Incompleteness, Undecidability, and Uncertainty

[J_B_]

The LHS has the mathematical rigour, the RHS doesn't.

Yes, I know. That was clear. I'm only asking about the RHS.

The use of mathematics in physics is far less rigorous because proof is not a criteria as it is in mathematics.
Physics like science in general stands or falls through observation and experiment.
The fact GR has passed every test thrown at it over the last hundred odd years, the educated guess of the RHS was quite justifiable.

That's fine. I'm not asking if the RHS has passed every test thrown at it. I'm asking if it's been proven that it can't be derived from a set of axioms.

 

[essentialsaltes]

With respect to the different spaces that result and the physical phenomena they are meant to describe, are all still valid? i.e. do all choices describe the physical phenomena of space-time?

I refer you back to xkcd. Mathematics is not an attempt to describe the physical phenomena of space-time.

 

[J_B_]

Mathematics is not an attempt to describe the physical phenomena of space-time.

What is the purpose of mathematics in physics?

 

[Speedwell]

What is the purpose of mathematics in physics?

To create predictive models of natural phenomenon--much the same as in any other science.

 

[J_B_]

To create predictive models of natural phenomenon--much the same as in any other science.

Predictive, but not in a way that describes those predictions.

 

[sjastro]

Yes, I know. That was clear. I'm only asking about the RHS.



That's fine. I'm not asking if the RHS has passed every test thrown at it. I'm asking if it's been proven that it can't be derived from a set of axioms.

I have already answered your question which is emphatically no.
If it is good enough to be supported by observation and experiment the question of whether it can be proven by axioms is irrelevant.

A more concrete example is the “derivation” of the RHS is the very antithesis of a mathematical proof (which in this case is not a proof by contradiction) because you start off with the conclusion and work backwards.

By working backwards one applies the weak field limit to the Einstein field equations;

to find they break down to the Newtonian (Poisson) equation describing the gravitational potential Φ.

Hence the introduction of the energy matter tensor Tuv is consistent with the Poisson equation being the weak field limit.
So while mathematicians would probably go into convulsions over this method, it's OK with physicists because it is not meant to be mathematically rigorous.

 

[J_B_]

I have already answered your question which is emphatically no.
If it is good enough to be supported by observation and experiment the question of whether it can be proven by axioms is irrelevant.

I must have missed the emphatic no, but it was clear this time. You answered my follow up question as well. If it could be derived, it would be irrelevant.

That surprises me. I would think the one who could derive it (if that's possible) would earn a name for himself. It would also seem to open up all kinds of possibilities to use further derivation to explore further possibilities.

I'm sure you'll object, but the way you talk about this again reminds me of experiences I've had with colleagues that I have dismissed as "mere curve fitting". IMO data that fits some arbitrary equation with no explanation why the equation fits is next to worthless. You might as well just stick to the data.

But I'm apparently a misinformed outlier. Despite my tone, I do thank you for the conversation.

 

[sjastro]

I must have missed the emphatic no, but it was clear this time. You answered my follow up question as well. If it could be derived, it would be irrelevant.

That surprises me. I would think the one who could derive it (if that's possible) would earn a name for himself. It would also seem to open up all kinds of possibilities to use further derivation to explore further possibilities.

The construction of the matter energy tensor Tuv is based on diverse concepts such as special relativity and fluid mechanics.
It is not derived on a fundamental level such as using axioms.

The author behind this video on how the energy matter tensor is constructed is a genius for not only explaining the construction in a convincing simplistic fashion but the subtle connection to zombies in The Living Dead.

I'm sure you'll object, but the way you talk about this again reminds me of experiences I've had with colleagues that I have dismissed as "mere curve fitting". IMO data that fits some arbitrary equation with no explanation why the equation fits is next to worthless. You might as well just stick to the data.

Both Kepler's laws of planetary motion and the Balmer series of emission lines in the hydrogen spectrum originated as "mere curve fitting" of the data without any explanation as to how.
With the evolution of Newtonian gravity and quantum mechanics both empirical formulae were derived mathematically from first principles; not from axioms but from ideas which included the central force laws for gravity and electromagnetism respectively.
This is how science works.

 

[FrumiousBandersnatch]

Well, yes. But so are the basic axioms of arithmetic. They seem reasonable (inferentially) but we cant really prove them. In fact they are so inferentially strong that we feel justified in using them as axioms to build a whole edifice of mathematics.

I agree, but we can't do mathematics without them; they are what mathematics is about.

If the laws of physics varied among places/times we'd be totally at sea, with no stability to reasoning from our observations. Stellar spectra? Spikes at various frequencies could now imply anything.

Of course, if the laws varied randomly, we probably wouldn't be here at all. I was thinking along the lines of consistent changes over cosmological timescales and distances, e.g. gravity, or the speed of light, or the strength of electromagnetism, or the influence of an 'aether', etc., all of which have been seriously considered at various times.

But we have chosen to assume the "axiom of legal consistency" (or whatever they call it) probably because practically speaking right now 3 variable science (1 what material stuff exists, and 2 what laws govern here, and 3 what laws might also govern there) is simply beyond us. Also the principle of parsimony come into play.

Well, that depends on the laws themselves (i.e. the universe itself) - Einstein had to break the principle of parsimony in his equations to obtain what was thought to be a more parsimonious static universe rather than a dynamic universe because his equations otherwise implied either an expanding or contracting spacetime...

 

[Speedwell]

Predictive, but not in a way that describes those predictions.

You might say, it's the difference between instrumentality and ontology.

 

[FrumiousBandersnatch]

JFYI, I found an interesting article on axioms.

"Axioms are not self-evident truths in any sort of rational system, they are unprovable assumptions whose truth or falsehood should always be mentally prefaced with an implicit ``If we assume that...''. Remembering that ultimately ``assume'' can make an ass out of u and me, as my wife (a physician, which is a very empirical and untrusting profession) is wont to say. They are really just assertions or propositions to which we give a special primal status and exempt from the necessity of independent proof."​

 

[SelfSim]

"Axioms are not self-evident truths in any sort of rational system, they are unprovable assumptions whose truth or falsehood should always be mentally prefaced with an implicit ``If we assume that...''. Remembering that ultimately ``assume'' can make an ass out of u and me, as my wife (a physician, which is a very empirical and untrusting profession) is wont to say. They are really just assertions or propositions to which we give a special primal status and exempt from the necessity of independent proof."​

Which is precisely why science, on the other hand, starts with no assumptions .. (eg: such as 'things existing').

 

[FrumiousBandersnatch]

Which is precisely why science, on the other hand, starts with no assumptions .. (eg: such as 'things existing').

I think, in practice, scientists have tended to start with certain assumptions - which are not necessary, have often been overturned, and can be counter-productive...

 

[SelfSim]

I think, in practice, scientists have tended to start with certain assumptions - which are not necessary, have often been overturned, and can be counter-productive...

Then they weren't thinking like scientists before they started were they .. (?)

 

[FrumiousBandersnatch]

Then they weren't thinking like scientists before they started were they .. (?)

They think like humans

 

[sjastro]

lf our model assumes flat space, and a test falsifies flat space, we choose a different mathematical system (different axioms).

Euclid's parallel postulate is not satisfactory under any circumstance as it tells us nothing whether spacetime is a manifold or locally flat.
The converse is also a problem where curved space is embedded in flat space.

Cosmologists instead use a technique familiar to surveyors.
Surveyors use triangulation where the sum of the angles of a triangle equals 180 degrees in Euclidean space.
Cosmologists use the thermal variations in the cosmic radiation background to determine the geometry of the observable universe where the angular dimension of the thermal variation forms one side of the triangle.

In a closed or spherical universe the sum of the angles is greater than 180 degrees, in an open or hyperbolic universe it is less than 180 degrees.
The observable universe is found to be flat.

​

 

[durangodawood]

...Of course, if the laws varied randomly, we probably wouldn't be here at all. I was thinking along the lines of consistent changes over cosmological timescales and distances, e.g. gravity, or the speed of light, or the strength of electromagnetism, or the influence of an 'aether', etc., all of which have been seriously considered at various times...

Ok, but still we proceed under the axiomatic assumption that the laws of physics are consistent and build the whole edifice of cosmology starting with that assumption. Science treats this consistency as an axiom basically. Perhaps it could be another way. But thats not whats happening.

 

[FrumiousBandersnatch]

Ok, but still we proceed under the axiomatic assumption that the laws of physics are consistent and build the whole edifice of cosmology starting with that assumption. Science treats this consistency as an axiom basically. Perhaps it could be another way. But thats not whats happening.

I'm simply suggesting that such an assumption is not necessary - observations can establish local laws and we can test the hypothesis that they apply further afield.

 

[durangodawood]

I'm simply suggesting that such an assumption is not necessary - observations can establish local laws and we can test the hypothesis that they apply further afield.

Im pretty sure the assumption of consistency is necessary to make conclusions from cosmological data, maybe not in principle, but at least in practice where our point of view is limited to near earth.

For example, differing redshift among various supernovae that led to the finding of accelerated expansion of the universe. That only works if we hold the laws of physics constant across the various times and positions of these data points. If not then we have to throw the whole finding out.

 

[FrumiousBandersnatch]

Im pretty sure the assumption of consistency is necessary to make conclusions from cosmological data, maybe not in principle, but at least in practice where our point of view is limited to near earth.

For example, differing redshift among various supernovae that led to the finding of accelerated expansion of the universe. That only works if we hold the laws of physics constant across the various times and positions of these data points. If not then we have to throw the whole finding out.

But observing the consistency of the redshifted spectra compared with local star spectra can establish that the laws of physics are the same...