Department of Humanities and Social Sciences
In this essay, I consider the ontological status of spacetime from the points of view of the standard tensor formalism and three alternatives: twistor theory, Einstein algebras, and geometric algebra. I briefly review how classical field theories can be formulated in each of these formalisms, and indicate how this suggests a structural realist interpretation of spacetime.
This essay is concerned with the following question: If it is possible to do classical field theory without a 4-dimensional differentiable manifold, what does this suggest about the ontological status of spacetime from the point of view of a semantic realist? In Section 2, I indicate why a semantic realist would want to do classical field theory without a manifold. In Sections 3, 4, and 5, I indicate the extent to which such a feat is possible. Finally, in Section 6, I indicate the type of spacetime realism this feat suggests.
2. Manifolds and Manifold Substantivalism
In classical field theories presented in the standard tensor formalism, spacetime is represented by a differentiable manifold M and physical fields are represented by tensor fields that quantify over the points of M. To some authors, this has suggested an ontological commitment to spacetime points (e.g., Field 1989, Earman 1989). This inclination might be seen as being motivated by a general semantic realist desire to take successful theories at their face value, a desire for a literal interpretation of the claims such theories make (Earman 1993, Horwich 1982). Arguably, the most literal interpretation of classical field theories motivated in this way is manifold substantivalism. Unfortunately for the semantic realist, however, manifold substantivalism succumbs to the hole argument. While spacetime realists have been prolific in constructing versions of spacetime realism that maneuver around the hole argument, all such versions subvert in one form or another the semantic realist's basic desire for a literal interpretation. But what about interpretations of classical field theories formulated in formalisms in which the manifold does not appear? Perhaps spacetime realism can be better motivated in such formalisms while at the same time remaining true to its semantic component.
3. Manifolds vs. Twistors
In this section, I indicate that, for certain classical field theories, the twistor formalism (see, e.g., Penrose and Rindler 1986) is expressively equivalent to the tensor formalism. This holds for theories describing:
(a) spinor fields in compactified, complexified Minkowski spacetime (CMc) that satisfy the null shear-free geodesic equation;
(b) zero rest mass fields in CMc;
(c) anti-self-dual Yang-Mills fields in CMc;
(d) anti-self-dual metric fields that satisfy the Einstein equations.
I indicate how these results follow from a general procedure known as the Penrose Transformation and suggest that the concept of spacetime that arises for these field theories is very different, under a literal interpretation, from the one that arises in the tensor formalism.
4. Manifolds vs. Einstein algebras
In this section, I indicate the extent to which the points of a differentiable manifold can be non-trivially reconstructed from an Einstein algebra (see, e.g., Geroch 1972, Heller and Sasin 1995, Bain preprint).
5. Manifolds vs. Geometric algebra
In this section, I indicate how certain field theories can be recast using geometric algebra (see, e.g., Hestenes and Sobczyk 1984) and the extent to which the geometric algebra formalism is non-trivially expressively equivalent to the tensor formalism. (I also indicate the extent to which geometric algebra might be considered more "fundamental" than the twistor and tensor formalisms.)
6. Spacetime as Structure
In this section, I suggest that if one's spacetime realism is based in part on a literal interpretation of classical field theories, then manifold substantivalism need not be the only option. Manifold substantivalism is a literal interpretation of just one way of formulating a classical field theory. To the extent that there are other ways in which manifolds do not appear, manifold substantivalism is not forced upon our literal interpretations. The fact that these alternative formulations differ, upon literal interpretation, at what might be called the level of "individuals-based" ontology suggests that what we should literally interpret is the structure that all such expressively equivalent formulations have in common. In particular, if our spacetime realism is motivated by an underlying field realism, then we should ontologically commit to the structure that is minimally required to support the mathematical representation of fields. This "spacetime structuralism" can then be seen as a version of structural realism.
Bain, J. (preprint), 'Einstein Algebras and the Hole Argument', forthcoming in Philosophy of Science. Available at philsci-archive.pitt.edu/archive/00001052/.
Earman, J. (1993), 'Underdeterminism, Realism and Reason', in P. French, T. Uehling, Jr., and H. Wettstein (eds.), Midwest Studies in Philosophy, XVIII, Notre Dame: University of Notre Dame Press, pp. 19-38.
Earman, J. (1989), World Enough and Spacetime, Cambridge: MIT Press.
Field, H. (1989), Realism, Mathematics, and Modality, Oxford: Blackwell Publishers.
Geroch, R. (1972), 'Einstein Algebras', Communications of Mathematical Physics 26, 271-275.
Heller, M. and W. Sasin (1995), 'Sheaves of Einstein Algebras', International Journal of Theoretical Physics 34, 387-398.
Hestenes, D. and G. Sobczyk (1984), Clifford Algebra to Geometric Calculus, Dordrecht: D. Reidel.
Horwich, P. (1982), 'Three Forms of Realism', Synthese 51, pp. 181-201.
Penrose, R. and W. Rindler (1986), Spinors and Spacetime, Vol. 2: Spinor and Twistor Methods in Spacetime Geometry, Cambridge: Cambridge University Press.