Axiomatic Systems & Modern Physics

[Atran]

Hi, I'm not sure if this is the right place to post this.
Are other different axiomatic systems used for QM & GR?
I mean, as I'm currently a high school student, all theorems I've learned are proved by the same set of axioms.
I wonder if another applicable axiomatic system is used for non-classical physics.

Thanks for any valuable answer...

 

[mathman]

QM and GR are inconsistent with each other, so presumably they couldn't start with the same axioms.

Bit of advice from a mathematician: Don't treat physics like mathematics. Ultimately physics depends on the real world, while mathematics comes from a consistent set of axioms.

 

[K^2]

Solid advice. A good theory will be mathematically consistent with itself, but its postulates still come from observation.

 

[Atran]

Bit of advice from a mathematician: Don't treat physics like mathematics. Ultimately physics depends on the real world, while mathematics comes from a consistent set of axioms.

Solid advice. A good theory will be mathematically consistent with itself, but its postulates still come from observation.

I know that, for every axiomatic system, there is a logical framework.
We describe reality logically and therefore we are continuously seeking for an axiomatic system which with more accuracy applies to our reality.
If Gödel's incompleteness theorem is true, then is it possible to achieve a complete physics theory?
What if an observed phenomena could be explained by two different sets of axioms, and if so, which one would you recommend?

 

[K^2]

Problem is that we can't tell that a theory fails to describe something until we try it. That means, no theory can ever be proven to be correct.

With that in mind, yes, it's possible to build a theory that's complete, but it will probably be useless. Something like 1=1 comes to mind. That's about as complete as you can make it, and in terms of mathematical absolutes, no better than any other theory we may develop. But it doesn't describe the real world very well. Or at all. From perspective of physics, that's a bit of a problem.

 

[Cleonis]

Are other different axiomatic systems used for QM & GR?
I mean, as I'm currently a high school student, all theorems I've learned are proved by the same set of axioms.
I wonder if another applicable axiomatic system is used for non-classical physics.

In general: it's very rare for a physics theory to be axiomatized.

For instance, there are no axioms for classical electrodynamics. (If I remember correctly someone tried, and came up with a system of 6 axioms or so. I don't know whether that was a sufficient system.)

In the few cases where there is a small set of "laws" the main purpose of those "axioms" is to be evocative. Compared to axioms in mathematics they really serve quite a different purpose.

As I said, the purpose of the "axioms" is to be evocative, to captivate, I don't think there is any mathematical rigor to it. It's about focussing attention on what is regarded as most fundamental in the theory.

 

[Andy Resnick]

Ultimately physics depends on the real world, while mathematics comes from a consistent set of axioms.

Solid advice. A good theory will be mathematically consistent with itself, but its postulates still come from observation.

In general: it's very rare for a physics theory to be axiomatized.

For instance, there are no axioms for classical electrodynamics.

I'm not a mathematician, but I am fairly certain that continuum mechanics (which includes GR, electrodynamics and thermodynamics) has been fully axiomatized (AFAIK, by Truesdell and Noll), at least in the sense Hilbert meant.

However, it is true that the axioms of continuum mechanics alone do not completely specify the physical behavior of an arbitrary system: for that, *constitutive relations* must be specified (by oberservation or measurement).

But I'm not able to claim this is a fact as well as a real mathematician could.