I What if a scientific theory is not testable?

[timmdeeg]

Supposed a future theory of Quantum Gravity reconciles Quantum Mechanics with the Theory of General Relativity, is mathematically self-consistent and makes falsifiable predictions which are not testable though.

Can we ever trust a scientific theory which is self-consistent but not testable?
Does self-consistency exclude the case that there exists another approach (aiming to resolve the same problem) which is self-consistent too?

Sorry, I am not sure if this is the right forum for that.

 

[PeterDonis]

I am not sure if this is the right forum for that.

It is now since I have moved the thread to the Quantum Interpretations and Foundations forum.

 

[hilbert2]

It's possible that the theory is testable in some way that is not immediately obvious. Quantum gravity in the very early universe could have left traces in the cosmic microwave background and other things, and this could exclude some theories.

 

[PeterDonis]

makes falsifiable predictions which are not testable though

I assume by "not testable" you mean "not testable at current or near term foreseeable levels of technology", since if you mean "not testable in principle, no matter how much technology advances", the theory would not be falsifiable.

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

My answer would be no: until we have some way of getting experimental data that tests the theory's predictions, we can't really trust them.

Does self-consistency exclude the case that there exists another approach (aiming to resolve the same problem) which is self-consistent too?

I don't see why it would.

 

[timmdeeg]

Quantum gravity in the very early universe could have left traces in the cosmic microwave background and other things, and this could exclude some theories.

Hmm perhaps, but I think the black hole singularity problem is different. Predictions of a theory resolving this problem would hardly be testable.

 

[timmdeeg]

I assume by "not testable" you mean "not testable at current or near term foreseeable levels of technology", since if you mean "not testable in principle, no matter how much technology advances", the theory would not be falsifiable.

Yes thanks, falsifiable is the determining criterion. But supposed the predictions refer exclusively to the Planck regime then it's hardly imaginable that to falsify them is ever possible.

Perhaps another possibility: Quantum Gravity predicts that starting from Planck decreasing density and temperature ends up with a known state of matter e.g. the quark-gluon plasma which we investigate by heavy ion collisions (so we have some knowledge about it). We then have overlapping of current with new physics and can have confidence.

Another question in this context. Are there any theoretical expectations that at the Planck regime matter isn't composed of jet unknown particles but rather exist's as a field (having in mind the decay of the Inflaton field)?

 

[MathematicalPhysicist]

Supposed a future theory of Quantum Gravity reconciles Quantum Mechanics with the Theory of General Relativity, is mathematically self-consistent and makes falsifiable predictions which are not testable though.

Can we ever trust a scientific theory which is self-consistent but not testable?
Does self-consistency exclude the case that there exists another approach (aiming to resolve the same problem) which is self-consistent too?

Sorry, I am not sure if this is the right forum for that.

Why bother worrying about such a theory?
The key words are: "mathematically self-consistent" which is impossible to prove, thus such a theory is impossible regardless if you can find some empirical evidence.

 

[ZapperZ]

To the OP: But isn't this a more general question on whether any scientific theory should be accepted if there are no experimental evidence? Isn't this the same issue that Sabine Hossenfelder has written about?

But let's just look at the FACTS, shall we? Physics is littered with theories that are "mathematically self-consistent", but are not accepted because they do not agree with experiment. When the OPERA experiment detected something that was thought to be superluminal, a flood of theories, preprints, etc. appeared to explain such a result. Many of those are "mathematically self-consistent", IF we start with the premise that there are such superluminal particles in what OPERA was detecting. You can't fault them mathematically.

But, as we all know, the OPERA result was incorrect, due to their instrumentation. It means that all those mathematically self-consistent theories are wrong! It means one very clear thing: a mathematically self-consistent theory is NOT a valid means to consider if a theory is valid or not! This is because it has been shown and proven that such a theory can be invalid.

This is not just about black holes and quantum gravity. It is about what we mean for something in SCIENCE to be valid.

Zz.

 

[PeterDonis]

supposed the predictions refer exclusively to the Planck regime then it's hardly imaginable that to falsify them is ever possible

"Hardly imaginable" is not the same as "impossible in principle". Most of our technology today would have been "hardly imaginable" to a person a few centuries ago.

Quantum Gravity predicts that starting from Planck decreasing density and temperature ends up with a known state of matter e.g. the quark-gluon plasma which we investigate by heavy ion collisions (so we have some knowledge about it). We then have overlapping of current with new physics and can have confidence.

This would be an example of a testable theory, provided that its predictions were different from the predictions of other competing theories of gravity. But if, for example, your quantum gravity theory just said "things work the same as classical GR" in a particular regime, that wouldn't help in testing your quantum gravity theory because there would be nothing to distinguish its predictions from those of our current theories.

Are there any theoretical expectations that at the Planck regime matter isn't composed of jet unknown particles but rather exist's as a field (having in mind the decay of the Inflaton field)?

The distinction you are trying to draw doesn't exist. "Particles" are not separate kinds of things from "fields". "Particles" are just particular kinds of states of quantum fields. When the inflaton field transferred its energy during reheating, the energy wasn't transferred to "particles"; it was transferred to other fields (the Standard Model quantum fields).

 

[MathematicalPhysicist]

To the OP: But isn't this a more general question on whether any scientific theory should be accepted if there are no experimental evidence? Isn't this the same issue that Sabine Hossenfelder has written about?

But let's just look at the FACTS, shall we? Physics is littered with theories that are "mathematically self-consistent", but are not accepted because they do not agree with experiment. When the OPERA experiment detected something that was thought to be superluminal, a flood of theories, preprints, etc. appeared to explain such a result. Many of those are "mathematically self-consistent", IF we start with the premise that there are such superluminal particles in what OPERA was detecting. You can't fault them mathematically.

But, as we all know, the OPERA result was incorrect, due to their instrumentation. It means that all those mathematically self-consistent theories are wrong! It means one very clear thing: a mathematically self-consistent theory is NOT a valid means to consider if a theory is valid or not! This is because it has been shown and proven that such a theory can be invalid.

This is not just about black holes and quantum gravity. It is about what we mean for something in SCIENCE to be valid.

Zz.

How do you prove a physical theory is self-consistent mathematically?

 

[ZapperZ]

How do you prove a physical theory is self-consistent mathematically?

Ask the OP what he/she meant by that, since that was the phrase being used first.

To me, it means no logical error.

Zz.

 

[MathematicalPhysicist]

Ask the OP what he/she meant by that, since that was the phrase being used first.

To me, it means no logical error.

Zz.

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.

 

[ZapperZ]

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.

So why are you arguing it with me?

Zz.

 

[timmdeeg]

Why bother worrying about such a theory?
The key words are: "mathematically self-consistent" which is impossible to prove, thus such a theory is impossible regardless if you can find some empirical evidence.

And vice versa? Is a theory which makes falsifiable predictions necessarily mathematically self-consistent?

 

[MathematicalPhysicist]

And vice versa? Is a theory which makes falsifiable predictions necessarily mathematically self-consistent?

Not necessarily.
Your question is more about philosophy.

 

[timmdeeg]

But, as we all know, the OPERA result was incorrect, due to their instrumentation. It means one very clear thing: a mathematically self-consistent theory is NOT a valid means to consider if a theory is valid or not! This is because it has been shown and proven that such a theory can be invalid.

A good example, thanks.

 

[vis_insita]

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.

You can prove consistency by constructing a model of that theory. If the theory is inconsistent, no model exists.

You are probably thinking of the impossibility of proving the consistency of some set of axioms from the very same axioms, which is a different thing entirely.

 

[timmdeeg]

The distinction you are trying to draw doesn't exist. "Particles" are not separate kinds of things from "fields". "Particles" are just particular kinds of states of quantum fields. When the inflaton field transferred its energy during reheating, the energy wasn't transferred to "particles"; it was transferred to other fields (the Standard Model quantum fields).

Thanks, here I had a wrong notion, the notion that the inflaton field decays into matter and that's it. Could one understand "transferred to other fields" in the sense of a phase transition? The heat generated during reheating reminds me of the crystallisation enthalpy due to the liquid-solid phase transition.

 

[MathematicalPhysicist]

You can prove consistency by constructing a model of that theory. If the theory is inconsistent, no model exists.

You are probably thinking of the impossibility of proving the consistency of some set of axioms from the very same axioms, which is a different thing entirely.

If Physics as a whole is to be considered one big model, then you cannot prove its consistency within it.
And as far as I can tell Physics is built out of models that may contradict each other but perhaps consistent on their own.

 

[vis_insita]

If Physics as a whole is to be considered one big model, then you cannot prove its consistency within it.
And as far as I can tell Physics is built out of models that may contradict each other but perhaps consistent on their own.

By "model" I meant a set of things which fulfills every statement made in the theory. The existence of such a set, even if it is completely trivial, proves the consistency of the theory. (Note that in this sense it is quite meaningless to talk about (proving) the "consistency of models" or about different models contradicting each other.)

Of course, different theories used in physics may contradict each other, in which case they cannot have a common model. In particular they cannot both be true in reality (the "model" of ultimate interest in physics). But that doesn't exclude the possibility of proving the consistency of each individual theory.

 

[Auto-Didact]

If Physics as a whole is to be considered one big model, then you cannot prove its consistency within it.
And as far as I can tell Physics is built out of models that may contradict each other but perhaps consistent on their own.

Indeed, the canon of physics consists of a patchwork of different theories, both in type as in content, which however share some underlying mathematical threads whose exact relation continues to elude us. These threads de facto make it so that we can speak coherently about physics as a connected subject in the first place, instead of merely refer to a vague disjoint set of theories about natural phenomena.

These threads are or represent the general principles and laws of physics which are true of all physical theories and/or have essentially kept the same form since Newton invented physics, e.g. conservation laws in their different reformulations: they are essentially the core subject of physics so much so that entire theories of pure and applied mathematics and mathematical methods are built around them and form much of the classification of different fields in mathematics.

A concrete example of such a thread is calculus, which actually isn't much more than an abstraction of the basic structure of a canonical physical theory; this explains why abstractions of calculus such as vector calculus, covariant calculus, differential geometry and functional analysis remain very close in spirit to calculus and therefore also why their respective applications in physics as the mathematical structure underlying the theories also remain very close to older physical theories, regardless of what is claimed to be part of the physical theory outside of these core mathematical structures.

In any case, these threads all have in common that they have a purely mathematical form and that they actually form the historical connection between different canonical theories in physics. Moreover, they also suggest how to extend physical theories based on analogies of how other physical theories were extended; this is necessarily a process of trial and error because not every theory can be generalized in exactly the same way in order to find the correct generalization, a fact which is extremely well-known in practice.

The fact that all of the above isn't necessarily directly capturable or stateable within some known and conventionally accepted mathematical framework is what has led in the 20th century both to the rejection of this classic point of view of what physics is as well as to the unhealthy dominance of some theories and/or frameworks within physics over all of physics; e.g. the naive and premature insistence of QT as the be-all-end-all theory of physics while clearly foundationally it is nor can be no such thing, but I digress.

Sometimes however, even all suggested generalizations that are at hand turn out to be wrong; in such a case genuine creativity is actually necessary in order to find the correct generalization in a infinitude of possibilities because it requires not just the creation of a new theory but a new form of mathematics; this makes the process of theoretical physics de facto an art form. It is safe to say that theoretical physics currently finds itself within this predicament and that the obvious pathways of choice don't seem to be productive.

 

[ZapperZ]

If Physics as a whole is to be considered one big model, then you cannot prove its consistency within it.
And as far as I can tell Physics is built out of models that may contradict each other but perhaps consistent on their own.

What model of physics "contradict" one another? And let's not get into "classical physics" contradicting "quantum mechanics" or "relativity" crap, because they don't, not when one can merge into the other upon suitable limits.

I just wish that whenever someone make generalized statement such as this, that it is accompanied by concrete examples without someone like me asking for it. Notice that in my reply to the OP, I bought up specific example of the OPERA result. I just didn't wave my hand in the air and proclaimed that there mathematically-consistent theories have been proven to be invalid without any evidence to support such a statement.

Zz.

 

[EPR]

I think he was referring to Goedel's incompletness theorem which places a bound on what can be known and modeled from the "inside" of reality.

 

[vis_insita]

What model of physics "contradict" one another? And let's not get into "classical physics" contradicting "quantum mechanics" or "relativity" crap, because they don't, not when one can merge into the other upon suitable limits.

General Relativity and Newtonian Gravity predict different values for the perihelion precession of Mercury. That is a contradiction between these two theories, is it not? That a suitable limit makes this discrepancy disappear in favor of the wrong Newtonian value is completely beside the point.

 

[hilbert2]

General Relativity and Newtonian Gravity predict different values for the perihelion precession of Mercury. That is a contradiction between these two theories, is it not? That a suitable limit makes this discrepancy disappear in favor of the wrong Newtonian value is completely beside the point.

But the orbital motion of Moon around Earth doesn't have to be calculated with GR, the Newtonian result is almost the same because the velocities and gravitational forces are small enough. And even when calculating it with general relativity, you usually ignore small things like loss of energy to gravitational waves, which would make the calculation too difficult.

 

[vis_insita]

But the orbital motion of Moon around Earth doesn't have to be calculated with GR, the Newtonian result is almost the same because the velocities and gravitational forces are small enough. And even when calculating it with general relativity, you usually ignore small things like loss of energy to gravitational waves, which would make the calculation too difficult.

True, of course. But the fact that GR and Newtonian physics agree on some facts in some situations doesn't show that they don't contradict each other. They are in contradiction if they disagree on some facts, which they do.

 

[Auto-Didact]

What model of physics "contradict" one another?

Example: Netwonian dynamics and Maxwellian electrodynamics contradict each other mathematically. This conflict was resolved by making mathematically explicit the underlying core of the conflict, which was that there was another form of dynamics which implicitly underlies the latter, namely SR, and then demonstrating that the former is in fact a limiting case of SR.

This strategy of explicitizing some underlying core and then demonstration of some theory being a limiting case is usually called 'unification', but it is merely one specific form of unification which can and has been generalized in many ways, i.e. the specific strategy is an element in a set of strategies called 'unification'. In any case, the strategy has been used successfully to resolve other analogous conflicts between other theories, but there are a few notorious cases where it has failed so far despite all attempts.

An infamous example is of course the open problem of theoretical physics that QT, in any of its various formulations (including QFT), and GR mathematically contradict each other; the reason why the above strategy seems to fail in this particular case is that the contradiction between the two theories is not merely mathematical, but also actually a conflict in their very axioms and principles, i.e. their underlying core mathematical structures are mathematically incompatible if left as is.

Unification in the face of two incompatible mathematical frameworks is nothing new in mathematics or physics. The correct strategy - obtained by looking at similar resolved cases in the past and making the correct analogy - is then usually to reformulate the axioms and/or principles in a manner such that they all can be cast into a single mathematical framework and then to reconstruct both older theories as special or limiting cases within this new framework.

Most attempts so far, including e.g. string theory and LQG, only try to unify the mathematics of both theories without modifying any of the principles or axioms; it is generally becoming accepted wisdom in the practice of theoretical physicists that this is why these attempts at unification, while partially successful, aren't sufficient to constitute a full and proper resolution of the conflict between QT and GR and therefore are ultimately judged as either failures or merely works in progress.

John Baez, Lucien Hardy and Lee Smolin have each independently described explicit research methodologies for solving open problems in foundational physics. Baez' methodology involves using the mathematical framework of (n-)category theory; see Baez' blog, this paper or this book. Hardy has offered a new constructivist framework which I have described in this post. Smolin's approach is described in his latest book (see this thread); if I remember correctly, Smolin's approach is based largely on Einstein's philosophy of physics which explicitizes the difference between constitutive theories and principle theories.

 

[ZapperZ]

General Relativity and Newtonian Gravity predict different values for the perihelion precession of Mercury. That is a contradiction between these two theories, is it not? That a suitable limit makes this discrepancy disappear in favor of the wrong Newtonian value is completely beside the point.

you are confusing different ACCURACY with contradiction.

Zz.

 

[vis_insita]

you are confusing different ACCURACY with contradiction.

No, I'm not. I claim that the Newtonian prediction is accurate but wrong, not true but inaccurate.

 

[ZapperZ]

No, I'm not. I claim that the Newtonian prediction is accurate but wrong, not true but inaccurate.

no, I mean that Newtonian mechanics, IN PRINCIPLE, is an approximation to general and special relativity.

Zz.

 

[timmdeeg]

Ask the OP what he/she meant by that, since that was the phrase being used first.

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?

 

[vis_insita]

no, I mean that Newtonian mechanics, IN PRINCIPLE, is an approximation to general and special relativity.

I understand that. Again, I claim that "A approximates B" doesn't show that there is no contradiction between A and B. On the contrary. It only means that A and B agree on some facts, namely those which are stated vaguely enough that the difference A-B doesn't matter. On the other hand they disagree on facts which are stated with more precision than |A-B|.

An example of the latter statement would be "The perihelion precession of Mercury is 574″ ± 5'' per century". This statement follows (if I'm not mistaken) from GR but contradicts Newtonian Physics, since |A-B| = 43'' per century in this case.

 

[vis_insita]

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.

@MathematicalPhysicist gave one above:

Consistency means you cannot prove both X and not-X in the theory.

In this sense inconsistent theories are pretty much worthless, since everything follows from them.

 

[hilbert2]

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 $n$.

 

[ZapperZ]

I understand that. Again, I claim that "A approximates B" doesn't show that there is no contradiction between A and B. On the contrary. It only means that A and B agree on some facts, namely those which are stated vaguely enough that the difference A-B doesn't matter. On the other hand they disagree on facts which are stated with more precision than |A-B|.

An example of the latter statement would be "The perihelion precession of Mercury is 574″ ± 5'' per century". This statement follows (if I'm not mistaken) from GR but contradicts Newtonian Physics, since |A-B| = 43'' per century in this case.

But how can something which is only an approximation, be considered to be contradictory, when it is, BY DEFINITION, should give a different result? You are taking something only to the first order, while another description, you are taking it to the 2 or 3rd order! It just means that the first description is not AS ACCURATE as the first.

I do not consider this as a contradiction. Maybe you do. But it is not in my book.

Zz.

 

[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.

 

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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.