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I am very much not a mathematician. I took physics at university, but that was 20 years ago. The mathematics here is beyond me. So, I realise I'm probably conf

I am very much not a mathematician. I took physics at university, but that was 20 years ago. The mathematics here is beyond me. So, I realise I'm probably confusing theoretical mathematics with a practical science. But, still. If this was a real, true result, couldn't someone simply take a sphere of, say, gold, and recut it to be two spheres of gold? Then keep doing it until they're richer than Croesus? Or, to be more serious, what value are results like this if they are obviously false in the real world? Also, if they plainly don't apply to the real world, isn't that a really good sign that the mathematics is actually incorrect? Or, have I completely misunderstood the article?

gdsimoes 11y ago
You would need to cut the spheres in very strange pieces and since spheres of gold are made of atoms you can't do that in the real world.
- summerdown2 11y ago
  Why not, assuming you make the spheres big enough? Let's say not gold, but a moon. I mean, let's do a thought experiment with superhuman tech and a big enough sphere that atoms are small enough to make the slices right. I guess I just don't understand why this doesn't violate conservation of mass.
  - ColinWright 11y ago
    Because you really, really can't make the slices small enough. And your point about violating the conservation of mass is the whole point of the article. When we try to come up with a mathematical model of what "volume" means there are certain properties we want it to have. One of them is that any arbitrary collection of point - not atoms, but mathematical points - can have a volume associated with it. The Banach-Tarski theorem shows that such a requirement is impossible. You're not alone if you think this is all nonsense - so did Feynman. However, many clever people not only believe that this is relevant, but also useful and insightful. The article is trying to give a sense of why that's the case. I want to write a sister article to this to help people like you come to understand what's going on, but I'm having trouble finding people who are willing to engage with me on it. They usually just find that it offends their sense of reality and reject it all. I think that's a shame, because unless people like me can come to understand what others find so objectionable we can never learn how to help people understand why this is interesting, useful, adn relevant.
    - summerdown2 11y ago
      I think the problem is there's mathematics on one hand and my experience of the real world on the other. I either didn't do any set theory, or I've forgotten it. So it looks just like squiggles on the page to me when I try to understand the maths. So, all I'm left with is trying to relate what the words might mean in real-world terms. And coming up confused. I don't think it's your fault. I just lack the grounding to see both sides of the picture. One thought, though. I'm not trying to reject it because it offends my sense of reality. I'm trying to use my sense of reality (which is the only tool I have) to understand it. For example, if someone came up to me at work and said, "I've just worked out how to cut a sphere up into bits and reassemble it as two spheres the same size," I'd say, "ok, then, show me." If all they could do is make marks on paper I couldn't understand I'd think they'd got the paper wrong, not reality. Which is the dissonance here. I guess, speaking personally, if you want someone like me to understand it you'd need to really, really explain the maths (as if to a simpleton!), or explain what's happening in real-world terms and why it wouldn't work yet is still valid. Does that help?
      - monochromatic 11y ago
        This book gives a good explanation of a construction, at a reasonably non-technical level. http://www.amazon.com/The-Pea-Sun-Mathematical-Paradox/dp/1568813279 http://www.amazon.com/The-Pea-Sun-Mathematical-Paradox/dp/15...
      - ColinWright 11y ago
        But what we're saying here is this. We use sets to model the real world. They do remarkably well, and the math we've developed to work on them includes calculus to make bridges that stay up, fluid dynamics that make aeroplanes fly, and discrete math that helps us understand routing, scheduling, and all sorts of stuff. We use the real line to model distances. In the real world we are limited as to the accuracy we can use, but modelling those limitations is nasty. It's easier to assume that things are continuous. In its turn, we make choices that make working with these models easier, and they turn out to be amazingly useful. But then we start poking the dusty corners. The choices we make in the development of the theory have consequences, and math is about exploring both the choices, and their consequences. So we can choose that between every two numbers there's another number. We can choose that there's a number whose square is 2. We can choose that the sum of the reciprocals of squares : 1+1/4+1/9+1/16+1/25+1/36+ ... : is a real number. And we can choose that there is no smallest positive number. That has a consequence. If you believe that there is no smallest positive number, then 0.99999... has to equal 1. You can't have one without the other. So we can talk about "the length of a line." Then we can talk about the "length" of a set of points on a line. Then we start to find that these simplistic models, these obvious and natural choices, even though they are amazingly useful have some unexpected consequences. Does that help you to understand the context? I'd be really interested in developing an agreed dialog about this. Will you send me an email?
  - Someone 11y ago
    As far as we know, you cannot cut a fractal such as the Mandelbrot set out of anything physical, either. The details get infinitely fine, and that's not possible in physical objects (as far as we know)
  - digama 11y ago
    Conservation of mass is a consequence of the fact that matter is not infinitely divisible. The math on this works out too: there is an easy "measure" on three dimensional objects in the real world - just count the atoms. (This is known as a "counting measure" in measure theory.) It is the presence of infinities in the real numbers that causes this measure to fail (you will end up measuring the B-T sets, and regular objects like spheres as well, to have infinite "mass", so that 2*infinity = infinity is no longer as surprising a result).
- SilasX 11y ago
  Why would the strangeness matter? The article insists that it's not an issue of infinitely small pieces, so what's the actual inherent barrier to doing this on real objects? Is it an issue of continuity that's not related to the pieces being arbitrarily small? I still don't see where this refutes the traditional line about the theorem being an artifact of uncountable sets.
  - monochromatic 11y ago
    It's not that the cuts are strange, it's that they involve infinitely small details. You can't cut physical atoms like that, and so the construction fails in the real world.
    - SilasX 11y ago
      Then his article just leaves me more confused than before. I was already familiar with the traditional resolution of BT as "well, that's just an artifact if the weirdness when you have uncountably many points". But now the author insists that "oh no, you get the same paradox with finite pieces". And yet on every probe of that point, it comes back to an issue of infinities. So what's wrong with the traditional explanation? And how does this article justify a "finite version" of the partition.
      - ColinWright 11y ago
        I don't understand what you think the "traditional resolution" might be. In the Banach-Tarski theorem you are partitioning a 3-dimensional solid ball into finitely many pieces. Because the ball has uncountably many points, those pieces will have uncountably many points.[0] Does that help? [0] Actually that only shows that at least one of the pieces must have uncountably many points, but in the theorem we find that at least four pieces must have uncountably many points.
      - monochromatic 11y ago
        The article doesn't try to explain how the construction works. The construction is done with finitely many pieces (I think five pieces is the limit), but those pieces have infinitesimally small details.
Sharlin 11y ago
It is a real, true, result, in a space that is based on different rules than our physical reality. A ball in R^3 is composed of a countably infinite number of zero-dimensional points. It's an abstract concept that doesn't really have much to do with a physical ball that consists of a finite number of atoms. Fundamentally, math is something that describes a large collection of possible worlds. Our universe is just a small subset of all the things that could be, and when using math to understand reality we have to remember that.
- dietrichepp 11y ago
  I'm pretty sure you mean "uncountably infinite" instead of "countably infinite".
  - Sharlin 11y ago
    Yep, thanks!
- reagency 11y ago
  Not "possible", " imaginable". "Possible" is begging the question.
CJefferson 11y ago
Have you ever heard stories like "Hilbert's Hotel"? Given an infinite hotel (each room labelled with a positive integer), you can empty half your rooms by moving the person in room x to the room 2*x, which gives you all the odd rooms to fill with a new infinity of people, effectively doubling the size of your infinite hotel? This is very similar -- except it works over the real numbers in 3D space. In particular the cuts are "inifinitely fine". Given any point p, and any distance d, there will be a point less than distance d from p which is in a different "slice". Imagine (while this isn't in banack-tarski) making a "slice" which is all numbers of the form 1/x for all integers. Clearly this "slice" could really exist, but mathematics can of course define it and operate on it.
- mikro2nd 11y ago
  Rudy Rucker explored this thoroughly in his novel "White Light" (which is where I first learned of the Banach-Tarski theorem) whilst having with much fun with infinities.
skybrian 11y ago
It's finitely many pieces, but not in the way you think. Each supposed piece isn't something you could carve with a knife, even if we defined "carving" as a mathematical operation. It's just a set of points with coordinates that obey a certain rule. In mathematics you can say things like "the set of all points in the sphere with irrational coordinates" and think of that as one "piece" of the sphere but it doesn't correspond to anything in the real world. I'm not a mathematician either, but I think the consequence is just that mathematical points are rather weird and admitting any infinite set of points (however disconnected) as a volume produces weird results. What we think of as a "piece" isn't just an infinite set of points.
ajuc 11y ago
The result IS true, just like "2i*2i=-4" is true. It just doesn't apply to physical reality, at least not in the most straightforward way (just like you can't have 2i chests with 2i apples in each chest).
- tripzilch 11y ago
  What if you have 2 iPhones?

gdsimoes 11y ago

You would need to cut the spheres in very strange pieces and since spheres of gold are made of atoms you can't do that in the real world.

summerdown2 11y ago
Why not, assuming you make the spheres big enough? Let's say not gold, but a moon. I mean, let's do a thought experiment with superhuman tech and a big enough sphere that atoms are small enough to make the slices right. I guess I just don't understand why this doesn't violate conservation of mass.
- ColinWright 11y ago
  Because you really, really can't make the slices small enough. And your point about violating the conservation of mass is the whole point of the article. When we try to come up with a mathematical model of what "volume" means there are certain properties we want it to have. One of them is that any arbitrary collection of point - not atoms, but mathematical points - can have a volume associated with it. The Banach-Tarski theorem shows that such a requirement is impossible. You're not alone if you think this is all nonsense - so did Feynman. However, many clever people not only believe that this is relevant, but also useful and insightful. The article is trying to give a sense of why that's the case. I want to write a sister article to this to help people like you come to understand what's going on, but I'm having trouble finding people who are willing to engage with me on it. They usually just find that it offends their sense of reality and reject it all. I think that's a shame, because unless people like me can come to understand what others find so objectionable we can never learn how to help people understand why this is interesting, useful, adn relevant.
  - summerdown2 11y ago
    I think the problem is there's mathematics on one hand and my experience of the real world on the other. I either didn't do any set theory, or I've forgotten it. So it looks just like squiggles on the page to me when I try to understand the maths. So, all I'm left with is trying to relate what the words might mean in real-world terms. And coming up confused. I don't think it's your fault. I just lack the grounding to see both sides of the picture. One thought, though. I'm not trying to reject it because it offends my sense of reality. I'm trying to use my sense of reality (which is the only tool I have) to understand it. For example, if someone came up to me at work and said, "I've just worked out how to cut a sphere up into bits and reassemble it as two spheres the same size," I'd say, "ok, then, show me." If all they could do is make marks on paper I couldn't understand I'd think they'd got the paper wrong, not reality. Which is the dissonance here. I guess, speaking personally, if you want someone like me to understand it you'd need to really, really explain the maths (as if to a simpleton!), or explain what's happening in real-world terms and why it wouldn't work yet is still valid. Does that help?
    - monochromatic 11y ago
      This book gives a good explanation of a construction, at a reasonably non-technical level. http://www.amazon.com/The-Pea-Sun-Mathematical-Paradox/dp/1568813279 http://www.amazon.com/The-Pea-Sun-Mathematical-Paradox/dp/15...
    - ColinWright 11y ago
      But what we're saying here is this. We use sets to model the real world. They do remarkably well, and the math we've developed to work on them includes calculus to make bridges that stay up, fluid dynamics that make aeroplanes fly, and discrete math that helps us understand routing, scheduling, and all sorts of stuff. We use the real line to model distances. In the real world we are limited as to the accuracy we can use, but modelling those limitations is nasty. It's easier to assume that things are continuous. In its turn, we make choices that make working with these models easier, and they turn out to be amazingly useful. But then we start poking the dusty corners. The choices we make in the development of the theory have consequences, and math is about exploring both the choices, and their consequences. So we can choose that between every two numbers there's another number. We can choose that there's a number whose square is 2. We can choose that the sum of the reciprocals of squares : 1+1/4+1/9+1/16+1/25+1/36+ ... : is a real number. And we can choose that there is no smallest positive number. That has a consequence. If you believe that there is no smallest positive number, then 0.99999... has to equal 1. You can't have one without the other. So we can talk about "the length of a line." Then we can talk about the "length" of a set of points on a line. Then we start to find that these simplistic models, these obvious and natural choices, even though they are amazingly useful have some unexpected consequences. Does that help you to understand the context? I'd be really interested in developing an agreed dialog about this. Will you send me an email?
- Someone 11y ago
  As far as we know, you cannot cut a fractal such as the Mandelbrot set out of anything physical, either. The details get infinitely fine, and that's not possible in physical objects (as far as we know)
- digama 11y ago
  Conservation of mass is a consequence of the fact that matter is not infinitely divisible. The math on this works out too: there is an easy "measure" on three dimensional objects in the real world - just count the atoms. (This is known as a "counting measure" in measure theory.) It is the presence of infinities in the real numbers that causes this measure to fail (you will end up measuring the B-T sets, and regular objects like spheres as well, to have infinite "mass", so that 2*infinity = infinity is no longer as surprising a result).
SilasX 11y ago
Why would the strangeness matter? The article insists that it's not an issue of infinitely small pieces, so what's the actual inherent barrier to doing this on real objects? Is it an issue of continuity that's not related to the pieces being arbitrarily small? I still don't see where this refutes the traditional line about the theorem being an artifact of uncountable sets.
- monochromatic 11y ago
  It's not that the cuts are strange, it's that they involve infinitely small details. You can't cut physical atoms like that, and so the construction fails in the real world.
  - SilasX 11y ago
    Then his article just leaves me more confused than before. I was already familiar with the traditional resolution of BT as "well, that's just an artifact if the weirdness when you have uncountably many points". But now the author insists that "oh no, you get the same paradox with finite pieces". And yet on every probe of that point, it comes back to an issue of infinities. So what's wrong with the traditional explanation? And how does this article justify a "finite version" of the partition.
    - ColinWright 11y ago
      I don't understand what you think the "traditional resolution" might be. In the Banach-Tarski theorem you are partitioning a 3-dimensional solid ball into finitely many pieces. Because the ball has uncountably many points, those pieces will have uncountably many points.[0] Does that help? [0] Actually that only shows that at least one of the pieces must have uncountably many points, but in the theorem we find that at least four pieces must have uncountably many points.
    - monochromatic 11y ago
      The article doesn't try to explain how the construction works. The construction is done with finitely many pieces (I think five pieces is the limit), but those pieces have infinitesimally small details.

Sharlin 11y ago

It is a real, true, result, in a space that is based on different rules than our physical reality. A ball in R^3 is composed of a countably infinite number of zero-dimensional points. It's an abstract concept that doesn't really have much to do with a physical ball that consists of a finite number of atoms. Fundamentally, math is something that describes a large collection of possible worlds. Our universe is just a small subset of all the things that could be, and when using math to understand reality we have to remember that.

dietrichepp 11y ago
I'm pretty sure you mean "uncountably infinite" instead of "countably infinite".
- Sharlin 11y ago
  Yep, thanks!
reagency 11y ago
Not "possible", " imaginable". "Possible" is begging the question.

CJefferson 11y ago

Have you ever heard stories like "Hilbert's Hotel"? Given an infinite hotel (each room labelled with a positive integer), you can empty half your rooms by moving the person in room x to the room 2*x, which gives you all the odd rooms to fill with a new infinity of people, effectively doubling the size of your infinite hotel? This is very similar -- except it works over the real numbers in 3D space. In particular the cuts are "inifinitely fine". Given any point p, and any distance d, there will be a point less than distance d from p which is in a different "slice". Imagine (while this isn't in banack-tarski) making a "slice" which is all numbers of the form 1/x for all integers. Clearly this "slice" could really exist, but mathematics can of course define it and operate on it.

mikro2nd 11y ago
Rudy Rucker explored this thoroughly in his novel "White Light" (which is where I first learned of the Banach-Tarski theorem) whilst having with much fun with infinities.

skybrian 11y ago

It's finitely many pieces, but not in the way you think. Each supposed piece isn't something you could carve with a knife, even if we defined "carving" as a mathematical operation. It's just a set of points with coordinates that obey a certain rule. In mathematics you can say things like "the set of all points in the sphere with irrational coordinates" and think of that as one "piece" of the sphere but it doesn't correspond to anything in the real world. I'm not a mathematician either, but I think the consequence is just that mathematical points are rather weird and admitting any infinite set of points (however disconnected) as a volume produces weird results. What we think of as a "piece" isn't just an infinite set of points.

ajuc 11y ago

The result IS true, just like "2i*2i=-4" is true. It just doesn't apply to physical reality, at least not in the most straightforward way (just like you can't have 2i chests with 2i apples in each chest).

tripzilch 11y ago
What if you have 2 iPhones?