I was supposed to be doing quantum in the library, but I accidentally picked up a book on mathematical physics and read it for an hour. Then a couple days later I was daydreaming about how I would lecture on vectors to my summer camp kids, and I suddenly got extremely nervous about something. As in, I had a total crisis of faith about the rigor of our mathematical description of the physical universe, which led me to start this post on the physics forums. I really hope someone answers so I can sleep in peace again.
As for the rest of life, well, it's going.
Total nerd-out
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I remember a physics TA talking about how a plane has the same number of points as a line, and that someone generated a proof by creatively using a line to fill an entire plane without crossing itself. About the same time someone told me that one Turing Machine can simulate 2 Turing Machines, and by inducting therefore N Turing Machines as well. This seems like a problem of similar nature to the cardinality one.
I think the issue is the same as the paradox presented by my HS calculus teacher when we learned how to calculate surfaces and volumes of rotation. There's that one cone-shaped surface with a finite volume but an infinite surface area. If you filled that volume with a finite amount of paint how could it not hold enough to coat its own (infinite) surface? Wow, crazy! This is a physically unrealistic situation though, since you'd have to paint the surface with particles of some sort, which take up 3d space.
This makes me think that the issue of all dimensions having the same cardinality is mostly a bookkeeping problem, although I can't figure out where the error is. Maybe it's that since space is quantized (I think) the definition of a line or a plane with an infinite number of points is physically unrealistic. But it seems equally paradoxical that you can use a thought experiment to 'prove' the quantization of real things.
Oh well. Life goes on.
do you think we can a way for having sex with one girlfriend to simulate having sex with N girlfriends simultaneously? that would make accomplishing my life goals so much easier.
Re: quantization of space. I'm not sure, but it does seem like there are problems if it isn't. No one has ever told me space is quantized, but I've never taken super-sexy insane string-M-supersymmetric whatever physics.
Here's another paradox if space is not quantized. Conduct a measurement on a particle using an operator with continuous eigenvalues (position or momentum, for example). The wave function collapses to one of the eigenstates. But if you want to describe which eigenstates, you need a number picked out from the continuum. That is, you need infinite precision. The universe then has infinite entropy/information.
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