What is space? What is time? Do they exist independently of the thingsand processes in them? Or is their existence parasitic on those thingsand processes? Are they like a canvas onto which an artist paints;they exist whether or not the artist paints on them? Or are they akinto parenthood; there is no parenthood until there are parents andchildren? That is, is there no space and time until there are thingswith spatial properties and processes with temporal durations?

These questions have long been debated and continue to be debated. Thehole argument arose when these questions were asked in the context ofmodern spacetime physics, and in particular in the context ofEinstein’s general theory of relativity. In that context, spaceand time are fused into a single entity, spacetime, and we inquireinto its status. One view is that spacetime is a substance: a thingthat exists independently of the processes occurring within it. Thisis spacetime substantivalism. The hole argument seeks to show thatthis viewpoint leads to unpalatable conclusions in a large class ofspacetime theories. In particular, it seeks to show that spacetimesubstantivalism leads to a failure of determinism, meaning that acomplete specification of the state of the universe at a given time,alongside the laws of nature of the theory under consideration (e.g.,the laws of general relativity, which are Einstein’s fieldequations), fails to determine uniquely how the universe will evolveto the future. It also presents a verificationist dilemma, for itappears to lead to the unexpected conclusion that there are factsabout the world which we can never know. Although these problemsare neither logically contradictory nor refuted by experience, manywould nevertheless regard them as being unpalatable.

Thereby, the hole argument provides a template for the analysis ofwhat are known as ’gauge redundancies’ in physicaltheories: i.e., the existence of surplus mathematical structure insuch theories. We learn from it that the identification of surplusmathematical structure cannot be achieved by any a priori or purelymathematical rule. Some physical grounds are needed. In line with theabove, the hole argument provides two grounds that can be used: (i)verifiability—changes in the candidate surplus structure make nodifference to what can be verified in observation; and (ii)determinism—the laws of the theory are unable to fix thecandidate surplus structure.

The hole argument was invented for slightly different purposes byAlbert Einstein late in 1913 as part of his quest for the generaltheory of relativity. It was revived and reformulated in the moderncontext by John³ = John Earman \(\times\) John Stachel\(\times\) John Norton.

See Stachel (2014) for a review that covers the historical aspects ofthe hole argument and its significance in philosophy and physics. Itis written at a technically more advanced level than this article. Foranother recent review of the hole argument, see Pooley (2021).

• 1. Modern Spacetime Theories: A Beginner’s Guide
• 2. The Freedom of General Covariance
• 3. The Preservation of Invariants
  • 3.1 Invariants
  • 3.2 Invariants and Observables
• 4. What Represents Spacetime?
• 5. The Price of Spacetime Substantivalism
• 6. Unhappy Consequences
• 7. The Hole Argument in Brief
• 8. The History of the Hole Argument
  • 8.1 Einstein Falls into the Hole…
  • 8.2 …and Climbs out Again
• 9. Responses to the Hole Argument
  • 9.1 Reject substantivalism
  • 9.2 Accept the indeterminacy
  • 9.3 Metric essentialism
  • 9.4 Sophisticated substantivalism
  • 9.5 Mathematical/formalist responses
• 10. Broader Significance of the Hole Argument
  • 10.1 A Limit to Scientific Realism
  • 10.2 The Hole Argument and the Quantization of Gravity
  • 10.3 The Hole Argument and Gauge Freedoms
• Bibliography
• Academic Tools
• Other Internet Resources
  • Preprints
  • Other Resources
• Related Entries

1. Modern Spacetime Theories: A Beginner’s Guide

Virtually all modern spacetime theories are now built in the same way.The theory posits a manifold of events and then assigns furtherstructures to those events to represent the content of spacetime. Astandard example is Einstein’s general theory of relativity. Asa host for the hole argument, we will pursue one of its best knownapplications, the expanding universes of modern relativisticcosmology.

This one example illustrates the core content of the hole argument.With only a little further effort, the argument can be made moreprecise and general. This will be done concurrently in these notes,^[1] intended for readers with some background in differential geometryand general relativity.

Here are the two basic building blocks of modern, relativisticcosmology: a manifold of events and the fields defined on it.

Manifold of Events. Consider our universe, whichrelativistic cosmologies attempt to model. Its spacetime is theentirety of all space through all time. The events of this spacetimeare generalizations of the dimensionless points of ordinary spatialgeometry. A geometric point in ordinary spatial geometry is just aparticular spot in the space and has no extension. Correspondingly, anevent in spacetime is a particular point in a cosmological space at aparticular moment of time.

So far, all we have defined is a set of events. To be afour-dimensional manifold, the set of events must be a little bit moreorganized. In a real spacetime, we have the idea that each event sitsin some local neighborhood of events; and this neighborhood sitsinside a larger neighborhood of events; and so on. That extraorganization comes from the requirement that we can label the eventswith four numbers—or at least we can do this for anysufficiently small chunk of the manifold. These labels form coordinatesystems. The fact that four numbers are just sufficient to label theevents makes the manifold four-dimensional. We can now pick outneighborhoods of some event as the set of all points whose spacetimecoordinates differ from our starting event by at most one unit; or twounits; or three units; etc.. That gives us the nested neighborhoods ofevents. Figure 1 illustrates how a set of events may be made into atwo dimensional manifold by assigning “\(x\)” and“\(y\)” coordinates to the events.

Figure 1. Forming a manifold ofevents

Metrical Structure and Matter Fields. In specifyingthat events form a four dimensional manifold, there is still a lot wehavenot said about the events. We have not specified whichevents lie in the future and past of which other events, how much timeelapses between these events, which events are simultaneous withothers so that they can form three dimensional spaces, what spatialdistances separates these events, and many more relatedproperties.

These additional properties are introduced by specifying the metricfield. To see how this field provides that information, imagine acurve connecting a given pairs of events in spacetime. The informationabout times elapsed and spatial distances is given by the timeselapsed and distances along such curves. See Figure 2:

Figure 2. The function of the metricfield.

That information could be supplied by a huge catalog that specifiesthe spatial or temporal distance between pairs of events along curvesconnecting them. Such a huge catalog would be massively redundant,however. If we know the distance from \(A\) to \(B\) andfrom \(B\) to \(C\) along some curve, then we know thedistance from \(A\) to \(C\) along that curve as well. Theminimum information we need is the temporal and spatial distancebetween each event and all those (loosely speaking) infinitesimallyclose to it. That information is what the metric field provides. It isa “field” since that information belongs just to oneevent. We can then piece together temporal and spatial distance alongany curve just by summing all the distances between successiveinfinitesimally close points along the curve.

The matter of the universe is represented by matter fields. Thesimplest form of matter—the big lumps that makegalaxies—can be represented by worldlines that trace out thehistory of each galaxy through time. In standard models, the galaxiesrecede from one another and this is represented by a spreading apartof the galactic worldlines as we proceed to later times. See Figure3:

Figure 3. Galaxies in an expandinguniverse.

2. The Freedom of General Covariance

Newton’s first law states that force-free bodies move onstraight-line trajectories through spacetime. Clearly, this law asstated cannot be true in all coordinate systems, for imagine such asystem accelerating with respect to one in which the law holds true:in the accelerating coordinate system, the force-free bodies will alsoappear to accelerate! So Newton’s first law is not—to usea piece of terminology introduced by Einstein in his quest for thegeneral theory of relativity in the 1910s—‘generallycovariant’. When Einstein finally arrived at his completedtheory of general relativity in 1915, one of its novel features wasits general covariance: unlike Newton’s first law in itsformulation above, if the Einstein field equations of generalrelativity hold in one coordinate system, then they hold true in allcoordinate systems related to that original system by smooth butotherwise arbitrary transformations. General relativity was, indeed,the first spacetime theory in which one was free to use arbitraryspacetime coordinate systems in this way. This feature is now sharedby virtually all modern formulations of spacetime theories, includingmodern versions of special relativity and (perhaps surprisingly!)Newtonian spacetime theory (on the latter, see e.g. Friedman(1983)).

In its original form, general covariance was understood“passively”; that is, as a freedom to describe structuresin spacetime by means of arbitrarily chosen coordinate systems. Thatfreedom is closely related to another freedom, known as“active” general covariance. According to active generalcovariance, we are licensed to spread structures like metric fieldsover the manifold in as many different ways as there are coordinatetransformations, and the resulting redistributions of said fields onthe manifold will preserve solutionhood of the theory underconsideration. Passive general covariance is not equivalent to activegeneral covariance defined in this way, for a theory can be passivelygenerally covariant and yet arbitrary smooth redistributions of thefields on the manifold will in general fail to preserve solutionhood:the details are not important here, but see Pooley (2017) for furtherdiscussion. (For a more extensive account of the relationship betweenactive and passive covariance, see also the supplementary document:Active and Passive Covariance.)^[2] What is important is that every ‘local’ spacetime theory(to use terminology of Earman & Norton (1987)) is activelygenerally covariant, and so will be subject to a version of the holeargument—this includes both special relativity and certainformulations of Newtonian mechanics. (One view is that this goes toofar, and that general relativity is distinct from many other spacetimetheories in that its spacetime geometry has become dynamical and it isonly in such theories that the hole argument should be mounted: forfurther discussion, see Earman (1989, Ch.9, Section 5), Stachel(1993), and Iftime and Stachel (2006).)

In what follows, we’ll focus on active general covariance, whichis indeed a property possessed by general relativity (as well as otherlocal spacetime theories). The essential manipulation of the holeargument involves exercising this freedom. Figure 4 illustrates oneway that we might spread the metrical structure and matter fields overthe manifold of spacetime events:

Figure 4. One way to spread metric andmatter over the manifold.

Figure 5 illustrates a second way:

Figure 5. Another way to spread metricand matter over the manifold.

We shall call the transformation between the two spreadings a“hole transformation.” The dotted region is The Hole. Thefirst distribution of metric and matter fields is transformed into thesecond in a way that

• leaves the fields unchanged outside the hole;
• within the hole they are spread differently;
• and the spreadings inside and outside the hole join smoothly.^[3]

Importantly, both the pre- and post-transformed states in Figure 5 aresolutions of Einstein’s field equations of generalrelativity—meaning that they both represent possible ways forthe world to be, according to the theory.

3. The Preservation of Invariants

3.1 Invariants

The two different spreadings share one vital characteristic upon whichthe hole argument depends: the two spreadings agree completely on allinvariant properties.

These invariant properties are, loosely speaking, the ones that areintrinsic to the geometry and dynamics, such as distance along spatialcurves and time along worldlines of galaxies, the rest mass of thegalaxy, the number of particles in it, as well as a host of otherproperties, such as whether the spacetimes are metrically flat orcurved.

The invariant properties are distinguished from non-invariantproperties. Best known of the non-invariant properties are thosedependent on a particular choice of coordinate system. For example,just one event in a two-dimensional Euclidean space lies at the originof a coordinate system, that is, at \(x=0,\) \(y=0\). Butwhich event that is changes when we change our coordinate system. So“being at the origin” is not an invariant. The spatialdistance between two points, however, is that same no matter whichcoordinate system is used to describe the space. It is aninvariant.

While the fields are spread differently in the two cases, they agreein all invariant properties; so, in invariant terms, they are thesame.

3.2 Invariants and Observables

There is a special relationship between the invariants of a spacetimetheory and its observables, i.e. those quantities which are accessibleto observational verification:

All observables can be reduced to invariants.

For example, if one makes a journey from one galaxy to another, allobservables pertinent to the trip will be invariants. These includethe time elapsed along the journey, whether the spaceship isaccelerating or not at any time in its journey, the age of the galaxyone leaves at the start of the trip and the age of the destinationgalaxy at the end and all operations that may involve signaling withparticles or light pulses.

Therefore, since the two spreadings or distributions of metric andmatter fields of a hole transformation agree on invariants, they alsoagree on all observables. They are observationallyindistinguishable.

4. What Represents Spacetime?

Recall our original concern: we want to know whether we can conceiveof spacetime as a substance—that is, as something that existsindependently of the material events which unfold within it. To dothis, we need to know what in the above structures representsspacetime. One popular answer to that question is that the manifold ofevents represents spacetime. This choice is natural since modernspacetime theories are built up by first positing a manifold of eventsand then defining further structures on them. So, on a realist view ofphysical theories on which they are to be understood‘literally’, it is very natural to regard the manifold asbeing an independently existing structure that bears properties, andwhich thereby plays the role of a container just as we expect thatspacetime does.^[4]

One might wonder whether some of the further structures defined on themanifold represent further properties of spacetime rather than what iscontained within spacetime. In particular, the metric field containsimportant information on spatial distances and times elapsed. Oughtthat not also to be considered a part of the containing spacetime asopposed to what is contained within spacetime?

Against this line of thought, some would argue that general relativitymakes it hard to view the metric field simply as being part of thecontaining spacetime. For, in addition to spatial and temporalinformation, the metric field also represents the gravitational field.Therefore it also carries energy and momentum—the energy andmomentum of the gravitational field (although a notorious technicalproblem in general relativity precludes identifying the energy andmomentumdensity of the gravitational field at any particularevent in spacetime). This energy and momentum is freely interchangedwith other matter fields in spacetimes. To carry energy and momentumis (the thought goes) a natural distinguishing characteristic ofmatter contained within spacetime.

So, the metric field of general relativity seems to defy easycharacterization. We would like it to be exclusively part of spacetimethe container, or exclusively part of matter the contained. Yet itseems to be part of both. But in any case, the crucial point to notehere (contrary to some historical writing on the hole argument) isthat one need not settle this issue in order to get the hole argumentoff the ground! As long as one is dealing with a theory which isactively generally covariant in the sense articulated above, the holeargument will rear its head, as we will now see.

5. The Price of Spacetime Substantivalism

So far we have characterized the substantivalist doctrine as the viewthat spacetime has an existence independent of its contents. Thisformulation conjures up powerful if vague intuitive pictures, but itis not clear enough to be deployed in the context of theinterpretation of physical theories. If we represent spacetime by amanifold of events, how do we characterize the independence of itsexistence? Is it the counterfactual claim that were there no metric ormatter fields, there would still be a manifold of events? Thatcounterfactual is automatically denied by the standard formulationwhich posits that all spacetimes have at least metrical structure.That seems too cheap a refutation of manifold substantivalism. Surely,there must be an improved formulation. Fortunately, we do not need towrestle with finding it. For present purposes we need only consider aconsequence of the substantivalist view and can set aside the task ofgiving a precise formulation of that view.

In their celebrated debate over space and time, Leibniz taunted thesubstantivalist Newton’s representative, Clarke, by asking howthe world would change if East and West were switched. For Leibnizthere would be no change since all spatial relations between bodieswould be preserved by such a switch. But the Newtonian substantivalisthad to concede that the bodies of the world were now located indifferent spatial positions, so the two systems were physicallydistinct.

Correspondingly, when we spread the metric and matter fieldsdifferently over a manifold of events, we are now assigning metricaland material properties in different ways to the events of themanifold. For example, imagine that a galaxy passes through some eventE in the hole. After the hole transformation, this galaxy might notpass through that event. For the manifold substantivalist, this mustbe a matter of objective physical fact: either the galaxy passesthrough E or not. The two distributions represent two physicallydistinct possibilities.

Figure 6. Does the galaxy pass throughevent \(E\)?

That is, manifold substantivalists must (it seems) deny an equivalenceinspired by Leibniz’ taunt which is thus named after him:^[5]

Leibniz Equivalence. If two distributions of fieldsare related by a smooth transformation, then they represent the samephysical systems.

The supplementary documentVisualizing Leibniz Equivalence Through Map Projections illustrates the essential idea of Leibniz Equivalence through ananalogy with different map projections of the Earth’ssurface.

6. Unhappy Consequences

We can now assemble the pieces above to generate unhappy consequencesfor the manifold substantivalist. Consider the two distributions ofmetric and material fields related by a hole transformation. Since themanifold substantivalist denies Leibniz Equivalence, thesubstantivalist must hold that the two systems represent distinctphysical systems. But the properties that distinguish the two are veryelusive. They escape both (a) observational verification and (b) thedetermining power of cosmological theory.

(a)Observational verification. The substantivalist mustinsist that it makes a physical difference whether the galaxy passesthrough event \(E\) or not. But we have already noticed that thetwo distributions are observationally equivalent: no observation cantell us if we are in a world in which the galaxy passes through event\(E\) or misses event \(E\).

It might be a little hard to see from Figure 6 that the twodistributions are observationally equivalent. In the firstdistribution on the left, the middle galaxy moves in what looks like astraight line and stays exactly at the spatial midpoint between thegalaxies on either side. In the second distribution on the right, allthat seems to be undone. The galaxy looks like it accelerates intaking a swerve to the right, so that it moves closer to the galaxy onits right.

These differences that show up in the portrayal of Figure 6 are allnon-invariant differences. For the right hand distribution, the galaxydoes veer to the right in the figure, but at the same time, distancesbetween events get stretched as well, just as they get stretched inthe various map projections shown in the supplement,Visualizing Leibniz Equivalence Through Map Projections. So the galaxy always remains at the spatial midpoint of the galaxieson either side; it just doesn’t look like it is at the spatialmidpoint from the way the figure is drawn.

Similarly, an acceleration vector along the galaxy’s worldlinedetermines whether the galaxy is accelerating. That accelerationvector is an invariant. So, if the galaxy in the left handdistribution has a zero acceleration vector, then the correspondinggalaxy in the right hand distribution will also have a zeroacceleration vector. Remember, a hole transformation preservesinvariants. So if a galaxy is unaccelerated in the left handdistribution, it is also unaccelerated in the right handdistribution.

(b)Determinism. The physical theory of relativisticcosmology is unable to pick between the two cases. This is manifestedas an indeterminism of the theory. We can specify the distribution ofmetric and material fields throughout the manifold of events,excepting within the region designated as The Hole. Then the theory isunable to tell us how the fields will develop into The Hole. Both theoriginal and the transformed distribution are legitimate extensions ofthe metric and matter fields outside The Hole into The Hole, sinceeach satisfies all the laws of the theory of relativistic cosmology.The theory has no resources which allow us to insist that one only isadmissible.

It is important to see that the unhappy consequence does not consistmerely of a failure of determinism. We are all too familiar with suchfailures and it is certainly not automatic grounds for dismissal of aphysical theory. The best known instance of a widely celebrated,indeterministic theory is quantum theory, where, in the standardinterpretation, the measurement of a system can lead to anindeterministic collapse onto one of many possible outcomes. Less wellknown is that it is possible to devise indeterministic systems inclassical physics as well. Most examples involves oddities such asbodies materializing at unbounded speed from spatial infinity, socalled “space invaders.” (Earman, 1986a, Ch. III; see alsodeterminism: causal) Or they may arise through the interaction of infinitely many bodiesin a supertask (Supertasks). More recently, an extremely simple example has emerged in which asingle mass sits atop a dome and spontaneously sets itself into motionafter an arbitrary time delay and in an arbitrary direction (Norton,2003, Section 3).

The problem with the failure of determinism in the hole argument isnot the fact of failure but the way that it fails. If we deny manifoldsubstantivalism and accept Leibniz Equivalence, then the indeterminisminduced by a hole transformation is eradicated. While there areuncountably many mathematically distinct developments of the fieldsinto the hole, under Leibniz Equivalence, they are all physically thesame. That is, there is a unique development of the physical fieldsinto the hole after all. Thus the indeterminism is a directconsequence of the substantivalist’s metaphysical commitments.Similarly, if we accept Leibniz Equivalence, then we are no longertroubled that the two distributions cannot be distinguished by anypossible observation. They are merely different mathematicaldescriptions of the same physical reality and so should agree on allobservables.

So, the anti-substantivalist conclusion invited by the hole argumentis this. We can load up any physical theory with objects or properties(here: spacetime events) that cannot be fixed by observation. If theirinvisibility to observation is not already sufficient warning thatthese properties are illegitimate, then finding that they visitindeterminism onto a theory that is otherwise deterministic ought tobe warning enough. Therefore, such objects or properties (again, herespacetime events) should be discarded along with any doctrine thatrequires their retention.

7. The Hole Argument in Brief

In sum, the hole argument amounts to this:^[6]

1. If one has two distributions of metric and material fields relatedby a hole transformation, then manifold substantivalists must maintainthat the two systems represent two distinct physical systems (i.e.,they must deny Leibniz Equivalence).
2. This physical distinctness transcends both observation and thedetermining power of the theory since:
   • The two distributions are observationally identical.
   • The laws of the theory cannot pick between the two developments ofthe fields into the hole.
3. Therefore, the manifold substantivalist advocates a problematicmetaphysics which should be discarded.

8. The History of the Hole Argument

8.1 Einstein Falls into the Hole…

The hole argument was created by Albert Einstein late in 1913 as anact of desperation when his quest for his general theory of relativityhad encountered what appeared to be insuperable obstacles. Over theprevious year, he had been determined to find a gravitation theorythat was generally covariant in the sense introduced above. He hadeven considered essentially the celebrated generally covariantequations he would settle upon in November 1915 and which now appearin all the textbooks.

Unfortunately, Einstein had at least initially been unable to see thatthese equations were admissible. Newton’s theory of gravitationworked virtually perfectly for weak gravitational fields. So it wasessential that Einstein’s theory revert to Newton’s inthat case. But try as he might, Einstein could not see that hisequations and many variants of them could properly mesh withNewton’s theory. In mid 1913 he published a compromise: a sketchof a relativistic theory of gravitation that was not generallycovariant. (For further details of these struggles, see Norton(1984).)

His failure to find an admissible generally covariant theory troubledEinstein greatly. Later in 1913 he sought to transform his failureinto a victory of sorts: he thought he could show that no generallycovariant theory at all is admissible. Any such theory would violatewhat he called the Law of Causality—we would now call itdeterminism. He sought to demonstrate this remarkable claim with thehole argument.

In its original incarnation, Einstein considered a spacetime filledwith matter excepting one region, the hole, which was matter-free. (Soin this original form, the term “hole” makes more sensethan in the modern version.) He then asked if a full specification ofboth metric and material fields outside the hole would fix the metricfield within. Since he had tacitly eschewed Leibniz Equivalence,Einstein thought that the resulting negative answer sufficient to damnall generally covariant theories.

8.2 …and Climbs out Again

Einstein struggled on for two years with his misshapen theory oflimited covariance. Late in 1915, as evidence of his errors mountedinexorably, Einstein was driven to near despair and ultimatelycapitulation. He returned to the search for generally covariantequations with a new urgency, fueled in part by the knowledge thatnone other than David Hilbert had thrown himself into analysis of histheory. Einstein’s quest came to a happy close in late November1915 with the completion of his theory in generally covariantform.

For a long time it was thought that Hilbert had beaten Einstein byfive days to the final theory. New evidence in the form of the proofpages of Hilbert’s paper now suggests he may not have. Moreimportantly, it shows clearly that Hilbert, like Einstein, at leasttemporarily believed that the hole argument precluded all generallycovariant theories and that the belief survived at least as far as theproof pages of his paper. (See Corry, Renn and Stachel 1997.)

While Einstein had tacitly withdrawn his objections to generallycovariant theories, he had not made public where he thought the holeargument failed. This he finally did when he published what JohnStachel calls the “point-coincidence argument.” Thisargument, well known from Einstein’s (1916, p.117) review of hisgeneral theory of relativity, amounts to a defence of LeibnizEquivalence. He urges that the physical content of a theory isexhausted by the catalog of the spacetime coincidences it licenses.For example, in a theory that treats particles only, the coincidencesare the points of intersection of the particle worldlines. Thesecoincidences are preserved by transformations of the fields. Thereforetwo systems of fields that can be intertransformed have the samephysical content; they represent the same physical system.

Over the years, the hole argument was deemed to be a trivial error byan otherwise insightful Einstein. It was John Stachel (1980) whorecognized its highly non-trivial character and brought thisrealization to the modern community of historians and philosophers ofphysics. (See also Stachel, 1986.) In Earman and Norton (1987), theargument was recast as one that explicitly targets spacetimesubstantivalism. For further historical discussion, see Howard andNorton (1993), Janssen (1999), Klein (1995) and Norton (1987). Athorough, synoptic treatment in four volumes is Renn (2007); for ahistory of the philosophical revival of the hole argument, seeWeatherall (2020).

For an account of the appropriation and misappropriation ofEinstein’s point-coincidence argument by the logicalempiricists, see Giovanelli (2013).

9. Responses to the Hole Argument

There are at least as many responses to the hole argument as authorswho have written on it. In this section, we regiment the literature byconsidering five broad classes of response to the argument since itwas revitalised in the philosophical literature of the 1980s. In thecourse of scrutinizing the argument, by now virtually all its aspectshave been weighed and tested.

9.1 Reject substantivalism

One line of thought simply agrees that the hole argument makesacceptance of Leibniz Equivalence compelling. It seeks to make moretransparent what that acceptance involves by trying to find a singlemathematical structure that represents a physical spacetime systemrather than the equivalence class of intertransformable structureslicensed by Leibniz Equivalence. One such attempt involves the notionof a “Leibniz algebra” (see Earman, 1989, Ch. 9, Sect. 9).It is unclear that such attempts can succeed. Just asintertransformable fields represent the same physical system, thereare distinct but intertransformable Leibniz algebras with the samephysical import. If the formalisms of manifolds and of Leibnizalgebras are intertranslatable, one would expect the hole argument toreappear in the latter formalism as well under this translation. (SeeRynasiewicz, 1992.)

Another approach seeks to explain Leibniz Equivalence and demonstratethe compatibility of general relativity with the hole argument throughthe individuation of spacetime points by means of “Diracobservables” and an associated gauge fixing stipulation (Lusannaand Pauri, 2006). More generally, we may well wonder whether theproblems faced by spacetime substantivalism are artifacts of theparticular formalism described above. Bain (1998, 2003) has exploredthe effect of a transition to other formalisms (including but notlimited to Leibniz algebras).

9.2 Accept the indeterminacy

An alternative response to the hole argument is to accept thatgenerally covariant theories of space and time such as generalrelativity are indeterministic. Perhaps (the thought goes) thisindeterminism is not troubling, because it is an indeterminism onlyabout which objects instantiate which properties and not about whichpatterns of properties are instantiated. It is not obvious, however,that this is sufficient to defuse worries about indeterminism: at thevery least, if another response to the argument were available, theywould seem to be preferable.

A related response here is toredefine determinism, and toargue that, in the relevant sense, theories such as general relativityare deterministic after all, in spite of the hole argument.Modifications to the definition of determinism in light of the holeargument have been explored by Belot (1995b), Brighouse (1994, 2020),Butterfield (1989), Melia (1999), and Pooley (2021).

9.3 Metric essentialism

In his own highly original response to the hole argument, Maudlin(1990) urges that each spacetime event carries its metrical propertiesessentially; that is, it would not be that very event if (afterredistribution of the fields) we tried to assign different metricalproperties to it. As a result, although there appears to be a class ofdistinct possible worlds associated with each class ofintertransformable solutions of general relativity (or whatever otheractively generally covariant theory in which one is interested), infact only one such world is metaphysically possible, and thereby thehole argument is vitiated. Teitel (2019) has explored a refinedversion of this essentialist response but concludes that it fails toimprove on standard modal responses to the hole argument. Butterfield(1989) portrays intertransformable systems as different possibleworlds and uses counterpart theory to argue that at most one canrepresent an actual spacetime. (For an updated version ofButterfield’s appeal to counterpart theory, see Butterfield& Gomes (2023a, 2023b).)

9.4 Sophisticated substantivalism

Maudlin’s metrical essentialism is an example of‘sophisticated substantivalism’. This term wasintroduced—in a somewhat pejorative sense—by Belot &Earman (2001) to refer to a class of views according to which it islegitimate after all for a substantivalist to deny that systemsrelated by hole transformation represent distinct possibilities,thereby side-stepping the hole argument. Another version ofsophisticated substantivalism is anti-haecceitist substantivalism,according to which physical spacetime points do not possesstrans-world identities. The distinctness of worlds in which thematerial content of the universe is shifted from its distribution inthe actual world presupposes such identities. Thus the apparentindeterminism of general relativity and other actively generallycovariant theories is avoided by denying such identities. Thisposition is currently a popular response to the hole argument: fordiscussion e.g. Hoefer (1996) and Pooley (2006b); for some concernsthat the position is metaphysically obscure, see Dasgupta (2011).

Another related version of sophisticated substantivalism has it thatspacetime is better represented not by the manifold of events alonebut by some richer structure, such as the manifold of events inconjunction with metrical properties. (See, for example, Hoefer,1996.) What motivates this escape is the idea that the manifold ofevents lacks properties essential to spacetime. For example, there isno notion of past and future, of time elapsed or of spatial distancein the manifold of events. Thus one might be tempted to identifyspacetime with the manifold of eventsplus some furtherstructure that supplies these spatiotemporal notions. Thereby, thethought might go, it is the metrical structure which individuatesspacetime points non-rigidly. This escape from the hole argumentsometimes succeeds and sometimes fails. In certain important specialcases, alternative versions of the hole argument can be mountedagainst the view: see Norton (1988).

9.5 Mathematical/formalist responses

Perhaps the simplest possible challenge to the hole argument maintainsthat Leibniz Equivalence is a standard presumption in the modernmathematical physics literature and suggests that even entertainingits denial is a mathematical blunder unworthy of serious attention.But in response: while acceptance of Leibniz Equivalence is widespreadin the physics literature, it is not alogical truth that canonly be denied on pain of contradiction. That it embodies non-trivialassumptions whose import must be accepted with sober reflection isindicated by the early acceptance of the hole argument by none otherthan David Hilbert himself. (SeeSection 8.2 above.) If denial of Leibniz Equivalence is a blunder so egregiousthat no competent mathematician would make it, then our standards forcompetence have become unattainably high, for they must exclude DavidHilbert in 1915 at the height of his powers.

The question has nevertheless been reopened recently by Weatherall(2018), who aargues that intertransformable mathematical structuresare taken in standard mathematical practice to be the same structure.Thus they should represent the same physical system, precluding thedenial of Leibniz Equivalence. Curiel (2018) argues for a similartriviality conclusion as Weatherall but on a different basis: there isnophysical correlate to the hole transformation in standardphysical practice. For a response to arguments of this kind, defendingthe view that this kind of ‘mathematical structuralism’ isinsufficient to block the hole argument, see Pooley & Read (2021);likewise, Roberts (2020) has responded that Nature—notmathematical practice—should decide whether two mathematicalstructures represent the same physical system.

Recently, some authors have argued that modifying the foundations ofmathematics from the set-theoretic orthodoxy would be sufficient toblock the hole argument: see Ladyman & Presnell (2020) andDougherty (2020) for discussions of such arguments in the particularcontext of homotopy type theory. Separately, Halvorson and Manchak(2021) have argued that because there is a unique metric-preservingmap (‘isometry’) relating two solutions of generalrelativity related by the hole transformation, the hole argument isthereby blocked; for sceptical responses to this argument, see Menon& Read (2023).

Belot (2018) argues against a single decision univocally in favor orcontrary to Leibniz Equivalence. While allowing that holetransformations relate systems that are physically the same, he arguesthat in some sectors of general relativity, some transformations thatpreserve the metric may relate physically distinct systems.

10. Broader Significance of the Hole Argument

The hole argument has had a broader significance in the philosophy ofscience, pertaininginter aliato realism about theoreticalentities, to theories of quantum gravity, and to the issue of gaugefreedoms in our physical theories. We discuss all three of theserelations in this section; there is also a supplementary documentwhich expands upon the third.

10.1 A Limit to Scientific Realism

The hole argument has presented a new sort of obstacle to the rise ofscientific realism. According to that view, we should read theassertions of our mature theories literally. So, if general relativitydescribes a manifold of events and a metrical structure, then that isliterally what is there in the view of the strict scientific realist.To think otherwise, it is asserted, would be to leave the success ofthese theories an unexplained miracle. If spacetime does not reallyhave the geometrical structure attributed to it by general relativity,then how can we explain the theory’s success?

Appealing as this view is, the hole argument shows that some limitsmust be placed on our literal reading of a successful theory. Or atleast that persistence in such literal readings comes with a highprice. The hole argument shows us that we might want to admit thatthere is something a little less really there than the literal readingsays, lest we be forced to posit physically real properties thattranscend both observation and the determining power of ourtheory.

10.2 The Hole Argument and the Quantization of Gravity

One of the most tenacious problems in modern theoretical physics isthe quantization of gravity. While Einstein’s 1915 generaltheory of relativity produced a revolutionary new way of thinking ofgravitation in terms of the curvature of spacetime, it is generallyagreed now that it cannot be the final account of gravity. The reasonis that it is still a classical theory. It does not treat matter inaccord with the quantum theory.

The problem of bringing quantum theory and general relativity togetherin a single theory remains unsolved. (SeeQuantum Gravity.) There are many contenders, notably string theory and loop quantumgravity. One of the issues that has been raised is that the holeargument has shown us (so goes the claim) that no successful theory ofquantum gravity can be set against an independent, containerspacetime. John Stachel was an early proponent of this outcome of thehole argument. See Stachel (2006). This issue has often been raised byloop quantum gravity theorists specifically as a criticism of stringtheoretic approaches, for string theoretic approaches have such abackground spacetime. See Rovelli and Gaul (2000) and Smolin (2006);for philosophical engagement with these issues of “backgroundindependence”, see Pooley (2017) and Read (2023).

In a related development, Gryb and Thébault (2016) have arguedthat the problem of the hole argument and the “problem oftime” of quantum gravity are essentially the same, givensuitable assumptions. For more, seeProblem of Time in the article on quantum gravity.

10.3 The Hole Argument and Gauge Freedoms

The hole argument has played a role in the growing recognition inphilosophy of physics of the importance of gauge transformations:transformations acting on certain degrees of freedom in our physicaltheories which are supposed to have no correlates in physical reality.The analysis of the hole argument provides philosophers of physicswith a convenient template when they are trying to decide whether ornot something is a gauge freedom. As mentioned in the introduction, welearn from the hole argument that the identification of surplusmathematical structure cannot be achieved by any a priori or purelymathematical rule (at least on the assumption that themathematical/formalist responses to the hole argument discussed abovedo not succeed). Rather, some physical grounds are needed. The holeargument provides two grounds that can be used: (i)verifiability—changes in the candidate surplus structure make nodifference to what can be verified in observation; and (ii)determinism—the laws of the theory are unable to fix thecandidate surplus structure. For more detailed discussion on the holeargument and gauge freedoms, see the supplementThe Hole Argument as a Template for Analyzing Gauge Freedoms.

Bibliography

• Bain, Jonathan, 1998,Representations of Spacetime: Formalismand Ontological Commitment, Ph.D. Dissertation, Department ofHistory and Philosophy of Science, University of Pittsburgh.
• –––, 2003, “Einstein Algebras and the HoleArgument,”Philosophy of Science, 70:1073–1085.
• Belot, Gordon, 1995a, “Indeterminism and Ontology,”International Studies in the Philosophy of Science, 9:85–101.
• –––, 1995b, “New Work for CounterpartTheorists”,British Journal for the Philosophy ofScience, 46(2): 185–195.
• –––, 1996,Whatever is Never and Nowhere isNot: Space, Time and Ontology in Classical and Quantum GravityPh. D. Dissertation, Department of Philosophy, University ofPittsburgh.
• –––, 1996a, “Why General RelativityDoes Need an Interpretation,”Philosophy ofScience, 63 (Supplement): S80–S88.
• –––, 2018, “Fifty Million Elvis FansCan’t be Wrong,”Noûs, 52:946–981.
• Belot, Gordon and Earman, John, 2001, “Pre-Socratic QuantumGravity”, in C. Callender and N. Huggett (eds.),PhysicsMeets Philosophy at the Planck Scale, Cambridge: CambridgeUniversity Press. pp. 213–255.
• Brighouse, Carolyn, 1994, “Spacetime and Holes,” in D.Hull, M. Forbes and R. M. Burian (eds.),PSA 1994, Volume 1,pp. 117–125.
• –––, 2020, “Confessions of a (Cheap)Sophisticated Substantivalist”,Foundations of Physics, 50: 348–359.
• Butterfield, Jeremy, 1988, “Albert Einstein meets DavidLewis,” in A. Fine and J. Leplin (eds.),PSA 1988,Volume 2, pp. 56–64.
• –––, 1989, “The Hole Truth,”British Journal for the Philosophy of Science, 40:1–28.
• Brading, Katherine and Castellani, Elena (eds.), 2003,Symmetries in Physics:Philosophical Reflections, Cambridge:Cambridge University Press, pp. 334–345.
• Corry, Leo, Renn, Juergen, and Stachel, John, 1997, “BelatedDecision in the Hilbert-Einstein Priority Dispute,”Science, 278: 1270–73.
• Curiel, Erik, 2018, “On the Existence of SpacetimeStructure,”British Journal for the Philosophy ofScience, 69: 447–483.
• Dasgupta, Shamik, 2011, “The Bare Necessities”,Philosophical Perspectives, 25: 115–160.
• Dougherty, John, 2020, “The Hole Argument, Take\(n\)”,Foundations of Physics, 50:330–347.
• Earman, John, 1986, “Why Space is not a Substance (At LeastNot to First Degree),”Pacific Philosophical Quarterly,67: 225–244.
• –––, 1986a,A Primer on Determinism,Dordrecht: Reidel.
• –––, 1989,World Enough and Space-Time:Absolute Versus Relational Theories of Space and Time, Cambridge,MA: MIT Bradford.
• –––, 2003, “Tracking down gauge: an ode tothe constrained Hamiltonian formalism”, in K. Brading and E.Castellani (eds.),Symmetries in Physics: PhilosophicalReflections, Cambridge: Cambridge University Press, pp.140–162.
• Earman, John and Norton, John D., 1987, “What PriceSpacetime Substantivalism,”British Journal for thePhilosophy of Science, 38: 515–525.
• Einstein, Albert, 1916, “The Foundation of the GeneralTheory of Relativity,” in H.A. Lorentz et al.,The Principleof Relativity, New York: Dover, 1952, pp. 111–164.
• Friedman, Michael, 1983.Foundations of Space-Time Theories:Relativistic Physics and Philosophy of Science,Princeton, NJ:Princeton University Press.
• Giovanelli, Marco, 2013 “Erich Kretschmann as aProto-Logical-Empiricist: Adventures and Misadventures of thePoint-Coincidence Argument,”Studies in History andPhilosophy of Modern Physics, 44: 115–134.
• Gomes, Henrique and Butterfield, Jeremy, 2023a, “The HoleArgument and Beyond, Part I: The Story so Far”.
• –––, 2023b, “The Hole Argument and Beyond,Part II: Treating Non-isomorphic Spacetimes”.
• Gryb, Sean and Thébault, Karim P. Y., 2016,“Regarding the ‘Hole Argument’ and the‘Problem of Time’,”Philosophy of Science,83: 563–584.
• Halvorson, Hans and Manchak, J.B., “Closing the HoleArgument”,British Journal for the Philosophy ofScience, 2021.
• Healey, Richard, 1999, “On the Reality of GaugePotentials,”Philosophy of Science, 68:432–55.
• Hoefer, Carl, 1996, “The Metaphysics of Space-TimeSubstantivalism,”Journal of Philosophy, 93:5–27.
• Hoefer, Carl and Cartwright, Nancy, 1993, “Substantivalismand the Hole Argument,” in J. Earmanet al. (eds.),Philosophical Problems of the Internal and External Worlds: Essayson the Philosophy of Adolf Gruenbaum, Pittsburgh: University ofPittsburgh Press/Konstanz: Universitaetsverlag Konstanz, pp.23–43.
• Howard, Don and Norton, John D., 1993, “Out of theLabyrinth? Einstein, Hertz and the Goettingen Answer to the HoleArgument,” in John Earman, Michel Janssen, John D. Norton(eds.),The Attraction of Gravitation: New Studies in History ofGeneral Relativity Boston: Birkhäuser, pp. 30–62.
• Iftime, Mihaela and Stachel, John, 2006, “The Hole Argumentfor Covariant Theories,”General Relativity andGravitation, 38: 1241–1252.
• Janssen, Michel, 1999, “Rotation as the Nemesis ofEinstein’s ‘Entwurf’ Theory,” in HubertGoenneret al. (eds.),Einstein Studies: Volume 7. TheExpanding Worlds of General Relativity, Boston: Birkhaeuser, pp.127–157.
• Jammer, Max, 1993,Concepts of Space: The History of Theoriesof Space in Physics, third enlarged edition, New York: Dover,Chapter 6. “Recent Developments.”
• Klein, Martin J.et al. (eds.), 1995,The CollectedPapers of Albert Einstein: Volume 4. The Swiss Years: Writing,1912–1914, Princeton: Princeton University Press.
• Ladyman, James and Presnell, Stuart, 2020, “The HoleArgument in Homotopy Type Theory”,Foundations ofPhysics, 50: 319–329.
• Lusanna, Luca and Pauri, Massimo, 2006 “Explaining LeibnizEquivalence as Difference of Non-inertial Appearances: Dis-solution ofthe Hole Argument and Physical Individuation of Point-Events,”Studies in History and Philosophy of Modern Physics, 37:692–725
• Liu, Chuang, 1996, “Realism and Spacetime: Of ArgumentsAgainst Metaphysical Realism and Manifold Realism,”Philosophia Naturalis, 33: 243–63.
• –––, 1996a, “Gauge Invariance,Indeterminism, and Symmetry Breaking,”Philosophy ofScience, 63 (Supplement): S71–S80.
• Leeds, Stephen, 1995, “Holes and Determinism: AnotherLook,”Philosophy of Science, 62: 425–437.
• Macdonald, Alan, 2001, “Einstein’s HoleArgument,”American Journal of Physics, 69:223–25
• Maudlin, Tim, 1989, “The Essence of Spacetime,” in A.Fine and J. Leplin (eds.),PSA 1988, Volume 2, pp.82–91.
• –––, 1990, “Substances and Spacetimes:What Aristotle Would have Said to Einstein,”Studies in theHistory and Philosophy of Science, 21: 531–61.
• Melia, Joseph, 1999, “Holes, Haecceitism and Two Conceptionsof Determinism”,British Journal for the Philosophy ofScience, 50(4):639–664.
• Menon, Tushar and Read, James, 2023, “Some Remarks on RecentMathematical-cum-formalist Responses to the HoleArgument”,Philosophy of Science.
• Muller, Fred A., 1995, “Fixing a Hole,”Foundations of Physics Letters, 8: 549–562.
• Mundy, Brent, 1992, “Spacetime and Isomorphism,” in D.Hull, M. Forbes and K. Okruhlik (eds.),PSA 1992, Volume 1,pp. 515–527.
• Norton, John D., 1984, “How Einstein found his FieldEquations: 1912–1915,”Historical Studies in thePhysical Sciences, 14: 253–316; reprinted in Don Howard andJohn Stachel (eds.),Einstein and the History of GeneralRelativity: Einstein Studies, Volume 1, Boston: Birkhäuser,1989, pp. 101–159.
• –––, 1987, “Einstein, the Hole Argumentand the Reality of Space,” in John Forge (ed.),Measurement,Realism and Objectivity, Dordrecht: Reidel, pp. 153–188.
• –––, 1988, “The Hole Argument,” inA. Fine and J. Leplin (eds.),PSA 1988, Volume 2, pp.56–64.
• –––, 1989, “Coordinates and Covariance:Einstein’s view of spacetime and the modern view,”Foundations of Physics, 19: 1215–1263.
• –––, 1992, “The Physical Content ofGeneral Covariance,” in J. Eisenstaedt and A. Kox (eds.),Studies in the History of General Relativity (Volume 3:Einstein Studies), Boston: Birkhauser, pp. 281–315.
• –––, 1992a, “Philosophy of Space andTime,” in M.H. Salmonet al.,Introduction to thePhilosophy of Science, Englewood Cliffs, NJ: Prentice-Hall;reprinted Hackett Publishing, pp. 179–231.
• –––, 1993, “General Covariance and theFoundations of General Relativity: Eight Decades of Dispute,”Reports on Progress in Physics, 56: 791–858.
• –––, 2003, “Causation as FolkScience,”Philosophers’ Imprint, 3 (4) [available Online].
• –––, 2003a, “General Covariance, GaugeTheories, and the Kretschmann Objection,” in K. Brading and E.Castellani (eds.),Symmetries in Physics: PhilosophicalReflections, Cambridge: Cambridge University Press, pp.110–123.
• –––, 2019, “Ontology of Space and Time:The Hole Argument”, inEinstein for Everyone:available online.
• Pooley, Oliver, 2002,The Reality of Spacetime, D.Phil.thesis, University of Oxford.
• –––, 2006a, “A Hole Revolution, or are WeBack Where We Started?”,Studies in History and Philosophyof Modern Physics, 37(2): 372–380.
• –––, 2006b, “Points, Particles, andStructural Realism,” in D. Rickleset al.(eds),The Structural Foundations of Quantum Gravity, Oxford: OxfordUniversity Press, pp. 83–120.
• –––, 2013, “Substantivalist andRelationalist Approaches to Spacetime,” in R. W. Batterman(ed.),The Oxford Handbook of Philosophy of Physics, Oxford:Oxford University Press, pp. 522–586.
• –––, 2017, “Background Independence,Diffeomorphism Invariance and the Meaning of Coordinates”, in D.Lehmkuhl, G. Schiemann and E. Scholz (eds.),Towards a Theory ofSpacetime Theories, Basel: Birkhäuser, pp.105–144.
• –––, 2021, “The Hole Argument”, inE. Knox and A. Wilson (eds.),The Routledge Companion toPhilosophy of Physics, London: Routledge, pp. 145–159.
• Pooley, Oliver and Read, James, 2021, “On the Mathematicsand Metaphysics of the Hole Argument”,British Journal forthe Philosophy of Science.
• Read, James, 2023,Background Independence in Classical andQuantum Gravity,Oxford: Oxford University Press.
• Renn, Juergen,et al. (eds.), 2007,The Genesis ofGeneral Relativity: Sources and Interpretations, (Boston Studiesin the Philosophy of Science, Volume 250), 4 Volumes, Berlin:Springer.
• Rickles, Dean, 2005, “A New Spin on the HoleArgument,”Studies in History and Philosophy of ModernPhyscis, 36: pp. 415–34.
• Roberts, Bryan W., 2020, “Regarding ‘LeibnizEquivalence’”,Foundations of Physics 50:250–269.
• Rovelli, Carlo and Gaul, Marcus, 2000, “Loop Quantum Gravityand the Meaning of Diffeomorphism Invariance”, in JerzyKowalski-Glikman (ed.),Towards Quantum Gravity, Berlin:Springer, pp. 277–324.
• Rynasiewicz, Robert, 1992, “Rings, Holes andSubstantivalism: On the Program of Leibniz Algebras,”Philosophy of Science, 45: 572–89.
• –––, 1994, “The Lessons of the HoleArgument,”British Journal for the Philosophy ofScience, 45: 407–436.
• –––, 1996, “Is There a Syntactic Solutionto the Hole Problem,”Philosophy of Science, 64(Proceedings): S55–S62.
• –––, 2012, “Simultaneity, convention, andgauge freedom”Studies in History and Philosophy of ModernPhysics, 43: 90–94.
• Smolin, Lee, 2006, “The Case for BackgroundIndependence”, in Dean Rickles, Steven French and Juha T. Saatsi(eds.),The Structural Foundations of Quantum Gravity,Oxford: Oxford University Press. pp. 196–239.
• Stachel, John, 1980, “Einstein’s Search for GeneralCovariance,” in Don Howard and John Stachel (eds.),Einsteinand the History of General Relativity (Einstein Studies, Volume1), Boston: Birkhäuser, 1989, pp. 63–100. [This paper wasfirst paper read at the Ninth International Conference on GeneralRelativity and Gravitation, Jena.]
• –––, 1986, “What can a Physicist Learnfrom the Discovery of General Relativity?,”Proceedings ofthe Fourth Marcel Grossmann Meeting on Recent Developments in GeneralRelativity, R. Ruffini (ed.), Amsterdam: North-Holland, pp.1857–1862.
• –––, 1993, “The Meaning of GeneralCovariance,” in J. Earmanet al. (eds.),Philosophical Problems of the Internal and External Worlds: Essayson the Philosophy of Adolf Gruenbaum, Pittsburgh: University ofPittsburgh Press/Konstanz: Universitaetsverlag Konstanz, pp.129–160.
• –––, 2006, “Structure, Individuality andQuantum Gravity”, in Dean Rickles, Steven French and Juha T.Saatsi (eds.),The Structural Foundations of Quantum Gravity,Oxford: Oxford University Press. pp. 53–82.
• –––, 2014 “The Hole Argument and SomePhysical and Philosophical Implications,”Living Reviews inRelativity, 17(1):available online.
• Teller, Paul, 1991, “Substances, Relations and ArgumentsAbout the Nature of Spacetime,”The PhilosophicalReview, 100(3): 363–97.
• Teitel, Trevor, 2019, “Holes in Spacetime: Some NeglectedEssentials,”Journal of Philosophy, forthcoming,preprint available online.
• Weatherall, James O., 2018, “Regarding the ‘HoleArgument’,”British Journal for the Philosophy ofScience, 69: 329–350,preprint available online.
• –––, 2020. “Some Philosophical Prehistoryof the (Earman-Norton) Hole Argument”,Studies in Historyand Philosophy of Modern Physics, 70: 79–87.
• Wilson, Mark, 1993, “There’s a Hole and a Bucket, DearLeibniz,” in P. A. French, T. E. Uehling and H. K. Wettstein(eds.),Philosophy of Science, Notre Dame: University ofNotre Dame Press, pp. 202–241.

Academic Tools

	How to cite this entry.
	Preview the PDF version of this entry at theFriends of the SEP Society.
	Look up topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO).
	Enhanced bibliography for this entryatPhilPapers, with links to its database.

Other Internet Resources

Preprints

• Iftime, Mihaela, 2006, “Gauge and the Hole Argument,” [Preprint at arXiv.org]
• Lyre, Holger, 1999, “Gauges, Holes, and their ‘Connections’,” [Preprint at arXiv.org]

Other Resources

• John Norton’s “Goodies” Pages

Related Entries

determinism: causal |Einstein, Albert: philosophy of science |general relativity: early philosophical interpretations of |physics: symmetry and symmetry breaking |substance

Acknowledgments

We are grateful to Erik Curiel, Robert Rynasiewicz and Edward N. Zaltafor helpful comments on earlier drafts; and to the subject editorGuido Bacciagaluppi for suggestions for revisions.