Abstract
As developed in Chap. 5, the attempts to fcapture the ontology of space and spacetime using the conceptual apparatus of substantivalism and relationism—that space is, respectively, either an independently existing entity or a relation among material substances—have proved to be quite problematic since the sophisticated versions of both substantivalism and relationism seem to be identical in the context of general relativity (GR), our best theory on the large scale structure of space. But, whether space is a substance or a relation is just one of the many conceptual distinctions that have entered the modern debate on the nature of space. In addition to substantivalism and relationism, one also finds references to realism versus anti-realism, platonism versus nominalism, and background independence versus background dependence. In this chapter, we will begin the examination of these new dichotomies, their impact on the traditional substantival/relational distinction, and their relationship to one of the least discussed, but quite central, components of the spatial ontology debate, namely, how the philosophy of mathematics factors into the philosophy of space and time.
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Notes
- 1.
See, Chakravartty 2011, who raises problems for using the traditional mind-based, or linguistic-conventional, type of nominalism in conjunction with scientific realism (SR): i.e., if a body’s properties are mind-dependent, then scientific realism is undermined (since SR claims that a mind-independent reality explains the success of our best scientific theories). Chakravartty’s seems to advocate a version of a trope theory to overcome this difficulty and uphold SR; i.e., he claims that a thing’s properties can allow different mind-based interpretations without undermining the reality of the property (2011, 170–171). However, the traditional nominalist could simply accept that certain interpretations of an object’s properties are true, and others false (which thereby upholds SR as well), but without reifying the object’s properties as real particular things (a la trope theory). In fact, Chakravartty’s view may constitute a trope-based version of the type of (neo-Meinongian) nominalism advanced in this chapter. See, also, footnotes 2 and 3.
- 2.
Immanent realism differs from a trope theory in that the former accepts abstract objects (so that, e.g., all square objects instantiate “squareness”, the latter being the very same abstract object in all square bodies), whereas trope theorists reject abstract objects (hence, all square bodies have a property of squareness, the latter being a particular that stands in relations of “resemblance” with other square particulars in other square material objects). In addition, Sepkoski (2007, 16) suggests a three part division of mathematical ontology that is close to the three part division offered in our investigation, although he equates the view that we have labeled “immanent realism” with Aristotle’s position as regards the existence of substantial forms.
- 3.
We have employed the term “truth-based nominalism”, which is adapted from Pincock’s (2012) concept, “truth-value nominalism”, so that Pincock’s own usage is kept separate from our interpretation. Alternatively, truth-based nominalism can be called “liberal nominalism”, a designation used in Slowik (2005a). In Balaguer (2009), the view that we have defined as truth-based nominalism is categorized as “neo-Meinongianism”, and he includes a number of the authors discussed in this chapter as among its defenders, e.g., Salmon 1998, Azzouni 2004, Priest 2005, and Bueno 2005.
- 4.
However, Newstead and Franklin’s view seems to be a mixture of our truth-based nominalism and immanent realism (2012, 83–84). Overall, there are many options for constructing an hypothesis that lies between the extremes of fictionalism and traditional platonism (where abstract objects are non-spatial in Plato’s sense). Examining all of these potential views, and how they might differ from one another, is beyond the scope of this chapter.
- 5.
In particular, Pincock (in Balaguer et al. 2013, 271–272) has doubts concerning the hypothesis entitled, “heavy duty platonism” (see below), although our analysis treats this concept as equivalent to truth-based nominalism. The problem probably resides in the difficulty in determining just what Field intends by this concept, as will be further explained below.
- 6.
The view of structure advocated in Brading and Landry (2006), called “minimal structuralism”, would seem to be in accord with both truth-based nominalism and the ESR view advocated in this monograph (and the ESR-L version that will be developed in Chap. 8). “What we call minimal structuralism is committed only to the claim that the kinds of objects that a theory talks about are presented through the shared structure of its theoretical models and that the theory applies to the phenomena just in case the theoretical models and the data models share the same kind of structure. No ontological commitment—nothing about the nature, individuality or modality of particular objects—is entailed” (2006, 577). If one interprets “objects” as including substantival space, bodies, fields, etc., then this characterization of structure could apply to the conception of spacetime structure advocated in our investigation.
- 7.
Another question concerns what counts as a spacetime theory. If any use of geometric relations were to qualify as the type of structuralist spacetime theory considered above, then nearly all physical theories would trivially count given an actual or potential geometric formulation. A case in point is Fresnel’s wave optics, cited in Chap. 5, which utilizes a trigonometric relationship to relate the intensities of reflected and refracted light that pass through mediums of different optical density (Worrall 1989, 119). One could classify such theories as spacetime theories, of course, but the designation seems more apt for the most basic dynamical theories that correlate inertial motion and force against a particular spacetime backdrop (e.g., Newtonian, Leibnizian, etc.).
- 8.
The Quine-Putnam Indispensability thesis is the best known philosophical debate related to the interface of mathematics and empirical reality, although the thesis does not really concern the interaction between the two realms; rather, the contention is that one should be a realist about mathematics if one is also a realist about scientific laws, objects, etc.
- 9.
It should be noted that the problem of incongruent counterparts (Kant’s handed objects) could stand as a counter-argument against the claim that spacetime lacks causal powers. On the other hand, one could always invoke a maneuver similar to that advocated in Sklar (1985, 234–248), which accepts an “intrinsic” property of handedness (i.e., a primitive property defined via continuous rigid motions) to avoid the commitment to substantival space. On the whole, there are a variety of strategies that the relationist can offer to undermine the inference that space “causes” the change in handedness. That is, while the background spatial structure is obviously relevant to determining an object’s handedness, the claim that space causes a change in handedness is as dubious as the claim that inertial structure is the cause of the observed non-inertial forces of a (non-uniformly) accelerating body.
- 10.
See, Teller (1991), Sklar (1990), Bricker (1990), and Azzouni (2004, 196–212), for similar arguments about the causal irrelevance of spacetime structures for explaining accelerated motions and effects. An important early critique is Einstein (1923, 112–113), who labels absolute space as “a fictitious cause” since rotation with respect to absolute space is not “an observable fact of experience”. Furthermore, with respect to the interrelationship between the metric and matter fields in GR, this interrelationship does not explain why non-inertial forces are associated with accelerated motion; rather, the interrelationship is relevant to explaining the metric curvature.
- 11.
Since truth-based nominalists reject platonism, they do not claim, of course, that mathematical entities and their relationship enter into spacetime theories. But, spacetime structures are not just the physical relationships between the physical objects, either, since (as argued above) causation is not a spacetime relationship (whereas causation is probably the most prevalent physical relationship among physical objects). Spacetime structures, like all other mathematical structures, are unique in this regard: systems can exemplify the structures, but the structures are neither identical to the systems, nor (for the nominalists) can they exists in the absence of systems.
- 12.
Some of Newton’s descriptions might challenge this conclusion, such as his contention that God “constitutes” space (N 91), if one takes that term as synonymous with “is”. Likewise, one may strive to read such terms as “effects”, “affection”, “attribute”, “modes”, and “consequences”, as Newton’s somewhat clumsy way of describing a fictionalist elimination of space in favor of God’s being. Yet, Clarke’s straightforward denial that “space is consequence of God” equates with “space is God” poses a significant challenge, as does the tenor of Newton’s Des Maizeaux drafts, which rejects Clarke’s use of the term “property” in favor of “modes” and “consequences”; likewise, he does not object to Clarke’s claim that “space is not God”. In short, a fictionalist interpretation faces severe obstacles given this evidence.
- 13.
One of the most straightforward declarations of nominalism can be found in the New Essays, where numbers and extension are compared (see also Chap. 3): “[I]n conceiving several things at once one conceives something in addition to the number, namely the things numbered; and yet there are not two pluralities, one of them abstract (for the number) and the other concrete (for the things numbered). In the same way, there is no need to postulate two extensions, one abstract (for space) and the other concrete (for body)” (NE II.iv.5).
- 14.
In more detail, although a monad’s extended secondary matter would not arise if God prevented it, with secondary matter equating with bodily extension at the macrolevel, the unextended primary matter would remain at the microlevel. See Chap. 4 for more on the primary/secondary, and primitive/derivative force, distinctions.
- 15.
While the origins of Leibniz’ view remain uncertain, it is highly likely that this particular dispute between Leibniz and Clarke stems, at least in part, from a more basic dichotomy concerning the definition of ratios in (Euclid’s) geometry, a dichotomy that found many important advocates in the seventeenth century. One school of thought, represented by Wallis, accepts that ratios are quantities that can be explained purely algebraically using an intricate exponent method; whereas others, such as Barrow, argue that ratios are not quantities, hence algebraic relationships fail in this regard (i.e., a geometric conception of quantity is primary). What is intriguing is that, like Wallis, Leibniz defends the primacy of algebra over geometry (e.g., AG 251–252), while Clarke and Newton side with Barrow in defending the primacy of geometry over algebra. In short, the foundations debate in the philosophy of mathematics, i.e., whether algebra or geometry is more fundamental, or at least superior, may be the origins of this particular aspect of the Leibniz-Clarke correspondence—and, in turn, it may ultimately reflect a more basic difference in their respective visions of the deep metaphysics of spatial ontology: Newton and Clarke’s spatially extended God conception versus Leibniz’ non-spatial God/monad hypothesis. More historical work is required on the origin of Leibniz’ view, but see Jesseph (2016) for an analysis of the ratio dispute between Wallis and Barrow.
- 16.
In the L.III.5 quote above, the contrast between “space as absolute being/reality” and “space as a real truth” could be taken to correspond to, respectively, a commitment to abstract geometric objects versus the truths of geometry. This inference gains support given Leibniz’ rejection of abstract objects in general (see footnote 13). Yet, leaving aside abstract objects, Newton’s conception of space is actually similar to Leibniz’ theory: both accept nominalism, with the main difference being the ontological foundation of that nominalist spatial structure, God (Newton/Clarke) versus matter and God (Leibniz). Since God is central to the ontology of space for both Newton and Leibniz, the L.III.5 quote can also be viewed as the difference between positing a conception of space that is either entirely independent of matter (Newton/Clarke) or dependent on matter for instantiating the God-grounded truths of spatial geometry (Leibniz).
- 17.
The analogy developed here is a bit loose, as a commentator has pointed out in an earlier version of this material. Since a wavefunction is an element of a Hilbert space, and is thus not defined on a Hilbert space in the same way that a metric is defined on a manifold, a more correct analogy would be between a wavefunction as an element of a Hilbert space and a manifold point as an element of a manifold. However, substituting an analogy between the wavefunction/Hilbert space and a point/manifold, rather than employing an analogy between the wavefunction/Hilbert space and a metric/manifold, works just as well for our purposes. On a related note, Dieks correctly points out that, as regard non-relativistic QM, “the Hilbert space formalism does not start from a space-time manifold in which particles are located. The quantum state is given by a vector in Hilbert space, and has in general no special relation to specific space-time points. Rather, ‘position’ is treated in the same way as ‘spin’ or other quantities that are direct particle properties: all the quantities are ‘observables’, represented by Hermetian operators in Hilbert space” (Dieks 2001a, 16).
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Slowik, E. (2016). The ‘Space’ at the Intersection of Physics, Metaphysics, and Mathematics. In: The Deep Metaphysics of Space. European Studies in Philosophy of Science. Springer, Cham. https://doi.org/10.1007/978-3-319-44868-8_7
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