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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

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?
  - ColinWright 11y ago
    I've replied to your email, but to reply here to your specific question: > immerse it in water, in a bathtub ready > to overflow, and start manipulating the > pieces. > At what point does the water level rise? A lovely question. The answer is that the "water" isn't "water", it has to be the same infinitely fine "mush" that the ball is made of. As a result, as you move the pieces out so the "water" ends up forming non-measurable holes to fit them into, and the complement is also non-measurable. So in the same way as the balls kind of "fold out" to become two balls, so the water kind of "folds in" and takes up less space than it used to, exactly balancing the actions of the pieces.
defen 11y ago
You could make the same argument against the real numbers. Almost all of them cannot even be written down or described, so it's hard to say how they map to the real-world. But the concept of real numbers is extremely useful for a large number of mathematical proofs.
- reagency 11y ago
  Which proofs depend on an infinitely discontinuous subset, though?
  - ColinWright 11y ago
    It's not that we rely on such things, but that these things are unavoidable consequences of choices we make that seem to be perfectly reasonable, and result in useful math. Phrasing in the other way: As we develop math that we find useful and powerful, we find that we have to make choices. Those choices have consequences, and sometimes as we explore the consequences we find that really strange things happen. We can go back and make different choices, but in practice we tend to find that no matter what choices we make there are odd and hairy things that result.
  - ebola1717 11y ago
    The natural way to encode an infinite binary tree into the real numbers is as the Cantor set, an uncountable totally disconnected set.
reagency 11y ago
Uncountable sets are not reality. They are unicorns. They are a pretend construction that cannot be mapped 1:1 to anything in the universe, by definition. "Uncountable" means "outside the realm of real world algorithms and physics" . they are like "god of the gaps", a name for what we call the beyond the edge of the universe, which we can never reach. But we can assign some structure to it, roughly as an extension of reality, to imagine what it could be like. Sort of like heaven. Sort of.

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?
- ColinWright 11y ago
  I've replied to your email, but to reply here to your specific question: > immerse it in water, in a bathtub ready > to overflow, and start manipulating the > pieces. > At what point does the water level rise? A lovely question. The answer is that the "water" isn't "water", it has to be the same infinitely fine "mush" that the ball is made of. As a result, as you move the pieces out so the "water" ends up forming non-measurable holes to fit them into, and the complement is also non-measurable. So in the same way as the balls kind of "fold out" to become two balls, so the water kind of "folds in" and takes up less space than it used to, exactly balancing the actions of the pieces.

defen 11y ago

You could make the same argument against the real numbers. Almost all of them cannot even be written down or described, so it's hard to say how they map to the real-world. But the concept of real numbers is extremely useful for a large number of mathematical proofs.

reagency 11y ago
Which proofs depend on an infinitely discontinuous subset, though?
- ColinWright 11y ago
  It's not that we rely on such things, but that these things are unavoidable consequences of choices we make that seem to be perfectly reasonable, and result in useful math. Phrasing in the other way: As we develop math that we find useful and powerful, we find that we have to make choices. Those choices have consequences, and sometimes as we explore the consequences we find that really strange things happen. We can go back and make different choices, but in practice we tend to find that no matter what choices we make there are odd and hairy things that result.
- ebola1717 11y ago
  The natural way to encode an infinite binary tree into the real numbers is as the Cantor set, an uncountable totally disconnected set.

reagency 11y ago

Uncountable sets are not reality. They are unicorns. They are a pretend construction that cannot be mapped 1:1 to anything in the universe, by definition. "Uncountable" means "outside the realm of real world algorithms and physics" . they are like "god of the gaps", a name for what we call the beyond the edge of the universe, which we can never reach. But we can assign some structure to it, roughly as an extension of reality, to imagine what it could be like. Sort of like heaven. Sort of.