A Revision of Set-Theory Based Formalization of Epistemic Structural Realism

Epistemic structural realism (ESR) is a view in philosophy of science that all we can know about the unobservable world is its structure1. Variants of ESR differ in terms of what is unobservable, but the basic idea of attainable knowledge (i.e. structures) is shared. Typically, structures are formalized in a set-theoretic style, with premises in arguments embedded in the definition.

In this essay, the formal description of ESR will be presented at first to demonstrate its implications and limitations, from which an attempt to find a weaker but reasonable version will be discussed.

Before moving on, usage of basic notations should be clarified:

s.t. is the abbreviation of “subject to" or “so that".

iff is the abbreviation of “ if and only if " and denoted as .

A -ary relation defined on a set is a subset of , where is a positive integer and is the -times Cartesian product of . is mostly denoted as , and is omitted if it is not of interest.

A structure consisting a set and a set of relations defined on is denoted as a tuple . and is also denoted as , when its meaning is clear from the context.

1Motivation and definition of homomorphism

By intuition, if knowledge of the structure of the world is possible, a correspondence between the strucure of the world and that of the theory or experience should be availbale. The set-theory based formalization of such correspondence, isomorphism, is generally stated as follows.

Definition (Isomorphism). Suppose that and are structures, and there exist a map and a map (which are both denoted as for convenience) satisfying that:

(I1)

For all -ary and , iff , where and ;

(I2)

and are bijective;

then is an isomorphism between and .

A few properties can be derived trivially from the definition. Firstly, an inverse is uniquely determined since is bijective. Secondly, isomorphism is reflexive, symmetric and transitive, therefore setting an equivalence relation on structures. Thirdly, following the aforesaid ones, isomorphic structures are identical in some sense, if their intensions are ignored. Such properties make isomorphism an appealing depiction of the ideal state of theories concerning a field of interest.

However, if the existence of any theory isomorphic to the world is not promised or impossible at all, the central claim of ESR would be more implausible, since it provides too bold an “upper bound" for what knowledge we can attain. For this reason, it is now necessary to remind us of how this is argued. A closer look at part of the definition, (I2), reveals its connection with Mirroring Relations Principle (MR) in the argument of indirect epistemic structural realism (IESR) by Russell.

Relations between percepts correspond to relations between their are non-perceptual causes, in a manner which preserves their logico-mathematical properties. (MR) 2

Unfortunately, there is doubt about what “logico-mathematical properties" really refers to. If we formaize it as (I1), isomorphism is not established because (I2) is independent of this claim; on the other hand, if we formaize it as the union of (I1) and (I2), MR would imply, by rendering (I2) back to natural language, that our perception is as rich as the world in both content and relations, which makes MR far less acceptable.

Meanwhile, when isomorphism is the only case to be considered, all theories would turn out to be either completely correct (isomorphic to the world) or completely incorrect (not isomorphic), without intermediate space for evolving theories. That is, it seems unnatural for a theory to be improved from being completely incorrect to being completely correct.

Given discussion above, it might be plausible to take a step back from isomorphism to a more conservative position in the framework of ESR by simply removing (I2) and leaving (I1) untouched, as the following definition does.

Definition (Homomorphism). Suppose that and are structures. is an homomorphism from to iff there exist a map and a map satisfying (I1).

We shall explore and interpret several trivial or non-trivial consequences of replacing isomorphism with homomorphism in the next section.

2Consequences of homomorphism

Compatibility with isomorphism

By definition, an isomorphism is inherently a homomorphism, while the inverse does not hold, which means formalization by homomorphism is a generalization of isomorphism, possibly capturing more reliable theories other than the perfect ones.

Trivial homomorphism

It should be noted that, homomorphism defined in common algebraic objects may provide no information, where the map named as trivial homomorphism can be made between two arbitary objects. For instance, mapping any to the identity is a trivial group homomorphism.

While there exist various forms of homomorphisms, such cases are avoided in strucures by virture of (I1). Alternatively, consider replacing (I1) by (I1*):

For all -ary and , if , then , where and .

The modified definition would make it possible to naturally construct a trivial homomorphism, by setting a “blackhole" object and a “blackhole" relation, and mapping all elements and relations to it.

Forms of homomorphisms, “approximate truth" and comparison among theories

Suppose that and are structures of the world and a theory respectively. If the homomorphism is not bijective, it could be one of the following:

1.

Surjective but not injective, or surjective homomorphism.

2.

Injective but not surjective, or injective homomorphism.

3.

Neither injective nor surjective.

Case (2) and (3) are tough but less important, since a theory is not likely to have a greater cardinality of theoretic objects than “objects of the world", and 3 could be viewed as the redundant part, and thus omitted.

With regard to case (1), although the theory equipped with a surjective homomorphism does not preserve all details of , i.e. there exist different elements and relations mapped to identical ones, it is nevertheless structure-preserving: in essence, is only confusing elements from the same equivalent class of objects and relations of the world. This idea is achieved by defining an equivalence relation on (and similarly on ), following the common practice in algebra.

Definition (Equivalence relation). Suppose that is a homomorphism. For all , iff , and is denoted as .

If is an isomorphism, every equivalent class is a singleton with exactly one element due to the bijective map. When , notice that technically, selecting a representative from each class creates a substructure isomorphic to the theory, but such selection would not be esaily justified once the field of interest is set. This confusion could be attributed to defects of the theory, or aspects of the world that do not interact with our mind separately, and not reflected by independent percepts. In both cases, a quotient strucure equipped with a canonical projection is isomorphic to the theory with a proper definition similar to theorems of homomorphism in algebra.

Definition (Quotient strucure). Suppose that are structures and is a homomorphism. The quotient strucure of with respect to is defined as , where and . The map , is called a canonical projection.

If or , for any relation that enters,justifying the construction of . Similarly, for relations ,justifying the construction of . Finally, the proof of being isomorphic to is trivial and omitted4, while the proposition itself is vital for formally illustraing how theories may work imperfectly by collapsing elements and relations of the world.

Through investigation, we find out that every homomorphism from to is informative, and (1) is an almost ideal case where we can construct a quotient structure isomorphic to the given theory. With all the premises, now it should be more convincing that a theory equipped with a homomorphism (thus being structure-preserving but imperfect) could be interpreted as an approximate truth within the framework of ESR.

Another noticeable feature is that such theory is not unique even up to isomorphism (which also agrees with the conception of approximate truth), permitting partial comparison among theories derived from the partial order of sets.

Definition (Partial order of theories). Suppose that are structures of theories, and , are surjective homomorphisms. is weaker than iff there exists a surjective map s.t. for all , and , and denoted as .

It is easy to verify the definition by checking axioms of partial order (reflexive, anti-symmetric, and transitive), and that itself is the only maximum. While all theories with surjective homomorphisms serve as approximate truths, they jointly form a tree, where nodes on the same chain comparable to each other. Following this approach, we may also imagine a sequence of theories converging to and define it as “eventually true", of which the details are already out of scope.

3Conclusion

Definition of homomorphism inspired by algebraic strucures are raised and discussed, to capture potentially more reliable theories of the world than the standard set-theoretic formalization of ESR based on isomorphism does. More importantly, its derivants provide a formal and intuitive description of approximate truths and comparison among theories, if we accept at least modestly the claim of ESR.

Despite its merits (and also how the definition is developed in the first section), the problem that whether the existence of a theory equipped with a homomorphism is warranted, remains unsolved. Even without the assumption of a bijective map, it would depend on an argument for structure-preserving maps. In another aspect, set theory is probably not an appropriate language in terms of the formalization of ESR, with insufficient tools to handle bare sets and relations, therefore leaving much room for further improvement perhaps by category theory or formal logic.

1.

Frigg, R., & Votsis, I. (2011). Everything you always wanted to know about structural realism but were afraid to ask. European Journal for Philosophy of Science, 1(2), 227–276.

2.

Quoted by Frigg & Votsis (2001), ibid.

3.

, where , to be precise.

4.

By considering the underlying sets of structures.