5 ms·

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 s

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?
    - summerdown2 11y ago
      > I'd be really interested in developing an agreed dialog about this. Will you send me an email? I've sent you an email, as requested. From the sound of it, though, the answer to my puzzlement, is that it doesn't apply to this universe. In which case, I don't think it's a problem at all. I'm quite happy to imagine mathematicians doing work on geometries that don't map to the real world, for example. Mostly, then, is this just a question of presentation? I mean, if you said "this does not apply to the real world, but to some fictitious mathematical assumption that assumes things can be divided up infinitely," I don't think anyone would have an issue with it. It sounds like it's a problem to most people because it's presented as if it's a real-world result. Or, at least, that seems to be the impression I get, given the other comments. What I mean is, is there's a question of misdirection here? I.e. the theorem is presented as this non-intuitive thing that can't possibly be the case in the real world, then when someone asks what would happen if you tried it in the real world, the answer is "it's assuming some things that aren't true for the real world." Because if that's the case, I'm not sure I see the paradox. Assuming a weird set of ideas, I would expect you can come up with weird answers. Here's a question, though, because something is nagging at me. And I'm going to assume this is a universe of infinite points and no atoms (as I understand it, at least). Let's say we have a sphere of volume 4/3 pi r^3 = 100 Now you do your cuts, but don't reassemble yet. The sphere is still the original sphere, with all the shapes it has been cut into still in virtually their original spaces. The total volume still has to be 100, right? I mean they all still fit into the original space. So now, you immerse it in water, in a bathtub ready to overflow, and start manipulating the pieces. At what point does the water level rise?
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).

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?
  - summerdown2 11y ago
    > I'd be really interested in developing an agreed dialog about this. Will you send me an email? I've sent you an email, as requested. From the sound of it, though, the answer to my puzzlement, is that it doesn't apply to this universe. In which case, I don't think it's a problem at all. I'm quite happy to imagine mathematicians doing work on geometries that don't map to the real world, for example. Mostly, then, is this just a question of presentation? I mean, if you said "this does not apply to the real world, but to some fictitious mathematical assumption that assumes things can be divided up infinitely," I don't think anyone would have an issue with it. It sounds like it's a problem to most people because it's presented as if it's a real-world result. Or, at least, that seems to be the impression I get, given the other comments. What I mean is, is there's a question of misdirection here? I.e. the theorem is presented as this non-intuitive thing that can't possibly be the case in the real world, then when someone asks what would happen if you tried it in the real world, the answer is "it's assuming some things that aren't true for the real world." Because if that's the case, I'm not sure I see the paradox. Assuming a weird set of ideas, I would expect you can come up with weird answers. Here's a question, though, because something is nagging at me. And I'm going to assume this is a universe of infinite points and no atoms (as I understand it, at least). Let's say we have a sphere of volume 4/3 pi r^3 = 100 Now you do your cuts, but don't reassemble yet. The sphere is still the original sphere, with all the shapes it has been cut into still in virtually their original spaces. The total volume still has to be 100, right? I mean they all still fit into the original space. So now, you immerse it in water, in a bathtub ready to overflow, and start manipulating the pieces. At what point does the water level rise?

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).