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The Point of the Banach-Tarski Theorem

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Re: The Point of the Banach-Tarski Theorem

#21

Post by defrost »

moloch-hai wrote: post #20
ttctciyf wrote: post #15

Earlier quoted context omitted.

I know what you mean, but Euler's identity is a counterexample to what you literally said, isn't it?

That is a symbolic manipulation. Wherever it comes down to actual numbers, you use adequate approximations to infinite summations for x and iy. Even nominally exact rational values are often idealizations of measurements: your house has no actual right angles, but eh, close enough.

I absolutely agree with you from the outset . . .

. . . and then I start thinking about that damn Bailey–Borwein–Plouffe formula.

Although ( handwave ) all the transcendentalness is magically manipulated away leaving a simple computation for the immediate reveal of any arbitrary n-th (hexadecimal) digit of π

But surely that's over simplification?

Is this magic by wizards?

Re: The Point of the Banach-Tarski Theorem

#22

Post by moloch-hai »

defrost wrote: post #21
moloch-hai wrote: post #20

Earlier quoted context omitted.

That is a symbolic manipulation. Wherever it comes down to actual numbers, you use adequate approximations to infinite summations for x and iy. Even nominally exact rational values are often idealizations of measurements: your house has no actual right angles, but eh, close enough.

I absolutely agree with you from the outset . . . . . . and then I start thinking about that damn Bailey–Borwein–Plouffe formula. Although ( handwave ) all the transcendentalness is magically manipulated away leaving a simple computation for the immediate reveal of any arbitrary n-th (hexadecimal) digit of π But surely that's over simplification? Is this magic by wizards?

You have an abstract definition of a sequence of digits, and an algorithm that reveals an actual digit from among them. It is surprising that you don't need to approximate an infinite series to get it, but everybody agrees that digit would show up there if you did one. You would still need to do an infinite amount of work to get the rest of them.

It is magic of a kind by wizards of a kind, or anyway indistinguishable from it.

Re: The Point of the Banach-Tarski Theorem

#23

Post by gpm »

Someone wrote: post #17
ttctciyf wrote: post #15

Earlier quoted context omitted.

I know what you mean, but Euler's identity is a counterexample to what you literally said, isn't it?

I don’t think so. Programs that use numbers approximate e and π. It’s only programs that manipulate symbols that can use their exact values.

All programs manipulate symbols. All bits are is symbols.

It's just sometimes we use those symbols to represent a subset of the integers (e..g. with the popular binary notation)... or a subset of the rationals (e.g. with the popular floating point notation)... or a subset of the reals that happens to include a transcendental number because we decided that some symbol (or combination of symbols) represents some particular transedental number.

Re: The Point of the Banach-Tarski Theorem

#24

Post by ttctciyf »

moloch-hai wrote: post #20
ttctciyf wrote: post #15

Earlier quoted context omitted.

I know what you mean, but Euler's identity is a counterexample to what you literally said, isn't it?

That is a symbolic manipulation. Wherever it comes down to actual numbers, you use adequate approximations to infinite summations for x and iy. Even nominally exact rational values are often idealizations of measurements: your house has no actual right angles, but eh, close enough.

> Wherever it comes down to actual numbers

Wait, are you saying e and pi aren't "actual numbers"?

Re: The Point of the Banach-Tarski Theorem

#25

Post by camjw »

moloch-hai wrote: post #20
ttctciyf wrote: post #15

Earlier quoted context omitted.

I know what you mean, but Euler's identity is a counterexample to what you literally said, isn't it?

That is a symbolic manipulation. Wherever it comes down to actual numbers, you use adequate approximations to infinite summations for x and iy. Even nominally exact rational values are often idealizations of measurements: your house has no actual right angles, but eh, close enough.

It feels to me like you're redefining what a number is to be very different to what anyone with a maths background would say a number is. Essentially you're saying that neither e nor pi are numbers?

Re: The Point of the Banach-Tarski Theorem

#26

Post by soVeryTired »

Mathoverflow has some good discussion of Banach-Tarski [0]. The top-rated posts argue that

- The 'problem' with B-T may be related to our notion of "space" (i.e. point-set topology) rather than any issues with the axiom of choice

- Most practical uses of the axiom of choice could get by with the axiom of countable choice, under which B-T doesn't hold

- Mathematics need not directly model the 'real' world or match our physical intuition

[0] https://mathoverflow.net/questions/260057/axiom-of-choice-ba...

Re: The Point of the Banach-Tarski Theorem

#27

Post by defrost »

moloch-hai wrote: post #22
defrost wrote: post #21

Earlier quoted context omitted.

I absolutely agree with you from the outset . . . . . . and then I start thinking about that damn Bailey–Borwein–Plouffe formula. Although ( handwave ) all the transcendentalness is magically manipulated away leaving a simple computation for the immediate reveal of any arbitrary n-th (hexadecimal) digit of π But surely that's over simplification? Is this magic by wizards?

You have an abstract definition of a sequence of digits, and an algorithm that reveals an actual digit from among them. It is surprising that you don't need to approximate an infinite series to get it, but everybody agrees that digit would show up there if you did one. You would still need to do an infinite amount of work to get the rest of them. It is magic of a kind by wizards of a kind, or anyway indistinguishable…

Just to be clear, for any third party spectators, the surprise isn't that more work is required to get more digits .. the WTF moment for some is that no matter how large N is there's no need to do an increasing amount of work as N grows (there's no cost to "skipping to (not quite) the end").

It's another of those intuition challenging moments in math.

Re: The Point of the Banach-Tarski Theorem

#28

Post by gerdesj »

karmakaze wrote: post #12

I'm glad to have found this post. I discovered the Banach-Tarski theorem via Vsauce[0]. It was interesting but I couldn't get the significance of it. It either didn't seem like an unexpected result or too esoteric to appreciate. There's phrasing in the post that could be misunderstood (later clarified) but can leave unclarity from assumed understanding of the earlier description. > In R3, given a solid ball B of radi…

It is a remarkable theorem but you can't take a snooker ball and turn it into two snooker balls without being Paul Daniels (UK magician).

This is an excellent example of language going astray and confusing mathematic rigour with some sort of "reality". You can loosely model a "sphere in R3" with a snooker ball in errr the universe thingie which is probably R3ish or perhaps R3T1 or whatevs. Besides someone has spilt a whole pint of Guinness on the table and the balls are chipped.

Even if we get our match balls out that are not chipped, and are jolly shiney, they are still subject to things like Mr Planck's constant and the fine structure of matter.

The point is quite literally in the title - don't balls it up!

Re: The Point of the Banach-Tarski Theorem

#29

Post by moloch-hai »

ttctciyf wrote: post #24
moloch-hai wrote: post #20

Earlier quoted context omitted.

That is a symbolic manipulation. Wherever it comes down to actual numbers, you use adequate approximations to infinite summations for x and iy. Even nominally exact rational values are often idealizations of measurements: your house has no actual right angles, but eh, close enough.

> Wherever it comes down to actual numbers Wait, are you saying e and pi aren't "actual numbers"?

I am not a Platonist, or any sort of theist.

e and pi arise in axiomatic systems we use to approximate our world. They never appear in nature. They do appear in formulas we find to approximate details of our world.

I say "actual numbers" to mean "numbers that refer to actual quantities or measures that can be taken". You might calculate that a stick must be exactly 1/pi meters long, but you will make the stick no better than 113/355 meters long.

Re: The Point of the Banach-Tarski Theorem

#30

Post by dkural »

soVeryTired wrote: post #26

Mathoverflow has some good discussion of Banach-Tarski [0]. The top-rated posts argue that - The 'problem' with B-T may be related to our notion of "space" (i.e. point-set topology) rather than any issues with the axiom of choice - Most practical uses of the axiom of choice could get by with the axiom of countable choice, under which B-T doesn't hold - Mathematics need not directly model the 'real' world or match our…

Thank you for the summary and the link. I'd also add that our physical intuition also need not match the "real world", it's a model "good enough" for evolutionary purposes that fails in myriad ways.
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