Abstract
The Hole Argument was originally formulated by Einstein and it haunted him as he struggled to understand the meaning of spacetime coordinates in the context of the diffeomorphism invariance of general relativity. This argument has since been put to philosophical use by Earman and Norton (Br J Philos Sci 515–525, 1987) to argue against a substantival conception of spacetime. In the present work I demonstrate how Earman and Norton’s Hole Argument can be extended to exclude everything and not merely substantival manifolds. These casualties of the hole demonstrate that the Hole Argument hinges essentially on our notion of determinism and not on the diffeomorphic freedom of general relativity. Just as Earman and Norton argue that we should not let our metaphysics run roughshod over the structure of our physical theories, so I will argue that, in particular, we should not uncritically allow our metaphysics to dictate what our physical theories must determine. The central conviction which drives the arguments of this paper is that deterministic theories are not required to determine for future moments what they cannot determine for any present or past moments. I provide two arguments to the effect that a physically informed notion of determinism does not require general relativity to determine substantival facts. Consequently the Hole Argument cannot be used against substantival spacetime. The position that I advocate is an instance of “sophisticated determinism.”
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Notes
See John Norton’s review article [3] for a detailed discussion.
For a precise treatment of this fact, see Earman and Norton [1].
Each non-trivial diffeomorphism generates a new mathematical metric (\(G^*\)) and matter fields (\(T^*\)) physically equivalent to the originals. To be more precise, if \(\langle M, G, T\rangle \) is a model of GR then \(\langle M, G^*, T^* \rangle \) is also a model giving the same solution to Einstein’s field equations. In general, since there are infinitely many diffeomorphisms which are designed to shift the fields (G,T) only after a certain moment of time (assuming a 3 + 1 split of spacetime), each model of the past will have infinitely many GR-models representing substantivally distinct possible futures. Though these models represent different substantival futures they are physically indistinguishable from the perspective of GR.
To be sure, initial value problems or “Cauchy problems” in mathematics are not usually stated in terms of determinism but in terms of solving some differential equation.
I am not hereby endorsing these other terms or the metaphysics which suggests them. I am only providing a rough translation guide between the use of these terms in the literature. The guide is meant to help translate the use of words in this limited literature and is not meant to provide an account of the meaning of these alternative terms.
Whether or not Aristotle was right with respect to his hierarchy of being, or if Plato was right, our current physics would not know it. Aristotle takes material particulars as the bedrock of reality and Plato takes abstract universals as the bedrock. However, the predictions of GR are invariant under this difference. We can mix up the levels and GR does not stop making empirically adequate predictions.
If a certain assumption is necessary in order for a physical theory to be conceptually coherent, I will refrain from calling that assumption metaphysical.
Which so happens to be Saint Cunegunda or ‘Kinga’.
Hoefer [8] puts a similar argument to a different use.
For the sake of the following argument, persons are meant to be three-dimensional and not four.
Note that I am not claiming that a person gets a new soul, for that would be incoherent given our construction. This mapping reassociates souls with things not persons.
I intend this to be tongue-in-cheek, but perhaps I should explain. In considering diffeo-metaphysms, we are not doing mathematical physics and thus need not be tied down by smoothness requirements.
In other words, the person associated with the cat-object now may still be associated with the cat-object then, or it may be associated with the dog-object at the window then.
See [10] The point is, that which we call a thing is not metaphysically innocent. Metaphysical assumptions go into how we carve up the world.
This is exactly Brighouse’s point [11].
Throughout this paper, I have attempted to make use of the term ‘physical’ and not ‘empirical’. I assume, however, that empirical content is sufficient for physical content.
Any set of points will do so long as they are appropriately related to some diffeomorphically invariant model of GR.
Earman [16] concedes to Maudlin that this might be an uninteresting form of indeterminism. I argue further that it is no form of determinism at all.
If the truth of the Axiom of Choice is required for the truth of GR or some other physical theory, which I doubt, then one can use the Continuum Hypothesis, souls, angles, Fregian numbers qua the extension of concepts to make the same point.
In cases where classical gauge degrees of freedom are quantized and then, for whatever reason, become empirically salient, we have moved into a new theoretical context and have to re-evaluate what we take to be physical.
This situation holds for general relativity as well. The theory cannot determine for any past, present or future moment where any object is located but it can determine whether or not it moves along a geodesic (straight line).
Under a relational interpretation, neither question is well defined since they make reference to objects which do not exist.
These consequences will turn out to be false in the long run as orbiting objects eventually plummet to the Earth.
From what I can determine, the last paper to discuss sophisticated determinism is Brighouse [2].
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Acknowledgements
I thank Nick Huggett, John Norton, Carl Hoefer, Sam Fletcher and Jim Weatherall, either for comments on earlier drafts of this paper or for a helpful discussion on some of its core ideas. I also thank the participants of the 2015 philosophy of science meeting in Dubrovnik as well as the southern California philosophy of physics group.
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Norton, J. The Hole Argument Against Everything. Found Phys 50, 360–378 (2020). https://doi.org/10.1007/s10701-019-00258-y
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DOI: https://doi.org/10.1007/s10701-019-00258-y