericson2314 wrote: post #11

I highly recommend https://twitter.com/andrejbauer/status/1428471658088738818 and follow ups. Yes, this stuff is fishy, and yes we can blame ZFC which is a bad formalization in comparison to what we've developed since. But the real scandal is why does our definition of geometry "leak" the underlying set theory it's built atop so much? Surely it's bad to have such a leaky abstraction in pure math! The series goes on t…

> But who's going to work without AC (other than crazy HoTT people)? Yeah... Those crazy HoTT people, trying to actualize the goal of putting mathematics on an actually firm foundation and removing the rest of the gobblygook handwaved into the religion of math as opposed to the pure logic it represents... Also you can use HoTT WITH AC / law of excluded middle... It's just not there by default and there are some reall…

Andreij Bauer is one of those HoTT people, so this is quite tongue-in-cheek.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#22

Post by jhgb »

desperate wrote: post #3

To me this is proof that infinity is something only present in our math and not in the universe. Infinity is a nice approximation but it feels like wishful thinking that our universe or anything in it is infinite. Happy to hear disagreements tho.

> infinity is something only present in our math and not in the universe

This is true of all mathematical objects. The number 7 doesn't exist in the universe either. It's not a physical object.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#23

Post by bobthechef »

paulpauper wrote: post #15

I don't understand the paradox. Obviously if you dissaemble or scamble something u can reararange it?

When's the last time you came across something that you could disassemble and could then reassemble into two things identical to the first thing you disassembled?

It is natural to suspect that foundational axioms are somewhere flawed.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#24

Post by [deleted] »

jhgb wrote: post #22
desperate wrote: post #3

To me this is proof that infinity is something only present in our math and not in the universe. Infinity is a nice approximation but it feels like wishful thinking that our universe or anything in it is infinite. Happy to hear disagreements tho.

> infinity is something only present in our math and not in the universe This is true of all mathematical objects. The number 7 doesn't exist in the universe either. It's not a physical object.

[deleted]

Re: Banach-Tarski and the Paradox of Infinite Cloning

#26

Post by tsimionescu »

canjobear wrote: post #18
tsimionescu wrote: post #6

Earlier quoted context omitted.

Well, infinity is inherently unscientific, as there is no way to scientifically differentiate between an infinite quantity and a really huge quantity (same for infinitesimals), in finite time. What this means is that a universe that contains infinities is, even in theory, entirely indistinguishable (in finite time) from an universe that contains really large/small but finite quantities.

I don’t see how this makes infinity “unscientific.” Infinity is part of the language of mathematics. It’s no more scientific or unscientific than the definition of matrix multiplication. Also wouldn’t your argument also apply to zero? You can never know if a quantity is zero as opposed to some enormously small epsilon that you haven’t detected yet. Is zero “unscientific?”

Well, I can say that there are precisely 0 african elephants in the room with me right now, so no, 0 and other integers don't have this problem. Similarly, the rationals are clearly realizable with perfect precision.

The reals however are a different problem, and it's not scientifically possible to prove that the ratio between the length and radius of any object is exactly pi (that it is a perfect circle). However, it's also impossible to prove scientifically that it is 3 or 3.14 or any other number.

Now my use of "unscientific" is more of a hyperbole or click-bait. I thought I explained my actual claim pretty well - that you can't measurably/scientifically distinguish between a universe that contains actual infinities and one that only contains some arbitrarily large numbers.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#27

Post by Sinidir »

Can someone correct me if im wrong?

What i see here is a splitting of the set of points in the sphere? However the set of points in the sphere is not really the sphere. A point has no volume so no matter how many you add together you don't get something with a volume. This seems more akin to splitting the natural numbers into odd and even numbers which are all equally large.

The language that i see in this article and elsewhere however is suggesting that we actually duplicated the sphere (doubled the volume).

This seems incorrect.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#28

Post by pacifist »

Sinidir wrote: post #27

Can someone correct me if im wrong? What i see here is a splitting of the set of points in the sphere? However the set of points in the sphere is not really the sphere. A point has no volume so no matter how many you add together you don't get something with a volume. This seems more akin to splitting the natural numbers into odd and even numbers which are all equally large. The language that i see in this article an…

A sphere in the mathematical sense is the 2D shell on the surface of the sphere you're thinking of. So, no height, no volume.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#29

Post by Ericson2314 »

mjw1007 wrote: post #20
Ericson2314 wrote: post #11

I highly recommend https://twitter.com/andrejbauer/status/1428471658088738818 and follow ups. Yes, this stuff is fishy, and yes we can blame ZFC which is a bad formalization in comparison to what we've developed since. But the real scandal is why does our definition of geometry "leak" the underlying set theory it's built atop so much? Surely it's bad to have such a leaky abstraction in pure math! The series goes on t…

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 foundations, and it is up to usflawed humans to decide what we like to work with best.

The earlier intuitionists like Brouwer were polemicists, perhaps because they felt very alone. Now there is a good network of CS-mathematician hybrids to keep everyone feeling more sane.

Here we see the dual track that you can question your foundational choices and your higher level abstractions (point-set topology vs locales which are distilled to being purely order-theoretic) concurrently. It's nice to take the same skepticism and interest in finding definitions the work with not alienste the working mathematician at multiple levels.

Because, for all the trepidation about abandoning ZFC, 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.

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