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In a number of publications, John Earman has advocated a tertium quid to the usual dichotomy between substantivalism and relationism concerning the nature of spacetime. The idea is that the structure common to the members of an equivalence class of substantival models is captured by a Leibniz algebra which can then be taken to directly characterize the intrinsic reality only indirectly represented by the substantival models. An alleged virtue of this is that, while a substantival interpretation of spacetime theories falls prey to radical local indeterminism, the Leibniz algebras do not.I argue that the program of Leibniz algebras is subject to radical local indeterminism to the same extent as substantivalism. In fact, for the category of topological spaces of interest in spacetime physics, the program is equivalent to the original spacetime approach. Moreover, the motivation for the program—that isomorphic substantival models should be regarded as representing the same physical situation—is misguided.

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After some background setting in which it is shown how Maudlin's (1989, 1990) response to the hole argument of Earman and Norton (1987) is related to that of Rynasiewicz (1994), it is argued that the syntactic proposals of Mundy (1992) and of Leeds (1995), which claim to dismiss the hole argument as an uninteresting blunder, are inadequate. This leads to a discussion of how the responses of Maudlin and Rynasiewicz relate to issues about gauge freedom and relativity principles.

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