The consequences of Goedel's first incompleteness theorum for physical theories

[jcsd]

I didn't know whether to put this one in maths or physics, but I'm interested on what people on PFs think are the consequences of Goedel's first incomplteness theorum are for physical theories in general and in particular for the idea of a single comphrehensive theory that can describe all physical situations (i.e. a TOE theory).

 

[mathman]

Goedel's theorem is about mathematics, not physics. Physcial models are "proven" by experimental evidence. It has no bearing on whether or not there is a single comprehensive theory - this ultimately will have to be shown consistent with nature.

 

[jcsd]

Well not quite Godel's proof was and still is seen as a blow to the idea of a physical theory with the fewest possible axioms that can describe everything.

My underlying assumption is that a theory that can describe everything would be a formal axiomatic system, in which case it would contain unprovable statements.

 

[HallsofIvy]

Then apparently you do not understand the difference between physics and mathematics. Physics theory are not based on axioms- they are based on experimental evidence. The test for truth of a mathematical theory is its consistency- exactly the point raised by Goedel. The test for truth of a physical theory is its confirmation by experiment- Goedel's theorems have nothing to do with that.

 

[jcsd]

Well then I don't think you understand physical theories as they do contain axioms, albeit ones that are derived from experimental data. For example in special relativity the fact that the speed of light is constant in all inertial refernce frames is axiomatic and not a result of the theory, unlike time dialtion which is a result of the theory. So a physical theory IS a formal axiomatic system.

 

[Integral]

A constant speed if light is based on Maxwell's equations, which are based on experimentation. So even though Einstein postulated a constant speed of light, it is based on experiment. After Maxwell derived the constant and computed the value of the propagation speed of Electro Magnetic waves he was shocked to find him self looking at the then current experimental value for the speed of light. This was the first valid evidence that light was a form of electro magnetic radiation. (Of course Maxwell's work was the first realization of the existence of electro magnetic radiation)

 

[Hurkyl]

The whole thing is a moot point anyways if the physical theory can't model the arithmetic of integers. There are (useful!) formal axiomatic systems in which every statement can be algorithmically proven true or false.

 

[Jonathan]

I have a thread on this in the Logic forum.

 

[jcsd]

Physical theories exist as abstract formal systems regardless of experimental evidence and therefore must contain senetnces, S such that neother S nor ~S can be proved using the formal system.

As for Maxwell's equations the interpretation at the time did not lead to a constant speed of light in all inertial reference frames, infact it was taken to mean that light must propgate through a medium (i.e. ether).

The effect of Goedel's theorum on physical theories was not something a plucked out of thin air, infact the thread was sparked by browsing through my maths dictionary which said that Goedel's theorum is seen as a blow to the idea of a physical theory with the fewest possible axioms.

 

[jcsd]

Originally posted by Hurkyl
The whole thing is a moot point anyways if the physical theory can't model the arithmetic of integers. There are (useful!) formal axiomatic systems in which every statement can be algorithmically proven true or false.

Hurkyl, my maths is mostly maths for physics, but surely Goedel's incompleteness theorum shows that no system can be complete?

 

[thermonuclear]

I think there is indeed a relation between Gödel’s theorem and some modern approaches to physics, like quantum causal histories and relational quantum mechanics. The point is rather intuitive: complete descriptions or self-measurement are not possible per definitio. This fact is considered in the formulation of the theory and not as a consequence of a mathematical therorem (or an idea at least) acting on them.

 

[HallsofIvy]

posted by jcsd
Physical theories exist as abstract formal systems regardless of experimental evidence and therefore must contain senetnces, S such that neother S nor ~S can be proved using the formal system.

Then they are models for physical theories, not physical theories.
It also, as Hurkyl pointed out, not follow that every formal system must not be complete. That is only true for formal systems that include the positive integers.

 

[jcsd]

Originally posted by HallsofIvy
Then they are models for physical theories, not physical theories.
It also, as Hurkyl pointed out, not follow that every formal system must not be complete. That is only true for formal systems that include the positive integers.

You see, I'm not a mathematician my undergrad training was in physics (though I've just decided to start a part-time undergrad. maths degree at the beginning of next year). Is it true only for systems with positive integers? My knowledge of Goedel's proof only comes from how it is layed out in my maths dictionary and it doesn't mention this.

 

[jcsd]

eh?

I've been to mathworld and does seem to say that the theorum only applies to number theory, but in other parts it suggests it applies to mathematical systems in general[?]

 

[Hurkyl]

Sorry, I was having computer troubles, so I made the dot post just to make sure I didn't forget about this thread!



Basically, Godel's theorem states that if a formal system can ask about its own consistency, then the system is inconsistent or the system cannot prove it is consistent (and is thus incomplete).


Number theory can encode logic, so any system that can encode number theory can then ask about its own consistency, and thus must be inconsistent or incomplete.

 

[quartodeciman]

I think it is a bogus question. What kind of crazy proposition would a theoretically-undecidable fact of physical reality be?

The purpose of a TOE is to integrate the (assumed) four fundamental interaction types into one deductive system, and provide some experimentally-testable propositions that indicate a level of success in the endeavor. Deriving every conceivable complicated proposition from it isn't in the gameplan. Likewise, there is no hope of performing every conceivable variation of every conceivable experiment, so we shouldn't expect to get absolutely-complete and absolutely-certain knowledge.