Model Completeness and Consistency
Perhaps you're consistent. ... But are you telling me everything?
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Do you expect a model to tell you everything about everything? I wouldn't think so, at least not without solving the self-referencing problem. But shouldn't it be able to tell you everything about what is supposed to make up the context of the model? Whatever the model is, shouldn't it be complete in that sense?
If you devise a representation of the stock market, for instance, it should include everything in the market environment pertinent to making trades, or pertinent to market prices. The idea should apply to broad theories like those of physics. In particular it should apply to quantum mechanics. But serious doubts had been raised about the theory's credibility.
Albert Einstein, for one, decried quantum mechanics, because it clashed with his belief that "God doesn't play dice with the world." The Einstein argument was that quantum mechanics is inconsistent and incomplete. The first because light waves can't be particles and particles can't be waves, and the second because it leaves out the fact that objects have proper states and move in real-space time. Objects, said Einstein, are positioned in space and can move from one place to another in a well-defined path. To think otherwise is ridiculous!
In Einstein's view -- as well as E. H. Walker and others-- physics can't be right until it is complete and consistent. Says Walker:
Our basic ideas must be
logical and complete, and they must fit the equations that so well quantify the physical world.More generally, can we prove that a system is complete and consistent? Also, can we show that physics, say, expresses everything there is to say about the real world? Can proofs be drawn from first principles without also generating assertions that contradict other proven statements -- i.e., that it remain consistent?
Quantum mechanics says you can't get the product of position and momentum with unlimited precision. (The product is limited by Plank's constant, well known in physics.) The measuring tool -- a light quantum, ultimately -- prevents the precision. The measured values can only be specified to within the span of the wavelength of the light quantum. If your telescope resolution of Mars was at best only a yard, you couldn't see objects on Mars that were shorter or wider than three feet and so you couldn't be conscious of them. It would then be meaningless to say an object has an exact position or an exact momentum -- it would be mere speculation. That's the way it is with QM.
In other words, what we can never see doesn't exist -- like the sound of a tree falling in the forest or the soccer ball when nobody's around to observe it. Only by measuring (observing) a system can we obtain information about it. According to Einstein, though, this whole idea is incomplete. It isn't reasonable to say the object doesn't have a real position or momentum. It goes against the very grain, and Einstein did everything to prove it was wrong-headed. But quantum mechanics has apparently won the day, by wearing down the opposition with solid experimental evidence.
There is the sense here that knowledge can only be gained through observation, that a rational approach without perception is mere "speculation." Contrary to this view, Einstein opposed QM on rational grounds and depended on rational intuition for his knowledge.
Even so, Walker says that Einstein's argument itself is flawed, that it's based on an incomplete theory of reality -- it leaves out an important ingredient, namely the observer. Einstein's world consists only of a bunch of masses interacting with each other in space-time. There's no room in the theory for someone to observe the goings on. Yet here we are! You and I! Never mind the objective world! It doesn't matter. It doesn't really exist! Not unless we observe it? Is that why we are here, to make sure the world exists?
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Can the model tell you everything? That is to say, can a theory tell you everything? Perhaps. But can it tell you everything and still be consistent? Ah! That's another concern, and the answer seems to be NO.
Simple models -- finite models -- can easily be shown to be both complete and consistent. The reason is, all of the statements of the model can be listed and checked one against the others -- constructively. But more complex models pose severe problems in that regard.
I refer to model, or theory, as a repository of information. Implicitly the theory contains a stack of information, in the form of statements that can be identified in different ways. In one mode or presentation it can be a collection or list of the statements. You show the system by showing its elements. If the system is very simple, this approach is perfectly adequate. But what do you do if you can't enumerate them?
In that case, the best you can do is to find a select group of statements in the collection from which you can derive all of the rest of them. Now that's a very tall order. It's referred to as deduction from given conditions. You select starting statements assumed to be true and deduce the rest of the system from them. That's been the practice in logic and math for centuries, and continues to be the procedure. The work has been confined mainly to logical or mathematical systems, but even relatively simple ones, like the arithmetic of real numbers, have proved daunting.
Keep in mind that any theory you devise involves statements that lie outside the theory. The theory is delineated within a language structure and its operations have to be explained in that structure. But the explanation itself is informal. It isn't part of a formal system like the one it describes. You might go ahead and formalize it, but you would then need to use informal language again to discuss it. The upshot is that you can never avoid the informal or non-systematic language. So there are always loose ends in the procedure.
The fact that statements from a system have to be selected to serve as a basis of discovery puts a heavy burden on selection. Questions can be raised about the legitimacy, utility, or adequacy of the basic statements. Among them are questions as to the completeness and consistency of the defining set. Do they do what they're supposed to do? Or does the deductive process have limitations? If so, what are they?
It was Godel who put the ceiling on our knowledge of theories. The declaration of limits came in the form of a theorem, which is referred to as the Godel Incompleteness Theorem. It deals with being able to decide the truth-value of each of the statements in a theory. This is heavy stuff, and it puts us right at the center of the furor in contemporary logic.
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