# Are there two theories that are mathematically identical but ontologically different?

I have no background in physics but there is a question that has been bothering me, so I'm asking you.

Are there at least 2 physical theories that are :

• Mathematically identical, which means that they would yield identical predictions for EVERY situation that these theories can cover, and therefore can not be compared through experimentation : the validity of one of them is equivalent to the validity of the other.
• Physically different, that is to say, based on a different spatio-temporal-whatever realities, whose differences are not only semantic.

If there are at least two theories that satisfy those requirements, it would mean that the "absolute", "metaphysical" reality can never be known. However, if we are capable of mathematically demonstrating that such theories can not mathematically exist, it would mean that absolute reality can be known.

When I say "mathematically identical", I am not speaking of theories that can not be experimented on, due to technological constraints (like atomism at the time when this was still debated) but really of theories that can theoretically not be compared, even by a Laplace demon.

Do you agree with my assumptions? If so is there such theories and/or a demonstration that they can not exist?

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The real question is, what do you mean by "the absolute metaphysical reality"? If it is the same thing that got me excited when I was a young man, then I can assure you, that the question will dissolve into irrelevance as you age. – CuriousOne Aug 12 '14 at 20:03
Would Newtonian vs Lagrangian vs etc count? – BMS Aug 12 '14 at 21:04
@BMS: Neither Lagrange nor Hamilton change the ontology of the theory, though. It seems to me, that time, distances, velocity, acceleration, mass, force, momentum, energy, action etc. mean exactly the same thing in all these representations, we are merely rewriting the equations to make them easier to handle. – CuriousOne Aug 12 '14 at 22:58
Consider that in Hamiltonian mechanics switching $q$ and $p$ around is allowed while in Newtonian mechanics $x$ and $\dot x$ cannot be interchanged like that. – Robin Ekman Aug 12 '14 at 23:17
I'm confused by the title saying metaphysically different and the question body saying physically different. – LarsH Aug 13 '14 at 18:34

Special relativity and Lorentz ether theory (LET). From the linked Wikipedia article:

Because the same mathematical formalism occurs in both, it is not possible to distinguish between LET and SR by experiment. However, in LET the existence of an undetectable aether is assumed and the validity of the relativity principle seems to be only coincidental, which is one reason why SR is commonly preferred over LET.

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Beat me to it! That "commonly preferred" is a massive understatement in that wiki article. The only people I've run across who prefer LET over special relativity are inevitably crackpots. – David Hammen Aug 12 '14 at 22:21
That one, however, is trivially decided by Occam's razor. If you don't need it (and it's not measurable), it clearly belongs in the garbage. – CuriousOne Aug 12 '14 at 22:53
+1, i like it even though not "commonly preferred" :)) – Nikos M. Aug 12 '14 at 23:04
@DavidHammen, i would be careful labeling people as crackpots, especially when they agree with all known experimental results rigorously. True, superfluous arguments may be superfluous, still is "general covariance" superfluous or not? Why would it be necessary, one could very well do physics not generally covariant? – Nikos M. Aug 12 '14 at 23:11

Copenhagen quantum mechanics and DeBroglie-Bohm quantum mechanics, mathematically they are equivalent, "metaphysicaly" or "epistemologicaly" they are quite different

update The reason they are equivalent is because they reach the same central equation (Schrödinger equation) but from different paths. So the rest computations and experimental results can be calculated the same and so on.

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I agree. What interests me more is, why are people still wasting their time on constructing ontologies for quantum mechanics? Do you have an insight? – CuriousOne Aug 12 '14 at 20:08
i have many insights, one is that people are trying to find alternative views which may lead to better understanding and deeper theories, the other is just plain politics reasons (and do not underestimate those) – Nikos M. Aug 12 '14 at 20:13
@CuriousOne, i may not be the best person to answer this right now. Nevertheless, ontology can affect the way (and range) the theory/description is applied, as such can provide for new experiments/refutations and infer more data. For example if one thinks that newtonian mechanics do not apply to planets but only to apples, will have another approach than one that said that the theory should apply to planets as well as apples. Newtonian theory per se is not changed only the range of apllication (which is related to ontology among others) – Nikos M. Aug 12 '14 at 23:01
For ones that do, they will most likely not be able to make sense of their favorite perspective too much to someone else, and if you asked them to explain the most common criticisms leveled of the theory, I would expect even less to come out of it. There is also not a clear consensus on the topic at all, at least within some communities (see e.g. arxiv.org/pdf/1301.1069v1.pdf). – Abel Molina Aug 17 '14 at 7:07
And one of the experiments that sparkled debate about the topic is on the recent side of things: en.wikipedia.org/wiki/Afshar_experiment. So this seems more to me like a topic where new ideas would be welcome, rather than the opposite. – Abel Molina Aug 17 '14 at 7:08

The other answers bring up some nice examples of theories in physics that are equivalent in prediction but different in interpretation. I just want to bring up a trivialized example to illustrate the flaw in the reasoning in the question.

Suppose I have two theories which each predict the same quantity, say the maximum temperature at a given location on Earth on a given day. To make this concrete, suppose the first uses sophisticated modelling of the atmosphere, weather processes, etc. Suppose the second one is a sophisticated statistical algorithm which makes its prediction based on a large sample of historical data, comparing temperature trends across decades and a wealth of other data.

And now suppose that both theories produce correct predictions. At first this seems implausible given the complexity of the system and the radically different approaches taken, but it's not so far-fetched that both produce correct predictions within the errors on those predictions.

How can you tell which theory is "correct" (probably more fair to ask which theory is "more correct than the other")? You could try to extend both theories to be more general and predict more quantities, and see which one starts to break down (or more likely, both break).

I think the more interesting underlying question is "Is there a unique theory that accurately predicts all physical processes?". First, though, one needs to settle the question of "Can a theory of everything exist?", see Gödel's incompleteness theorems (and also this) for some information on that topic.

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nice but the question asks about mathematicaly equivalent theories, which i think means also formally equivalent (or having same or similar formalism) – Nikos M. Aug 12 '14 at 23:44
@NikosM. but he goes on to define "mathematically equivalent" as "which means that they would yield identical predictions for EVERY situations that these theories can cover", which is what I went with. – Kyle Oman Aug 12 '14 at 23:45
ok, its fine with me, the OP will decide :)) – Nikos M. Aug 12 '14 at 23:46
It depends, the 2 examples you give does not seem to be identical. The first one is a mathematical theory while the second seems to be an experiment designed to validate it. – user3419556 Aug 13 '14 at 12:01
@user3419556 To me, the second example looks like a theory that uses the current and past states of the system and an algorithm to attempt to predict the future state of the system... this bears some striking resemblances to the first example... – Kyle Oman Aug 13 '14 at 17:21

N-dimensional classical gravity can be mapped onto N-1 dimensional quantum field theory.

Those worlds are very different. You'd think one or the other would have to be "true", but they're completely equivalent. If you think you live in one, the other is just a mathematical trick. But who's to say the other viewpoint isn't right and your perceptions are just being tricked?

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I'm not sure that this counts. To make this sort of holographic abstraction, you need to know the boundary surface, as well as the field projected upon it, so all of the information is preserved through the transformation. – KidElephant Aug 13 '14 at 15:52

Well, I can make a guess for the future. Suppose that we establish that a specific string theory incorporates the standard model, thus gives the same predictions and descriptions as the standard model, and includes quantization of gravity. This theory of everything (TOE) will be identical as far as observations and predictions go as the standard model and effective quantized gravity we now have as established. ( Big Bang Model).

The two theories have different metaphysical implications, standard model + effective quantized gravity aim to completely describe reality as we know it,no metaphysics, but string theory adds another six dimensions at least which could easily accommodate metaphysics in those dimensions.

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But thanks to a monk we have learned, that adding extra degrees of freedom when they are not needed to describe the data is not a good idea. This points out the fundamental difficulty for finding a unique TEO: we are always one experiment away from having to modify the last TEO, because there is always a chance that we will find deviations from its predictions that make a modification necessary. At most, we can find one for which we will never know (due to financial limitations with regards to experiment size), if it is violated, or not. – CuriousOne Aug 13 '14 at 18:51
String theory is not mathematically identical to the standard model. – Ben Crowell Aug 14 '14 at 3:37
@BenCrowell I think you are wrong. The group structure of the SM will have to be embedded into any TOE, including a string theoretical one. It is too well fitted. – anna v Aug 14 '14 at 4:11

Consider classical electrodynamics. Here you can introduce potentials $\phi$ and $\vec A$ from which you can derive the electric and magnetic field. However they are not uniquely defined by the electric and magnetic field, but different potentials can generate the same fields. Unlike electric and magnetic field, those potentials are usually not considered real, but just a mathematical tool to help describing electric and magnetic fields, because the observable behaviour is determined solely by the electric and magnetic fields.

But of course you can easily modify classical electrodynamics by just declaring those potentials as physical/ontological fields. Then the gauge freedom would mean that we cannot determine the actual values by observation, because as physical fields they would, of course, have to have unique, well defined values everywhere.

Now quantum mechanics gives this an interesting twist, through the Aharonov-Bohm effect: In this effect, a confined magnetic field affects the quantum phase of electrons where there's no electric or magnetic field, in a way visible through interference. Now if you assume that the potentials are actually physical fields, then the Aharonov-Bohm effect is completely local: You just integrate the vector potential along the way of the electron. However if you don't consider the potentials to be physical fields, then you'll find that Aharonov-Bohm is a non-local effect: The interference pattern of the electrons depends on the magnetic field in a region which the electrons never enter, not even in a quantum sense.

Note that also the Aharonov-Bohm effect is gauge invariant (indeed, we know that all of physics is); that is, you cannot distinguish different potentials leading to the same electric and magnetic fields by doing an Aharonov-Bohm experiment. Therefore also this experiment cannot distinguish between the two theories.

Also note that this "Aharonov-Bohm nonlocality" is independent from the nonlocality associated to Bell's inequality.

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