Computability: The Greatest Law of Physics
researchblogs.cs.bham.ac.uk
researchblogs.cs.bham.ac.uk
Also, as a Physicist and Computer Scientist, it explains my uneasiness when existence claims are made regarding infinite objects.
For example, we can describe the trajectory of a particle using Feynman path integrals, which involve infinite sums over all possible paths. That's fine.
Some will then treat this as a mechanism, i.e. claim that the reason particles have the trajectories they do, is because they literally are taking every possible path at once. This kind of reasoning sets my off my CS alarm bells, and articles like this provide justification for that.
To see why this leap of reasoning is flawed, consider the fact that we don't solve integrals by summing up an infinite number of infinitesimal quantities. To say that a human writing out a sequence of symbolic manipulations on a page literally is performing an infinite amount of computation is clearly false.
I think Computer Scientists are much more comfortable than Physicists with considering the role of calculation in a theory; i.e. in Physics, the calculations we perform about a system are utterly distinct from that system: whether those calculations are easy or hard says nothing about what the system is doing (e.g. we can easily calculate path integrals, which particles "solve" using an infinite amount of brute-force); the only physically-relevant details are the values. In CS, we focus on the performance of calculations; we cannot claim that a system behaves over time in some way unless we can show that calculating that behaviour can be done in that time.
[1] https://en.wikipedia.org/wiki/Mathematical_universe_hypothes...
[2] http://www.osti.gov/scitech/servlets/purl/28404/
For example, Champernowne's constant contains every number in its decimal expansion, and hence contains a complete description of our universe, all of the true laws of Physics and the outcome of every random quantum effect. However, it's not a very satisfactory "theory of everything", since it makes no predictions. (Note that I could have used pi instead, but its not yet known whether pi is a "normal" number [3]). This is the same idea as the Library of Babel [4].
On the other hand, if you start distinguishing between possible worlds in some way, then you can do real science. For example, the Boltzmann Brain idea describes ordered systems (like our Universe) emerging from disordered systems (like clouds of gas) by pure chance [5]. Such a statistical argument is useful, since we can reason about probability distributions over possible Universes. In this case, small pockets of order are vastly more likely to arise spontaneously than large ones (since we can consider a large pocket to be a contiguous collection of smaller pockets), hence we obtain predictions for all kinds of experiments: namely that we'll probably see a cloud of gas, rather than any ordered structure. Since we tend to see ordered structure, we can prove the hypothesis wrong experimentally.
A related idea, which is also relevant for this article, is that the Universe could be generated by a random computer program [7]. If we apply the same reasoning as with Boltzmann Brains, we would expect short programs to be vastly more likely than long programs. Since the length of a program determines how "random" its result is (in the Kolmogorov sense [8]), short programs would produce more ordered structure than long programs, hence this hypothesis make the opposite prediction to the Boltzmann Brain: i.e. that when we observe new places, we will tend to see the same kind of order as we have already observed elsewhere. So far, these predictions seem to hold ;)
[1] http://arxiv.org/abs/0912.5434 [2] https://en.wikipedia.org/wiki/Champernowne_constant [3] https://en.wikipedia.org/wiki/Normal_number [4] https://en.wikipedia.org/wiki/The_Library_of_Babel [5] https://en.wikipedia.org/wiki/Boltzmann_brain [6] http://www.preposterousuniverse.com/blog/2008/12/29/richard-... [7] http://arxiv.org/abs/quant-ph/0011122 [8] https://en.wikipedia.org/wiki/Kolmogorov_complexity
"and the laws of physics just fell out as a byproduct"
Why? Don't you need a theory for why this is?
My guess is that once you consider all of the symmetries that could exist (and there are many which have not yet been found; supersymmetry has long been posited but remains undiscovered), then our laws of physics are basically the only laws that can exist in a mathematically consistent way. Of course, that's just my hypothesis.
There's also the possibility that our universe is unnatural, which some physicists have recently been considering, but I'm skeptical. ("Unnatural" means that all of the constants in our universe that cannot be calculated from first principles are simply the result of random chance — i.e., only those universes with "finely-tuned" parameters capable of supporting life would have life in it to observe those parameters.)
>My guess is that once you consider all of the symmetries that could exist (and there are many which have not yet been found; supersymmetry has long been posited but remains undiscovered), then our laws of physics are basically the only laws that can exist in a mathematically consistent way.
If physical evolution is unitary, it may be phrased in terms of an S-matrix, and the possible symmetries of the S-matrix are well-known by the Coleman-Mandula theorem[0] and its supersymmetric generalization[1]. So, supersymmetry is possible, but, as you rightly point out, may or may not exist in the world. So mathematical consistency (unless what you mean is that quantum gravity might violate one of the assumptions of those theorems) is a lot less powerful than you guess.
[0] https://en.wikipedia.org/wiki/Coleman%E2%80%93Mandula_theore...
[1] https://en.wikipedia.org/wiki/Haag%E2%80%93Lopuszanski%E2%80...
I see you are of the belief that Bishnu singlehandedly created mathematics, and I respect that, but I consider it more likely it must have been the work of a union of gods. Ganesh, who I consider to be my patron God (no snark here), and the lord of Buddhi, surely had an important role to play there?