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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

#11

Post by Ericson2314 »

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 to show that by abandoning "points" — which pull all the funny set theory stuff into geometry/topology/whatever is the topic at hand, one can still have a classical foundation — e.g. with the axiom of choice and law of excluded middle — that makes mathematicians feel at ease, but also purge this Banach–Tarski gobbledygook.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#12

Post by tsimionescu »

turminal wrote: post #7
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.

> there is no way to scientifically differentiate between an infinite quantity and a really huge quantity (same for infinitesimals), in finite time That depends on the model of computation you pick, doesn't it?

Only if you want to think about models of computation that allow performing infinite operations in finite amounts of time, which I don't think are that interesting.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#13

Post by boxfire »

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 really nice things you get without it, so it's pretty much only the lazy crutch of mathematics since forever. If you see proof via excluded middle, consider it a code smell (and recall by the Curry-Howard correspondence the proof is essentially code)

Re: Banach-Tarski and the Paradox of Infinite Cloning

#14

Post by ithkuil »

tsimionescu wrote: post #6
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.

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.

What about negative numbers? Or complex numbers? They are only tools, which can be quite useful to build models of the world with predictive powers but shouldn't be confused for the underlying reality.

Even whole numbers are an abstraction that makes sense only when you can clearly define what is the thing you're counting.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#16

Post by creata »

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…

You're exaggerating the significance of HoTT. Mathematics is already on a pretty firm foundation. And I somehow doubt Bauer meant any ill intent with that line...[0][1]

[0]: https://en.wikipedia.org/wiki/Homotopy_type_theory#Special_Y...

[1]: http://math.andrej.com/2016/10/10/five-stages-of-accepting-c...

Re: Banach-Tarski and the Paradox of Infinite Cloning

#17

Post by mjw1007 »

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…

For the avoidance of doubt: Andrej Bauer is himself very much one of those crazy HoTT people.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#18

Post by canjobear »

tsimionescu wrote: post #6
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.

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?”

Re: Banach-Tarski and the Paradox of Infinite Cloning

#19

Post by bobthechef »

robot_no_419 wrote: post #8
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.

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 clarify: mathematics is a field that studies such things.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#20

Post by mjw1007 »

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 showing some of the places where the formalisation we're starting with isn't a great fit for some things we might hope to use it for.

I don't think I'd go as far as to say that makes the formalisation outright bad, but looking at alternate systems which don't admit Banach-Tarski-like results is surely a worthwhile way of spending time.

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