Fiber Bundle Gauge Theories and "Field's Dilemma"

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We propose a distinction between the physical and the mathematical parts of gauge field theories. The main problem we face is to uphold a strong and meaningful criterion of what is physical. We like to call it "Field's dilemma", referring to Hartry Field's nominalist proposal which we consider to be inadaequate. The resolution to the dilemma, we believe, is implicitly provided by the so-called fiber bundle formalism. We shall demonstrate, in detail, that the bundle structure underlying modern quantum and gravitational gauge field theories allows for a genuine distinction between the physically significant and the merely mathematical parts of these theories.