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A late entry: the Banach-Tarski theorem: http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

by hugh 18y ago

nickb 18y ago
Ahhh... you just caused a major flashback in my memory. I remember reading about Banach-Tarski paradox about 7 years ago or so and it gave me so many headaches. I think I read the paper explaining it trice and it still made no sense to me and on my second reading I was convinced there was an error in an equation somewhere.. I asked a math prof who's a brilliant theoretician and he explained it to me and it still made little sense. Then I read von Neumann's explanation and it finally started making some sense. The key to understanding it is, like with many other concepts, to ignore your intuition and trust the math behind it.
- andreyf 18y ago
  For me, the key to understanding was to realize that reality is only a figment of your imagination ;) On a more serious note, it was understanding the Axiom of Choice, or that our "intuitive" notions of reality aren't really consistent with themselves: http://en.wikipedia.org/wiki/Nonmeasurable_set http://en.wikipedia.org/wiki/Nonmeasurable_set
- DaniFong 18y ago
  I personally thought that the Banach-Tarski theorem was a mess, and these non-measurable sets had to go. So I started studying nonstandard-analysis: analysis with infinities and infinitesimals as genuine, arithmetically viable, quantities. Imagine my distress: the same problem occurs, Banach-Tarski still holds, unless you go into the rather more weird, Smooth Infinitesimal Analysis or Synthetic Differential Geometry; basically, treating geometry as made up of little discrete infinitesimally small points.
- yummyfajitas 18y ago
  I'd say trust your intuition more than the math, but I'm more physicist than mathematician. To prove Banach-Tarski, you need to assume Zermelo-Frankel set theory (just ordinary set theory) plus the uncountable axiom of choice. In my view, this means the uncountable axiom of choice is false. This actually breaks far less than you would expect. For most practical purposes, you only need the countable axiom of choice, which is the intuitively reasonable version. The main thing the loss of the uncountable version breaks is the Hahn-Banach theorem on spaces of dimension higher than countable infinity. This in turn breaks quantum field theory on free space (not in a box), but that's broken anyway.
ionfish 18y ago
I, for one, appreciate you referring to it as a theorem and not a paradox.
jodrellblank 18y ago
Reality can't be wrong, only our understanding or modeling of it can be wrong. So a paradox is either: - our "intuitive" mental model of the world which says one thing can't become two of the same thing without using any more material, is wrong - the mathematical model that says you can do that, is wrong - the mathematical framework on which the model is built, is limited. In other words, while I can't follow the maths, I would have no problem believing the mathematical weirdness is self consistent because I don't believe it would have any rational link to the real world anyway.
- jodrellblank 18y ago
  Thinking about it a bit more, that a mathematical model created by humans on a mathematical framework created by humans comes out with weird results sometimes - that fact is not even slightly weird or difficult to believe. That such a human model can be as accurate and precise as quantum mechanics is, now that's hard to believe!
- jerf 18y ago
  Your last paragraph seems to imply that you believe that somebody, somewhere is claiming this result is "real". In fact, nobody claims the theorem has any applicability to "reality", because the real world is quantized. You can't cut an electron in half. (So far as we know.) Everybody knows the theorem isn't real. Now, I'm not just being pedantic here, because the fundamental and obvious failure of the theorem to have any connection to the real world is at the heart of one of the philosophical questions about the nature of math it engenders, along with its good friend the Axiom of Choice. While there's nothing wrong with either thing mathematically, does it really matter when anything that invokes the Axiom of Choice basically just opted out of concrete reality entirely? Basically, I'm saying that the contents of your post are true, but trivial; there's no interesting argument about correspondence to reality to be made. The fun lies at the next level. I have no easy answers for you there. (Personally, I have no problem breaking math into "Things that Might Be Practically Useful" and "Things to Entertain Mathematicians", especially since the latter tends to migrate to the former at quite surprising times, but many people seem to have trouble with giving such a rich meaning to the single word "math".)

A late entry: the Banach-Tarski theorem: http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

by hugh 18y ago

A late entry: the Banach-Tarski theorem: http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

nickb 18y ago

Ahhh... you just caused a major flashback in my memory. I remember reading about Banach-Tarski paradox about 7 years ago or so and it gave me so many headaches. I think I read the paper explaining it trice and it still made no sense to me and on my second reading I was convinced there was an error in an equation somewhere.. I asked a math prof who's a brilliant theoretician and he explained it to me and it still made little sense. Then I read von Neumann's explanation and it finally started making some sense. The key to understanding it is, like with many other concepts, to ignore your intuition and trust the math behind it.

andreyf 18y ago
For me, the key to understanding was to realize that reality is only a figment of your imagination ;) On a more serious note, it was understanding the Axiom of Choice, or that our "intuitive" notions of reality aren't really consistent with themselves: http://en.wikipedia.org/wiki/Nonmeasurable_set http://en.wikipedia.org/wiki/Nonmeasurable_set
DaniFong 18y ago
I personally thought that the Banach-Tarski theorem was a mess, and these non-measurable sets had to go. So I started studying nonstandard-analysis: analysis with infinities and infinitesimals as genuine, arithmetically viable, quantities. Imagine my distress: the same problem occurs, Banach-Tarski still holds, unless you go into the rather more weird, Smooth Infinitesimal Analysis or Synthetic Differential Geometry; basically, treating geometry as made up of little discrete infinitesimally small points.
yummyfajitas 18y ago
I'd say trust your intuition more than the math, but I'm more physicist than mathematician. To prove Banach-Tarski, you need to assume Zermelo-Frankel set theory (just ordinary set theory) plus the uncountable axiom of choice. In my view, this means the uncountable axiom of choice is false. This actually breaks far less than you would expect. For most practical purposes, you only need the countable axiom of choice, which is the intuitively reasonable version. The main thing the loss of the uncountable version breaks is the Hahn-Banach theorem on spaces of dimension higher than countable infinity. This in turn breaks quantum field theory on free space (not in a box), but that's broken anyway.

ionfish 18y ago

I, for one, appreciate you referring to it as a theorem and not a paradox.

jodrellblank 18y ago

Reality can't be wrong, only our understanding or modeling of it can be wrong. So a paradox is either: - our "intuitive" mental model of the world which says one thing can't become two of the same thing without using any more material, is wrong - the mathematical model that says you can do that, is wrong - the mathematical framework on which the model is built, is limited. In other words, while I can't follow the maths, I would have no problem believing the mathematical weirdness is self consistent because I don't believe it would have any rational link to the real world anyway.

jodrellblank 18y ago
Thinking about it a bit more, that a mathematical model created by humans on a mathematical framework created by humans comes out with weird results sometimes - that fact is not even slightly weird or difficult to believe. That such a human model can be as accurate and precise as quantum mechanics is, now that's hard to believe!
jerf 18y ago
Your last paragraph seems to imply that you believe that somebody, somewhere is claiming this result is "real". In fact, nobody claims the theorem has any applicability to "reality", because the real world is quantized. You can't cut an electron in half. (So far as we know.) Everybody knows the theorem isn't real. Now, I'm not just being pedantic here, because the fundamental and obvious failure of the theorem to have any connection to the real world is at the heart of one of the philosophical questions about the nature of math it engenders, along with its good friend the Axiom of Choice. While there's nothing wrong with either thing mathematically, does it really matter when anything that invokes the Axiom of Choice basically just opted out of concrete reality entirely? Basically, I'm saying that the contents of your post are true, but trivial; there's no interesting argument about correspondence to reality to be made. The fun lies at the next level. I have no easy answers for you there. (Personally, I have no problem breaking math into "Things that Might Be Practically Useful" and "Things to Entertain Mathematicians", especially since the latter tends to migrate to the former at quite surprising times, but many people seem to have trouble with giving such a rich meaning to the single word "math".)