Diffeomorphism Invariance: General Relativity Spacetime Points Cannot Possess Haecceity.

Eliminative or radical ontic structural realism (ROSR) offers a radical cure—appropriate given its name—to what it perceives to be the ailing of traditional, object-based realist interpretations of fundamental theories in physics: rid their ontologies entirely of objects. The world does not, according to this view, consist of fundamental objects, which may or may not be individuals with a well-defined intrinsic identity, but instead of physical structures that are purely relational in the sense of networks of ‘free-standing’ physical relations without relata.

Advocates of ROSR have taken at least three distinct issues in fundamental physics to support their case. The quantum statistical features of an ensemble of elementary quantum particles of the same kind as well as the features of entangled elementary quantum (field) systems as illustrated in the violation of Bell-type inequalities challenge the standard understanding of the identity and individuality of fundamental physical objects: considered on their own, an elementary quantum particle part of the above mentioned ensemble or an entangled elementary quantum system (that is, an elementary quantum system standing in a quantum entanglement relation) cannot be said to satisfy genuine and empirically meaningful identity conditions. Thirdly, it has been argued that one of the consequences of the diffeomorphism invariance and background independence found in general relativity (GTR) is that spacetime points should not be considered as traditional objects possessing some haecceity, i.e. some identity on their own.

The trouble with ROSR is that its main assertion appears squarely incoherent: insofar as relations can be exemplified, they can only be exemplified by some relata. Given this conceptual dependence of relations upon relata, any contention that relations can exist floating freely from some objects that stand in those relations seems incoherent. If we accept an ontological commitment e.g. to universals, we may well be able to affirm that relations exist independently of relata – as abstracta in a Platonic heaven. The trouble is that ROSR is supposed to be a form of scientific realism, and as such committed to asserting that at least certain elements of the relevant theories of fundamental physics faithfully capture elements of physical reality. Thus, a defender of ROSR must claim that, fundamentally, relations-sans-relata are exemplified in the physical world, and that contravenes both the intuitive and the usual technical conceptualization of relations.

The usual extensional understanding of n-ary relations just equates them with subsets of the n-fold Cartesian product of the set of elementary objects assumed to figure in the relevant ontology over which the relation is defined. This extensional, ultimately set-theoretic, conceptualization of relations pervades philosophy and operates in the background of fundamental physical theories as they are usually formulated, as well as their philosophical appraisal in the structuralist literature. The charge then is that the fundamental physical structures that are represented in the fundamental physical theories are just not of the ‘object-free’ type suggested by ROSR.

While ROSR should not be held to the conceptual standards dictated by the metaphysical prejudices it denies, giving up the set-theoretical framework and the ineliminable reference to objects and relata attending its characterizations of relations and structure requires an alternative conceptualization of these notions so central to the position. This alternative conceptualization remains necessary even in the light of ‘metaphysics first’ complaints, which insist that ROSR’s problem must be confronted, first and foremost, at the metaphysical level, and that the question of how to represent structure in our language and in our theories only arises in the wake of a coherent metaphysical solution. But the radical may do as much metaphysics as she likes, articulate her theory and her realist commitments she must, and in order to do that, a coherent conceptualization of what it is to have free-floating relations exemplified in the physical world is necessary.

ROSR thus confronts a dilemma: either soften to a more moderate structural realist position or else develop the requisite alternative conceptualizations of relations and of structures and apply them to fundamental physical theories. A number of structural realists have grabbed the first leg and proposed less radical and non-eliminative versions of ontic structural realism (OSR). These moderate cousins of ROSR aim to take seriously the difficulties of the traditional metaphysics of objects for understanding fundamental physics while avoiding these major objections against ROSR by keeping some thin notion of object. The picture typically offered is that of a balance between relations and their relata, coupled to an insistence that these relata do not possess their identity intrinsically, but only by virtue of occupying a relational position in a structural complex. Because it strikes this ontological balance, we term this moderate version of OSR ‘balanced ontic structural realism’ (BOSR).

But holding their ground may reward the ROSRer with certain advantages over its moderate competitors. First, were the complete elimination of relata to succeed, then structural realism would not confront any of the known headaches concerning the identity of these objects or, relatedly, the status of the Principle of the Identity of Indiscernibles. To be sure, this embarrassment can arguably be avoided by other moves; but eliminating objects altogether simply obliterates any concerns whether two objects are one and the same. Secondly, and speculatively, alternative formulations of our fundamental physical theories may shed light on a path toward a quantum theory of gravity.

For these presumed advantages to come to bear, however, the possibility of a precise formulation of the notion of ‘free-standing’ (or ‘object-free’) structure, in the sense of a network of relations without relata (without objects) must thus be achieved.  Jonathan Bain has argued that category theory provides the appropriate mathematical framework for ROSR, allowing for an ‘object-free’ notion of relation, and hence of structure. This argument can only succeed, however, if the category-theoretical formulation of (some of the) fundamental physical theories has some physical salience that the set-theoretical formulation lacks, or proves to be preferable qua formulation of a physical theory in some other way.

F. A. Muller has argued that neither set theory nor category theory provide the tools necessary to clarify the “Central Claim” of structural realism that the world, or parts of the world, have or are some structure. The main reason for this arises from the failure of reference in the contexts of both set theory and category theory, at least if some minimal realist constraints are imposed on how reference can function. Consequently, Muller argues that an appropriately realist stucturalist is better served by fixing the concept of structure by axiomatization rather than by (set-theoretical or category-theoretical) definition.

Advertisement

Electronic Money versus Paper Currency: The Case of Central Banks’ Monetary Policies. Note Quote + ‘Adumbration’

If electronic money is issued through the conversion of banknotes or sight deposits, it does not change the money supply and price stability is not endangered. However, if electronic money is issued as a consequence of credit, private issuers have incentives to supply additional amounts of electronic money as long as the difference between the interest charged on the credit and the one paid on electronic money covers the credit risk premium, the provision of the payment service, and possibly also the cost of refinancing. Given the low marginal cost of producing electronic money, its issuance could in principle proceed until the interest rate charged on the credit extended for the provision of electronic money is equal to the credit risk premium. This, by lowering the level of interest rates, could in turn endanger the maintenance of price stability.

The risk of overissue would be limited by two factors which increase the costs of issuing electronic money, thereby limiting its supply: first, in a competitive environment, electronic money balances could be remunerated; second, and more importantly, a redeemability requirement could oblige the issuer to possess central bank money. An even stronger measure, which could be considered in the light of future developments in electronic money, would be to introduce a coverage requirement on electronic money, i.e. to request issuers of electronic money to cover part or all of their liabilities with base money. Another way to limit the risk of overissue would be to require rapid clearing of electronic money balances in central bank money.

Thus, it appears that there are several reasons to assume that the risk of overissue of electronic money can be contained. However the issuance of electronic money may have an impact on the conduct of monetary policy.

The question of overissue of electronic money is also related to the question of whether electronic money could endanger the unit-of-account role as incorporated in central bank money. If, in the absence of any regulation, certain electronic money products were to spread at a rapid rate, market views about the creditworthiness of issuers could be affected and electronic money products from different issuers could start to be traded at varying exchange rates. Therefore, the need to preserve the unit-of-account function of money is another argument for imposing a redeemability requirement on electronic money. Such a requirement would guarantee that the role of money in providing a common financial denominator for the whole economy will be maintained.

But, what are the challenges as far as central banks monetary policies are concerned? The danger from the high degree of substitute of currency in circulation with e-money is presented through the influence of decreasing central bank balance sheet, which means limitation of their positive influence on the monetary instruments. In order to avoid this negative influence in the near future, there is another possibility taken into consideration and that is the possibility for the central bank to impose reserve requirements on all issuers of electronic money.

The impact of the emergence of electronic money on monetary policy can mostly be expected in the following areas:

a)  Decrease in the control of central bank over money supply. Decreasing the central bank’s control of money supply depends on the degree of substitution of currency in circulation with e-money. The currency in circulation is part of monetary aggregates, and if it is decreased as a result of wider use of e-money, it will produce difficulties in measurement of monetary aggregates and of the control of money supply by the central bank. Possible solutions for limitations of this effect are for example, limitations of the use of e-money. But this will be in direct confrontation with the laws of technological progress and could produce negative external effects on the banking in general. Because of that, there is a need for intensified research on new opportunities to limit the adverse impacts of replacing cash with electronic money.

b)  Increase in the velocity of money. The influence of e-money over monetary policy can be seen through monetary aggregates, the ability of the central bank to control money supply. In the future impact should be seen through indicators related to monetary aggregates, like the velocity of money. With the use of e-money, transactions are relatively cheaper which allows increase in the number of transactions, and increase in the speed of money. Generally speaking, it will be useful, but only to the level that the central bank could control or measure the monetary aggregates.

c) Volatility in exchange rates. The change of the monetary multiplier is an important indicator. This indicator shows the share of currency in the money supply. As a result of e-money the currency decreases producing effects to multiplier.

d)  With the use of currency, the need for printing cash is decreasing, which influences the revenues of central banks.

e)  E-money has a characteristic of easy portability and affordability which offers a wide use in trade among countries. It is assumed that the user of e-money, motivated with the cheaper foreign currency transactions, will prefer the transactions to be in the most powerful currency. So, through PayPal and other services, users from the country with weaker currency will prefer to transfer their money in higher currency. So in this way the dollarization or euroisation will be a subject to clicking the mouse. But still, this situation can weaken the central bank’s control in the process of foreign currency exchange among the countries.

Still, all these analyses and facts about the influence of e-money on the monetary policy of the central bank will be part of future scenarios. Today, with the current use of e-money there is no room to worry on the part of central banks. Still, there is no broad willingness to accept e-money because of the expensive implementation, insuf ciently developed supervision, need for broader statistics and also the danger from hackers. In the future, we could expect decrease of the currency, but it will be without larger effects or without reasons for any dramatic changes in the monetary policy.

Ontological-Objects Categoric-Theoretic Physics. A Case of Dagger Functor. Note Quote.

Jonathan Bain’s examples in support of the second strategy are:

(i) the category Hilb of complex Hilbert spaces and linear maps; and

(ii) the category nCob, which has (n−1)-dimensional oriented closed manifolds as objects, and n-dimensional oriented manifolds as morphisms.

These examples purportedly represent ‘purely’ category-theoretic physics. This means that formal statements about the physical theory, e.g. quantum mechanics using Hilb, are derived using the category-theoretic rules of morphisms in Hilb.

Now, prima facie, both of these examples look like good candidates for doing purely category-theoretic physics. First, each category is potentially useful for studying the properties of quantum theory and general relativity respectively. Second, each possesses categorical properties which are promising for describing physical properties. More ambitiously, they suggest that one could use categorical tools to develop an approach for integrating parts of quantum theory and general relativity.

Let us pause to explain this second point, which rests on the fact that, qua categories, Hilb and nCob share some important properties. For example, both of these categories are monoidal, meaning that both categories carry a generalisation of the tensor product V ⊗ W of vector spaces V and W. In nCob the monoidal structure is given by the disjoint union of manifolds; whereas in Hilb, the monoidal structure is given by the usual linear-algebraic tensor product of Hilbert spaces.

A second formal property shared by both categories is that they each possess a contravariant involutive endofunctor (·)^†called the dagger functor. Recall that a contravariant functor is a functor F : C → D that reverses the direction of arrows, i.e. a morphism f : A → B is mapped to a morphism F (f ) : F (B) → F (A). Also recall that an endofunctor on a category C is a functor F : C → C, i.e. the domain and codomain of F are equal. This means that, given a cobordism f: A → B in nCob or a linear map L: A → B in Hilb, there exists a cobordism f^†: B → A and a linear adjoint L^† : B → A respectively, satisfying the involution laws f^† ◦ f = 1_A and f ◦ f^† = 1_B, and identically for L.

The formal analogy between Hilb and nCob has led to the definition of a type of quantum field theory, known as a topological quantum field theory (TQFT), first introduced by Atiyah and Witten. A TQFT is a (symmetric monoidal) functor:

T : nCob → Hilb,

and the conditions placed on this functor, e.g. that it preserve monoidal structure, reflect that its domain and target categories share formal categorical properties. To further flesh out the physical interpretation of TQFTs, we note that the justification for the term ‘quantum field theory’ arises from the fact that a TQFT assigns a state space (i.e. a Hilbert space) to each closed manifold in nCob, and it assigns a linear map representing evolution to each cobordism. This can be thought of as assigning an amplitude to each cobordism, and hence we obtain something like a quantum field theory.

Recall that the significance of these examples for Bain is their apparent status as purely category-theoretic formulations of physics which, in virtue of their generality, do not make any reference to O-objects (represented in the standard way, i.e. as elements of sets). We now turn to a criticism of this claim.

Bain’s key idea seems to be that this ‘generality’ consists of the fact that nCob and Hilb (and thus TQFTs) have very different properties (qua categories) from Set. In fact, he claims that three such differences count in favor of (Objectless):

(i) nCob and Hilb are non-concrete categories, but Set (and other categories based on it) are concrete.

(ii) nCob and Hilb are monoidal categories, but Set is not.

(iii) nCob and Hilb have a dagger functor, but Set does not.

We address these points and their implications for (Objectless) in turn. First, (i). Bain wants to argue that since nCob and Hilb ‘cannot be considered categories of structured sets’, nor can these categories be interpreted as having O-objects. If one is talking about categorical properties, this claim is best couched in the standard terminology as the claim that these are not concrete categories. But this inference is faulty for two reasons. First, his point about non-concreteness is not altogether accurate, i.e. point (i) is false as stated. On the one hand, it is true that nCob is not a concrete category: in particular, while the objects of nCob are structured sets, its morphisms are not functions, but manifolds, i.e. sets equipped with the structure of a manifold. But on the other hand, Hilb is certainly a concrete category, since the objects are Hilbert spaces, which are sets with extra conditions; and the morphisms are just functions with linearity conditions. In other words, the morphisms are structure-preserving functions. Thus, Bain’s examples of category-theoretic physics are based in part on concrete categories. Second and more importantly, it is doubtful that the standard mathematical notion of concreteness will aid Bain in defending (Objectless). Bain wants to hold that the non-concreteness of a category is a sufficient condition for its not referring to O-objects. But nCob is an example of a non-concrete category that apparently contain O-objects—indeed the same O-objects (viz. space-time points) that Bain takes to be present in geometric models of GTR. We thus see that, by Bain’s own lights, non-concreteness cannot be a sufficient condition of evading O-objects.

So the example of nCob still has C-objects that are based on sets, albeit morphisms which are more general than functions. However, one can go further than this: the notion of a category is in fact defined in a schematic way, which leaves open the question of whether C-objects are sets or whether functions are morphisms. One might thus rhetorically ask whether this could this be the full version of ‘categorical generality’ that Bain needs in order to defend (Objectless). In fact, this is implausible, because of the way in which such a schematic generality ends up being deployed in physics.

On to (ii): unfortunately, this claim is straightforwardly false: the category Set is certainly monoidal, with the monoidal product being given by the cartesian product.

Finally, (iii). While it is true that Set does not have a dagger functor, and nCob and Hilb do, it is easy to construct an example of a category with a dagger functor, but which Bain would presumably agree has O-objects. Consider the category C with one object, namely a manifold M representing a relativistic spacetime; the morphisms of C are taken to be the automorphisms of M. As with nCob, this category has natural candidates for O-objects (as Bain assumes), viz. the points of the manifold. But the category C also has a dagger functor: given an automorphism f : M → M, the morphism f^† : M → M is given by the inverse automorphism f⁻¹. In contrast, the category Set does not have a dagger functor: this follows from the observation that for any set A that is not the singleton set {∗}, there is a unique morphism f : A → {∗}, but the number of morphisms g : {∗} → A is just the cardinality |A| > 1. Hence there does not exist a bijection between the set of morphisms {f : A → {∗}} and the set of morphisms {g : {∗} → A}, which implies that there does not exist a dagger functor on Set. Thus, by Bain’s own criterion, it is reasonable to consider C to be structurally dissimilar to Set, despite the fact that it has O-objects.

More generally, i.e. putting aside the issue of (Objectless), it is quite unclear how one should interpret the physical significance of the fact that nCob/Hilb, but not Set has a dagger functor. For instance, it turns out that by an easy extension of Set, one can construct a category that does have a dagger functor. This easy extension is the category Rel, whose objects are sets and whose morphisms are relations between objects (i.e. subsets of the Cartesian product of a pair of objects). Note first that Set is a subcategory of Rel because Set and Rel have same objects, and every morphism in Set is a morphism in Rel. This can be seen by noting that every function f : A → B can be written as a relation f ⊆ A × B, consisting of the pairs (a, b) defined by f(a) = b. Second, note that – unlike Set – Rel does have a non-trivial involution endofunctor, i.e. a dagger functor, since given a relation R : A → B, the relation

R^† :B→A is just defined by bR^†a if and only if aRb.

By extending Set to Rel, we have obtained a category that is akin to nCob/Hilb in having a dagger functor, but which also appears to be at least as good a candidate as Set for codifying the standard notion of physical structure, since its morphisms are n-ary relations and they can be used to encode structure. Evidently, if Bain’s suggestion about the physical difference between these categorical properties is to amount to anything, then one must seek the right physical interpretation of such categories.

We have seen that (i)–(iii) will not help Bain defend (Objectless). But what is the notion of generality that lies behind these three points? Although Bain does not state it explicitly, we can extract it from the details of his argument and summarize it as follows:

Category theory offers us the resources to predicate properties (‘categori- cal properties’) of a category of C-objects without further specifying the properties of the C-objects, or indeed what types of objects these are (e.g. whether they are groups, manifolds, lattices, and so on). And such predications/properties are general in the sense that they are instanced by categories of specific C-objects, which provide instances of such categorical properties.

Eliminating Implicit Reference to Elements: Via Einsteinian Algebra. (3)

Previously, we highlighted the inadequacy of implicitly quantifying over elements and it is to circumvent, or circumnavigate this point that Jonathan Bain introduced his specific argument, to which we now turn here.

G3 above yields a special translation scheme that allows one to avoid making explicit reference to elements. The key insight driving the specific argument is that, if one looks at a narrower range of cases, a rather different sort of translation scheme is possible: indeed one that not only avoids making explicit reference to elements, but also allows one to generalize the C-objects in such a way that these new C-objects can no longer be considered to have elements (or as many elements) in the sense of the original objects. According to Bain, this shows that the

‘…correlates [of elements of structured sets] are not essential to the articulation of the relevant structure.’

Implicit reference to elements is thereby claimed to be eliminated. Bain argues by appealing to a particular instance of how this translation is supposed to work, viz. the example of Einstein algebras. 

The starting point for Bain’s specific argument is a category-theoretic version of the ‘semantic view of theories’ on which a scientific theory is identified with its category of models—indeed this will be the setup assumed in the following argument. Note two points. First, we are not using ‘model’ here in the strict sense of model theory, but rather to mean a mathematical structure that represents a physical world that is possible according to the theory. Second, this proposal is not to be confused with Lawvere’s category-theoretic formulation of algebraic theories (Lawvere). In the latter, models are functors between categories, whereas in the former, the models are just objects of some category (not necessarily a functor category) such as Top. Indeed, Lawvere’s proposal is much more closely related to—though not the same as—the Topological Quantum Field Theories. 

Here is our abstract reconstruction of Bain’s specific argument. Let there be two theories T1 and T2, each represented by a category of models respectively. T1 is the original physical theory that makes reference to O-objects.

S1: T1 can be translated into T2. In particular, each T1-model can be translated into a T2 model and vice versa.

S2: T2 is contained in a strictly larger theory (i.e. a larger category of models) T2∗. In particular, T2∗ is constructed by generalizing T2-models to yield models of T2∗, typically by dropping an algebraic condition from the T2-models. We will use T2′ to denote the complement of T2 in T2∗.

S3: T2′ cannot be translated back into T1 and so its models do not contain T1 -objects.

S4: T2′ is relevant for modeling some physical scenarios.

When taken together, S1-S4 are supposed to show that:

CS: The T1-object correlates in T2 do not play an essential role in articulating the physical structure (smooth structure, in Bain’s specific case) of T2∗ .

Let us defer for the moment the question of exactly how the idea of ‘translation’ is supposed to work here. The key idea behind S1–S4 is that one can generalize T2 to obtain a new – more general – theory T2∗, some of whose models do not contain T1-objects (i.e. O-objects in T1).

In Bain’s example, T1 is the category of geometric models of general relativity (GTR), and T2 is the category of Einstein algebra (EA) models of GTR (Einstein algebras were first introduced as models of GR in Geroch 1972). Bain is working with the idea that in geometric models of GTR, the relata, or O-objects, of GTR are space-time points of the manifold.

We now discuss the premises of the argument and show that S3 rests on a technical misunderstanding; however, we will rehabilitate S3 before proceeding to argue that the argument fails. First, S1: Bain notes that these space-time points are in 1-1 correspon- dence with ‘maximal ideals’ (an algebraic feature) in the corresponding EA model. We are thus provided with a translation scheme: points of space in a geometric description of GTR are translated into maximal ideals in an algebraic description of GTR. So the idea is that EA models capture the physical content of GR without making explicit reference to points. Now the version of S2 that Bain uses is one in which T2, the category of EAs, gets generalized to T2∗, the category of sheaves of EAs over a manifold, which has a generalized notion of ‘smooth structure’. The former is a proper subcategory of the latter, because a sheaf of EAs over a point is just equivalent to an EA.

Bain then tries to obtain S3 by saying that a sheaf of EAs which is inequivalent to an EA does not necessarily have global elements (i.e. sections of a sheaf) in the sense previously defined, and so does not have points. Unfortunately, he confuses the notion of a local section of a sheaf of EAs (which assigns an element of an EA to an open subset of a manifold) with the notion of a maximal ideal of an EA (i.e. the algebraic correlate of a spacetime point). And since the two are entirely different, a lack of global sections does not imply a lack of spacetime points (i.e. O-objects). Therefore S3 needs to be repaired.

Nonetheless, we can easily rehabilitate S3 is the following manner. The key idea is that while T1 (a geometric model of GTR) and T2 (the equivalent EA model) both make reference to T1-objects (explicitly and implicitly, respectively), some sheaves of EAs do not refer to T1-objects because they have no formulation in terms of geometric models of GTR. In other words, the generalized smooth structure of T2′ cannot be described in terms of the structured sets used to define ordinary smooth structure in the case of T1 and T2.

Finally, as regards S4, various authors have taken the utility of T2′ to be e.g. the in- clusion of singularities in space-time, and as a step towards formulating quantum gravity (Geroch).

We now turn to considering the inference to CS. It is not entirely clear what Bain means by ‘[the relata] do not play an essential role’ – nor does he expand on this phrase – but the most straightforward reading is that T1-objects are eliminated simpliciter from T2∗.

One might compare this situation to the way that the collection of all groups (analogous to T2) is contained in the collection of all monoids (analogous to T2∗): it might be claimed that inverses are eliminated from the collection of all monoids. One could of course speak in this way, but what this would mean is that some monoids (in particular, groups) have inverses, and some do not – a ‘monoid’ is just a general term that covers both cases. Similarly, we can see that CS does not follow from S1–S3, since T2∗ contains some models that (implicitly) quantify over T1-objects, viz. the models of T2, and some that do not, viz. the models of T2′.

We have seen that the specific argument will not work if one is concerned with eliminating reference to T1-objects from the new and more general theory T2∗. However, what if one is concerned not with eliminating reference, but rather with downgrading the role that T1-objects play in T2∗, e.g. by claiming that the models of T2′ have a conceptual or metaphysical priority? And what would such a ‘downgrading’ even amount to?

Object as Category-Theoretic or Object as Ontological: The Inadequacy of Implicitly Quantifying Over Elements. (2)

It will be convenient to use the term ‘object’ in two senses. First, as an object of a category, i.e. in a purely mathematical sense. We shall call this a C- object (‘C’ for category-theoretic). Second, in the sense commonly used in structural realist debates, and which was already introduced above, viz. an object is a physical entity which is a relatum in physical relations. We shall call this an O-object (‘O’ for ‘ontological’).

We will also need to clarify our use of the term ‘element’. We use ‘element’ to mean an element of a set, or as it is also often called, a ‘point’ of a set (indeed it will be natural for us to switch to the language of points when discussing manifolds, i.e. spacetimes.) This familiar use of element should be distinguished from the category-theoretic concepts of ‘global element’ and ‘generalized element’, which is introduced below.

Jonathan Bain’s first strategy for defending (Objectless) draws on the following idea: the usual set-theoretic representations of O-objects and relations can be translated into category-theoretic terms, whence these objects can be eliminated. In fact, the argument can be seen as consisting of two different parts.

In the first part, Bain attempts to give a highly general argument, in the sense that it turns only on the notion of universal properties and the translatability of statements about certain mathematical representations (i.e. elements of sets) of O-objects into statements about morphisms between C-objects. As Bain himself notes, the general argument fails, and he thus introduces a more specific argument, which is what he wishes to endorse. The specific argument turns on the idea of obtaining a translation scheme from a ‘categorical equivalence’ between a geometric category and an algebraic category, which in turn allows one to generalize the original C-objects. The argument is ‘specific’ because such equivalences only hold between rather special sorts of categories.

The details of Bain’s general argument can be reconstructed as follows:

G1: Physical objects and the structures they bear are typically identified with the elements of a set X and relations on X respectively.

G2: The set-theoretic entities of G1 are to be represented in category-theoretic language by considering the category whose objects are the relevant structured sets, and whose morphisms are functions that preserve ‘structure’.

G3: Set-theoretic statements about an object of a category (of the type in G2) can often be expressed without making reference to the elements of that object. For instance:

1. In any category with a terminal object any element of an object X can be expressed as a morphism from the terminal object to X. (So for instance, since the singleton {∗} is the terminal object in the category Set, an element of a set X can be described by a morphism {∗} → X.)

2. In a category with some universal property, this property can be described purely in terms of morphisms, i.e. without making any reference to elements of an object.

To sum up, G1 links O-objects with a standard mathematical representation, viz. elements of a set. And G2 and G3 are meant to establish the possibility that, in certain cases, category theory allows us to translate statements about elements of sets into statements about the structure of morphisms between C-objects.

Thus, Bain takes G1–G3 to suggest that: 

C: Category theory allows for the possibility of coherently describing physical structures without making any reference to physical objects.

Indeed, Bain thinks the argument suggests that the mathematical representatives of O- objects, i.e. the elements of sets, are surplus, and that category theory succeeds in removing this surplus structure. Note that even if there is surplus structure here, it is not of the same kind as, e.g. gauge-equivalent descriptions of fields in Yang-Mills theory. The latter has to do with various equivalent ways in which one can describe the dynamical objects of a theory, viz. field. By contrast, Bain’s strategy involves various equivalent descriptions of the entire theory.

Bain himself thinks that the inference from G1–G3 to C fails, but he does give it serious consideration, and it is easy to see why: its premises based on the most natural and general translation scheme in category theory, viz. redescribing the properties of C-objects in terms of morphisms, and indeed – if one is lucky – in terms of universal properties. 

First, the premise G1. Structural realist doctrines are typically formalized by modeling O-objects as elements of a set and structures as relations on that set. However, this is seldom the result of reasoned deliberation about whether standard set theory is the best expressive resource from some class of such resources, but rather the product of a deeply entrenched set-theoretic viewpoint within philosophy. Were philosophers familiar with an alternative to set theory that was at least as powerful, e.g. category theory, then O-objects and structures might well have been modeled directly in the alternative formalism. Of course, it is also a reasonable viewpoint to say that it is most ‘natural’ to do the philosophy/foundations of physics in terms of set theory – what is ‘natural’ depends on how one conceives of such foundational investigations.

So we maintain that there is no reason for the defender of O-objects to accept G1. For instance, he might try to construct a category such that O-objects are modeled by C-objects and structures are modeled by morphisms. For example, there are examples of categories whose C-objects might coincide with the mathematical representatives of O-objects. For instance, in a path homotopy category, the C-objects are just points of the relevant space, and one might in turn take the points of a space to be O-objects, as Bain does in his example of general relativity and Einstein algebras. Or he might take as his starting point a non-concrete category, whose objects have no underlying set and thus cannot be expressed in the terms of G1.

The premise G2, on the other hand, is ambiguous—it is unclear exactly how Bain wants us to understand ‘structure’ and thus ‘structure-preserving maps’. First, note that when mathematicians talk about ‘structure-preserving maps’ they usually have in mind morphisms that do not preserve all the features of a C-object, but rather the characteristic (albeit partial) features of that C-object. For instance, with respect to a group, a structure-preserving map is a homomorphism and not an isomorphism. Bain’s example of the category Set is of this type, because its morphisms are arbitrary functions (and not bijective functions).

However, Bain wants to introduce a different notion of ‘structure’ that contrasts with this standard usage, for he says:

(Structure) …the intuitions of the ontic structural realist may be preserved by defining “structure” in this context to be “object in a category”.

If we take this claim seriously, then a structure-preserving map will turn out to be an isomorphism in the relevant category – for only isomorphisms preserve the complete ‘structural essence’ of a structured set. For instance, Bain’s example of the category whose objects are smooth manifolds and whose morphisms are diffeomorphisms is of this type. If this is really what Bain has in mind, then one inevitably ends up with a very limited and dull class of categories. But even if one relaxes this notion of ‘structure’ to mean ‘the structure that is preserved by the morphisms of the category, whatever they happen to be’, one still runs into trouble with G3.

We now turn to the premise G3. First, note that G3 (i) is false, as we now explain. It will be convenient to introduce a piece of standard terminology: a morphism from a terminal object to some object X is called a global element of X. And the question of whether an element of X can be expressed as a global element in the relevant category turns on the structure of the category in question. For instance, in the category Man with smooth manifolds as objects and smooth maps as morphisms, this question receives a positive answer: global elements are indeed in bijective correspondence with elements of a manifold. This is because the terminal object is the 0-dimensional manifold {0}, and so an element of a manifold M is a morphism {0} → M. But in many other categories, e.g. the category Grp, the answer is negative. As an example, consider that Grp has the trivial group 1 as its terminal object and so a morphism from 1 to a group G only picks out its identity and not its other elements. In order to obtain the other elements, one has to introduce the notion of a generalized element of X, viz. a morphism from some ‘standard object’ U into X. For instance, in Grp, one takes Z as the standard object U, and the generalized elements Z → G allow us to recover the ordinary elements of a group G.

Second, while G3 (ii) is certainly true, i.e. universal properties can be expressed purely in terms of morphisms, it is a further – and significant – question for the scope and applicability of this premise whether all (or even most) physical properties can be articulated as universal properties.

Hence we have seen that the categorically-informed opponent of (Objectless) need not accept these premises – there is a lot of room for debate about how exactly one should use category theory to conceptualize the notion of physical structure. But supposing that one does: is there a valid inference from G1–G3 to C? Bain himself notes that the plausibility of this inference trades on an ambiguity in what one means by ‘reference’ in C. If one merely means that such constructions eliminate explicit but not implicit reference to objects, then the argument is indeed valid. On the other hand, a defense of OSR requires the elimination of implicit reference to objects, and this is what the general argument fails to offer – it merely provides a translation scheme from statements involving elements (of sets) to statements involving morphisms between C-objects. So, the defender of objects can maintain that one is still implicitly quantifying over elements.