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I asked a mathematician about what a wacky conclusion it is. He said that whenever you allow infinity, you get results like that. It relies on uncountably-infin

I asked a mathematician about what a wacky conclusion it is. He said that whenever you allow infinity, you get results like that. It relies on uncountably-infinite division of an object, which corresponds to no real-world experience anywhere in the universe. Real objects have, you know, atoms. We use real numbers a lot, but we are careful never to rely on their more extreme properties anywhere it would matter. In practice, in fact, we use floating-point numbers, not reals, when doing actual calculations, and use numerical analysis to stay well clear of nonsensical results. If you tried to rely on BT in a real calculation, you would find a lot of NaNs and Infs.

Aicy 4y ago
In the real world we have atoms, but atoms are disturbances in the wave function of the universe, which may not be discrete. We know atoms pop in and out of existence in matter anti-matter pairs. Maybe the universe is performing Banach Tarski under the hood, to make something from nothing? > we use floating-point numbers, not reals, when doing actual calculations I don't believe only calculations made by computers to be "actual calculations". Humans were calculating thousands of years before computers were invented, using integer or real numbers (e.g pi).
- moloch-hai 4y ago
  No human or computer has ever used an actual real, transcendental number in any computation. It would take infinitely long.
  - ttctciyf 4y ago
    I know what you mean, but Euler's identity is a counterexample to what you literally said, isn't it?
    - Someone 4y ago
      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.
      - gpm 4y ago
        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.
    - moloch-hai 4y ago
      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.
      - defrost 4y ago
        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?
        moloch-hai 4y ago
        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.
        defrost 4y ago
        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.
        moloch-hai 4y ago
        It's the wackiest result I know of, and seems overwhelmingly more profound than BT.
- deleted 4y ago
  [deleted]
- CJefferson 4y ago
  Banach Tarski, in some sense, really can't make sense in the universe because the Universe is... discreteish, as you can't get smaller than the Planck Lenth (1.6 x 10^(-35)), and while that's very small, as far as maths is concerned that means the number of "identifiable points" in a sphere of any finite size is itself finite, not even a countable infinity. I realise that's all very dodgy, but of course we are also bringing in quantum mechanics -- but I consider it enough to say Banach Tarski doesn't apply :)
  - lodi 4y ago
    My understanding is that Planck Length is not a limit to the smallest possible thing; it's a limit to the most precise possible measurement. So there could in fact be lots of things happening at sub-Planck Length scales, but we could never observe them.
whatshisface 4y ago
Atoms move according to path integrals which would be at least countably infinite in number even if time and space were discrete because they are allowed to loop back on themselves.
zokier 4y ago
> It relies on uncountably-infinite division of an object But the theorem claims finite division and not infinite? In R³, given a solid ball B of radius R it is possible to partition B into finitely many pieces such that those pieces can be reassembled to form two solid balls B1 and B2 each of radius R
- moloch-hai 4y ago
  Finitely many pieces, but on infinitely variable boundaries. It was clever to make the proof allow a finite number in that place. Without, it would have attracted no attention.
  - wnkrshm 4y ago
    It's still astounding to me that the 'fractal trick' starts working in R3 and not already in R2.
    - moloch-hai 4y ago
      Agreed, each added dimension adds wacky surprises. Packing spheres into a cube leaves you a pretty comprehensible space in the middle. Pack hyperspheres into a hypercube, and all hell breaks loose.
- drdec 4y ago
  Keep in mind that "pieces" might not mean the same thing you think of it as meaning.
deleted 4y ago
[deleted]
theemathas 4y ago
The main reason that we use real numbers is because we need real numbers to do calculus, at least in the normal way. And to do physics, we need to do calculus. You can keep approximating integrals as sums of very large number of terms, but it gets unwieldy to handle symbolically.
- moloch-hai 4y ago
  Real numbers almost always give the same result as more cumbersome methods, and we are mostly only interested in such results, so it is surprising when something wacky shows up to remind us of the tightrope we walk.

Aicy 4y ago

In the real world we have atoms, but atoms are disturbances in the wave function of the universe, which may not be discrete. We know atoms pop in and out of existence in matter anti-matter pairs. Maybe the universe is performing Banach Tarski under the hood, to make something from nothing? > we use floating-point numbers, not reals, when doing actual calculations I don't believe only calculations made by computers to be "actual calculations". Humans were calculating thousands of years before computers were invented, using integer or real numbers (e.g pi).

moloch-hai 4y ago
No human or computer has ever used an actual real, transcendental number in any computation. It would take infinitely long.
- ttctciyf 4y ago
  I know what you mean, but Euler's identity is a counterexample to what you literally said, isn't it?
  - Someone 4y ago
    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.
    - gpm 4y ago
      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.
  - moloch-hai 4y ago
    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.
    - defrost 4y ago
      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?
      - moloch-hai 4y ago
        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.
        defrost 4y ago
        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.
        moloch-hai 4y ago
        It's the wackiest result I know of, and seems overwhelmingly more profound than BT.
deleted 4y ago
[deleted]
CJefferson 4y ago
Banach Tarski, in some sense, really can't make sense in the universe because the Universe is... discreteish, as you can't get smaller than the Planck Lenth (1.6 x 10^(-35)), and while that's very small, as far as maths is concerned that means the number of "identifiable points" in a sphere of any finite size is itself finite, not even a countable infinity. I realise that's all very dodgy, but of course we are also bringing in quantum mechanics -- but I consider it enough to say Banach Tarski doesn't apply :)
- lodi 4y ago
  My understanding is that Planck Length is not a limit to the smallest possible thing; it's a limit to the most precise possible measurement. So there could in fact be lots of things happening at sub-Planck Length scales, but we could never observe them.

whatshisface 4y ago

Atoms move according to path integrals which would be at least countably infinite in number even if time and space were discrete because they are allowed to loop back on themselves.

zokier 4y ago

> It relies on uncountably-infinite division of an object But the theorem claims finite division and not infinite? In R³, given a solid ball B of radius R it is possible to partition B into finitely many pieces such that those pieces can be reassembled to form two solid balls B1 and B2 each of radius R

moloch-hai 4y ago
Finitely many pieces, but on infinitely variable boundaries. It was clever to make the proof allow a finite number in that place. Without, it would have attracted no attention.
- wnkrshm 4y ago
  It's still astounding to me that the 'fractal trick' starts working in R3 and not already in R2.
  - moloch-hai 4y ago
    Agreed, each added dimension adds wacky surprises. Packing spheres into a cube leaves you a pretty comprehensible space in the middle. Pack hyperspheres into a hypercube, and all hell breaks loose.
drdec 4y ago
Keep in mind that "pieces" might not mean the same thing you think of it as meaning.

deleted 4y ago

[deleted]

theemathas 4y ago

The main reason that we use real numbers is because we need real numbers to do calculus, at least in the normal way. And to do physics, we need to do calculus. You can keep approximating integrals as sums of very large number of terms, but it gets unwieldy to handle symbolically.

moloch-hai 4y ago
Real numbers almost always give the same result as more cumbersome methods, and we are mostly only interested in such results, so it is surprising when something wacky shows up to remind us of the tightrope we walk.