Philosophy of Mathematics x ‘Substance Versus Body Metaphysic’. Note Quote.


An obvious initial dilemma that faces the prospective Structural Realist (SR) theorist is the ontological status of the spacetime structures themselves. Do they exist as a sort of Platonic universal, independent of all physical objects or events in spacetime, or are they dependent on matter/events for their very existence or instantiation? This problem arises for the mathematician in an analogous fashion, since they also need to explicate the origins of mathematical structures (e.g., set theory, arithmetic). Consequently, a critic of the SR spacetime project might seem justified in regarding this dispute in the philosophy of mathematics as a re-emergence of the traditional substantivalist versus relationist problem, for the foundation of all mathematical structures, including the geometric spacetime structures, is once again either independent of, or dependent on, the physical.

Nevertheless, a survey of the various positions in the mathematical ontology dispute may work to the advantage of the SR spacetime theory, especially when the relevant mathematical and spacetime options are paired together according to their analogous role within the wider ontology debate. First, mathematical structuralism can be classified according to whether the structures are regarded as independent or dependent on their instantiation in systems (ante rem and in re structuralism, respectively), where a “system” is loosely defined as a collection of “objects” and their interrelationships. Ante rem structuralism, as favored by Resnik and Shapiro, is thus closely akin to the traditional “absolute” conception of spacetime, for a structure is held to “exist independent of any systems that exemplify it” (Shapiro). Yet, since “system” (and “object”) must be given a broad reading, without any ontological assumptions associated with the basis of the proposed structure, it would seem that substantivalism would not fit ante rem structuralism, as well. The structure of substantivalism is a structure in a substance, namely, a substance called “spacetime”, such that this unique substance “exemplifies” the structure (whereas ante rem structure exists in the Platonic sense as apart from any and all systems that exemplify it). The substantivalist might try to avoid this implication by declaring that their spacetime structures are actually closer in spirit to a pure absolutism, without need of any underlying entity (substance) to house the structures (hence, “substantivalism” is simply an unfortunate label). While this tactic may be more plausible for interpreting Newtonian spacetimes, it is not very convincing in the context of GTR, especially for the sophisticated substantivalist theories. Given the reciprocal relationship between the metric and matter fields, it becomes quite mysterious how an non-substantival, “absolute” structure, g, can be effected by, and effect in turn, the matter field, T. For the ante rem structuralist, mathematical structures do not enter into these sorts of quasi-causal interrelationships with physical things; rather, things “exemplify” structures (see also endnote 8). Accordingly, one of the initial advantages of examining spacetime structures from within a mathematical ontology context is that it drives a much needed wedge between an absolutism about quantitative structure and the metaphysics of substantivalism, although the two are typically, and mistakenly, treated as identical.

In fact, as judged against the backdrop of the ontology debate in the philosophy of mathematics, the mathematical structures contained in all spacetime theories would seem to fall within a nominalist classification. If, as the nominalists insist, mathematical structures are grounded on the prior existence of some sort of “entity”, then both the substantivalists and relationists would appear to sanction mathematical nominalism (with in re structuralism included among nominalist theories, as argued below): whether that entity is conceived as a unique non-material substance (substantivalism), physical field (metric-field relationism), or actual physical objects/events (relationism, of either the modal (R2) or strict (R1) type), a nominalist reading of mathematical structure is upheld. This outcome may seem surprising, but given the fact that traditional substantivalist and relationist theories have always based spatiotemporal structure on a pre-existing or co-existing ontology – either on a substance (substantivalism) or physical bodies (relationism) – a nominalist reading of spacetime structure has been implicitly sanctioned by both theories. Consequently, if both substantivalism and relationism fall under the same nominalist category in the philosophy of mathematics, then the deeper mathematical Platonist/nominalist issue does not give rise to a corresponding lower-level substantival/relational dichotomy as regards the basis of those spacetime structures (e.g., with substantivalism favoring a Platonic realism about mathematical structures, and relationism siding with a nominalist anti-realism). This verdict could change, of course, if a non-substantival “absolute” conception of spacetime becomes popular in GTR; but this seems unlikely, as argued above.

As there are a number nominalist reconstructions of mathematics, a closer examination of their content reveals that the different versions can be paired to different substantival and relational theories. For instance, a reductive (R1) spacetime relationism can be linked to some strict nominalist reconstructions of mathematics, as in Field’s attempt to treat mathematical objects and structures as entirely dispensable, or “fictional”. Field posits a continuum of spacetime points, conceived physically in the manner of a manifold substantivalist, in his effort to rewrite Newtonian gravitation theory along mathematically anti-realist lines. Modal (R2) relationists, like Teller, would not constitute the spacetime analogue of Field’s program, accordingly, since this form of relationism sanctions modal spacetime structures that can transcend the structures exhibited by the actually existing physical objects: e.g., the affine structure ∇ instantiated by a lone rotating body. Whereas Field requires an infinity of physical spacetime points (isomorphic to ℜ4 in order to capture the full content of the mathematician’s real numbers, the (R2) relationist can allow modal structures to serve this function, thus releasing the ontology of such extravagant demands. More importantly, if Field’s nominalist program is committed to manifold substantivalism, M then it is susceptible to the hole argument. All spacetime theories that utilize the metric in the identity of spacetime points, such as sophisticated substantivalism and metric-field relationism, would thereby incorporate a divergent set of structural scenarios (since they are not susceptible to the hole argument). Therefore, Field’s nominalist mathematics entails a spacetime structure that comprises a different SR theory than sophisticated substantivalism and metric-field relationism.

The mathematical equivalent of both sophisticated substantivalism and (R2) relationism is, rather, any of the less stringent nominalist theories that rejects Field’s strict nominalism, as in, e.g., Chihara in re structuralism. Much like the modal (R2) relationist theories surveyed above, the “minimal nominalists” do not allow structures to exist independently of the systems they exemplify, yet they do not believe that these structures can be dismissed as mere fictions, either. Contra Field, the minimal mathematical nominalists deny a purely instrumentalist construal of mathematical structures (while simultaneously rejecting a Platonic absolutism): they all insist, for instance, that mathematical structures cannot be excised from scientific theories without loss of valuable physical content. In re structuralism, moreover, employs “possible structures” as a means of avoiding a commitment to an infinite background ontology, a feature that helps to explain its common nominalist classification. The minimal nominalist theories often differ on how the mathematical structures are constructed from their basic ontology, as well as how to construe the truth-values of mathematical statements about mathematical structures. But these differences, such as in the kind of modality sanctioned, etc., can be quite subtle, and do not lend themselves to any drastic distinctions in type or basic intent. Hence, any attempt to rekindle the substantival/relational distinction among the competing minimal nominalist theories would seem implausible. A firm reliance on some form of modality and a non-instrumental construal of mathematical structures is the common, and crucial, similarity among these theories; and it is these aspects that are most important for the spacetime structuralist, whether of the sophisticated substantivalist or relationist variety.

These last observations are not meant to downplay the importance of the ongoing research in the philosophy of mathematics on the origin of structures, for it is always possible that substantial problems will arise for some of the minimal nominalist theories, thus eliminating them from contention. From the SR standpoint, in fact, the philosophy of mathematics would likely be considered a more proper arena for assessing the structures employed by spacetime theories, at least as opposed to the apparently unverifiable metaphysics of “substance versus body”. Not only has the traditional spacetime dichotomy failed to explain how these mathematical structures arise from their basic ontology, but, as we have seen, the underlying structures advocated by the sophisticated versions of both substantivalism and relationism are identical when judged within the wider philosophy of mathematics framework. Whether that foundational entity is called a substance or a physical object is irrelevant, and probably a conventional stipulation, since the real work, as judged from the mathematical perspective, concerns how the structures are constructed from the underlying entity—and the competing claims of substance or physical existent do not effect this mathematical construction. In essence, the only apparent difference between the sophisticated substantivalists and (R2) relationists are where those mathematical structures are located: either internal to the substance or field (for the substantivalists and metric-field relationists, respectively), or external to bodies/events (for non-field formulations of (R2) relationism, such as Teller’s). Needless to say, this internal/external distinction does not provide any information on how the mathematical structures are built-up; rather, it reveals the pervasive influence of the age-old substance/property dichotomy within the philosophy of science community, an unfortunate legacy that the SR theorist regards as hindering the advancement of the debate on spacetime theories.

Finally, since the competing minimal nominalist constructions are not being judged solely from a mathematical perspective, but from a scientific and empirical standpoint as well, a few words are in order on the relevance of empirical evidence in assessing spacetime structure. This issue will be addressed further in the remaining sections, but, in brief, it is unlikely that any physical evidence could provide a strong confirmation of any one of the competing nominalist theories. In effect, these nominalist constructions are only being utilized to explain the origins of the spacetime structures, such as M or g, that do appear in our best physical theories, with the important qualification that these nominalist constructions do not commit the physical theory to any problematic or meaningless physical outcomes (e.g., Field’s nominalism and the hole argument, as noted above). As for the sophisticated brands of both substantivalism and relationism, all of the minimal nominalist construction are apparently identical as regards their implications for possible spacetime scenarios and meaningful physical states. Unless other reasons are brought forward, the choice among the competing minimal nominalist constructions could thus be viewed as conventional, since it is difficult to conceive how empirical evidence could reach the deep mathematical levels where the differences in nominalist constructions of spacetime structures come into play.



Economics is the Science which Studies Human Behaviour as a Relationship Between Ends and Scarce Means which have Alternative Uses. Is Equilibrium a Choice? Note Quote.

What is the place of choice in equilibrium theory? Alfred Marshall and Leon Walras, who introduced competitive equilibrium theory, employed the theory of choice in terms of utility, analogously to the Austrian school. Enrico Barone and Karl Gustav Cassel (the latter introducing general equilibrium theory in the German speaking world. Walras-Cassel System) used demand and supply functions as starting data, disregarding the theory of choice. Pareto, on the one hand, argued that the two approaches are compatible. However, he discarded cardinal utility introducing the notion of preferences, i.e. ordinal utility, as sufficient foundations for the theory of choice, thus starting the modern analysis of choice. Pareto also suggested that these data can be derived directly from choices, so short-cutting the theory of choice (since choices are not to be explained) and anticipating the theory of revealed preferences. This theory is, perhaps, the point of maximal distance between equilibrium theory and the Austrian school. On the other hand, Pareto’s theory of economic efficiency, or Pareto-optimality, and all analysis connected with it (such as, for example, the theory of the core of an economy) requires at least individual preferences, an element which underlies choices and helps to explain them.

What was presented above is the present state of competitive equilibrium theory. Demand and supply functions are sufficient for determining prices and equilibrium allocations. These functions represent choices. In other words, theory of choice is not necessarily an integral element of competitive equilibrium theory, only a prerequisite. However, individual preferences and the theory of choice are required in order to define Pareto-optimal allocations and demonstrate the two theorems of welfare economics. Competitive equilibrium studies the compatibility of price-taking agents’ choices. Thus, it concerns choices without representing a theory about them. Nevertheless, such a theory is required if statements on Pareto-optimality and other relevant characteristics of competitive equilibrium are to be made.

In similar terms, the theory of choice is required by non-competitive equilibrium theory as well. For instance, game theory deeply analyzes strategic choices and in every non-competitive market equilibrium price-making agents’ choices have to be analyzed to a certain extent. However, this analysis differs from that given by the Austrian school. The difference lies in the aim of the two approaches. While the Austrian school is interested mainly in individual choices and their implications in as much as according to the famous Robbins’s definition, “economics is the science which studies human behaviour as a relationship between ends and scarce means which have alternative uses”, equilibrium theory, including game theory, is interested mainly in the compatibility of choices. That is, equilibrium theory is not so much a theory of intentional actions as a the theory of intentional interactions. The two approaches overlap but do not coincide, even if they share the same vision of society and main assumptions about the behavior of its components.

Maffeo Pantaleoni  did not accept Pareto’s new theory of choice. He continued to follow the Jevons-Menger-Walras hedonistic approach to utility and he identified economic theory with the theory of subjective value. Probably, there was a courious change of position beteween Pantaleoni and Pareto about the political significance of economic theory. On the hand, Pareto was initially reluctant to accept Walras’s economic theory (introduced to him by Pantaleoni) because he was too liberal for sharing Walras’s socialism. In fact, he accepted Walras’s economic theory, not at all Walras’s political and philosophical view. On the other hand, Pantaleoni seems to have refuted Pareto’s new theory also because of its focus on equilibrium instead of individual choice. This would limit the liberal doctrine he envisaged strictly connected to economics as the theory of the individual choice.

For instance, let us take into consideration the theory of non-cooperative games. Its focus is on strategic interdependence, i.e. those situations in which each agent chooses their action knowing the result also depends on the actions of other individuals and that those actions as well generally depend on theirs. Individual action (for better clarity, plan of actions, or strategy) is not simply determined selecting the option that maximizes one’s utility from the set of actions available to each agent. The agent under consideration knows that other agents actions could prevent him from performing their optimal action and reaching the desired result. Individual equilibrium actions are determined simultaneously, i.e. we can generally determine the choice of an individual only by determining also the choices of all other individuals. Both the Austrian school and (competitive) equilibrium theory isolate the individual agent’s choice. However, while the Austrian school does not analyze the compatibility of the actions chosen by individuals (compatibility is presumed), competitive equilibrium theory analyzes interactions, although those among price-taking agents are rather weak. Interaction is predominant in strategic situations, where choices cannot be analyzed without taking interdependence explicitly into account.

Geometric Structure, Causation, and Instrumental Rip-Offs, or, How Does a Physicist Read Off the Physical Ontology From the Mathematical Apparatus?


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.