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Banach-Tarski and the Paradox of Infinite Cloning

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Re: Banach-Tarski and the Paradox of Infinite Cloning

#41

Post by snet0 »

tsimionescu wrote: post #26
canjobear wrote: post #18

Earlier quoted context omitted.

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, i…

That's just because you use the word "exact", though. Exactitude doesn't exist in the universe as we understand it.

There's a difference between something not being instantiated in this universe and being unscientific, though.

If we produce a model of the universe that doesn't make a single incorrect prediction given all data available, and it predicts infinities to exist in some strange but quite real cases, is it unscientific?

Re: Banach-Tarski and the Paradox of Infinite Cloning

#42

Post by thaumasiotes »

ouid wrote: post #38
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…

"no matter how many you add together", is where this argument breaks down in ZFC. The sphere is indeed the union of all of the singletons consisting of its points, all of which are measure zero. Banach-Tarski is mainly considered "weird" because it describes a partition into so few pieces, and they are rearranged via rigid motions only. It is trivial to come up with bijections between compact finite dimensional manif…

> "no matter how many you add together", is where this argument breaks down

Interpret it as "adding more points will not necessarily increase the volume, no matter how many points you add". There are plenty of measure-0 sets containing as many points as the continuum does.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#43

Post by ajuc »

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 point has no volume so no matter how many you add together you don't get something with a volume

Not true. If you add uncountably many infinitesimal objects they can add up to noninfinitesimal object, that's how integration works in math, it's pretty confusing cause there's many kinds of infinity and they allow some unintuitive things to happen, but if they didn't worked we couldn't move (see Zeno paradox).

Banach-Tarski is formally correct, you add a finite number of sets with uncountably many points in each so you can get something with volume (depending on how they are positioned).

And yes - a line in math is just a set of points, same with a sphere (but it has 0 volume cause a sphere is just the "skin" without the insides) and a ball (which is what Banach-Tarski talks about). In fact every geometric object is just a set of points.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#44

Post by snet0 »

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.

Tell that to Plato.

Honestly I think that's a continuous claim, and comes down to differences in understanding. I can certainly have 7 of some object, does the 7-ness exist in the collection? Not really, but what about another phenomenon: colour? An object appears blue, and we say it is blue, and the blueness is due to physics, but it's a subjective delineation. A table is a delineation too, the leg is part of the table and the White House is not. In some sense, the table-ness category is just as real as the 7-ness category.

Of course you could just say that all that actually exists is some collection of particles/fields, but then you've abused all the words we're using until they stop being useful.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#45

Post by thaumasiotes »

thaumasiotes wrote: post #39
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…

> 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. It isn't incorrect. You're right that the number of points in the sphere does not equate to the volume of the sphere. But the Banach-Tarski theorem does in fact let you double the volume. It is considered to be of interest because it does the following: 1.…

Come to think of it, the fact that two spheres contain the same number of points as one sphere does would seem to be closely related to why it's possible to produce two spheres from one sphere just by rearranging the points.

You can obviously produce a large sphere from a small sphere by rearranging the points, as long as you're willing to handle one point at a time -- that's what scaling is. But that requires an uncountably infinite number of translations. The Banach-Tarski theorem says we can do the same thing in only a finite number of translations.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#46

Post by [deleted] »

thaumasiotes wrote: post #45
thaumasiotes wrote: post #39

Earlier quoted context omitted.

> 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. It isn't incorrect. You're right that the number of points in the sphere does not equate to the volume of the sphere. But the Banach-Tarski theorem does in fact let you double the volume. It is considered to be of interest because it does the following: 1.…

Come to think of it, the fact that two spheres contain the same number of points as one sphere does would seem to be closely related to why it's possible to produce two spheres from one sphere just by rearranging the points. You can obviously produce a large sphere from a small sphere by rearranging the points, as long as you're willing to handle one point at a time -- that's what scaling is. But that requires an unc…

[deleted]

Re: Banach-Tarski and the Paradox of Infinite Cloning

#47

Post by thaumasiotes »

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…

> 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 follow. You can view the Intermediate Value Theorem as something that motivates the definition of "continuous function", so that once you have the definition it had better conform to the theorem, sure.

But the Banach-Tarski theorem isn't like that. It's just a cool result of some other things that work well. It's not motivating anything or being motivated by anything.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#48

Post by Ericson2314 »

thaumasiotes wrote: post #47
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…

> 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'…

Banach-Tarski is quite arguably a red flag that all these non-measursble, non-open sets are barking up the wrong tree.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#49

Post by jhgb »

danparsonson wrote: post #34
jhgb wrote: post #22

Earlier quoted context omitted.

> 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.

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.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#50

Post by mjw1007 »

thaumasiotes wrote: post #47
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…

> 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'…

What I mean is: if you imagine someone drawing up a requirements document for the team assigned to the task of axiomatising geometry, and somebody asked "Do we want our model of geometry to support cutting up a ball into five pieces, moving the pieces rigidly, and reassembling them into two copies?", I think their first idea would be to answer "no".

So it isn't parallel to the intermediate value theorem, but opposite to it.

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