"The Hole Argument and Some Physical and Philosophical Implications"
John Stachel 
1 Introduction
1.1 Why should we care?
1.2 Summary: Where we are headed
1.3 Outline of the article
2 Early History
2.1 From the special theory to the search for a theory of gravity
2.2 From the equivalence principle to the metric tensor
2.3 From the metric tensor to the hole argument
2.4 From the hole argument back to general covariance
2.5 The point coincidence argument
2.6 From general covariance to Kretschmann’s critique
2.7 The Cauchy problem for the Einstein equations: from Hilbert to Lichnerowicz
3 Modern Revival of the Argument
3.1 Did Einstein misunderstand coordinate transformations?
3.2 Einstein’s vision and fiber bundles
4 The Hole Argument and Some Extensions
4.1 Structures, algebraic and geometric, permutability and general permutability
4.2 Differentiable manifolds and diffeomorphisms, covariance and general covariance
4.3 Fiber bundles: principal bundles, associated bundles, frame bundles, natural and gauge-natural bundles
4.4 Covariance and general covariance for natural and gauge-natural bundles
5 Current Discussions: Philosophical Issues
5.1 Relationalism versus substantivalism: Is that all there is?
5.2 Evolution of Earman’s relationalism
5.3 Pooley’s position: sophisticated substantivalism
5.4 Stachel and dynamic structural realism
5.5 Relations, internal and external, quiddity and haecceity
5.6 Structures, algebraic and geometric
5.7 Does “general relativity” extend the principle of relativity?
6 Current Discussions: Physical Issues
6.1 Space-time symmetry groups and partially background-independent space-times
6.2 General relativity as a gauge theory
6.3 The hole argument for elementary particles
6.4 The problem of quantum gravity
7 Conclusion: The Hole Argument Redivivus
A Nijenhuis on the Formal Definition of Natural Bundles
B Appendix to Section 4: Some Philosophical Concepts
B.1 Things: Synchonic states and diachronic processes
B.2 Open versus closed systems: Determinism versus causality
B.3 Properties: Intrinsic and extrinsic
B.4 Relations: Internal and external
B.5 Quiddity and haecceity

B Appendix to Section 4: Some Philosophical Concepts

This appendix discusses some concepts that may be unfamiliar to some readers, non-philosophers in particular; and some concepts that may be familiar, but are used here in unfamiliar ways.

B.1 Things: Synchonic states and diachronic processes

Every theory involves things,98 their properties, and the relations between them. A physical thing occupies some region of space. As for time, there are two main approaches. The diachronic approach treats a thing as a process, developing over time.

[A] thing, insofar as it is more than an instantaneous occurrence and has duration through time, is a process. This introduces some odd results in our ways of talking. For example, talking would be a process but we would hardly talk of it as a “thing”; similarly, it is not usual to talk of a rock or a human being as a process (Wartofsky, 1968, p. 332).

The synchronic approach characterizes a thing as it exists at one moment of time, often called its state at time t. A diachronic process is treated as a time-ordered sequence of such synchronic states.

B.2 Open versus closed systems: Determinism versus causality

A closed system is a collection of things not subject to any external influence. Determinism is associated with closed systems and implies fatalism: Nothing external can alter the course of a process in a closed system. An open system is a collection of things that are subject to external influences. Causality is associated with open systems and implies control: By manipulating these external factors, one can affect the course of a process occurring in an open system.

Most systems in the world are open systems; closed systems are the rare exception in nature. …If all systems were naturally closed, there would be no need for – nor indeed any possibility of – experiment. …[E]xperiments help us to formulate and test hypotheses about things, structures, and mechanisms (hereafter structures for short) that underlie the world of events…. [T]he laws that express the nature of these structures operate as tendential laws…For example, Galileo’s law of falling bodies asserts that gravity produces a tendency of all objects to fall to earth with the same acceleration. Due to the effect on it of air resistance, which varies with velocity, an object actually falls to earth with varying acceleration – or even none, if it reaches terminal velocity…. A body subject to gravity may even move upward if it is immersed in a liquid, the buoyant force of which on it is greater than the gravitational force…. [I]f one does not distinguish between events and underlying mechanisms, and hence fails to recognize that laws are tendential, it is hard to recognize …that the upward motion of an air bubble in water is just as much an expression of the law of gravity as the downward motion of a stone in air (Stachel, 2003, pp. 144–145).

We almost always construct models of bounded systems undergoing finite processes. Such a system is confined to a finite, bounded region of space-time. Its surroundings, or environment, consist of the contents of those parts of space-time external to this region that are capable of influencing the system. At the boundary of such a finite process, new data can be used to initiate a new process (preparation), fed into the system as it evolves (information), and extracted from it to record the final outcome of the process (registration).

Example: Skipping rope: If each end of a (finite) rope is held by one person, they can determine its shape at some initial time; their movement of the two ends will then help to change the evolution of the rope’s shape; and a photograph can be taken of its final shape at a later time just before they let go.

It is often assumed that, if an open system is enlarged sufficiently by adjoining to it enough of its environment, this will ultimately result in a closed system. But this is a philosophical assumption, not a physical result, even if the enlargement is cosmological in scope. Open cosmological models have been proposed; and the rejection of one model for physical reasons, such as the steady state universe, does not imply that all cosmological models must be closed.

From this discussion, it is clear that the concepts of things and processes are intertwined with those of space and time. As our views on the nature of space and time change, so must our concepts of things and processes.

B.3 Properties: Intrinsic and extrinsic

As mentioned above, each thing is characterized by its properties. Property is often defined as a one-place relation:

Properties will have to be counted among the relations, just as 1 is taken to be a natural number (Weyl, 1949Jump To The Next Citation Point, p. 4)

However, the word is used here without such a restriction. Rather, one must distinguish between intrinsic properties, which are indeed one-place relations; and extrinsic properties, which are not.

Intrinsic properties:
Those one place relations of an entity that serve to characterize its nature, essence or natural kind. An entity that does not possess these properties cannot be of the same kind as an entity that does.

Example: Electrons and protons differ in their charge and mass, which are intrinsic properties. Hence, electrons and protons in the same atom are of different natural kinds, while an electron on Earth and an electron on Mars are both of the same kind.

Extrinsic properties:
Those multi-place relations of an entity that do not serve to characterize its nature. Entities of the same natural kind will generally differ in their extrinsic properties.

Example: Its position, energy, and momentum are extrinsic properties of an elementary particle. They relate it to some independently-defined external frame of reference.

In other words, the intrinsic properties of a thing are independent of its relation to any other thing, while its extrinsic properties do depend on its relation(s) with other things.99

B.4 Relations: Internal and external

An N-place relation R involves N things 𝜗i in an ordered sequence: R (𝜗1, 𝜗1,...,𝜗N ). Although N = 1 is not excluded (see the discussion of properties), relations proper always involve two or more things, often called the relata.

Example: Parent and child are the relata in the two-place relation of parenthood. In general the order of the N things is important. Invariance under some permutation(s) of its relata is a special property of a relation.

One must distinguish between internal and external relations:100

Internal relations:
There are relations, in which the relata are primary and their relation is secondary: no essential property of the relata depends on the particular relation under consideration. In this case, we customarily speak of the relations between things.
External relations:
However, there are also relations, in which the relation is primary and the relata are secondary: at least one essential property of each of the relata must depend on the relation. In this case, one may speak of the things between relations (Stachel, 2002).

In order to make sense of the distinction between these two types of relations, one must assume that there is a distinction between the essential and non-essential or accidental properties of each thing. Such a distinction can be traced back to Aristotle:

[I]t is clear that each individual thing is one and the same with its essence, and not accidentally so, but because to understand anything is to understand its essence (Metaphysics, Book IV).

Until recently, it was assumed that all relations – internal and external – supervened on intrinsic properties that uniquely defined each of the relata. Hermann Weyl warned against such an assumption:

The (explicit or implicit) assumption that every relation must be based on [intrinsic] properties has given rise to much confusion in philosophy [the original German text is much stronger: “viel Unheil angerichtet” = “has wrought much havoc”]. A statement asserting, say, that one rose is differently colored from a second is indeed founded on the fact that one is red, the other yellow. But the relation ‘the point A lies on the left of B’ is not based on a qualitatively describable position of A alone and of B alone. The same holds for kinship relations among people (Weyl, 1949, p. 4).

B.5 Quiddity and haecceity

Two terms useful in discussing this question are quiddity and haecceity.

What characterizes all entities having the same nature, i.e., the same intrinsic properties.

Example: All electrons have the same quiddity, as do all goats.

What individuates entities of the same quiddity.

Example: An electron bound to an atom on Earth and an electron bound to an atom on Mars differ in their haecceity.

We can now reformulate the earlier observation: Up until the last century, it was assumed that each entity of some quiddity always had an intrinsic haecceity; i.e., it could always be individuated independently of any relations, into which it entered. This assumption is often referred to as Leibniz’s principle:

For there are never two things in nature that are perfectly alike and in which it is impossible to find a difference that is internal, or founded on an intrinsic denomination (G.W. Leibniz, “The Monadology” in Loemker, 1969).

Any further individuation of a thing through some of its relations was assumed to supervene on its intrinsic haecceity.

Example: As a biological organism, Bill Gates has the same quiddity as all human beings, but a haecceity based on his individuating his physical characteristics (e.g., fingerprints). He is further individuated socially by his relationship to other members of the IT community as the Chairman of Microsoft.

However, the advent of quantum statistics led to the recognition that elementary particles indeed have quiddity but no inherent haecceity. As noted above, every electron has the same mass, charge and spin, which fix its quiddity; but an individual electron is only singled out (to the extent that can be) by its relations to other entities, such as its effect on the emulsion of a photographic plate.

Einstein’s way of avoiding the hole argument can be similarly formulated: The points of space-time have the same quiddity, but no intrinsic haecceity. A space-time point is only singled out (to the extent that can be) by the unique physical properties of the fields at that point.

So both relativity and quantum theory lead to the same conclusion: Leibniz’s principle is not universally applicable. There is a category of entities with quiddity but no inherent haecceity. Given that both general relativity and quantum mechanics are based on such entities, it is difficult to believe that, in any theory purporting to underlie both relativity and quantum theory, inherent individuality would re-emerge in its fundamental entities, whatever they are (see Section 6.4).

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