The Banach–Tarski Paradox [video]

https://www.youtube.com/watch?v=s86-Z-CbaHA

by selectnull • 11 years ago

51 36 11 years ago

A couple of months ago I posted this[0]:

    The Point of the Banach-Tarski Theorem
        – not just a curiosity

That spawned a lot of discussion. Indeed, there are many many submissions[1], and some spawn considerable discussion, others are still-born.

The Banach-Tarski theorem is a lovely result, and I look forward to seeing what people say about this new presentation of it.

========

[0] https://news.ycombinator.com/item?id=9674286

[1] https://hn.algolia.com/?query=banach%20tarski&sort=byDate&pr...

gus_massa• 11 years ago

This video is much better than I expected. If you have 25 minutes you can see a friendly presentation of the proof.

[The connection with particle physics at the end I a little too much.]

An important detail is that this video is made by Vsauce. He usually has good videos, but sometimes the connections between the parts are too farfetched.

prezjordan• 11 years ago

It's so refreshing to see a video like this have over a million views. Really love the stuff Vsauce puts out - this might be his best yet.

roflmyeggo2• 11 years ago

It's his passion for the material that always keeps me watching. Listening to someone explain a topic that they find deeply intriguing is a pleasure. Reminds me of Feynman.

msherry• 11 years ago

Q: What's an anagram of "Banach-Tarski"?

A: "Banach-Tarski Banach-Tarski"

acconrad• 11 years ago

This is crazy as I was just thinking about uncountable infinity when I was on the bus this week, having no idea this was part of such a paradox.

The things you consider when you're not distracted by a cell phone...

baddox5• 11 years ago

His initial explanation of uncountable infinity isn't exactly correct, and I'm afraid it will give people the wrong idea. He says that the real numbers are uncountable, because even between 0 and 1 on the number line there are an infinite number of real numbers. But that is also true of the rational numbers, which are countable! After all, what is the smallest rational number larger than 0?

baddox5• 11 years ago

He does go on to explain diagonalization, which is of course a better way to demonstrate the uncountability of the reals.

egonschiele• 11 years ago

If anyone's curious on how it is possible to list all the rational numbers, it would go something like this:

0, 1, -1, 2, -2, 1/2, -1/2, 3, -3, 1/3, -1/3, 2/3, -2/3, 3/2, -3/2, 4 ...

at each step you list the numbers where the numerator and denominator are <= x. For example, if x = 2, we can count 1, -1, 2, -2, 1/2, and -1/2. Obviously it is an infinite list, but you can list them.

baddox5• 11 years ago

My favorite way to visualize the bijection is https://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree.

madez5• 11 years ago

I consider uncountability an artefact of a flawed approach to mathematics. I'd recommend looking into constructive mathematics based on intuitionistic logic. All fruitful insights based on other mathematics can be proven by it, from all of it's results we can easily extract methods and it is much less mystic. I think it is much more fun, too.

baddox5• 11 years ago

I'm aware of constructivism, but what is "intuitionistic logic?" I am under the impression that the real numbers (and their uncountability) are generally accepted by constructivists.

madez5• 11 years ago

Intuitionistic logic is nearly classical logic. That is, we start by assigning statements one of the two values ‘True‘ and ’False’. While truth in the classical sense is abstract, it is concretized in intuitionistic logic with the meaning of ’we can prove it’. We know that there are statements that can neither be proved nor disproved, so we cannot make use of the law of excluded middle "a statement is true or it is not". Add that we are consistent, that is "not (a and not a)" for all a, and then negation must not be the inverse of itself, because we'd be able to proof the law excluded middle otherwise. You see, it is basically classical logic with some minor adaptions to take into account what we’ve learned.

I understand under "constructive reals" the computable reals, and there are only countably many of them.

thetwiceler5• 11 years ago

They're usually used synonymously. Yes, Cantor's diagonal argument can still be used to show the uncountability of real numbers in a constructive setting.

ihm3• 11 years ago

I would take a look at Andrej Bauer's excellent talk "Stages of Accepting Constructive Mathematics"[0]. Essentially, constructive mathematics is mathematics done without the law of excluded middle ("for all propositions P, either P or not P"). Importantly, it does not necessitate that you deny the law of excluded middle.

Intuitionism (or intuitionistic mathematics) is sometimes used to mean the same thing as constructivism and sometimes refers to Brouwer's[1] practice of denying excluded middle by introducing an axiom along the lines of "all functions are computable" or "all functions are continuous" (which from some perspectives is almost the same thing).

[0] https://www.youtube.com/watch?v=zmhd8clDd_Y [1]: https://en.wikipedia.org/wiki/L._E._J._Brouwer

thetwiceler5• 11 years ago

This video is somewhat misleading. I appreciate the attempt at making Banach-Tarski accessible to a general audience, but it dwells on the wrong aspects of what makes Banach-Tarski interesting, making the construction look more like a magic trick with sleight-of-hand. I wish the video had at least mentioned the Axiom of Choice somewhere, as that is fundamentally what Banach-Tarski is about.

The sleight-of-hand comes in around 14:30 into the video, where we are told to create the sequence for an "uncountably infinite number of starting points." That's exactly the point where the construction is non-constructive, and the infamous Axiom of Choice is used. There is no construction - in the sense of constructive mathematics - that can achieve what is described at this point.

Banach-Tarski is not generally regarded as some deep fact about mathematics, a point the video mistakenly belabors. Rather, it is a consequence about particular axiomatizations of set theory which admit the Axiom of Choice. Banach-Tarski is only valid with the Axiom of Choice, and in fact that is the main interest in the paradox.

In my personal opinion, the Banach-Tarski paradox isn't much more enlightening than the simpler construction of the Vitali set (assuming the Axiom of Choice), which is a non-measurable set of real numbers (with Lebesgue measure, i.e., length).

Another part of the video I find misleading has to do with the hyper-dictionary, where he describes the hyper-dictionary by putting some parts of the dictionary "after" other parts which are infinitely long.

The putative applications of Banach-Tarski to physics are ridiculous. Uncountable sets are fundamentally unphysical. The Axiom of Choice serves mainly as a convenience to mathematicians when either a proof avoiding the Axiom of Choice would be more complicated, or so that mathematicians can state properties of objects which are set-theoretically larger than anything that can be relevant to physics anyways.

tome2• 11 years ago

Uncountable sets are fundamentally unphysical.

I think you didn't quite mean to say this, the real numbers being uncountable yet forming the basis for classical physics.

jjoonathan• 11 years ago

I think he did: you don't need real numbers to formulate classical physics. Everything measurable has finite precision so you can always get away without postulating that your limits actually converge to something.

Of course, the reality is that then you would wind up with awkward limit-taking machinery in your answers. Real numbers encapsulate that complexity so you might as well use them to simplify both the notation and manipulation of limits. But you don't need to.

mjh2539• 11 years ago

The set of real numbers is uncountable. There are infinitely many members.

The set of natural numbers is countable. There are infinitely many members.

It would be better to say that infinite sets are fundamentally unphysical, since there are no actual infinities. Talk about uncountable versus countable is tangential.

michaelochurch• 11 years ago

I have to commend Michael Stevens (VSauce) for the courage to take on Banach-Tarski at all. I was surprised that someone could make it that accessible inside of 20 minutes.

I do agree that a "there is mathematical dispute [++] over whether this operation is valid" disclaimer could have accompanied the part where AoC was used.

[++] Side note: "mathematical dispute" doesn't mean what many non-mathematicians think it does. AoC isn't as "controversial" as some make it out to be. There is no disagreement over whether it's "true" or "false" in any Platonic sense. Rather, there is nearly universal acknowledgement that a valid mathematical system exists with AoC and that a valid mathematical system exists without it... and that both have useful properties. Mathematical dispute over an axiom means that a valid mathematics exists with or without the axiom-- not necessarily that mathematicians hold strong and divergent opinions on whether to include it.

As for the physical interpretations, he's right in that nothing that we know about physics rules out a Banach-Tarski-like behavior at the subatomic level. That said, it's obviously impossible to Banach-Tarski an orange or a ball of gold, since we'd have to literally split every atom. A macroscopic Banach-Tarski event is almost certainly impossible and would be, even with some unforeseen physical capability that made it possible, extremely expensive in terms of energy due to mass-energy equivalence.

roflmyeggo2• 11 years ago

Vsauce makes a good point that we just aren't made to intuitively understand this type of stuff. Recognizing this fact, in my opinion, is a key driver in helping to wrap our minds around these concepts. It's important to understand that these concepts are valid in both our visible world and the hidden quantum world, the main difference is scale.

For example, I could never wrap my head around the fact that electrons can have multiple paths/histories simultaneously when travelling. The same is true of a baseball thrown in the air, the only difference is that on the visible scale that we are used to the chance of that baseball taking a different path/history is so small that it will never happen.

ihm3• 11 years ago

I actually think it's very possible to have an "intuitive understanding" of Banach Tarski (I would say I have one, but perhaps we disagree on what is mean by such an understanding).

My "intuitive understanding" of this comes via an intuitive understanding of a paradoxical decomposition of the free group and its Cayley graph, which is flashed briefly in the video here[0] but sadly not discussed at length.

[0] https://www.youtube.com/watch?v=s86-Z-CbaHA&feature=youtu.be...

amelius2• 11 years ago

Infinity is a concept, not a number. Confusing the two is what gets you into trouble. And, unfortunately, it is easy to confuse them because in mathematical notation, infinity is often used in place of a number.

grumpy-buffalo• 11 years ago

There are plenty of senses in which infinity IS a number -- or rather, many numbers. See e.g. the Wikipedia articles on cardinal numbers, ordinal numbers, hyperreal numbers, and surreal numbers.

amelius2• 11 years ago

There are plenty of senses in which [...]

Yes, perhaps. But still it IS not a number.

Calling infinity a number is a "hack" done by mathematicians.

anotheradhoc54• 11 years ago

from layperson's intuition - isnt this just a trick performed by extracting the extra elements from an infinity ? sort of the opposite to losing information through common scaling by zero ? 2 = 1 because (2)0 = (1)0 a matter of defining rules and staying consistent to them. the paradox arises from expecting a contrived model to manifest in physical reality

The Banach–Tarski Paradox [video]

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