Gödel and the dreams of a Final Theory
I’m going to venture out of my own field, which is probably a bad idea, but I think I have something somewhat sensible to say. I doubt the following is novel—I suspect it has been suggested by others well before me, at least in some formulation. Nonetheless, I don’t think I’ve ever seen the following argument stated in this way, so it’s probably worthwhile presenting it.
For starters let me come clean and state what my conclusion will be: We will never reduce physics to a single theory. There will be many possible candidates, all of which happen to be consistent with the best experimental resolution that we can get, but we will have no way of telling which is the ‘real’ one and which are mere candidates.
The major assumption of this argument will be a belief in the Church-Turing thesis. If you don’t buy that, then this discussion is essentially moot. If you do buy it (and I’m not going to delve into the Thesis here—there are good books on the subject out there), then the question becomes whether it implies anything about physics per say.
I argue that yes, it does. Given Gödel’s Incompleteness results, we know that any finite or countably infinite set of axioms is not sufficient to derive all the statements that are satifisable given the assumptions. Now the axioms in any mathematical system effectively act as constraints, specifying what can be done without inducing F -> T, in which case everything can be done and the whole system is rendered moot.
Making the transition to physics, instead of considering the axioms as the rules/laws/equations that govern the universe, instead consider them to be experimental observations. This view certainly satisfies the above notion of axioms as constraints. An experimental observation ‘constrains’ our view of what the universe can possibly be, and thus we search for theories that are consistent with the given constraints. If apples were to fly instead of to fall, then Newton’s laws would not be consistent with the observation, i.e. not satisfy the axioms.
So, going back to the Incompleteness results, we know that any set of such observations, axioms, will always under-specify the system. It will allow us to derive a great deal about what is true, but not everything that is true, and will allow us to rule out a great deal of what cannot be true, but not everything that cannot be true. An inverse way of looking at this is that there will be many systems (“Structures” in predicate logic) which satisfy our axioms/observations, yet these systems would not be isomorphic, i.e. they would have some differences between them. So, many theories of physics, all of which are consistent with the given observations.
Granted, the more experiments we do, and the better we do them, the more ‘axioms’ we’ll have. But we know that even if we add an infinite number of such axioms, we’ll never get everything, and so while we’ll get closer and closer to our single theory, we’ll never quite get to it. We’ll always have a space of possible theories (an infinite one in fact) which cannot be winnowed down.
And, presumably, the observations would get more and more esoteric, requiring higher and higher energies, to eliminate increasingly smaller fringes of the possible theories' space. We’re already seeing this a little bit, but given that we don’t yet have at least a single consistent picture of everything, then we’re obviously not there yet.
A trivial aside, although it may appear to not be trivial which is why I’m addressing it, is whether this space of possible theories would be over the actual structure of the theory, i.e. the equations themselves for example, or over the parameters/constants, etc. My view is that this distinction is rather superfluous. In fact, we already have the Final Theory—it’s called a Universal Turing Machine, or any Universal axiomatic system for that matter. The only caveat is that we still don’t know the right set of initial conditions to feed it, and we don’t know what set of computations correspond to an “electron”, to give an example. But the point here is that the parameter space can be arbitrarily large, and so can the model space. The bottom line is that we’ll still be stuck with a large number of competing theories, and never have a way to pin it down to one.
Chances are this won't matter much. We'll get most of the 'truths' we care about—in many cases we already have. And in most other cases, the real problem will be the computational cost associated with simulating a chaotic/undecidable system, as opposed to a shortcoming in our theory of fundamental physics. Perhaps this is why I've decided to pursue biology and not physics. (Or perhaps not.)
For starters let me come clean and state what my conclusion will be: We will never reduce physics to a single theory. There will be many possible candidates, all of which happen to be consistent with the best experimental resolution that we can get, but we will have no way of telling which is the ‘real’ one and which are mere candidates.
The major assumption of this argument will be a belief in the Church-Turing thesis. If you don’t buy that, then this discussion is essentially moot. If you do buy it (and I’m not going to delve into the Thesis here—there are good books on the subject out there), then the question becomes whether it implies anything about physics per say.
I argue that yes, it does. Given Gödel’s Incompleteness results, we know that any finite or countably infinite set of axioms is not sufficient to derive all the statements that are satifisable given the assumptions. Now the axioms in any mathematical system effectively act as constraints, specifying what can be done without inducing F -> T, in which case everything can be done and the whole system is rendered moot.
Making the transition to physics, instead of considering the axioms as the rules/laws/equations that govern the universe, instead consider them to be experimental observations. This view certainly satisfies the above notion of axioms as constraints. An experimental observation ‘constrains’ our view of what the universe can possibly be, and thus we search for theories that are consistent with the given constraints. If apples were to fly instead of to fall, then Newton’s laws would not be consistent with the observation, i.e. not satisfy the axioms.
So, going back to the Incompleteness results, we know that any set of such observations, axioms, will always under-specify the system. It will allow us to derive a great deal about what is true, but not everything that is true, and will allow us to rule out a great deal of what cannot be true, but not everything that cannot be true. An inverse way of looking at this is that there will be many systems (“Structures” in predicate logic) which satisfy our axioms/observations, yet these systems would not be isomorphic, i.e. they would have some differences between them. So, many theories of physics, all of which are consistent with the given observations.
Granted, the more experiments we do, and the better we do them, the more ‘axioms’ we’ll have. But we know that even if we add an infinite number of such axioms, we’ll never get everything, and so while we’ll get closer and closer to our single theory, we’ll never quite get to it. We’ll always have a space of possible theories (an infinite one in fact) which cannot be winnowed down.
And, presumably, the observations would get more and more esoteric, requiring higher and higher energies, to eliminate increasingly smaller fringes of the possible theories' space. We’re already seeing this a little bit, but given that we don’t yet have at least a single consistent picture of everything, then we’re obviously not there yet.
A trivial aside, although it may appear to not be trivial which is why I’m addressing it, is whether this space of possible theories would be over the actual structure of the theory, i.e. the equations themselves for example, or over the parameters/constants, etc. My view is that this distinction is rather superfluous. In fact, we already have the Final Theory—it’s called a Universal Turing Machine, or any Universal axiomatic system for that matter. The only caveat is that we still don’t know the right set of initial conditions to feed it, and we don’t know what set of computations correspond to an “electron”, to give an example. But the point here is that the parameter space can be arbitrarily large, and so can the model space. The bottom line is that we’ll still be stuck with a large number of competing theories, and never have a way to pin it down to one.
Chances are this won't matter much. We'll get most of the 'truths' we care about—in many cases we already have. And in most other cases, the real problem will be the computational cost associated with simulating a chaotic/undecidable system, as opposed to a shortcoming in our theory of fundamental physics. Perhaps this is why I've decided to pursue biology and not physics. (Or perhaps not.)
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