lady man 2007

Scientific StructuralismJames Ladyman and Bas C. van Fraassen

2007 The Aristotelian SocietProceedings of the Aristotelian Soci

IJAMES LADYMAN

ON THE IDENTITY AND DIVERSITY OF OBJECTS IN A STRUCTURE

The identity and diversitygrounded, and intrinsic grounded in haecceities, otion can be grounded in relative or weak discerniwith the denial of haecceiStructuralism implies contal individuation is in gention may be grounded in r

Introduction. According als cannot both individuis because individuation irelation As such, indimetaphysical sense of exgoes on: Certainly, it seeto include self-individuwhich entities have this stropes, and individual sudidates. Presumably Lowin virtue of what a thing its principle of individuathe identity and diversitymust appeal to what is aldividuals cannot individsion is widely acceptedpresuppose numerical div

1 I first heard this pithy formulationot endorse it. The issue was fa(1963; 1965), Chappell (1964), ayety Supplementary Volume LXXXI

23

of individual objects may be grounded or un-or contextual. Intrinsic individuation can ber absolute discernibility. Contextual individua-relations, but this is compatible with absolute,bility. Contextual individuation is compatibletism, and this is more harmonious with science.extual individuation. In mathematics contextu-eral primitive. In physics contextual individua-elations via weak discernibility.

I

to Jonathan Lowe, two different individu-ate, or help to individuate, each other. Thisn the metaphysical sense is a determinationviduation is an explanatory relation in theplanatory (2003, p. 93, his emphasis). Hems that any satisfactory ontology will haveating elements, the only question beingtatusspace-time points, bare particulars,bstances all being among the possible can-es implicit argument is as follows: to say

is the thing it is and not any otherto givetionis thereby to explain the facts about of it and other related things; explanationsready given, therefore relations between in-uate those same individuals. This conclu- and is often expressed thus: Relationsersity and so cannot account for it.1

n from Jeremy Butterfield in conversation, though he maymously discussed by Russell (1911), and also by Allairend Meiland (1966); see MacBride (2006).

IJAMES LADYMAN24


Philosophers of a more empiricist bent disagree with Lowe aboutthe need for self-individuating elements and look instead to theproperties of objects to ground their individuality. Most havethought that the properties in question must be restricted to intrinsicproperties and spatiotemporal relations, and have assumed thatfacts about the identity and diversity of individual objects must beontologically and conceptually prior to their relations to other indi-viduals in a structure. Hence, they agree with Lowe that individualobjects cannot individuahave what John Stachel (

The ante rem structuraart Shapiro among othephysics defended by Steboth metaphysical positirelational structure is onual objects. There are them, the most essential oform of realism about thof the world, whereas anmathematical reality. Nosies about the identity anhave arisen within the de

2 Stachel criticizes Putnam for asshis infamous model-theoretic arg3 There are two versions of mathmathematical structures exist indand an eliminativist position accoare disguised generalizations abopp. 14950). For an excellent survcates of realist structuralism in th(1997) and Shapiro (1997). RecenOntic structural realism was intstructural realism, and has since2003a, 2003b). Both are developmby Worrall (1989) to defend scGowers (2000) historical survey othe thought of Cassirer, Schlick, in regarding structural realism as ern physics as well as a solution tEsfeld (2004), M. Esfeld and V. Lrelationship between ontic structuby Psillos (forthcoming), Busch (realism and further references ca(2007).yety Supplementary Volume LXXXI

te or help individuate each other but must2005) calls intrinsic individuality.2

lism about mathematics defended by Stew-rs, and the ontic structural realism aboutven French and myself among others, areons.3 They have in common the idea thattologically more fundamental than individ-of course important differences betweenf which is that ontic structural realism is ae modal (causal or nomological) structurete rem structuralism is only concerned withnetheless, similar metaphysical controver-d diversity of individuals within a structurebates about each doctrine. It is therefore of

uming that individuality must be intrinsic in the context ofument.ematical structuralism: a realist view according to whichependently of being instantiated by a concrete structure;rding to which statements about mathematical structures

ut sets of objects that exemplify them (see Shapiro, 1997,ey see Reck and Price (2000). The most well known advo-e philosophy of mathematics are Parsons (1990), Resnikt critiques include Hellman (2005) and MacBride (2005).

roduced by Ladyman (1998) in contrast with epistemic been elaborated by French and Ladyman (for example,

ents of ideas introduced into the contemporary literatureientific realism against the pessimistic meta-induction.f structural realism discusses how structuralism figures in

Carnap and Russell. Others who follow Ladyman (1998)a metaphysical response to the ontological import of mod-o the problem of theory change include: Bain (2004; MS),am (forthcoming), Lyre (2004), Stachel (2002; 2006). Theral realism and ante rem structuralism has been explored2003) and Pooley (2006). A full discussion of structuraln be found in chapters 2 and 3 of Ladyman and Ross

ON THE IDENTITY AND DIVERSITY OF OBJECTS IN A STRUCTURE 25


some importance that the range of possible positions on these issuesbe clarified.4 Once this is done it is apparent that the argumentsagainst the structuralist idea of what Stachel calls contextual indi-viduality collapse into expressions of prejudice. Esoteric posits suchas tropes and bare particulars seem an embarrassment of metaphys-ical riches to a naturalistic philosopher interested in the metaphysicsof mathematics and science (and self-individuating spacetime pointsare mismatched with the diffeomorphism invariance of General Rel-ativity). Yet individuatioan option for either matther. For both the ante realist the identity and drelational structure of whave self-individuating elof the other individuals inparts.

The next section explain the context of contemplains how relations mayobjects in a structure. Intheory illustrate the issueality grounded in relationics. Hence, it is argued tungrounded or primitivebetween primitive identilast section notes the keymathematical and physic

Intrinsic Individuality. It

(i) The existence ofindependent of o

4 Structuralism has also become ptialism. This is the doctrine that texhaust their natures. See Shoemtheory of properties, and Bird (2turalist. Harte (2002) discusses ayety Supplementary Volume LXXXI

n by qualitative intrinsic properties is nothematical or physical objects in general ei-m structuralist and the ontic structural re-iversity of objects is dependent on thehich they are parts. Individuals need notements, nor be individuated independently the relational structures of which they are

ins briefly how intrinsic individuality faresporary physics. The following section ex- be used to ground identity and diversity of section four some examples from graphs and show that even contextual individu-s does not obtain in general in mathemat-

hat the ante rem structuralist must accept contextual individuality. The relationshipty and haecceitism is then considered. The issues concerning the relationship betweenal structure.

II

is usually assumed that:

an individual in spacetime is ontologicallyther individuals (except for those that are

opular in metaphysics recently in the form of causal essen-he causal relations that properties bear to other propertiesaker (1980), Mumford (2004), which adopts a structural007), whose theory of dispositions is in some ways struc-n interesting Platonic form structuralism.

IJAMES LADYMAN26


its proper parts).5 Facts about the numerical identity and di-versity of individuals are determined independently of theirrelations to each other.

(ii) Each has some properties that are intrinsic to it.6

Haecceity, the property of primitive self-identity or primitive this-ness, is posited to answer the general question as to what groundsfacts about the numerical identity and diversity of individuals.7 Ifeach and every fundamethen their individuality isnot inclined to invoke suof individuation offers emality while only quantifreach of natural science. to be explained by the possess, then it would sediscernibles (PII), restrictinvolving properties are some properties (perhapsdistinguish each thing froindividuality of physicalqualitative terms.

Quantum Mechanics ature of space, time and mthat describes the world a

5 Einstein in a letter to Max Bornp. 7 and Hagar, 2005, p. 757) safrom everyday thinking. He alsosequent developments seem to haand Ross (2007, ch. 1) call for th6 It is also often claimed that all temporal relations supervene onnience). Ladyman and Ross (20Humean supervience. It may alsoseem to be extrinsic to individual7 The notion of haecceity originawas revived in the guise of primitthought to engender an infinite rcan be primitively the particular hviduality of objects be primitive, dis also reasonable to question whground anything.yety Supplementary Volume LXXXI

ntal individual possesses such a property intrinsic. However, many philosophers arech metaphysical devices. The bundle theory

piricists a way of accounting for individu-ying over properties that are within theIf the identity and individuality of objects isempirically accessible properties that theyem that the Principle of the Identity of In-ed so that only qualitative and not identityin its scope, must be true. If so there are including spatio-temporal properties) thatm every other thing, and the identity and objects can be accounted for in purely

nd General Relativity teach us that the na-atter are not in keeping with a metaphysicss composed of self-subsistent distinguisha-

(Born, ed., 1971, pp. 1701; quoted in Maudlin, 2002,ys that the idea of independently existing objects comes regards it as a necessary presupposition of physics. Sub-

ve proved him wrong in this second speculation. Ladymane abandonment of such everyday thinking in metaphysics.the relations between individuals other than their spatio- the intrinsic properties of the relata (Humean superve-07, ch. 3) argue that contemporary physics undermines be argued that all the properties of fundamental physics objects.tes with Duns Scotus but in contemporary metaphysics itive thisness by Adams (1979). Positing haecceities may beegress for what individuates each haecceity? If haecceitiesaecceities that they are, then why not allow that the indi-ispensing with the need for haecceities in the first place? Itether such metaphysical posits ever genuinely explain or



ble individuals. In the case of Quantum Mechanics the problem isthat there are quantum states of many particles that attribute exact-ly the same properties to each of them. For example, the famous sin-glet state of two fermions, such as electrons, attributes to the pairthe relation that their spins in any given direction are opposite toeach other, but does not attribute a definite spin in any direction toeither particle alone. Given that they may also be attributed exactlythe same spatial wavefunction, as when they are both in the first or-bit of an atom for exampusually formulated. ThisSteven French and Michcles are not individuals, individuation that applikind of empirically translike.

In the case of Generalgeneral covariance of tmeans that any spacetimphism (a infinitely differself) are in all observabphysical properties are exlationships between geomof spacetime are intrinsicer, and it makes no differlong as the overall strucapparent by the so-calledomorphic models are regbreakdown of determiniabandonment of the ascpoints (1996, p. 11). Byviduality.

Of course, these concompel us to abandon thed objects each possessindividuated quantum parpossess all their metricalphysical possibilities. Hoof such a world would beyety Supplementary Volume LXXXI

le, then clearly such particles violate PII as leads to a dilemma that was articulated byael Redhead (1988); either quantum parti-or they are individuals but the principle ofes to them must make reference to somecendent haecceity, bare particularity or the

Relativity the problem is more subtle. Thehe field equations of General Relativitye model and its image under a diffeomor-entiable, bijective map of the model to it-le respects equivalent to one another; allpressed in terms of generally covariant re-etrical objects. In other words, the points

ally entirely indiscernible one from anoth-ence if we swap their properties around soture remains the same. This is made more hole argument, which shows that if diffe-arded as physically distinct then there is a

sm. Carl Hoefer argues in response for theription of primitive identity to space-time

primitive identity he means intrinsic indi-

siderations from physics do not logicallye idea of a world of intrinsically individuat-g intrinsic properties. Transcendentally in-ticles and spacetime points that essentially properties (see Maudlin, 1990) are meta-wever, physics tells us that certain aspects unknowable in principle.

IJAMES LADYMAN28


Frank Jackson (1998), Rae Langton (1998) and David Lewis(forthcoming) advocate a world of unknowable intrinsic natures.Jackson refers to Kantian physicalism (pp. 234), Langton toKantian Humility, and Lewis to Ramseyian Humility. Jackson ar-gues that science only reveals the causal/relational properties ofphysical objects: We know next to nothing about the intrinsic na-ture of the world. We know only its causal cum relational nature(p. 24). Langton argues that science only reveals the extrinsic prop-erties of physical objectstures, and hence the intrinaccessible. Jackson pointhe natures of objects awith their relational or makes a mystery of whata naturalist unknowableare merely philosophicachine whose workings dofeld (2004, pp. 61416), a gap between epistemolo

Given that there is noworld must be composedintrinsic natures, and givsure on such a view, it mought to deny that quantviduals.8 I used to take tas realism about structurvidual quantum objects vival of the weak form2003a, 2003b and 2006structural realism ought individuals, but ought insindividuals it is the relatiilarly, Oliver Pooley (200ual spacetime points caGeneral Relativity, if it isand diversity is grounde

8 In fact Jonathan Lowe (2005, pralist metaphysics.yety Supplementary Volume LXXXI

. Both then argue that their intrinsic na-insic nature of the world, are epistemicallyts out that this inference can be blocked if

nd their intrinsic properties are identifiedextrinsic properties, but argues that this it is that stands in the causal relations. To intrinsic natures and things in themselvesl toys; idle wheels in a metaphysical ma- not affect the observable world. Like Es-

Ladyman and Ross (2007) take it that suchgy and metaphysics is unacceptable. a priori way of demonstrating that the of intrinsically individuated objects withen that our best physics puts severe pres-ight seem that the naturalistic philosopherum particles and spacetime points are indi-his line and defend ontic structural realisme combined with eliminativism about indi-and spacetime points. Simon Saunders re- of the identity of indiscernibles (see his) convinced me that the defender of onticnot to insist that quantum particles are nottead to emphasize that in so far as they are

ons among them that account for this. Sim-6) argues that eliminativism about individ-n be avoided without any tension with accepted that the facts about their identityd in relations they bear to each other. His

. 78) adopts this position despite his avowedly non-natu-



sophisticated substantivalism allows that spacetime points be indi-viduated relationally and not independently of the metric field. Thismeans embracing contextual individuality grounded in relationalstructure.

III

Grounding Identity and ics of ante rem structuraabout the identity and dmined by their places imathematical objects whobjects are numerous thThe field of complex nphism, which is to say thof the complex plane ontmap, f, is structure-presein the domain, and for x*y=z, f(x)*f(y)=f(x*y)R, xRy implies that f(x)Rplex number of the fornumber xiy. So, i is 8(10)i=8+10i, and sis completely preserved bcomplex numbers are rproperties then i and i each other. Hence, it is aabout mathematical objetinct but indiscernible mdiffer only haecceitisticalthat there is no more toproperties must be aband

9 This objection is due to Burges(2004; 2005).10 It is easy to check that this opmatical structures with non-triviaby the integers under addition wispace which possess various symmyety Supplementary Volume LXXXI

Diversity in Properties and Relations. Crit-lism have claimed that the idea that factsiversity of mathematical objects are deter-n structures cannot account for cases ofich violate standard PII.9 Examples of suche most well-known being that of i and i.umbers admits of a non-trivial automor-ere is a structure-preserving bijective map

o itself other than the identity map. (Here arving just in case, for any objects, x and yany operation between them, *, such that=f(z), and for any relation between them,f(y).) The map in question takes each com-m x+iy and replaces it by the complexmapped to i and 810i is mapped too on.10 The structure of the complex planey this mapping so if the properties of the

estricted to their relational mathematicalpossess all and only the same properties asrgued that the structuralist who is a realistcts faces a dilemma; either apparently dis-athematical objects are identical, or they

ly. If the latter horn is grasped then the idea mathematical objects than their structuraloned and ante rem mathematical structur-

s (1999) and Kernen (2001) and is pressed by MacBride

eration is structure preserving. Other examples of mathe-l automorphisms (symmetries) include the group formed

th the map z goes to z, and the set of points in Euclideanetries including, for example, arbitrary rotations.

IJAMES LADYMAN30


alism collapses into traditional Platonism. If the former horn isgrasped then the mathematical structuralist is contradicting thepractice of mathematicians which treats the mathematical objects inquestion as being distinct. As Fraser MacBride (2005, p. 582) putsit, is seems that ante rem mathematical structuralism is either oldnews or bad news.

The standard philosophical example of a structure that admits ofa non-trivial automorphism, and hence violates standard PII, is thatof Max Blacks two intrapart in empty space (19the same world. Similarlof the non-trivial automHowever, Simon Saundeclaim that PII is false fointroducing Quines (196formulations of PII, nam

Two objects are absoin one free variable whicFor example, ordinary pbecause they occupy diffely discernible mathematisquare of itself and i is no

Two objects are relatimula in two free variablMoments in time are relaisfy the earlier than remathematical objects whare relatively discerniblespace with an orderingpoints, x and y, if they axy or yx but not both

Finally, two objects artwo-place irreflexive relaare obviously only weakthe singlet state and i anthat the entities involved

11 Ladyman (2005) deploys Saundagainst the identity problem. Foryety Supplementary Volume LXXXI

insically identical spheres that are a mile52). Permutation of the spheres results in

y, the singlet state of two fermions admitsorphism that permutes the two particles.rs (2003a; 2003b; 2006) challenges the

r fermions in such entangled states by re-0, p. 230; 1976) distinction between threeely absolute, relative and weak.11

lutely discernible if there exists a formulah is true of one object and not the other.hysical objects are absolutely discerniblerent positions in space and time. Absolute-cal objects include i and 1 since 1 is thet.

vely discernible just in case there is a for-es that applies to them in one order only.tively discernible since any two always sat-lation in one order only. An example ofich are not absolutely discernible but which include the points of a one-dimensional relation, , since, for any such pair ofre they are not the same point then either.e weakly discernible just in case there istion that they satisfy. Blacks two spheresly discernible. So too are two fermions ind i. All these examples have in common stand in some irreflexive (but symmetric)

erss version of PII to defend mathematical structuralism discussion see Ketland (2006) and MacBride (2006).



relation to each other.The weak notion of individuality advocated by Saunders (accord-

ing to which weak discernibility is sufficient for individuality) seemscoherent. It would be question-begging to deny the sufficiency ofweak discernibility merely because stronger forms of discernibilityare sometimes available. Note however that while Saunderss viewvindicates an ontology of individuals in the context of QuantumMechanics, it is a thoroughly structuralist one in so far as objectsare not assumed to be inrelations in which they stated.

In the context of philohave followed Russell inthere could be individualties, but whose individuaindividuals:

it is impossible thanothing but the termsthey are to be anythithey must differ from from sounds. What Deinition by means of thmade always indicatesof their own. (Russell,

The argument is that witically prior to the relatiometric relations that arerelata. Contemporary phinfluenced in accepting tPaul Benacerraf (1965) wcalled must be individuastrual of abstract objectsacerraf, an object withidentified with any objecstructure and could not t

The metaphysical artiproperties must have intions is one that ante reyety Supplementary Volume LXXXI

dividuated independently of the nexus ofand. Rather they are contextually individu-

sophy of mathematics, many philosophers arguing that it is incoherent to supposes which dont possess any intrinsic proper-lity is conferred by their relations to other

t the ordinals should be, as Dedekind suggests, of such relations as constitute a progression. Ifng at all, they must be intrinsically something;other entities as points from instants, or coloursdekind intended to indicate was probably a def-e principle of abstractionBut a definition so

some class of entities having a genuine nature 1903, p. 249)

hout distinct individuals that are metaphys-ns, there is nothing to stand in the asym- supposed to confer individuality on theilosophers of mathematics have been mosthis insistence on intrinsic individuation by

ho argued that objects to be properly sols, and that therefore a structuralist con- like numbers must fail. According to Ben- only a structural character could be

t in the appropriate place in any exemplaryherefore be an individual.cle of faith to the effect that objects andtrinsic natures prior to entering into rela-m structuralists can simply reject. Charles

IJAMES LADYMAN32


Parsons opposes Kernen (2001) and others who demand meta-physical accounts of objecthood in terms of haecceity, self-individu-ating elements, or intrinsic natures:

There is a reason for my resistance, and this is that the structuralistview of mathematical objects coheres with a rather thin conceptionof what an object is, that the most general concept of object derivesfrom formal logic, that we are speaking of objects when we use the ap-paratus of singular terms, identity and quantification. This thin con-ception has a traditionFrege, Carnap and Quimplications. It could ject is a formal concep

Parsons includes the call the scope of what Kernuse of irreflexive relationof indiscernibles is repeaan authority here. Howeis may turn out to be evenLeitgeb and Ladyman (foory that violate even weasection suggest that, at leor diversity of objects or for by anything other thversity of objects or placbe viewed as integral comas, for example, the succthe structure of natural sponding cases in physicsfinal section.

12 Quine has no problem with imviduation in his critique of Davidicativity is more plausible in epist(forthcoming).13 Button (2006) mentions similthem; his arguments are discussedarrives at similar conclusion to thyety Supplementary Volume LXXXI

behind it, whose principal representatives areine; it is particularly Quine who has pressed itsbe described as the view that the concept of ob-t. (Parsons, 2004, p. 75)

for a principle of individuation, and PII, inen illicitly requests. On the other hand, thes to formulate a weak form of the identitytedly endorsed by Quine whom he cites asver, the right conception of what an object thinner than Quine and Saunders argue.12

rthcoming) consider cases from graph the-k PII.13 The examples discussed in the nextast in the case of mathematics, the identityplaces in a structure is not to be accountedan the structure itself. The identity and di-es in a structure are relations that ought toponents of that structure in the same way

essor relation is an integral component ofnumbers. Whether or not there are corre- is an open question briefly discussed in the

predicative definitions while rejecting impredicative indi-sons theory of events. This is peculiar; insisting on pred-emological rather than ontological contexts. See Horsten

ar examples, while drawing dissimilar conclusions from in Leitgeb and Ladyman (forthcoming). Ketland (2006)e latter as does Shapiro (forthcoming).



IV

Lessons from Graph Theory. Graphs are mathematical structuresthat contain only two kinds of entity, namely nodes and edges be-tween nodes. Graphs may be undirected or directed depending onwhether the relation between two nodes of their being joined by anedge is presupposed to be a symmetric or an asymmetric relation re-spectively. Graphs are unlabelled or labelled. In an unlabelled graph,different nodes are ind(which is why unlabelledists). Labelled graphs aretional assignment of lingwhich nodes become disttaken in isolation.14 Howtwo nodes in a labelled gis regarded as the same gdepending on whether ornodes and leaves the grapwhether or not there is a

Graph theory clearly eative discernibility, withoa directed graph with twexample does not admitother directed graphs donode has a directed edge and where each node maand the resulting structuin this graph are relativeence of a non-trivial autoonly weakly discernible nodes are not absolutely

14 In the famous Black (1952) patinguishing between the two globis directly analogous to the case oacter points out that to use ratherway of fixing their reference is avin graph theory that such labels c15 Simple graphs, as characterizenodes may be connected by morenect more than two nodes. It is nodes to themselves (loops). Thesyety Supplementary Volume LXXXI

istinguishable if considered in isolation graphs are of special interest to structural- unlabelled graphs that come with an addi-uistic or numerical labels to their nodes, byinguishable by means of their labels even ifever, it should be noted that if the labels ofraph with no edges are permuted the resultraph. Graphs are symmetric or asymmetric not there is a function that rearranges theh structure unchanged, in other words, on

non-trivial graph automorphism.15

xemplifies the different versions of PII. Rel-ut absolute discernibility, is exemplified byo nodes and one edge joining them. This

of a non-trivial automorphism. However, such as the three node graph where eachcoming towards it and going away from it,y be mapped to its downstream neighbour

re is the same as the original. All the nodesly discernible from each other, so the exist-morphism is not sufficient for there beingobjects in a structure. Note also that the discernible and so if anything other than

per one of the two characters in the dialogue suggests dis-es by calling one Castor and the other Pollux; this casef a labelled edgeless graph with order two. The other char- than mention these names one must presuppose that someailable which begs the question at issue. It is just assumedan be deployed.d here, can be generalized to multigraphs, in which two than one edge, or hypergraphs, where one edge may con-also possible to consider graphs with edges that connecte complications do not play any role in the following.

IJAMES LADYMAN34


haecceities accounts for the facts about the identity and diversity ofthe nodes then it must be relations that do so. (The existence of anon-trivial automorphism is of course sufficient for the failure ofabsolute discerniblity.)

The case of weak discernibility, without absolute or relative dis-cernibility, is exemplified by the following unlabelled graph G withtwo nodes and one edge. This is the graph-theoretic counterpart ofBlacks two-spheres universe (or the complex field substructure con-sisting of the imaginary umions):

G obviously admits of a metric graph. However, since they stand in the ipressed x is connected todiscernibility of Identicadistinct.16

If the standard graph-tis applied to G the result

For a graph theorist, Gis.17 Indeed, according tobelled (simple, loopless) (A large part of graph (classes of) unlabelled grspect to the debate aboutjects in a structure. Sinceleaves the graph unchan

16 The Distinctness of Discernibleand the contrapositive of PII is th17 Shapiro (1997, p. 115), considebe regarded as corresponding to sponding reference class is the coence class of G is the collectionedges is constitutive of the latter, eShapiros 2-pattern could just as wyety Supplementary Volume LXXXI

nits i and i, or the singlet state of two fer-

non-trivial automorphism and so is a sym-the two nodes in G are weakly discerniblerreflexive (though symmetric) relation ex- y by an edge (in G). Therefore, by the In-ls (which is not contentious), they are

heoretic operation of taking away an edgeing graph G looks as follows:

is just as much an unlabelled graph as G graph theory there are precisely two unla-graphs with two nodes, namely, G and G.theory is devoted to the enumeration ofaphs.) G is of particular interest with re- the identity and diversity of individual ob- permuting the two nodes of G obviouslyged, G allows for non-trivial automor-

s is the contrapositive of the Indiscernibility of Identicals,e Discernibility of the Distinct.rs patterns of small cardinal numbers. His 2-pattern canthe above graph G, however, while in his case the corre-llection of cardinal numbers, in the present case the refer- of unlabelled graphs. While the possible occurrence ofdges do no play any role whatsoever in the former. Hence,ell be identified with G rather than G.



phisms. Yet there is no irreflexive relation that may be used toground the identity or difference of the nodes in accordance withthe weak version of PII, nor is there any need for one; to the graphtheorist the two nodes in G are perfectly respectable mathematicalobjects that are distinct from each other.18 The fact that G consistsof two nodes is simply part of what G is being part of its graph-the-oretic structure. The analogue of the structuralist slogan about nat-ural numbers (as stated, for example, in MacBride, 2005, p. 583):There is no more to therelations they bear to eadifference of nodes is inca graph bear to each othe

Graphs such as G graphs that contain at leaof the 156 possible unlathat are not even weakpoint can be made abouttwo distinct but isomorpwords, two isomorphic leads from a node withinit. Consider, for example

G is symmetric and theway of grounding the difbrute structural fact abonents that are structurallgeneral, the identity and counted for by anything

Before considering thetive identity, it is worthweak PII is satisfied is notion of whether the factsuals in a structure can brelations among them (ex

18 There are many other exampleven weakly discernible that havsuggests the example of the Kleinyety Supplementary Volume LXXXI

individual nodes in themselves than thech other is true provided that identity andluded among the relations that the nodes inr.

are not exceptional; all other unlabelledst two isolated nodes (for example, 11 out

belled graphs with 6 nodes) include nodesly discernible. Furthermore, an analogous all unlabelled graphs which include at leasthic and unconnected components, in othersubgraphs for which there is no edge that one of the subgraphs to a node outside of, the following graph G:

re is no irreflexive relation, nor any otherference between its two components. It is aut this graph that it includes two compo-y indistinguishable if taken in isolation. Indiversity of nodes in a graph cannot be ac-than the graph structure itself. details of this return to the idea of primi- noting that the question of whether onlyt (as I used to think) the same as the ques-

about the identity and diversity of individ-e accounted for by and only by the (other)cluding haecceities). As noted above, there

es of mathematical structures with elements that are note a less trivial structure. Philip Welch (in correspondence) 4-group.

IJAMES LADYMAN36


are directed graphs in which the nodes admit of relative but not ab-solute discernibility, and hence where only relations can account forthe diversity of the nodes. On the other hand, there are also cases ofabsolute discernibility where relations can account for the identityand diversity of the nodes. The following undirected graph is asym-metric:

Each node satisfies a struas follows. Let a node beach node it is related tonode is the number of no

(3), (134), (234

Hence, each node in thisof the others. Yet clearlybetween the nodes determsity of the individual nodfor intrinsic as opposed individuality is compatib

According to the aboving grounded in qualitatprimitive identity is assocderided as purely metaphated with haecceitism (thin respect of the permutatcauses problems in the inmutation invariance of qvariance of models of Gethese problems is what mplace (Ladyman, 1998).

19 See Lewis (1986).yety Supplementary Volume LXXXI

cture description that no other node doese described by a list of numbers, one for, and where the number assigned to that

des to which it is related.

), (34), (1233), (4)

graph is absolutely discernible from each in this example facts about the relationsine the facts about the identity and diver-es. Absolute discernibility is not sufficient

to contextual individuality, and contextualle with the traditional form of PII.e, individuality is primitive, rather than be-ive properties or relations. The notion ofiated with the theory of haecceity that wasysical above. Worse, haecceities are associ-e claim that there are worlds differing solelyion of individuals19), and the latter doctrineterpretation of physics because of the per-uantum states and the diffeomorphism in-neral Relativity. Indeed, the need to avoidotivated ontic structural realism in the firstRecall also that MacBride (2005) argues



that irreducible identity facts amount to old news because they turnstructuralism into traditional Platonism with haecceities.

However, primitive contextual individuality is different to primi-tive intrinsic individuality (whether or not the latter is construed interms of haecceities), for only the latter and not the former implieshaecceitism. If individuation is intrinsic, and not grounded in quali-tative properties but is either ungrounded or grounded in haeccei-ties, then the identity of an individual objects is determinate in othercounterfactual situationsdiscernible object gives rlan (1975, p. 722) definesense to ask questions abindependently of their printrinsic individuality anviduality is contextual thtalk of the same object inIn the mathematical caseless unlabelled graph, ansame graph results whichstructurally similar indiviexactly the same structuin G above does not resuis true even if the graph marked above, permutatboth particle physics andsituations that differ onlyas distinct. Contextual inist, and primitive conteceitism and so is not old

20 Also if we have a graph with there is no fact of the matter aboone that was removed and replaceof one node. This makes unlabeldiachronic as well as synchronic cussion in Lowe, 2004).yety Supplementary Volume LXXXI

, and permuting an object with another in-ise to another state of affairs. Indeed Kap-s haecceitism as the doctrine that it makesout the transworld identity of individuals

operties and relations; he equates primitived haecceitism. On the other hand, if indi-en there is in general no reason to regard another relational structure as intelligible. this is clear. Consider the three node edge-d the operation of removing a node. Theever node we remove.20 Permuting exactlyduals in a mathematical structure results inre. For example, permuting the two nodeslt in a new graph (and as noted above, this

is labelled). In the physical case, as was re-ion invariance of individuals is a feature of spacetime physics. Physicists do not regard in virtue of the permuation of individualsdividuality is the only option for a natural-xtual individuality does not imply haec-news.

two nodes and one is removed and then a node is addedut which of the nodes in the graph we end up with is thed and which was left on its own in the intermediary graph

led graphs a good model for discussing issues concerningidentity in quantum mechanics (see, for example, the dis-

IJAMES LADYMAN38


V

The Relation Between Mathematical and Physical Structuralism. Insum, philosophical theories of the identity and diversity of individu-al objects in a structure, can be distinguished according to whetherindividuation is posited to be grounded or ungrounded, and intrin-sic or contextual. Intrinsic individuation can be grounded in haec-ceities, or absolute discernibility based on intrinsic qualitativeproperties. Intrinsic identplies haecceitism. Contextions but this is compdiscernibility. Contextuagrounded is compatible more harmonious with malism in general implies ccal case contextual indstructure. In physics conrelations via weak discerThe weak indiscernibilityto be sufficient for denyThe existence of mathemcernible may be one imand physical objects, or iclaim that elementary bofermions and like nodes gregated (although not and Krause, 2006 for disponding formal framew

Another philosopher wphysics of physical realitis Dipert (1997). He advthat the world is an asymabout the numerical idgrounded in absolute discdescriptions, whereas in mit of exactly the same

21 Bird (2007) also thinks that theture that represents the relationshtions is necessary to avoid the reyety Supplementary Volume LXXXI

ity, whether grounded or ungrounded, im-tual individuation can be grounded in rela-atible with absolute, relative or weakl individuation whether grounded or un-with the denial of haecceitism, and this isathematical and physical theory. Structur-

ontextual individuation. In the mathemati-ividuation is in general primitive in atextual individuation may be grounded innibility for fermions and spacetime points. of elementary bosons is taken by Saundersing them the status of individual objects.atical objects that are not even weakly dis-portant difference between mathematical

t may be taken as a reason for disputing thesons are not objects. After all, bosons, likein an unlabelled edgeless graph, can be ag-enumerated; see Teller, 1995, and Frenchscussion and the development of a corre-ork).ho has applied graph theory to the meta-

y in defence of a broadly structuralist viewocates contextual individuality, but claimsmetric graph because he believes that factsentity and diversity of objects must beernibility in the form of a unique structuresymmetric graphs there are nodes that ad-structure descriptions.21 In the light of the

absence of non-trivial automorphisms in the graph struc-ip between dispositions and their stimuli and manifesta-

gress of pure powers.



above it seems that the failure of all forms of discernibility is con-ceptually possible. Carnap (1928), in his famous example of a rail-track network (cf. 14 of his Aufbau), acknowledges the possibilitythat the world might present itself as allowing for non-trivial auto-morphisms, in which case there would be two places in the networkwhich would have to be regarded as indistinguishable by all scientif-ic means. It remains an open question whether the empirical worldhas such a structure.

The following worry nmatical theories is relevastructural aspects of theries are relevant to ontoltinguish physical and mdelivered in Leiden in 19the problem with radical

It must imply: what hunknown qualitative fthis, the contrast betwappeared. Thus, fromtion, any difference bdisappears. It seems thism at all! For if thereBut for those who do external or prior poinbe entirely re-conceive

The essence of van Fraasstween mathematical (unin(instantiated/concrete) ststructural terms.22 He recontext of Fock space fothat there is cannot merebecause then there wouldpied and a cell being unotalks of occupation numthe cell must be a non-str

22 There is an analogy here with th23 A similar complaint is made byreason to think that ontic structuthe world is mathematical.yety Supplementary Volume LXXXI

ow arises: If only the structure of mathe-nt to ontology in mathematics, and only

mathematical formalism of physical theo-ogy in physics, then there is nothing to dis-

athematical structure. In a paper first99, van Fraassen argues that the heart of

structuralism is this:

as looked like the structure of something witheatures is actually all there is to nature. But witheen structure and what is not structure has dis- the point of view of one who adopts this posi-etween it and ordinary scientific realism alsoen that, once adopted, not be called structural-

is no non-structure, there is no structure either.not adopt the view, it remains startling: from ant of view, it seems to tell us that nature needs tod. (van Fraassen, 2006, pp. 2923)

ens objection here is that the difference be-stantiated/abstract) structure, and physical

ructure cannot itself be explained in purelyiterates the point (2006, pp. 2934) in thermalism used in quantum field theory: allly be the structure of this space, he insists, be no difference between a cell being occu-ccupied. However, just because our theory

bers does not imply that what is occupyinguctural object, individual or not.23

e theory of universals and the problem of exemplification. Cao (2003). Saunders (2003c) points out that there is noral realists are committed to the idea that the structure of

IJAMES LADYMAN40


Physical structure exists, but what is it? What makes the world-structure physical and not mathematical? Ladyman and Ross (2007)advocate a kind of neo-positivism according to which when ques-tions like this arise it is time to stop; however, perhaps van Fraassenis right that I should have stopped earlier.24

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James Ladyman - On the Identity and Diversity of Objects in a StructureI. IntroductionII. Intrinsic IndividualityIII. Grounding Identity and Diversity in Properties and RelationsV. The Relation Between Mathematical and Physical StructuralismIV. Lessons from Graph TheoryReferences

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