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?

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