[During recent exchanges on this blog about the relationship between metaphysics and physics [see here and here], a forthcoming chapter by Arntzenius and Dorr was repeatedly mentioned. Jody Azzouni has offered us the following reflections on their approach.—ES]
Nominalists beware! It is not nominalistically benign to take objects that a nominalist is committed to, and then stipulate conveniently that such objects (or the “spaces” that they or their properties are located in) have all manner of intrinsic structure. But this is exactly what Arntzenius and Dorr (following Hartry Field) do on behalf of their “nominalism.”
Call Topologism the doctrine that there are no abstracta other than the points and regions of (one or another) space. Call Fieldian Topologism one or another program of showing that any scientific theory T which has among its existential commitments mathematical abstracta can be replaced with an equally useful theory T* which quantifies only over topological entities.
The classical text of Fieldian Topologism is, of course, Field’s (1980) “Science without Numbers.” A recent entry in the topic area is Arntzenius and Dorr’s forthcoming “Calculus as Geometry.” In both cases, regions and points of one “space” or another are quantified over. Arntzenius and Dorr are even willing to posit a continuum “mass space,” and to stipulate that each particle occupies a point therein. They attempt to allay the queasiness that any nominalist is bound to feel at this by suggesting we instead call these “mass properties” and stipulate that particles have such properties.
Arntzenius and Dorr justify this dubious procedure on the grounds that positing such topological structures nicely explain the properties that physical objects actually have. It will illustrate the problem to consider the “mass space” that they so blithely introduce. This additional space must be continuum-rich; further, its “points” must be continuum-structured (ordered, dense, etc.) in order for it to adequately support the mass function mathematically. And the space has these properties even if, as Arntzenius and Dorr admit, most of these mass-point locations are actually unoccupied by particles. In addition, the locations of this “mass space” aren’t located anywhere in relation to actual spacetime: they occur in an additional space posited purely for the mathematical needs of a physical theory of mass. Labeling it a space of mass properties (a set of continuum-ordered, etc., properties?) doesn’t escape the challenge: How isn’t this a positing of the very abstracta that nominalists want to avoid?
The style of inference to the best explanation, that Arntzenius and Dorr use to justify their positing of various spaces with rich intrinsic mathematical structure, has a long pedigree in philosophy. Indeed, one finds Plato using it to justify his claim that the fine-structure of the afterlife involves reincarnation. But the nominalist’s scruples at the end of the day are empiricist ones. The nominalist—like scientists generally—don’t posit metaphysical structure on grounds of sheer explanatory force. Newton notoriously avoided doing so to explain action at a distance. What’s called for are instrumental probes that detect the physical objects that scientific theories quantify over, and that track their properties. The other entities quantified over are then treated as purely mathematical in nature.
Arntzenius and Dorr neatly avoid collision with these truisms about scientific practice by packaging their meta-scientific comments entirely in terms of simplicity. That’s rhetorically useful because no one, scientists included, knows what simplicity is, especially when it’s supposed to be applied to scientific theories. But that’s not the real philosophic sleight of hand in utilizing talk of simplicity in this context; it’s that the sophisticated ways that scientific theories are (1) used in applications, (2) evidentially brought to bear on the world, and (3) are explored for ontological implications, are buried in a simplistic and false picture of scientific practice where one has numerous theories to choose from when theorizing about a phenomenon, and that one is (of course) committed to everything one’s theory quantifies over when selecting one of those theories to use. Scientists, we are told near the beginning of this paper, have not bothered to explore alternative theories which don’t utilize mathematical entities. Thus there is something for philosophers to do.
To see what is at stake for genuine nominalists, let me offer a non-scientific example first. Suppose you’re a nominalist. You claim (let’s say) that only certain concreta exist. Perhaps you think such concreta are macro-sized objects like the ones around us. Footballs, for example. You deny the existence of properties, as well as the kinds of abstracta routinely studied in mathematics. Furthermore, in claiming that certain concreta exist, you don’t mean that claim to licence all sorts of additional ontological commitments to arbitrary parts of those concreta. For example, the knowledgeable nominalist knows that the axiom of choice allows (in a purely mathematical sense) a handball to be mathematically divided into finitely many (nonmeasurable) pieces that when reassembled will produce a sphere the size of the Sun. The knowledgeable nominalist won’t allow that the intrinsic structure of handballs, whatever that empirically turns out to be, to entail that such parts of handballs exist in addition to the handball itself. What parts of a handball are real isn’t to be stipulated, but to be discovered to be real in the same way that the handball itself is; what properties a handball has aren’t to be stipulated for similar reasons.
To get more scientific about it, consider this example: The nucleus of any atom contains real subatomic particles with specific physical properties. Scientific facts about such particles and their properties has been established not merely on the basis of the simplicity of the scientific theories about such, but on the basis of actual physical probings of nuclei. Something similar has not occurred with the fine structure of spacetime. No instrumental device or set of empirical experiments has established the presence of spacetime points or the nature of their cardinality. The same point can be made about arbitrary regions of spacetime. Notice: Objections to Fieldian Topologism (as an attempted defence of nominalism) are not technically subtle, and not particularly deep—philosophically speaking. They are right on the surface.
To deny Fieldian Topologism philosophical content isn’t to deny that it offers technical challenges. Of course it does. The important question that proponents of Fieldian Topologism should explore carefully is how, if at all, these technical challenges correspond to real philosophical concerns. If, for example, it can be shown that points and regions of continuum-rich spaces are nominalistically acceptable, then the topic area will connect in that way to a possible vindication of nominalism. But this has not been shown by anyone attempting one or another version of Fieldian Topologism. Arntzenius and Dorr are particularly glib on the matter as I hope I’ve indicated above. Field tries harder in his 1980; his attempts were widely and successfully criticized. Especially noteworthy was the review of the book by David Malament, who showed that Field’s language was far too rich for nominalistic purposes. Arntzenius and Dorr fail to show that they’ve taken Malament’s lessons to heart; indeed, their citations don’t even indicate familiarity with it. (I might also note, parenthetically, my own 2009 “Evading truth commitments: the problem reanalyzed.”)
Nominalism is a serious matter. How challenging it is to sustain nominalism shouldn’t be hidden behind technical maneuvers. The Platonistic wolf, philosophers should be warned, has many disguises.
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