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I see many learned contribution about the role of a TOE, what it might do or not do, what kind of answer it might provide, and what not.

But I do not know what a TOE is, how I would recognize it if I met it in the street. Is it an axiomatic theory, a collection of equations, a sacred book, an incoherent dream, a good topic for fun discussions on a forum, or just a unification of QM and GR, which might not even tell us everything about either.

It would be nice for naive users like me to know what is being talked about, and whether it is the same for all who talk.

Personally, I naively took it to mean a complete description of the physical universe, which does remain very vague, and may not be meaningful.

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"A theory of everything (ToE) or final theory is a theory of theoretical physics that fully explains and links together all known physical phenomena, and predicts the outcome of any experiment that could be carried out in principle." Please explain why/how that comes short. –  Glen The Udderboat Sep 6 '13 at 21:57
    
Since everyone seems intent on discussing its properties, I am wondering how it is supposed to be expressed. How do I discuss the properties of anything if I do not know what it looks like. And, since the description is part of what it describes, the means for the description could be dependent of what is being described. How do you define "theory of theoretical physics" in a formally usable way? How stable has this definition been historically? (BTW, how was the beer at your party?) –  babou Sep 6 '13 at 22:35
    
See this. It's the proper definition, definitely. A quantised, unify-all theory. –  Dimensio1n0 Sep 7 '13 at 6:58
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3 Answers

The ToE in a way is the surpassing of

1) Current knowledge. 2) Current errors. 3) Current isolation.

Things that are yet to be learnt can put a halt on progress since change is required to move from the old to the new. Current errors can be tough to overcome if these errors have been overlooked for a significant time period. A common component can be viewed from numerous directions and thus be seen in numerous ways, thus be seen as separate components even though they are not separate. Again, this too can be tough to overcome if this separation or isolation has been overlooked for a significant time period.

Also, to see and understand the big picture concerning reality, yet be doing so while still being inside the very same reality of which you try to understand, another barrier must somehow be overcome.

In short, you may find the equation that takes the EVERYTHING into account, however the human mind may not still understand the equation due to the act of understanding meaning that the mind must wrap around that which it is to completely understand.

Thus the mind must wrap around the reality of which it resides within to thus completely understand it.

Thus we are confined to merely the title of the "THEORY" of everything instead.

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I think a good way to start is to understand what a theory is. A theory is a mathematical model. This sounds complicated, but it's generally just a set of equations that we solve to find out how our system behaves. So Newton's laws are a mathematical model that describe the motions of particles at velocites low compared to the speed of light.

So far at least, all our mathematical models are approximations that give accurate results only in a restricted range of circumstances. So Newton's laws apply only at speeds low compared to the speed of light and fail when the speeds approach $c$. Special relativity gives accurate results for all speeds, but fails when gravitational fields become high. General relativity fails (probably) when the density is so high that it predicts singularities.

The reason all our existing theories fail at some point is that we have made approximations to simplify the theory, and there will inevitably be circumstances under which our approximations aren't justified. It would be nice if we had a theory that didn't rely upon any approximations. Such a theory would never fail, so it would in principle describe everything in our universe and possible others. This would be a theory of everything. Note that you wouldn't actually attempt to use the theory to describe anything any more than we'd attempt to use QCD to understand hydrodynamics.

Initially it was thought that String Theory might be theory of everything, but then we discovered that it was (probably) a low energy approximation to M Theory, so it ends up being an effective theory just like all the others we have. I don't think enough enough is known about M Theory to know whether it is a possible TOE.

The appeal of a TOE is that because it doesn't require any approximations it would be a true representation of the universe(s) i.e. we'd understand how the universe really works (whatever real means in this context). Every now and then you hear a despairing cry that such a theory will never be possible and all we can achieve is increasingly good approximations to reality. I suspect most of us think this is unnecessarily pessimistic.

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Sorry, this is wrong. Firstly, a theory isn;'t a model. A model means a lot of fitting into experiments, but a theory should predict these results. Conventional String theories are not low-energy approximations, but more of compactifications of M-theory. We want M - theory, because it allows a description of all the other string theories. You shouldn't compromise accuracy for the sake of your answer being accessible to the layman. –  Dimensio1n0 Sep 7 '13 at 9:01
    
@DImension10AbhimanyuPS: theories are mathematical models e.g. the Standard Model and the Cosmological Standard Model. A mathematical model takes input and predicts what will happen as the system evolves. Maybe this is just terminology, but I think it's fairly standard. Compactification produces a low(er) energy effective theory, so string theory is an effective theory derived from M theory, just as the standard model is (probably) an effective theory derived from string theory. –  John Rennie Sep 7 '13 at 10:25
    
@DImension10 The informal use of the word model is often the opposite of its formal use in logic. Formally, it is the universe that is supposed to be a model of the theory formally describing it, and the role of experiments is to verify that it is indeed the case. If they fails to conform, it does not mean that the theory is unsound (logically), but simply that the universe is not a model for it: it is not described by the theory. Conversely, a theoretical statement that happens to be true of the universe will be invalid if it is not true for other interpretations (models) of the theory. –  babou Sep 7 '13 at 13:15
    
@babou that is the platonic view, that the theory exists before reality, and it throws models out that reality has to conform to or not. –  anna v Sep 7 '13 at 13:35
    
I did not say that theory exists before reality (whatever that is). I am only saying there are all kinds of theories, and some conform reality, i.e., have (some part of) reality as a model, while other do not. My statements are purely technical, carrying no philosophical or ideological implication, as far as I can tell ... and I certainly intend none. Despite the tone of my question, I am very seriously trying to understand what you are after and expect, and I am not sure you all agree on it. A theory may also have many models, one of them being reality. –  babou Sep 7 '13 at 15:03
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Personally, I naively took it to mean a complete description of the physical universe, which does remain very vague, and may not be meaningful.

I can describe for you the mathematical form of a theory of everything:

       #=0

Where # is a symbol which includes convoluted differential forms. This is the form all partial mathematical models of electromagnetism, mechanics, thermodynamics ... every type of mathematical model for physical systems has taken in the past and will appear in the future. They can be reduced to differential forms on the left and a zero on the right.

It will have its mathematical axioms that will make the solutions rigorous but it will have the physics postulates that have to be satisfied as the input in choosing a particular TOE from multitudes of similar ones.

The word "theory" in physics does not simply have the burden of mathematical rigorous existence. It could be very rigorous mathematically and irrelevant for the physics to be modeled. Mathematical theories become models for physical states.

The fact that it will be mathematically rigorous means that the solutions will describe correctly all known physical data and predict, given the boundary conditions, any new ones we could think about, in precise numbers. There is nothing vague about it.

We expect that the Standard Model, which describes almost completely all known up to now particle data, will naturally nest in the model of the TOE. There is nothing vague about this either. We expect gravity to be modeled naturally within TOE.

At the moment it seems that string theories offer all these options, but as there are thousands of possibilities, the final model has not been found yet, not even the class of models within string theories, which can be candidate for the embeddings necessary of the SM to assure consistency with existing data. If/when decided upon the predictions of the model will be tested for consistency with new data.

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Very clear, thanks. There is an answer to a ToE question giving Conway's Game Of Life as an example of ToE. It gathered 23 upvotes, but does not fit your definition. The Game Of Life has no equation in its definition and is a discrete universe. Hence, I was wondering how flexible the concept of a ToE can be. Considering that axiomatic systems may be used to enumerate theorems, I am wondering how dependent the concept of ToE is on what we know or ignore about physical computability. Would you know why my question was downvoted? –  babou Sep 10 '13 at 10:56
    
For the down vote: Some people dislike general philosophical type questions and some are just cranky :). Well, as for the TOE it depends what is the "everything" that is being described. For the physicist it is all the available data have to be described and new data predicted and found to validate the theory. A discrete universe, if you mean a countable one, contradicts important observations in our universe, like Lorentz invariance. –  anna v Sep 10 '13 at 12:23
    
I do not take it as a philosophical issue (cf my reaction to platonicism). There is real technical research on physical computability issues. I only know of it (I am handicapped by my ignorance of QM, though it is not about quantum computing, or relativity), but it seems they consider that a form of space discretization could be compatible with existing physics. –  babou Sep 10 '13 at 13:24
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