Monday, January 2, 2012

Implications of Godel's Theorems on a Physical "Theory of Everything"

Disclaimer: I'm no expert on physics. If I've mischaracterized anything in this article, please let me know so I can correct it.

I've been on something of a logic binge lately. For the past several days, I've had Godel's Incompleteness Theorems, Paraconsistent Logic, and more floating through my head. I've always been interested as to what the Incompleteness Theorems mean for the notion of a physical Theory of Everything. After writing my recent post on the subject, I feel that I have a much more solid understanding of the Theorems now than I ever have before, thus I feel somewhat confident in revisiting this idea with a fresh perspective. I'm a bit tired, so I'll keep things pretty simple.

Godel's Theorems consider the inherent limitations of axiomatic systems, and the goal of a Theory of Everything certainly fits under this category. Theoretical physicists have for many years searched for a minimal set of all-guiding universal laws which underlie all physical phenomena. So far, this quest has met little success. According to the Incompleteness Theorems, there are two obvious possibilities: Either the axioms of the universe are incomplete or they are inconsistent - presumably, they are not both.

What are the implications of an incomplete universe? An incomplete but consistent universe would have a set of physical laws, physical axioms, which determine the totality of all physical phenomena in the universe. However, within such a universe, there would exist some physical behaviors which could not be explained purely through deduction from the axioms, and there would be certain scientific questions which could never be answered by the laws of physics alone. Such a worldview might help to explain such phenomena as emergent properties in Biology, turbulence in Physics, and consciousness in psychology/neuroscience. However, an incomplete universe has deeply disturbing implications, as it seems to imply a universe in which the laws of cause and effect are not necessarily absolute.

What, then, are the implications of an inconsistent universe? In an inconsistent (or paraconsistent) universe, things can be both true and false at the same time. The rules of classical logic which are used throughout all realms of scientific inquiry become virtually moot, as the proof by contradiction and almost all forms of reductionism become useless. As horrible as it sounds, the phenomena of quantum mechanics almost seem to suggest exactly such a universe. Quantum logic works on the basis that a statement might be true, false, or both simultaneously, and it is exactly this sort of contradiction that underlies quantum physics. Nonetheless, the notion of a universe in which all classical logic fails is deeply unnerving, as the laws of such logic are deeply ingrained into much that we as humans do as a species. An inconsistent universe is scarcely any more comforting than an incomplete one.

These are usually the two possibilities put forth when considering Godel's Theorems' implications on Physics, but (and I'm sure that I'm not the first to suggest this), I feel there's a third: Perhaps the Incompleteness Theorems simply don't apply to Physics at all. "But how could that be?" you ask, "any effectively generated theory capable of expressing the natural numbers is subject to either incompleteness or inconsistency!"

...Well who in the hell ever decided that the Universe is capable of describing the natural numbers anyway? Assume a finite Universe. Then there is a finite number of atoms in the Universe. Or strings if you prefer. Or ham sandwiches (my personal and equally likely alternative to strings). Then it stands to reason that the Universe cannot express quantities greater than the number of subatomic particle-string-ham-sandwiches, and thus the Universe cannot express the natural numbers. Then perhaps the axioms underlying the Universe are weaker than Peano Arithmetic, and perhaps Godel's Incompleteness Theorems do not apply to them at all. Perhaps, assuming a finite Universe, we can have a complete, consistent physical reality after all. And perhaps this is actually a pretty strong reason to believe in a finite reality...Which raises some questions about what exactly all of this non-finite mathematics is good for (I'm looking at you, Large Cardinal Theory).

Just some thoughts.