Musings on the infinite

[steuard]

A few weeks ago, my friend and former colleague Sean Carroll was a guest on the Colbert Report to promote his book about the nature of time. Toward the end of the interview, they discussed the idea of the "multiverse", which Sean uses to refer to the (possibly) infinite number of "universe-sized" regions within the vast web of space and time where we live. The notion is that if we could somehow travel far enough (faster than light) to regions many times more distant than our telescopes can see, we could find countless independent "universes" that can never talk to each other at all. Some of them would be much like our own but others could be very different, maybe even with different laws of physics. Steven Colbert seemed quite interested:

Colbert: Am I in these other universes?

Carroll: There will be people very much like you.

Colbert: In these other universes, is it possible that my show's on at 11 and John Stewart is at 11:30?

Carroll: Maybe more often!

It's a cute exchange, and it's a variant on the old idea that "in an infinitely big universe, everything that could possibly happen must happen somewhere."

Trouble is, I don't know that I buy that argument, for rather subtle reasons. However we define them, the number of "independent universe-sized regions" of space and time is countably infinite: we could in principle come up with some way of labeling each one by an integer. But many sets (like the real numbers) are uncountably infinite: no matter how you try to label each real number by an integer, you'll miss the vast majority of them. The real numbers are just a much bigger infinity than the integers are. Going on, the set of all possible curves in space is a yet larger infinity. (Assuming space and time are continuous! If they turn out to be discrete, then the set of curves has the same infinite size as the real numbers.)

The thing is, the set of "everything that could possibly happen" is a lot more like the set of all curves than like the set of integers: if anything, it's a still larger infinity. So no matter how large our multiverse may be, it's mathematically impossible for every possible history to occur somewhere. Does that mean that our Steven Colbert (on at 11:30) is the only one? Quite possibly so. I'm not convinced that the multiverse idea opens up as many possibilities as people sometimes think.

Comments

[marphod.livejournal.com]

There are many places in physics where the rationals are insufficient, and the reals are required. Anything that has a square effect (magnetic field strength, distance covered in an inertial reference frame under acceleration, etc.)

[jon-leonard.livejournal.com]

There are lots of kinds of numbers between the rationals and the reals: Quadratic surds (stuff you get by adding the square root operator), algebraic numbers, etc. Admittedly neither of those are sufficient for physics, since pi keeps showing up, but it's still not clear to me that you need all the reals. Well, all the complex numbers. You can't really do quantum physics with only real numbers.

[steuard.livejournal.com]

Yeah, pi and e are two of the big sticking points that I'd worry about when trying to develop a rationals-only physics. Oh, and by "rationals" I should be clear that I was thinking "complex rationals". (Now I'm starting to worry about whether that's well-defined in the ways I'd need, though.)

Though to be honest, my original thought was actually just that the space-time manifold would be Q^{3,1} rather than R^{3,1}; I hadn't followed that up in my head to conclude that the whole formalism would have to be based only on rationals. I'm still not sure.

[steuard.livejournal.com]

I guess what I'm thinking of here isn't "can you find a closed-form solution to an arbitrary equation of physics with just rationals" but rather "would it be possible to write a fully predictive theory with just the rationals". So I'm picturing less "find the speed of a block sliding down an inclined plane" and more "compute the scattering amplitude for this process in particle physics". And for that, all I really need to have in hand is the Lagrangian of the system (which can usually be written as a polynomial function of position and momentum) and some fancy integration (which I suspect could be sensibly defined over the rationals if one put in some effort).