Empty space and fields

Before I move on to the Leibniz-shift argument, I should first deal briefly with a potential difficulty faced by relationalism, which is sometimes glossed over in

61 I mention this aspect briefly in the next subsection, and consider its bearing on the substantivalism-relationalism debate.

the literature. Namely, that the requirement that there be no spatial and temporal vacua might seem to be too stringent a condition for the relationalist to meet. There are two questions to be asked here: (1) does she need to meet the condition? (2) if the answer to (1) is no, how are such vacua to be constructed relationally? This is closely connected with the question of how fields impact upon the debate between relationalists and substantivalists, for the ability to support fields is thought to provide good reason for adopting substantivalism (cf. Earman [1989], p. 173; and [Field, 1980]). Yet, at the same time, field theory is supposed to provide a response to the relationalist's problem concerning empty space since fields are continua.

There appear to be empty regions of space. If there aren't, then the, at least conceptual, possibility of producing a vacuum remains (for example, Hooke's experiments). There are at least two options open to the relationalist: (1) to dig her heels in and deny that such vacua are (metaphysically or physically) possible; or (2) to admit possible but non-actual spatial and/or temporal relations.

In the time period of the Leibniz-Clarke debate, the question of what is space and what is matter was at least unambiguous. As Earman nicely puts it, "the participants of the debate had the luxury of knowing what they were talking about" ([1989], p. 18). Generally, space (in the sense of vacuum) would simply be defined by a lack of material or mass.

Advocating (1) would imply plenism, and could perhaps be achieved nowadays with a field ontology; for fields extend to cover all of space. In Leibniz's day, the former option would have been hard to uphold on physical grounds, since experiments had been conducted to produce vacua and the field picture was unknown. It could only be underwritten by metaphysical principles. The latter option could be cashed out in terms of counterfactuals expressing what relations would be instantiated if objects were placed in the evacuated region. Hence, for a relationalist in the days before field theory, a full characterisation would have to involve plenism or else include possible spatiotemporal relations—for more on this issue see Sklar ([1974], p. 170). Earman ([1989] §6.6) has Leibniz down as a plenist. Unless one has an ontology of possibilia at one's disposal already, the inclusion of possible spatiotemporal relations would appear to be as bad as the spacetime points that relations were supposed to replace, for the disposal of spacetime points was seen as scoring one over the substantivalist in terms of ontological economy. Perhaps a case could be made for bolstering the modal relationalists position by pointing out that possibilia are not things? That depends upon one's position in modal metaphysics, for some would argue that they are things. But quite regardless of this, even if we allow possibilia, a problem remains concerning what they are possibilities of. Presumably objects, and the possibilities concern where they might have been placed, over there for example. But then how is "there" defined, for the relationalist, if not by the object that is located at it? There is the prospect of circularity: the possible objects appear to require spatial positions in order fulfill their role, yet spatial positions require objects according to relationalism.

The impact of modern physics radically alters the debate's compass. In particular, the notion of 'vacuum' qua 'empty space' prima facie makes little sense in such the context of field theory, for the field is defined at every point of space(time). Moreover, field theories constitute our best scientific description of the world. In quantum field theory, the situation is intensified, since the vacuum generically possesses mass-energy. For this reason, I will assume that some variant of the former option is the better one for the relationalist in the context of modern physics, though not a lot rests on this in what follows—see [Saunders and Brown, 1991] for a collection of essays on the notion of vacuum in modern physics. There are other problems too, as Butterfield explains:

classical physics has introduced the electromagnetic field endowed with energy and momentum; and relativity has identified mass and energy. Furthermore, in general relativity there can be material in the sense of mass-energy in a region where there is not only no material object of an ordinary sort like a chair, but also no field apart from the metric-gravitational field. ([Butterfield, 1984], p. 104)

Coupled with quantum field theory, this rather messes up the nice cleanly formulated debate that was taking place in the time-frame of the Leibniz-Clarke correspondence—I return to this problem again in §4.1.

The concept of a field can be used to the substantivalist's advantage too. For example, Hartry Field ([1980], p. 35) argues that since "a field is usually described as an assignment of some property ...to each point of space-time, this obviously assumes that there are space-time points" (cf. Earman [1989], pp. 154-9). Teller also writes that "[t]he idea of a field enters as the idea that values of physical quantities can be attributed to the space-time points" ([1990], p. 53). The idea is clearly the Quinean one that ineliminable quantification over a type of object entails an ontological commitment to them (cf. Butterfield [1984], p. 101). This is indeed one way to describe a field; but it is not the only way. For example, Belot describes an alternative description in which fields are simply "extended objects whose parts stand in determinate spatial relations to one another" ([2000], p. 584). Hence, it is not obvious that the quantification over points is ineliminable, though the burden of proof is quite definitely rested upon the relationalist to come up with an empirically adequate theory that dispenses with points and involves only kosher material objects—this has tended to be the main problem with relational accounts.

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