Incompleteness, Undecidability, and Uncertainty

[FrumiousBandersnatch]

Well, just that Euclidean space is *not* an axiom of Science.

Axioms of science would be more like methodological naturalism.

Quite - I am wondering whether we need a concept of 'nature' before we can do science, or is 'nature' a framework we propose post-facto to frame our observations? I suppose it comes down to where science begins...

The supernatural is an alternative framework that is not derived scientifically, but phenomena can transition from supernatural to natural if they can be shown to meet the criteria of scientific models in the natural framework.

Apologies, I'm just thinking aloud here!

 

[FrumiousBandersnatch]

Right, not an axiom of science, but assumptions, stated or not, in a particular model.
I wouldn't quite call them axioms, but they are assumptions used in building a particular model.

Euclidean space is an assumption of Newtonian mechanics.
'flat' Minkowski space is an assumption of special relativity.
Riemannian space is an assumption of general relativity.

I guess jumping back to #4 if that's where it started, Gödel's Theorem is never going to be relevant to science. Mathematical axiomatic systems are nothing but their axioms. Everything else follows as a necessary logical consequence. That's not what science is, even if it has assumptions.

Yes - it seems to me that mathematics and logic are explorations of the implications of the chosen axioms.

 

[J_B_]

Sure. Could you clarify the difference you see between an assumption and an axiom?

I guess jumping back to #4 if that's where it started, Gödel's Theorem is never going to be relevant to science. Mathematical axiomatic systems are nothing but their axioms. Everything else follows as a necessary logical consequence. That's not what science is, even if it has assumptions.

I think you added this after I asked my question. It's a very good point - or at least it sounds good at first. But as I parse it, I start to ask, "OK, what additional thing does science have?"

Can we first restrict ourselves to sciences that extensively use mathematical models (e.g. physics)? In that case, would you say the additional thing is that the arbiter of science is the experiment, not the logic? If the experiment falsifies the hypothesis, it doesn't matter if the logic was consistent?

But ... if that happens ... wouldn't the axioms be changed? Aren't we seeking a state where the logic holds and the test is successful?

 

[FrumiousBandersnatch]

I would offer this again. For science to operate it requires certain assumptions that are deeper than any one fields reigning paradigm. I think its fair to call those assumptions "axioms".

This is what I'm questioning - whether those assumptions are necessary for science to operate (I'm not including the assumptions of reasoning itself).

One I offered previously was: the laws of nature are stable across accessible time and space. Dispense with that and theres no way to generalize from repeat observations at all.

Can't that be an inference from observational experience? If we see the same patterns repeating across time and space, and our predictions that they are stable across time and space are fruitful, then we are justified (pace Hume) in making that inference. And if the laws of nature varied in some consistent manner across time and space, we would have a harder job, but it seems to me that we could, in principle, discover what they were and how they varied.

(There's our uncertainty. I dont think there's a way for scientific investigation to demonstrate the validly of that axiom with certainty).

Yup.

 

[sjastro]

Physics by virtue of the fact is more mathematical than any of the other the sciences is therefore "more axiomatic".
This is not to say it is in the same league as pure mathematics.
Physicists are free to use poetic license when it comes to the maths as shown in Einstein's field equations.

Einstein stated the left hand side of the field equations were built using marble while the right hand side were made of straw.

 

[J_B_]

Physicists are free to use poetic license when it comes to the maths as shown in Einstein's field equations.

Yet, were it falsified, the search would go on. Are you saying that as long as it's not falsified, no one desires to find the axiomatic basis of the RHS?

I know of engineering situations where that got people into trouble. Specifically, in the example I'm thinking of, they overlooked maintaining a mathematical dimensionality.

 

[FrumiousBandersnatch]

Given you are a biologist, I'd be curious about your thoughts on the two papers referenced.

If you mean the two papers on the problem of trying to pin down a genome or produce truly genetically identical organisms - because of mutations; I already commented - they seem to be explicit demonstrations of what should be obvious to working transmission or molecular geneticists.

 

[J_B_]

If you mean the two papers on the problem of trying to pin down a genome or produce truly genetically identical organisms - because of mutations; I already commented - they seem to be explicit demonstrations of what should be obvious to working transmission or molecular geneticists.

OK.

 

[Tinker Grey]

Physics by virtue of the fact is more mathematical than any of the other the sciences is therefore "more axiomatic".
This is not to say it is in the same league as pure mathematics.
Physicists are free to use poetic license when it comes to the maths as shown in Einstein's field equations.

Einstein stated the left hand side of the field equations were built using marble while the right hand side were made of straw.

Reminds me of Purity

 

[essentialsaltes]

In that case, would you say the additional thing is that the arbiter of science is the experiment, not the logic?

That is one big additional thing yes, empirical data. Another would be... I dunno... brute facts of existence. Like things. Masses.

An inverse square law in an empty 3D Euclidean universe is a flawless physical model. But not a model of reality.

In nonnegative integer arithmetic once you have axioms like
#1: 0 is a number
#2: Every number has a successor

Then all the numbers poof into existence. You don't have to add things to the model like adding planets into a solar system model.

But ... if that happens ... wouldn't the axioms be changed?

That's one reason why I resist calling them axioms. Mathematical axioms can't be changed. Certainly not by empirical observations. We don't pick a really large number and test whether it has a successor. Euclidean geometry hasn't been falsified by observation. Both Euclidean and non-Euclidean geometry are valid mathematical systems.

 

[J_B_]

That is one big additional thing yes, empirical data. Another would be... I dunno... brute facts of existence. Like things. Masses.

I'll accept that. The addition of things, yes. Bodies, yes.

Masses ... no. Mass requires a theoretical description ... well, depending on your usage I guess. Of course "mass" had a colloquial meaning prior to physics, and physics appropriated the word because the colloquial meaning was close to what the scientist intended. Still, though, the mass of physics requires theory.

That's one reason why I resist calling them axioms. Mathematical axioms can't be changed.

Sure they can. Axiom: Given a line A and a point not on the line, there is one and only one line B parallel to A.

Axiom: Given a line A and a point not on the line, there is no line B parallel to A. I just changed it. And both result in valid mathematical systems. If it is truly an axiom, there is always an alternative ... at least for me. I'm not a monist.

Certainly not by empirical observations. We don't pick a really large number and test whether it has a successor.

A valid point, but not what I meant. If our model assumes flat space, and a test falsifies flat space, we choose a different mathematical system (different axioms).

 

[sjastro]

Yet, were it falsified, the search would go on. Are you saying that as long as it's not falsified, no one desires to find the axiomatic basis of the RHS?

I know of engineering situations where that got people into trouble. Specifically, in the example I'm thinking of, they overlooked maintaining a mathematical dimensionality.

I suspect your definition of falsified is very different from that of a scientist.
The equations are falsifiable as they make predictions which are testable.
The simplest form of the equations is to set the RHS equal to zero which corresponds to the absence of external gravitational and electromagnetic fields.
From these corresponding vacuum equations, a solution is the Schwarzschild metric which has explained the perihelion advance of Mercury's orbit to the gravitational bending of light.

When the RHS is non zero it becomes a gravitational theory which is applied to cosmological models.
The field equations with a non zero RHS is a generalization of the classical Poisson equation since Newtonian gravity is a low order approximation of general relativity.

 

[sjastro]

Reminds me of Purity

I use this to stir physicists.

 

[J_B_]

I suspect your definition of falsified is very different from that of a scientist.
The equations are falsifiable as they make predictions which are testable.
The simplest form of the equations is to set the RHS equal to zero which corresponds to the absence of external gravitational and electromagnetic fields.
From these corresponding vacuum equations, a solution is the Schwarzschild metric which has explained the perihelion advance of Mercury's orbit to the gravitational bending of light.

When the RHS is non zero it becomes a gravitational theory which is applied to cosmological models.
The RHS is a generalization of the classical Poisson equation since Newtonian gravity is a low order approximation of general relativity.

That doesn't answer my question, so let me ask it this way: What does it mean that the RHS is made of straw?

 

[essentialsaltes]

Sure they can. Axiom1:

Axiom 2:

You may choose one or the other, but neither has primacy. They are equally valid. Neither has been shown incorrect. One didn't break like a lightbulb and need to be replaced.

 

[J_B_]

You may choose one or the other, but neither has primacy. They are equally valid. Neither has been shown incorrect. One didn't break like a lightbulb and need to be replaced.

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?

 

[sjastro]

That doesn't answer my question, so let me ask it this way: What does it mean that the RHS is made of straw?

I thought it was self evident.
To a pure mathematician the LHS is very familiar, composed of second rank Ricci tensor, a Riemann curvature scalar and a metric tensor.
The RHS has been made up.
It is based on a brilliant physical insight of expressing energy as a tensor, there is no pure mathematical derivation of the tensor T.

 

[J_B_]

The RHS has been made up.
It is based on a brilliant physical insight of expressing energy as a tensor, there is no pure mathematical derivation of the tensor T.

That's what I thought, but then you seemed to indicate otherwise. Is it proven it can't be derived, or is it simply that no one has?

 

[durangodawood]

This is what I'm questioning - whether those assumptions are necessary for science to operate (I'm not including the assumptions of reasoning itself).

Can't that be an inference from observational experience? If we see the same patterns repeating across time and space, and our predictions that they are stable across time and space are fruitful, then we are justified (pace Hume) in making that inference. And if the laws of nature varied in some consistent manner across time and space, we would have a harder job, but it seems to me that we could, in principle, discover what they were and how they varied.

Yup.

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.

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.

I suppose if the laws vary consistently we could somehow in principle "triangulate" the variance of laws vs the material conditions we're observing. 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.

 

[sjastro]

That's what I thought, but then you seemed to indicate otherwise. Is it proven it can't be derived, or is it simply that no one has?

Perhaps I haven't been clear enough.
The LHS has the mathematical rigour, the RHS doesn't.
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.