Monday, 18 May 2015

Kurt Gödel's Theorems & Physics

It's often asked whether or not Kurt Gödel's theorems can be applied outside mathematics. John Horgan certainly applies them to the theories of physics. Or, more accurately, he writes that

“Kurt Gödel's incompleteness theorem denies us the possibility of constructing a complete, consistent mathematical description of reality” (6).

Clearly there's a jump here from Gödel's mathematical incompleteness theorems to physical reality. Or, more accurately, from Gödel's theorems to a “consistent mathematical description of reality”. Is that jump justified?

Well, for a start, physics is utterly dependent on mathematics. Thus if all descriptions of reality in physics involve mathematics, and mathematics is subject to Gödel's theorems, then that must pass over to the descriptions of reality which are offered by physicists. In other words, if a mathematical system must be incomplete (or not entirely provable), then that description of reality must be incomplete (or not entirely provable). The two must fall and rise together.

More meat is put on this idea of whether or not Gödel's theorems are applicable to theories about reality when John Horgan says that the “British physicist John Barrow argued that Gödel's incompleteness theorem undermines the very notion of a complete theory of nature” (69). We move again from mathematical systems to the incompleteness of a “complete theory of nature”. It can be said here that Barrow is simply transferring the incompleteness of mathematics to the incompleteness of a “complete theory of nature”. Again, does the former necessarily pass over to the latter?

In more detail: Godel established that "any moderately complex system of axioms inevitably raises questions that cannot be answered by the axioms". Then Horgan moves onto to say that the “implication is that any theory will always have loose ends”.

Many scientists accept this application of Godel's theorems to physics, including Moravec, Roger Penrose and Freeman Dyson. The latter says:

Since we know the laws of physics are mathematical, and we know that mathematics is an inconsistent system, it's sort of plausible that physics will also be inconsistent.” (254)

Thus what we have here is a logical argument:

i) Physics is mathematical.

ii) Mathematics is an inconsistent system.

iii) Therefore physics must be an inconsistent system (or simply incomplete).

The only problem here is seeing the entirety of mathematics as a single system (which itself incorporates systems). Perhaps it is.

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