What if a scientific theory is not testable?

[ZapperZ]

He/she searching the web found "mathematically self-consistent" in the sense of a necessary attribute of physical theories quite often. I am not aware of an unambiguous definition though.

Eg. Thermodynamics, what does "mathematically self-consistent" mean if empirical evidence of the laws is not a criterion?

Or QM, I think it is mainly agreed that it's formalism is "mathematically self-consistent". But how to prove it mathematically?

I'm sorry, but is there ANYTHING in science that can be proven to the same degree as mathematics? The starting point in any science and in physics in particular, are not derived. No one derived the symmetry principles that gave us all the conservation laws. You don't "prove" physical principles or theories. You verify them via experimental agreement to the degree that it can be tested.

Zz.

 

[vis_insita]

But how can something which is only an approximation, be considered to be contradictory, when it is, BY DEFINITION, should give a different result?

Whether there is a contradiction or not is a purely logical question. If GR implies "The perihelion precession of Mercury is 574″ ± 5'' per century", while Newtonian gravity implies that this is not the case, then there is as clear a contradition between both theories as can be.

Of course knowing both theories and how they relate to each other we expect exactly that difference. But only the existence of that difference matters, not whether it surprises us.

 

[ZapperZ]

Whether there is a contradiction or not is a purely logical question. If GR implies "The perihelion precession of Mercury is 574″ ± 5'' per century", while Newtonian gravity implies that this is not the case, then there is as clear a contradition between both theories as can be.

Of course knowing both theories and how they relate to each other we expect exactly that difference. But only the existence of that difference matters, not whether it surprises us.

And I do not consider that to be a contradiction, because GR merges back into Newtonian gravity at some point. So how can it contradicts itself?

See, you obviously do not know a lot about condensed matter physics, because I can point out NUMEROUS models that are contradicting one another. Case in point: the mechanism for superconductivity in cuprate superconductors. For the longest time, two competing models were running neck-and-neck: the spin-fluctuation mechanism versus the phonon coupling. And get this. Both models produces many of the SAME results that agree with experimental measurements!

Now THAT is what I call a "contradiction", because the difference here is NOT simply because one is an approximation of the other. The differences are at the FUNDAMENTAL level, because those coupling channels are not the same beast.

But here's the thing about research-front areas of science: such contradictions are NORMAL. When something is still being actively researched, competing models and theories are a normal part of the activity! This is because you are trying to figure out what exactly is in the black box that you still haven't been able to open. Many different descriptions can fit what have been observed. As more and more are known, theories that do not fit the observations will fall to the wayside. That is how science progresses!

Newtonian mechanics and Special/General Relativity are not at odds with one another, because we KNOW the connection between the two. Applying small-angle approximation of a pendulum when it is clearly no longer oscillating at a small angle is NOT the fault of the theory, but the fault of the person who applied the theory where it shouldn't be applied to!

Zz.

 

[vis_insita]

And I do not consider that to be a contradiction, because GR merges back into Newtonian gravity at some point. So how can it contradicts itself?

I did not say or imply that GR contradicts itself. The Newtonian limit of GR is not GR itself.

 

[ZapperZ]

I did not say or imply that GR contradicts itself. The Newtonian limit of GR is not GR itself.

The Newtonian limits are within GR and SR. So I consider Newtonian mechanics to be a subset of GR/SR.

Zz.

 

[timmdeeg]

Consistency means you cannot prove both X and not-X in the theory.
I don't see how one can prove a theory is self-consistent, you can only prove that it's not self-consistent by proving a claim and its negation.

Unfortunately I have been overlooking that. This statement is very clear, thanks!

EDIT How about proving that no negation of any claim exists?

 

[vis_insita]

The Newtonian limits are within GR and SR. So I consider Newtonian mechanics to be a subset of GR/SR.

As I tried to make clear above, the relevant relation is that of logical implication. In that sense neither theory is a subset of the other. To derive Newtonian Mechanics from SR, e.g, you need an additional assumption, like $v \ll c$, which is not itself implied by SR (nor Newtonian Mechanics for that matter).

So, I think my point still stands: The Newtonian limit of GR produces a theory, some implications of which are in contradiction with GR itself. I gave an example above. I don't think you have addressed it adequately yet.

 

[ZapperZ]

As I tried to make clear above, the relevant relation is that of logical implication. In that sense neither theory is a subset of the other. To derive Newtonian Mechanics from SR, e.g, you need an additional assumption, like $v \ll c$, which is not itself implied by SR (nor Newtonian Mechanics for that matter).

So, I think my point still stands: The Newtonian limit of GR produces a theory, some implications of which are in contradiction with GR itself. I gave an example above. I don't think you have addressed it adequately yet.

So you think that when I simplified the pendulum differential equation to the small angle approximation, that that solution is DIFFERENT and is no longer part of the full-blown differential equation? By your definition, ANY simplification and specification of any theory to a particular case is now divorced from the original general description. Does that make sense to you?

I mean, you can categorize things any way you want, but from what I have dealt with, this is still considered a part of the BIGGER, MORE GENERAL description! And that is how *I* defined it.

Zz.

 

[vis_insita]

So you think that when I simplified the pendulum differential equation to the small angle approximation, that that solution is DIFFERENT and is no longer part of the full-blown differential equation?

I think the solution to the approximate equation is different from the solution to the exact equation, if this is what you are asking.

By your definition, ANY simplification and specification of any theory to a particular case is now divorced from the original general description. Does that make sense to you?

No, but it is also not my definition. I think you misunderstood something. I just claimed that applying both Newtonian Physics and GR to the same particular case may give mutually contradictory predictions (at least in some situations e.g. Mercury). I did not imagine this statement to be that controversial to be honest.

 

[timmdeeg]

I'm sorry, but is there ANYTHING in science that can be proven to the same degree as mathematics? The starting point in any science and in physics in particular, are not derived. No one derived the symmetry principles that gave us all the conservation laws. You don't "prove" physical principles or theories. You verify them via experimental agreement to the degree that it can be tested.

Yes, I fully agree.

But I didn't talk about proving science. I talked about how to prove mathematical self-consistency of a physical theory, see also post #41. Perhaps I'm misunderstanding your point.

 

[vis_insita]

But I didn't talk about proving science. I talked about how to prove mathematical self-consistency of a physical theory, see also post #41. Perhaps I'm misunderstanding your point.

I tried to answer that question above. I think you can prove consistency by constructing simple toy models of your theory. If such models exist then the theory must be consistent. Because no model can satisfy both statements X and not-X.

The easiest way to accomplish this may be to just find an explicit solution to the fundamental equations of your theory. That solution can even be completely trivial.

 

[timmdeeg]

I tried to answer that question above. I think you can prove consistency by constructing simple toy models of your theory. If such models exist then the theory must be consistent. Because no model can satisfy both statements X and not-X.

The easiest way to accomplish this may be to just find an explicit solution to the fundamental equations of your theory. That solution can even be completely trivial.

Ok, thanks.

 

[DrChinese]

You always ignore some insignificant things when calculating something in physics. For instance, in the case of Earth-Moon system, it's usually ignored that energy is gradually lost to viscous dissipation by the tidal effect (this will change the orbit period and average Earth-Moon distance on a large time scale). And some things are completely irrelevant, like seeing the Moon as a quantum wavepacket of mass $\approx 7.3 \times 10^{22}$ kilograms and the most likely total Earth-Moon energy corresponding to a hydrogenic orbital of some very large principal quantum number $n$.

[Agreeing with this...]

Rather than thinking of GR as "true" and trying to fit Newtonian gravity into the "false" column: think of these as being "more useful" or "less useful" in particular situations. When you have every value that can be plugged into a GR prediction (without assumption), you will get a more accurate answer than with Newtonian gravity.

But much of the time, you won't have that much information available (other than by assumption); and there is essentially no difference in the utility of the resulting predictions. The "less accurate" one is faster and easier to calculate, which can be useful too. Which is why it is used more often than GR.

Going back to the OP: Theories are intended to be useful models of some group of patterns (and pattern exceptions). If you specify that there is no experimental confirmation possible for a new theory, then you are also saying that it provides NO new predictive power. It therefore lacks any utility. I would call such a theory "ad hoc".

 

[ZapperZ]

I think the solution to the approximate equation is different from the solution to the exact equation, if this is what you are asking.
No, but it is also not my definition. I think you misunderstood something. I just claimed that applying both Newtonian Physics and GR to the same particular case may give mutually contradictory predictions (at least in some situations e.g. Mercury). I did not imagine this statement to be that controversial to be honest.

Applying small-angle approximation to the pendulum when the angle of oscillation isn't small also produces "contradictory" result! But you are applying the rules for an apple to an orange!

The Newtonian result differs because the approximation is no longer as accurate! Thus, my example of the small-angle approximation.

I'm sorry, but this is going nowhere, and I'm tired of repeating myself.

Zz.

 

[Auto-Didact]

Applying small-angle approximation to the pendulum when the angle of oscillation isn't small also produces "contradictory" result! But you are applying the rules for an apple to an orange!

The Newtonian result differs because the approximation is no longer as accurate! Thus, my example of the small-angle approximation.

I'm sorry, but this is going nowhere, and I'm tired of repeating myself.

Zz.

Approximation isn't the same thing as logical equivalence, this is why you are getting confused. An approximation is de facto not the same thing as the unapproximated thing - regardless of what the small angle approximation, perturbation theory or any other mathematical method says.

You are literally equivocating two distinct things, namely the approximation and the unapproximated thing, based on the indistinguishability at some level of precision between the two; this is just logically inconsistent reasoning.

More true and certainly more justifiable would be to say that the one approximates the other with some level of accuracy and/or precision and the two only become equivalent in some specific limit; this is of course exactly what is stated in mathematics textbooks on approximation techniques as well as in the physics literature.

 

[vis_insita]

Applying small-angle approximation to the pendulum when the angle of oscillation isn't small also produces "contradictory" result!

Yes, I know, which is why I called both solutions "different".

The Newtonian result differs because the approximation is no longer as accurate!

It doesn't matter why it differs, only that it differs.

 

[Auto-Didact]

Applying small-angle approximation to the pendulum when the angle of oscillation isn't small also produces "contradictory" result!

Any proper textbook on dynamical systems shows how small angle approximations are essentially pathological mathematical methods regardless of the smallness of the angle. The even better textbooks then also go on to demonstrate how analyses based on perturbation theory tend to result in fundamentally mathematically inconsistent equations when directly compared to the exact equations, as well as go on to show the mathematical breakdown of perturbation theory itself.

 

[ZapperZ]

Any proper textbook on dynamical systems shows how small angle approximations are essentially pathological mathematical methods regardless of the smallness of the angle. The even better textbooks then also go on to demonstrate how analyses based on perturbation theory tend to result in fundamentally mathematically inconsistent equations when directly compared to the exact equations, as well as go on to show the mathematical breakdown of perturbation theory itself.

This is all meaningless. Unless you can show me explicitly why making a small-angle approximation is mathematically faulty, and that the full solution itself does not converge to the small-angle approximation, then what you have described are human personal preference.

Zz.

 

[Auto-Didact]

This is all meaningless. Unless you can show me explicitly why making a small-angle approximation is mathematically faulty, and that the full solution itself does not converge to the small-angle approximation, then what you have described are human personal preference.

Zz.

The reason for the mathematical faultyness of not only the small angle approximation, but practically all approximation techniques is essentially that they are used in the following context: they are attempts to linearize a nonlinear system of equations in order to find a solution, for example by replacing a second order equation by a first order equation which is easier to solve.

In general, second order equations cannot be reduced to first order equations, because the former requires two inputs (e.g. $x$ and $\dot{x}$) while the latter only requires one input (which if linearly expanded would determine the second). All of this becomes immediately apparent when the state space of a system is checked using stability analysis methods from bifurcation theory. In fact, non-perturbative analysis was created for this very reason.

 

[PeterDonis]

Any proper textbook on dynamical systems shows how small angle approximations are essentially pathological mathematical methods regardless of the smallness of the angle. The even better textbooks then also go on to demonstrate how analyses based on perturbation theory tend to result in fundamentally mathematically inconsistent equations when directly compared to the exact equations, as well as go on to show the mathematical breakdown of perturbation theory itself.

Please give specific references to these textbooks you speak of.

 

[Auto-Didact]

Please give specific references to these textbooks you speak of.

My personal favourite:
Strogatz, S. (1994). Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering (studies in nonlinearity).

 

[timmdeeg]

Going back to the OP: Theories are intended to be useful models of some group of patterns (and pattern exceptions). If you specify that there is no experimental confirmation possible for a new theory, then you are also saying that it provides NO new predictive power. It therefore lacks any utility. I would call such a theory "ad hoc".

Then as I understand it one could call the string theory "ad hoc", at least from the current point of view.

 

[DennisN]

Can we ever trust a scientific theory which is self-consistent but not testable?

Generally speaking, I'd say a definite no to that.

More specifically, I'd say since there a degrees of "testable", there are also quite naturally degrees of "trust" in scientific theories, e.g.

• Experimentally verified physics is very testable, and is thus very trustworthy.
• There are various theoretical concepts that are expected from established theories, but are, or have been considered hard to observe (e.g. gravitational waves which now have been detected).
• There are leading hypotheses to observed phenomena (e.g. galaxy rotations) that still hasn't been observed (e.g. dark matter).
• And then there are various ideas that are not yet uniquely associated with any known observed phenomena (e.g. string theory, loop quantum gravity).

 

[timmdeeg]

Generally speaking, I'd say a definite no to that.

More specifically, I'd say since there a degrees of "testable", there are also quite naturally degrees of "trust" in scientific theories, e.g.

Whereby your "no" seems to imply "not testable in principle", as @PeterDonis has pointed out in post #4.

But I'm still not sure what "in principle" means.
Supposed a theory of quantum gravity predicts properties of the state of matter in the Planck regime or in a black hole. Can we test these predictions "in principle"? Wouldn't this require a particle accelerator of the size of the milky way. Or do we say well we can't exclude that a yet unknown future technology could test Planckian signatures somehow.
But then it's still different regarding the state of matter in the interior of a black hole, no chance.

And then there are various ideas that are not yet uniquely associated with any known observed phenomena (e.g. string theory, loop quantum gravity).

There seem to be 5 consistent superstring theories. The superstring theory describes about $10^{500}$ vacuum states. Do we have any reason to believe that these are testable in principle? If not why are so many talented physicists searching in this field?

 

[PeterDonis]

then it's still different regarding the state of matter in the interior of a black hole, no chance

Yes, that's the difference.

 

[Mary Conrads Sanburn]

[. . . ]

Can we ever trust a scientific theory which is self-consistent but not testable?

[ . . . ]

Theory: In science, a well-substantiated explanation of some aspect of the natural world that can incorporate facts, laws, inferences, and tested hypotheses.

 

[timmdeeg]

Theory: In science, a well-substantiated explanation of some aspect of the natural world that can incorporate facts, laws, inferences, and tested hypotheses.

And to add one more detail, that predicts falsifiable test results.

 

[Ian J Miller]

My answer may be a little contentious, but my view is that while the mathematics in a theory must not have clear mistakes, mathematics does not make physics; it merely describes it. A mathematical equation is simply a statement in a somewhat condensed form, thus F = ma simply says the force on an object is given by its mass multiplied by the perceived acceleration. That applies always, although in fairness the statement relating to the Lagrangian in the standard model of particle physics would be somewhat cumbersome. However, the point I am trying to make is truth is not determined by whether the mathematics are consistent, but whether they are correct physically.

If a theory makes predictions, then it is falsifiable, although there is the caveat that theory may make a prediction of what happens, but concede that there are additional effects, i.e. the theory gives, say, an expectation result that should follow statistically if enough samples are taken, or alternatively, it might be unfalsifiable because it says this is what happens first but something else might follow or some overlooked effect might apply additionally. In each case, a result that does not fit might not falsify. Quantum mechanics gives a good example of the first. If I predict the position of one electron fired at two slits, the Uncertainty Principle and the diffraction effects will mean I cannot predict where that electron will be, but the physics are perfectly sound. For the second, I can have an equation that is correct for the trajectory of an arrow, but if you overlook the wind, an erroneous landing cannot falsify Newtonian dynamics.

Finally, there is another aspect. I have a theory relating to planetary formation and biogenesis, and I have made over seventy predictions. Some of them are conditional, because they are the "expectation" sort, but in general they all have problems of not being able to be applied because so far you cannot carry most of them out. For example, one of them predicts what you will find if you go and dig reasonably deeply at the bottom of Hellas Planitia on Mars. I regard that as a useful prediction that justifies what led to it being a theory. Obviously, it may not be correct, but that is irrelevant to whether it is a theory. One day it may be tested. However, if the theory is incapable of making any prediction that could be tested, or will not lead to something that could be tested I regard it as not a theory, but more fiction.