Einstein Algebra and General Theory of Relativity Preserve the Empirical Structure of the Theories

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In general relativity, we represent possible universes using relativistic spacetimes, which are Lorentzian manifolds (M, g), where M is a smooth four dimensional manifold, and g is a smooth Lorentzian metric. An isometry between spacetimes (M,g) and (M,g′) is a smooth map φ : M → M′ such that φ ∗ (g′) = g, where φ∗ is the pullback along φ. We do not require isometries to be diffeomorphisms, so these are not necessarily isomorphisms, i.e., they may not be invertible. Two spacetimes (M,g), (M′,g′) are isomorphic, if there is an invertible isometry between them, i.e., if there exists a diffeomorphism φ : M → M′ that is also an isometry. We then say the spacetimes are isometric.

The use of category theoretic tools to examine relationships between theories is motivated by a simple observation: The class of models of a physical theory often has the structure of a category. In what follows, we will represent general relativity with the category GR, whose objects are relativistic spacetimes (M,g) and whose arrows are isometries between spacetimes.

According to the criterion for theoretical equivalence that we will consider, two theories are equivalent if their categories of models are “isomorphic” in an appropriate sense. In order to describe this sense, we need some basic notions from category theory. Two (covariant) functors F : C → D and G : C → D are naturally isomorphic if there is a family ηc : Fc → Gc of isomorphisms of D indexed by the objects c of C that satisfies ηc ◦ Ff = Gf ◦ ηc for every arrow f : c → c′ in C. The family of maps η is called a natural isomorphism and denoted η : F ⇒ G. The existence of a natural isomorphism between two functors captures a sense in which the functors are themselves “isomorphic” to one another as maps between categories. Categories C and D are dual if there are contravariant functors F : C → D and G : D → C such that GF is naturally isomorphic to the identity functor 1C and FG is naturally isomorphic to the identity functor 1D. Roughly speaking, F and G give a duality, or contravariant equivalence, between two categories if they are contravariant isomorphisms in the category of categories up to isomorphism in the category of functors. One can think of dual categories as “mirror images” of one another, in the sense that the two categories differ only in that the directions of their arrows are systematically reversed.

For the purposes of capturing the relationship between general relativity and the theory of Einstein algebras, we will appeal to the following standard of equivalence.

Theories T1 and T2 are equivalent if the category of models of T1 is dual to the category of models of T2.

Equivalence differs from duality only in that the two functors realizing an equivalence are covariant, rather than contravariant. When T1 and T2 are equivalent in either sense, there is a way to “translate” (or perhaps better, “transform”) models of T1 into models of T2, and vice versa. These transformations take objects of one category – models of one theory—to objects of the other in a way that preserves all of the structure of the arrows between objects, including, for instance, the group structure of the automorphisms of each object, the inclusion relations of “sub-objects”, and so on. These transformations are guaranteed to be inverses to one another “up to isomorphism,” in the sense that if one begins with an object of one category, maps using a functor realizing (half) an equivalence or duality to the corresponding object of the other category, and then maps back with the appropriate corresponding functor, the object one ends up with is isomorphic to the object with which one began. In the case of the theory of Einstein algebras and general relativity, there is also a precise sense in which they preserve the empirical structure of the theories.

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

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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)

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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. 

Algebraic Representation of Space-Time as Esoteric?

If the philosophical analysis of the singular feature of space-time is able to shed some new light on the possible nature of space-time, one should not lose sight of the fact that, although connected to fundamental issues in cosmology, like the ‘initial’ state of our universe, space-time singularities involve unphysical behaviour (like, for instance, the very geodesic incompleteness implied by the singularity theorems or some possible infinite value for physical quantities) and constitute therefore a physical problem that should be overcome. We now consider some recent theoretical developments that directly address this problem by drawing some possible physical (and mathematical) consequences of the above considerations.

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Indeed, according to the algebraic approaches to space-time, the singular feature of space-time is an indicator for the fundamental non-local character of space-time: it is conceived actually as a very important part of General Relativity that reveals the fundamental pointless structure of space-time. This latter cannot be described by the usual mathematical tools like standard differential geometry, since, as we have seen above, it presupposes some “amount of locality” and is inherently point-like. The mathematical roots of such considerations are to be found in the full equivalence of, on the one hand, the usual (geometric) definition of a differentiable manifold M in terms of a set of points with a topology and a differential structure (compatible atlases) with, on the other hand, the definition using only the algebraic structure of the (commutative) ring C(M) of the smooth real functions on M (under pointwise addition and multiplication; indeed C(M) is a (concrete) algebra). For instance, the existence of points of M is equivalent to the existence of maximal ideals of C(M). Indeed, all the differential geometric properties of the space-time Lorentz manifold (M,g) are encoded in the (concrete) algebra C(M). Moreover, the Einstein field equations and their solutions (which represent the various space-times) can be constructed only in terms of the algebra C(M). Now, the algebraic structure of C(M) can be considered as primary (in exactly the same way in which space-time points or regions, represented by manifold points or sets of manifold points, may be considered as primary) and the manifold M as derived from this algebraic structure. Indeed, one can define the Einstein field equations from the very beginning in abstract algebraic terms without any reference to the manifold M as well as the abstract algebras, called the ‘Einstein algebras’, satisfying these equations. The standard geometric description of space-time in terms of a Lorentz manifold (M,g) can then be considered as inducing a mathematical representation of an Einstein algebra. Without entering into too many technical details, the important point for our discussion is that Einstein algebras and sheaf-theoretic generalizations thereof reveal the above discussed non-local feature of (essential) space-time singularities from a different point of view. In the framework of the b-boundary construction M = M ∪ ∂M, the (generalized) algebraic structure C corresponding to M can be prolonged to the (generalized) algebraic structure C corresponding to the b-completed M such that CM = C, where CM is the restriction of C to M; then in the singular cases, only constant functions (and therefore only zero vector fields) can be prolonged. This underlines the non-local feature of the singular behaviour of space-time, since constant functions are non-local in the sense that they do not distinguish points. This fundamental non-local feature suggests non-commutative generalizations of the Einstein algebras formulation of General Relativity, since non-commutative spaces are highly non-local. In general, non-commutative algebras have no maximal ideals, so that the very concept of a point has no counterpart within this non-commutative framework. Therefore, according to this line of thought, space-time, at the fundamental level, is completely non-local. Then it seems that the very distinction between singular and non-singular is not meaningful anymore at the fundamental level; within this framework, space-time singularities are ‘produced’ at a less fundamental level together with standard physics and its standard differential (commutative) geometric representation of space-time.

Although these theoretical developments are rather speculative, it must be emphasized that the algebraic representation of space-time itself is “by no means esoteric”. Starting from an algebraic formulation of the theory, which is completely equivalent to the standard geometric one, it provides another point of view on space-time and its singular behaviour that should not be dismissed too quickly. At least it underlines the fact that our interpretative framework for space-time should not be dependent on the traditional atomistic and local (point-like) conception of space-time (induced by the standard differential geometric formulation). Indeed, this misleading dependence on the standard differential geometric formulation seems to be at work in some reference arguments in contemporary philosophy of space-time, like in the field argument. According to the field argument, field properties occur at space-time points or regions, which must therefore be presupposed. Such an argument seems to fall prey to the standard differential geometric representation of space-time and fields, since within the algebraic formalism of General Relativity, (scalar) fields – elements of the algebra C – can be interpreted as primary and the manifold (points) as a secondary derived notion.