2 Purely epistemic interpretations of gauge symmetries were endorsed by some of the main actors in the history of gauge theories, like for instance Dirac. This seems to imply that the general covariance of general relativity is nothing but a mere formal feature (that is, an artifact of the chosen formulation) with no physical content. The kernel of Kretschmann's objection is that pre-relativistic theories like Newtonian gravity can also be recasted in generally covariant terms. The idea that gauge symmetries have per se no physical content arose for the first time in the context of the Kretschmann's objection against the physical scope of the principle of general covariance in general relativity (Norton, 1992, 1993, 2003). the functions on the reduced phase space obtained by “quotienting out” the gauge symmetries (Dirac, 1964 Henneaux & Teitelboim, 1994)). The main argument that grounds this epistemic stance is that a physical state of affairs can be completely described by means of gauge invariant quantities (such as for instance the so-called Dirac observables, i.e. coordinate-independent) state of affairs.
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1 According to (what we shall call) the epistemic interpretations of gauge symmetries, the latter would be nothing but a mere mechanism by means of which we can get rid of the “ surplus structure” (Redhead, 2003) or descriptive redundancy associated with the different possible re-coordinatizations of a unique physical (i.e. The understanding of the empirical and/or ontological status of gauge symmetries is probably the main philosophical problem posited by the gauge theories of the fundamental interactions. We shall argue that this homotopic-theoretic understanding of Yang-Mills theories paves the way toward an ontological interpretation of gauge symmetries, that is an interpretation according to which gauge symmetries-far from being nothing but a “ surplus structure” resulting from a descriptive redundancy-rely on the intrinsic geometric structures of Yang-Mills theories. We shall revisit in this framework the articulation (heuristically established in the framework of the so-called gauge argument) between gauge symmetries and the mathematical notion of connection. We shall apply the homotopic reconceptualization of equalities to the equalities between the base points and the fibers of the fiber bundle associated to a Yang-Mills theory.
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From a philosophical perspective, we shall argue that the homotopic paradigm relies a) on a rejection of Leibniz's Principle of the Identity of Indiscernibles and b) on a constructivist understanding of propositions as types of proofs. This mathematical paradigm was mainly developed in the framework of (higher) category theory and homotopy type theory and relies on a groupoid-theoretical understanding of equality statements of the form a = b. We shall analyze the intrinsic geometric structure of Yang-Mills theory from the standpoint provided by (what we shall call) the homotopic paradigm.