Belot, Gordon. Geometric Possibility.

Roberts, John T.

BELOT, Gordon. Geometric Possibility. Oxford: Oxford University Press, 2011. x + 197 pp. Cloth, $50.00--The central philosophical debate about space is between substantivalism and relationalism. Substantivalists hold that space exists and is composed out of parts (points or extended regions) that exist independently of whether they are occupied, and that bodies stand in geometric relations to one another derivatively, in virtue of the geometric relations of the regions they occupy. Relationalists hold that bodies are geometrically related directly, and that space does not enjoy an independent existence, but is some kind of abstraction from the geometric relations of bodies. (The standard example of a controversy between substantivalists and relationalists is the dispute between the substantivalists Newton and Clarke and the relationalist Leibniz, but in a fascinating historical appendix, Belot shows that the history of the dispute goes back much further.) Relationalists may be distinguished into conservative relationalists, who hold that all that is true about space must supervene on the geometric relations of actual bodies, and modal relationalists, who hold that possible relations involving merely possible bodies are relevant as well. (Leibniz is usually taken to be a modal relationalist, and in another appendix, Belot argues compellingly that he was.) This lucid and exciting book is an exploration of the prospects for developing a detailed and coherent version of modal relationalism.

Belot assumes that a defensible relationalism must supply truth conditions for statements such as "space is infinite," "space is N-dimensional," "space is hyperbolic" and so forth. For substantivalists, this is easy: It is true that space is hyperbolic, for instance, just in case the metrical relations among those obligingly existent individuals the spatial points instantiate the right sort of structure. For relationalists, matters are not so easy. It would be one thing if we could take it for granted that matter forms a plenum, for in that case a relationalist could simply identify the geometrical structure of space with that of matter. But if matter is not a plenum, then the geometrical structure exemplified by all the material objects vastly underdetermines the structure of space. As Belot demonstrates, it will not help to suggest that the geometry of space is the simplest geometrical structure within which the geometry of the actually existing matter can be embedded--for in general, there is no unique simplest one. So relationalists must appeal to something above and beyond the actual geometrical structure of matter in order to find truth conditions for propositions about the structure of space; hence the appeal of modal relationalism, which says that these truth conditions are about possible, but not necessarily actual, arrangements of material bodies.

We might ask, however, "possible" in what sense? Most of Belot's book is about the range of possible answers to this question. Anyone who agrees that Newtonian physics and general relativity are both metaphysically possible must say that the relevant possibility is stronger than metaphysical--in a Newtonian world, it is metaphysically possible for the world to be globally hyperbolic (since there is a metaphysically possible general-relativistic world where it is), but in the sense of "possible" relevant to the modal relationalist, it is not possible relative to a Newtonian world for matter to be arranged in a globally hyperbolic way. Belot argues that nomic modality is not what is wanted either: There are cases where the range of geometric possibilities outstrips the range of nomic possibilities, and vice versa. So, Belot concludes, geometrical possibility is sui generis.

Belot identifies three attractive features in an account of geometrical possibility: An account is grounded if it implies that geometric possibility supervenes on the actual material configuration; it is ambitious if it recognizes all the qualitatively distinct possibilities substantivalism does (for example, it recognizes many worlds containing only a single motionless particle, differing on the structure of the surrounding empty space); it is metric if it says that the geometrical features of material objects are exhausted by their distance relations. No account can have all three, so modal relationalists will have to sacrifice at least one. Belot explores three accounts of geometric possibility, each of which dispenses with one of the three features while keeping the other two. The three accounts have much in common with three of the leading approaches to nomic modality, namely the Humean best-system account of laws, the primitivist account, and the scientific essentialist account. Belot argues that the account that dispenses with ambition--the one that corresponds to the best-system approach to nomic modality--is subject to terrible difficulties. (Along the way, he develops a novel and interesting objection to the best-system account of laws that deserves serious attention from advocates of that account.) The other two, he thinks, are more promising, though each faces challenges, and in the end there may not be much to choose between them.--John T. Roberts, University of North Carolina.