The benefits of the various structuralist approaches in the philosophy of mathematics is that it allows both the mathematical realist and anti-realist to use mathematical structures without obligating a Platonism about mathematical objects, such as numbers – one can simply accept that, say, numbers exist as places in a larger structure, like the natural number system, rather than as some sort of independently existing, transcendent entities. Accordingly, a variation on a well-known mathematical structure, such as exchanging the natural numbers “3” and “7”, does not create a new structure, but merely gives the same structure “relabeled” (with “7” now playing the role of “3”, and visa-verse). This structuralist tactic is familiar to spacetime theorists, for not only has it been adopted by substantivalists to undermine an ontological commitment to the independent existence of the manifold points of M, but it is tacitly contained in all relational theories, since they would count the initial embeddings of all material objects and their relations in a spacetime as isomorphic.
A critical question remains, however: Since spacetime structure is geometric structure, how does the Structural Realism (SR) approach to spacetime differ in general from mathematical structuralism? Is the theory just mathematical structuralism as it pertains to geometry (or, more accurately, differential geometry), rather than arithmetic or the natural number series? While it may sound counter-intuitive, the SR theorist should answer this question in the affirmative – the reason being, quite simply, that the puzzle of how mathematical spacetime structures apply to reality, or are exemplified in the real world, is identical to the problem of how all mathematical structures are exemplified in the real world. Philosophical theories of mathematics, especially nominalist theories, commonly take as their starting point the fact that certain mathematical structures are exemplified in our common experience, while other are excluded. To take a simple example, a large collection of coins can exemplify the standard algebraic structure that includes commutative multiplication (e.g., 2 x 3 = 3 x 2), but not the more limited structure associated with, say, Hamilton’s quaternion algebra (where multiplication is non-commutative; 2 x 3 ≠ 3 x 2). In short, not all mathematical structures find real-world exemplars (although, for the minimal nominalists, these structures can be given a modal construction). The same holds for spacetime theories: empirical evidence currently favors the mathematical structures utilized in General Theory of Relativity, such that the physical world exemplifies, say, g, but a host of other geometric structures, such as the flat Newtonian metric, h, are not exemplified.
The critic will likely respond that there is substantial difference between the mathematical structures that appear in physical theories and the mathematics relevant to everyday experience. For the former, and not the latter, the mathematical structures will vary along with the postulated physical forces and laws; and this explains why there are a number of competing spacetime theories, and thus different mathematical structures, compatible with the same evidence: in Poincaré fashion, Newtonian rivals to GTR can still employ h as long as special distorting forces are introduced. Yet, underdetermination can plague even simple arithmetical experience, a fact well known in the philosophy of mathematics and in measurement theory. For example, in Charles Chihara, an assessment of the empiricist interpretation of mathematics prompts the following conclusion: “the fact that adding 5 gallons of alcohol to 2 gallons of water does not yield 7 gallons of liquid does not refute any law of logic or arithmetic [“5+2=7”] but only a mistaken physical assumption about the conservation of liquids when mixed”. While obviously true, Chihara could have also mentioned that, in order to capture our common-sense intuitions about mathematics, the application of the mathematical structure in such cases requires coordination with a physical measuring convention that preserves the identity of each individual entity, or unit, both before and after the mixing. In the mixing experiment, perhaps atoms should serve as the objects coordinated to the natural number series, since the stability of individual atoms would prevent the sort of blurring together of the individuals (“gallon of liquid”) that led to the arithmetically deviant results. By choosing a different coordination, the mixing experiment can thus be judged to uphold, or exemplify, the statement “5+2=7”. What all of this helps to show is that mathematics, for both complex geometrical spacetime structures and simple non-geometrical structures, cannot be empirically applied without stipulating physical hypotheses and/or conventions about the objects that model the mathematics. Consequently, as regards real world applications, there is no difference in kind between the mathematical structures that are exemplified in spacetime physics and in everyday observation; rather, they only differ in their degree of abstractness and the sophistication of the physical hypotheses or conventions required for their application. Both in the simple mathematical case and in the spacetime case, moreover, the decision to adopt a particular convention or hypothesis is normally based on a judgment of its overall viability and consistency with our total scientific view (a.k.a., the scientific method): we do not countenance a world where macroscopic objects can, against the known laws of physics, lose their identity by blending into one another (as in the addition example), nor do we sanction otherwise undetectable universal forces simply for the sake of saving a cherished metric.
Another significant shared feature of spacetime and mathematical structure is the apparent absence of causal powers or effects, even though the relevant structures seem to play some sort of “explanatory role” in the physical phenomena. To be more precise, consider the example of an “arithmetically-challenged” consumer who lacks an adequate grasp of addition: if he were to ask for an explanation of the event of adding five coins to another seven, and why it resulted in twelve, one could simply respond by stating, “5+7=12”, which is an “explanation” of sorts, although not in the scientific sense. On the whole, philosophers since Plato have found it difficult to offer a satisfactory account of the relationship between general mathematical structures (arithmetic/”5+7=12”) and the physical manifestations of those structures (the outcome of the coin adding). As succinctly put by Michael Liston:
Why should appeals to mathematical objects [numbers, etc.] whose very nature is non-physical make any contribution to sound inferences whose conclusions apply to physical objects?
One response to the question can be comfortably dismissed, nevertheless: mathematical structures did not cause the outcome of the coin adding, for this would seem to imply that numbers (or “5+7=12”) somehow had a mysterious, platonic influence over the course of material affairs.
In the context of the spacetime ontology debate, there has been a corresponding reluctance on the part of both sophisticated substantivalists and (R2, the rejection of substantivalist) relationists to explain how space and time differentiate the inertial and non-inertial motions of bodies; and, in particular, what role spacetime plays in the origins of non-inertial force effects. Returning once more to our universe with a single rotating body, and assuming that no other forces or causes, it would be somewhat peculiar to claim that the causal agent responsible for the observed force effects of the motion is either substantival spacetime or the relative motions of bodies (or, more accurately, the motion of bodies relative to a privileged reference frame, or possible trajectories, etc.). Yet, since it is the motion of the body relative to either substantival space, other bodies/fields, privileged frames, possible trajectories, etc., that explains (or identifies, defines) the presence of the non-inertial force effects of the acceleration of the lone rotating body, both theories are therefore in serious need of an explanation of the relationship between space and these force effects. The strict (R1) relationists face a different, if not less daunting, task; for they must reinterpret the standard formulations of, say, Newtonian theory in such a way that the rotation of our lone body in empty space, or the rotation of the entire universe, is not possible. To accomplish this goal, the (R1) relationist must draw upon different mathematical resources and adopt various physical assumptions that may, or may not, ultimately conflict with empirical evidence: for example, they must stipulate that the angular momentum of the entire universe is 0.
All participants in the spacetime ontology debate are confronted with the nagging puzzle of understanding the relationship between, on the one hand, the empirical behavior of bodies, especially the non-inertial forces, and, on the other hand, the apparently non-empirical, mathematical properties of the spacetime structure that are somehow inextricably involved in any adequate explanation of those non-inertial forces – namely, for the substantivalists and (R2) relationists, the affine structure, that lays down the geodesic paths of inertially moving bodies. The task of explaining this connection between the empirical and abstract mathematical or quantitative aspects of spacetime theories is thus identical to elucidating the mathematical problem of how numbers relate to experience (e.g., how “5+7=12” figures in our experience of adding coins). Likewise, there exists a parallel in the fact that most substantivalists and (R2) relationists seem to shy away from positing a direct causal connection between material bodies and space (or privileged frames, possible trajectories, etc.) in order to account for non-inertial force effects, just as a mathematical realist would recoil from ascribing causal powers to numbers so as to explain our common experience of adding and subtracting.
An insight into the non-causal, mathematical role of spacetime structures can also be of use to the (R2) relationist in defending against the charge of instrumentalism, as, for instance, in deflecting Earman’s criticisms of Sklar’s “absolute acceleration” concept. Conceived as a monadic property of bodies, Sklar’s absolute acceleration does not accept the common understanding of acceleration as a species of relative motion, whether that motion is relative to substantival space, other bodies, or privileged reference frames. Earman’s objection to this strategy centers upon the utilization of spacetime structures in describing the primitive acceleration property: “it remains magic that the representative [of Sklar’s absolute acceleration] is neo-Newtonian acceleration
d2xi/dt2 + Γijk (dxj/dt)(dxk/dt) —– (1)
[i.e., the covariant derivative, or ∇ in coordinate form]”. Ultimately, Earman’s critique of Sklar’s (R2) relationism would seem to cut against all sophisticated (R2) hypotheses, for he seems to regard the exercise of these richer spacetime structures, like ∇, as tacitly endorsing the absolute/substantivalist side of the dispute:
..the Newtonian apparatus can be used to make the predictions and afterwards discarded as a convenient fiction, but this ploy is hardly distinguishable from instrumentalism, which, taken to its logical conclusion, trivializes the absolute-relationist debate.
The weakness of Earman’s argument should be readily apparent—since, to put it bluntly, does the equivalent use of mathematical statements, such as “5+7=12”, likewise obligate the mathematician to accept a realist conception of numbers (such that they exist independently of all exemplifying systems)? Yet, if the straightforward employment of mathematics does not entail either a realist or nominalist theory of mathematics (as most mathematicians would likely agree), then why must the equivalent use of the geometric structures of spacetime physics, e.g., ∇ require a substantivalist conception of ∇ as opposed to an (R2) relationist conception of ∇? Put differently, does a substantivalist commitment to whose overall function is to determine the straight-line trajectories of Neo-Newtonian spacetime, also necessitate a substantivalist commitment to its components, such as the vector d/dt along with its limiting process and mapping into ℜ? In short, how does a physicist read off the physical ontology from the mathematical apparatus? A non-instrumental interpretation of some component of the theory’s quantitative structure is often justified if that component can be given a plausible causal role (as in subatomic physics)—but, as noted above, ∇ does not appear to cause anything in spacetime theories. All told, Earman’s argument may prove too much, for if we accept his reasoning at face value, then the introduction of any mathematical or quantitative device that is useful in describing or measuring physical events would saddle the ontology with a bizarre type of entity (e.g., gross national product, average household family, etc.). A nice example of a geometric structure that provides a similarly useful explanatory function, but whose substantive existence we would be inclined to reject as well, is provided by Dieks’ example of a three-dimensional colour solid:
Different colours and their shades can be represented in various ways; one way is as points on a 3-dimensional colour solid. But the proposal to regard this ‘colour space’ as something substantive, needed to ground the concept of colour, would be absurd.