SR and differential geometry

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atyy

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It's not nearly as bad as an inconsistency, but it's certainly undesirable. I think we can define "inconsistent" as "it's possible to derive the negation of one of the axioms from the others". This would obviously be disaster, but I think it's also pretty bad to have a situation where "it's possible to derive one of the axioms from the others". Then we have an axiom that adds nothing to the theory, and I would certainly prefer to remove it. To include it would give people the impression that we either believe that it adds something significant, or don't understand that it doesn't.
Well, I'm thinking of something like: What's an a charged particle? Something that responds to an electric field. What's an electric field? Something that makes a charged particle move.

I like this MTW claim: Here and elsewhere in science, as stressed not least by Henri Poincare, that view is out of date which used to say, "Define your terms before you proceed." All the laws of physics, including the Lorentz force law, have this deep and subtle character, that they both define the concepyts they use and make statements about these concepts. Contrariwise, the absence of some body of theory, law and principle deprives one of the means properly to define or even to use concepts. Any forward step in human knowledge is truly creative in this sense: that theory, concept, law, and method of measurement - forever inseparable - are born into the world in union.
 

Fredrik

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Well, I'm thinking of something like: What's an a charged particle? Something that responds to an electric field. What's an electric field? Something that makes a charged particle move.
I would define all of those things in the mathematical model and leave the real-world concepts undefined. I don't think I like that MTW claim.
 

atyy

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I would define all of those things in the mathematical model and leave the real-world concepts undefined. I don't think I like that MTW claim.
I vacillate back and forth about this point. Let's say you want to find the Higgs boson, and you build the LHC. Clearly, to do the experiment, one needs the concept of "Switzerland" - otherwise experimentalists are not going to find their way to the experiment. According to MTW, "Switzerland" would be a concept definable from the Standard Model. At first that seems preposterous. On the other hand, we do have a vague way of going from the Standard Model to relativistic quantum mechanics to quantum mechanics to classical mechanics to "Switzerland" ....
 

Fredrik

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Yes, clocks, particle detectors and Switzerland all consist of particles of the types that the SM makes predictions about. But that doesn't mean that we have to use that fact when we define our theories (or that we can use it as an excuse to not define anything, or to use circular definitions). The "clocks" that we have to mention in the axioms of SR are defined by instructions on how to build them, and the terms in those instructions are defined by human language, human senses, and conscious experience. The last two of those have to be considered primitives, i.e. things we don't define. (That probably goes for some things in the language as well). The fact that we have to resort to talking about human language, senses and experiences is annoying as hell, but there's no way around it. All we can do is to try to make sure that these things don't introduce any more ambiguities into the theory than we can tolerate.
 
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