Friday, March 16, 2012

On the radical incompleteness of modern physics

Kurt Gödel published two theorems in 1930 that establish the inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The first of these famous "incompleteness" theorems (and here I quote from Wikipedia), "states that no consistent system of axioms whose theorems can be listed by an "effective procedure (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers.  For any such system, there will always be statements about the natural numbers (or irrational or complex numbers) that are true, but that are unprovable within the system." (Should one add axioms that make the truth or falsity of these statements provable, it will be possible to find more true statements that can not proved within the axiomatic structure of this new, enlarged axiomatic system.)The second incompleteness theorem, a corollary of the first, shows that such a system cannot demonstrate its own consistency. 

Something analogous seems to have happened to physics in the late 20th and early 21st centuries.

The explanatory and predictive power of modern physics has and is being repeatedly demonstrated at both the relativistic and sub-atomic scales. The theories of special and general relativity have been tested again and again and not found wanting. The same can be said of The Standard Theory of Quantum Mechanics—the missing link of the Standard Theory, the Higgs Boson, will it seems, soon be cemented into place. But all is not well: there seems to be some inconsistency between Relativity and the Standard Theory. Every attempt to unify these two theories has failed. Both of these are field theories, which is to say that every particle or object in the universe is connected with a field and what these theories describe (if that's the word) is these fields. You can feel the electro-magnetic field of a bar magnet if you put an iron object near it and you can feel the field of gravity every time you fall; you can see it in the motion of a pendulum—or of the moon; if there is such a thing as the Higgs boson, and it seems that there is, you can feel the presence of the Higgs field, which is what gives matter its mass, every time you push someone on a swing and feel the inertia of his or her body.

The trouble is that the equations of the Standard Theory can't be used to describe relativistic fields and the equations of relativity can't be used to describe electromagnetic fields, or indeed any of the other fields described in the Standard Theory. 

So, there's a massive disconnect between Relativity and Quantum Mechanics. For a while it seemed that String Theory (which I do not understand at all) might be able to bring Relativity and Quantum Mechanics together: the equations of Relativity and those of Quantum Mechanics show up quite naturally in the mathematics of string theory. There's just one hitch, and it is not a small one: the mathematics of string theory has not produced a single verifiable prediction since it was first invented more than twenty years ago. And that's not for lack of trying.

Just as Gödel demonstrated the fundamental incompleteness of mathematics, so it would seem that we must now face the fact that physics too is fundamentally incomplete—not because of some radical fault in our assumptions or our thinking or our experimental procedures, but because nature is more mysterious than we can possibly understand. It's not that we have somehow or other asked the wrong questions but that nature is so strange that we will never, ever know how to ask the right ones. Maybe there aren't any. I find that last possibility not only interesting and amusing but also satisfying.

And then there's the fact that about 96% of the universe is composed of dark matter and dark energy, which cannot be accounted for in any of our theories.

I don't want to be misunderstood: none of this has anything to do with religion; if you want to fill in the empty blanks with the word 'god' that's your business, not mine.

Gödel's incompleteness theorem did not slow down the development of logic and mathematics, even slightly.
The incompleteness of physics will have no effect on the work that is now going on at the frontiers of science in Astronomy or condensed-matter physics or superconductivity.

(Since my ignorance of these matters while not total, is profound, I would welcome comments or corrections from those who really know what they are talking about. I doubt, however, that I am seriously mistaken on my main point, the radical incompleteness of modern physics.)

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