Thursday, 5 November 2015

The Anthropic Principles as Categoricity Proofs

The Standard Model of particle physics has a number of physical constants which need to be determined by measurement and don't seem to predicted by any more fundamental theories. These are: the charge of an electron, the ratio of electron mass to proton mass (the 137 figure), the gravitational constant, the cosmological constant and a couple of others.

One of the many things that puzzles philosophically-inclined scientists is that there's not a lot of wiggle room for these constants. If the charge on the electron (and hence proton) is a lot higher, then electrons will bind so tightly to the nucleus that chemical reations won't happen. If the cosmological constant isn't 1.0 to a lot of decimal places, the universe would have a) expanded to quickly, or b) failed to expand at all. And so on. The puzzle is: how is it that this universe got created, with the fundamental constants at just the right values to create Nobel prize-winners, and not some other values that created a boring universe?

Something like this happens in mathematics. If you want an algebraically-complete (so that all polynomials of degree n have n roots) set of numbers which is also order-complete (so that every convergent sequence of numbers has a limit) that forms a field, then you can have the complex numbers. Or.... you can have the complex numbers. And if you want something different... you can't. Those requirements can be satisified only by the complex numbers and there's even a proof of it. Strictly, all models of those requirements are isomorphic. The theory is categorical - in second-order logic.

Mathematicians are not puzzled by this. In fact, they are rather pleased by it. The reason they aren't puzzled by it is because they have a proof of the uniqueness of the complex numbers. In all universes, and all civilisations, all algebraically-closed, complete number fields are isomorphic to the complex numbers. Why? Becuase proof.

To me, the lack of wiggle-room for the fundamental constants feels very similar. It says something like this: if we build a universe where the stable particles are electrons, neutrons, neutrinos, protons and photons (an e2n2p universe) then unless the fundamental constants are very close to the values in this universe, you get a boring universe. Why? Because proof.

The puzzle isn't about the values of the fundamental constants. It's why this universe is an e2n2p universe, and if there might be other ways of building molecules that don't use atoms made up of electrons, protons, neutrons, neutrinos and photons. What needs to be proved is: all universes must be e2n2p-universes or be boring.

You're going to remind me that there are quarks which make up neutrons and protons. Also other short-life hadrons, and muons, which also have short lives. For my purposes that doesn't matter. How the stable particles in a universe are constructed, or perhaps we should say, how the stable particles in a universe break down under high-energy collisions, isn't relevant. What's relevant is that the stable particles are what they are.

The Anthropic Principles are really a statement that a certain kind of theory is in a (possibly metaphorical) sense categorical. So what's really interesting is: can we build another universe out of stable particles that aren't isomorphic to the ones in this universe? And if we can, how much wiggle room is there for the values of the relevant fundamental constants? My guess is that, even if we can find an non-isomorphic set of fundamental particles, there won't be much wiggle-room.

(An "isomorphic set of particles"? Either treat it as a metaphor, or remember that fundamental particles correspond to generators of groups. So it would be the groups that were isomorphic.)

No comments:

Post a Comment