towards a field ontology

dialectica

Vol. 60, N° 1 (2006), pp. 5–27DOI: 10.1111/j.1746-8361.2005.01019.x

© 2005 The Author. Journal compilation © 2005 Editorial Board of

dialectica

Published by Blackwell Publishing Ltd., 9600 Garsington Road, Oxford, OX4 2DQ, UK and 350Main Street, Malden, MA 02148, USA

Christina Schneider

Towards a Field Ontology

Christina S

chneider

†

A

BSTRACT

The aim of the present article is to make the notion of an ontology of fields mathematicallyrigorous. The conclusion will be that couching an ontology in terms of mathematical bundlesand cross-sections (i.e. fields) both (1) captures many important intuitions of conventionalontologies, including the universal-particular paradigm, the connection of universals and their‘instantiations’, and the notion of ‘possibility’, and (2) makes possible the framing of ontologieswithout ‘substrata’, bare particulars, and primitive particularizers (a goal that trope ontologies,for example, have sought to attain).

A field ontology is one that takes mathematical bundles and the structures asso-ciated with them as its model and paradigm. The approach aims at what is heredubbed

ontological field theory

or simply

field theory

. The theory with which thisarticle is concerned is inspired by Campbell’s proposal to conceive of tropes asfields (Campbell 1990, 145), as well as by von Wachter’s more elaborate discus-sions (von Wachter 2000b and von Wachter 2000a). This article attempts toimprove on Campbell and von Wachter by making the notion of an ontology offields mathematically more precise.

The article is organized as follows: The first section formulates some thesesand methodological presuppositions that govern the enterprise to follow. Thesecond section presents some formal preliminaries. The third section makesCampbell’s and von Wachter’s intuition more precise. This reveals, among otherthings, that conceiving tropes under the paradigm of fields (cross-sections ofmathematical bundles), as Campbell and von Wachter do, is at variance with theunicategorial

credo

of trope theory. The fourth section presents the framework foran ontological field theory, paying particular attention to

possibilities

and reinter-preting the relevant formalism. The final section looks further afield.

1. Preliminary theses

Field ontology, as it is understood and formulated in this article, concerns fieldsas they are defined in mathematics rather than on a merely intuitive level. Fieldontology, as it is proposed here, is based upon two major ontological and philo-sophical theses.

†

Instit für Statistik, Ludwig-Maximiliams-Universität Munich, Germany; Email:[email protected]@uwyo.edu

6 Christina Schneider


dialectica

The first thesis is rather weak, it claims that universals – better, ‘genericaspects’ – should play an important role in ontology. The second thesis is a boldone and is, presumably, rejected by most philosophers: It is the claim that anontological framework should be

fundamentally

indeterministic; determinismshould hold – at most – only within specific regions of the ontological domain.

1

The question of the ontological status of ‘universals’ or generic aspects arises,because, given that two or more complex entities can exhibit the same feature(s),it must be determined how one and the same feature can be multiply particular-ized.

2

Historically, within the framework of substance ontologies, such featuresare presented as universals that are said to be

instantiated

in substances.

3

Asinstantiations they are particular – a universal is not instantiated twice at oneinstant in one substance. Yet because any one universal may be instantiated in two(or more) different substances, any one that is will be ‘wholly present’ in two (ormore) different regions of space or space-time; this appears to be incoherent.Those who – like proponents of individual accidents as well as trope theorists ofseveral types – dispense with universals (or generic aspects) as ontological entitiesin their own right face the task of explaining how two individuals may be (quali-tatively) exactly similar in one aspect.

4

Because field ontologies are different fromsubstance ontologies, the term ‘universal’ will not here be used. Instead, ‘genericaspects’ or ‘characteristics’ are the terms for – pretheoretically speaking – general(qualitative) aspects. In conjunction with generic aspects, there are particularaspects, which play a role somewhat similar to that played by ‘instantiations ofuniversals’ within the frameworks of substance ontologies. Particular aspects arebound to their generic aspects – unlike tropes, they cannot be conceived of beingentities in their own right; nor are there any generic aspects without their respec-

1

Determinism

is here understood in the sense put forward by Laplace. Roughly speak-ing, a world is

deterministic

in this sense if and only if at any instant of its history it has a uniquefuture as well as a unique past such that any alternative future or past is impossible. For detailcf. Earman 1986, 12.

Indeterminism

, as understood here, is the opposite of this understandingof determinism. A world is

indeterministic

iff there are instants of its history that have morethan one possible future and past. Of course, only one of them will be or has been actual.

2

The term ‘generic aspect’ is unusual. It is used here to indicate that these entities figurewithin a special ontological framework, that of

field ontology

. Intuitively, they share somefeatures with universals. This is discussed below.

3

Here, the term ‘substance ontology’ is used to indicate an ontological framework thathas two categories, one category being the theoretical counterpart to what are pretheoreticallyunderstood as individuals in the robust sense (such entities as animals and humans, but alsoartifacts and things); the other category is the counterpart to what are pretheoretically understoodas ‘properties’. Because these are two distinct categories, neither is reducible to the other. Thereare, of course, various substance ontologies.

4

Within trope theory, e.g. the relation of

exact resemblance

is meant to solve thisproblem. Accordingly, there are different tropes that resemble each other exactly. Their beingdifferent is taken to be primitive. This paper does not consider whether this is a good architec-tonic device.

Towards a Field Ontology 7


dialectica

tive particular aspects. The field ontology proposed here combines generic andparticular aspects without incurring any diffculties in explaining multiple instan-tiation. Because the two aspects are entangled, fields cannot adequately be char-acterized either as universals or as particular entities.

The second thesis asserts that no world containing individuals like humanbeings can be deterministic. Determinism is still a cherished dogma of manycontemporary philosophers, who hold that determinism is a consequence of ourbest scientific theories, and that philosophical theories, to agree with science,should also be deterministic. Because this article is not the place to argue for thecontrary view, the author relies on John Earman, who, in

A Primer on Determinism

(Earman 1986), concludes from his discussion of an impressive number of physicaltheories that determinism is neither a consequence nor an empirical result of anyof those theories. Instead, it is a methodological and/or metaphysical presupposi-tion that physical theories seek, often unsuccessfully, to confirm. If this is so, thereis no need for ontologies to be deterministic.

5

Ontologies may and indeed should,however, provide room for deterministic regions as well as for nondeterministicones. Their design should be adequate for both ontological options as well as forcombinations of them. The field ontology proposed here satisfies this requirement.

2. Formal preliminaries

Mathematical fields arise from

mathematical bundles

and other notions associatedwith them. These mathematical concepts must now be defined. It must be noted,however, that

not

all of the definitions to follow are of equal importance withrespect to the ontology considered in this article. Of minor importance are thenotions

topological space, topology

, and

continuity

. These serve mainly to makepossible the correct formulation of the mathematical background. With respect tothe philosophical issues, it is suffcient to note that they help to express the oldintuition

natura non facit saltus

. Readers are advised to focus on the interpreta-tionally and ontologically more important interplay of the

mappings

introducedbelow, viz., the bundle

p

and the associated cross-sections

s

(see definitions 2 and4); these express the key ideas.

Before the definitions are introduced, the question of their significance mustbriefly be considered. To what extent – if at all – are mathematical formalisms ortheories relevant with respect to clarifying and formulating philosophical, andespecially ontological, theories? This is a complicated and intricate question thatcannot be dealt with adequately in the context of the present article. Some com-ments, however, may be in place. The attitudes within philosophy towards math-ematical formalisms seem to range from unreflected acceptance to explicit

5

Of course, quantum theory is often held to be the paradigm of an indeterministictheory. But there are other such positions, e.g. Bohmian mechanics.



dialectica

suspicion or even rejection. A remarkable number of philosophers, especiallythose who devote themselves to ontological questions, will expect the author ofan article using such tools to explain in detail both how he proceeds and why.This expectation is legitimate. Whether this can adequately be done depends,however, on the aim of the particular article. There are two different ways ofresponding to the expectation. The first is to address and clarify this topic exclu-sively and (at least ideally) exhaustively. This could be done in an article that didnothing else. The other way is to use mathematical tools to discuss a specific andconcrete philosophical (ontological) topic. The question of the usefulness ofmathematical tools can then be discussed and clarified on the basis of such anapproach. In principle, neither way of responding excludes the other – on thecontrary, each implies the other. Pragmatically, however, it will scarcely be pos-sible to combine the two within a single article (articles are, after all, within theambit of finitude). The present article follows the second procedure: it

uses

mathematical tools to formulate an ontological suggestion.For a mathematical bundle to be defined, there must be two topological spaces.

A topological space is a

pair

of a set and a set of subsets thereof. The set of subsetsis called a

topology

. It is defined as follows:

Definition 1

Let M denote a set and

O

(M)

Ã

√

(M) a set of subsets of M

.

O

(

M

)

Ã

√

(M) is called a topology, if it satisfies the following three conditions

:

1. M

Œ

O

(M)

2. If O

1

, . . . ,

O

n

is a finite family of sets of

O

(M)

,

then

3. If (O

k

:

k

Œ

K) is a family (not necessarily finite or countable) of sets of

O

(M)

,

then

The elements of

O

(

M

) are called

open sets

.

6,7

6

O

(

M

) expresses a very general way of conceptualizing ‘distances’ between elementsin a set

M

. For example, let

M

be the set of real numbers. For each pair of real numbers

x

,

y

there is the distance

d

(

x

,

y

)

=

|

x

-

y

|, defined as usual. The set {

y

: |

x

–

y

|

<

r

}, which is the setof all reals

y

that have a distance from

x

less than r, is a typical open set of the topology definedby the distance d. (It is what is known from textbooks as an open interval of the real line.) Thereals y with |y - x| = r do not belong to the set. This may be seen as a motivation for calling theelements of topologies ‘open sets’. Topological spaces are written as (M,O(M)). Generally,several topologies can be defined on a set M, e.g. the coarsest topology that includes only Mand f, or the discrete topology that regards any singleton as an open set. Topologies may alsobe induced by relations of sorts, e.g. the resemblance relation, as is put forward and discussedby Mormann 1995.

7 In situations relevant for physics, topological spaces usually must be separableHausdorff-spaces. This expresses a condition concerning the topology and will not be problematized

« Œ ( )=in

iO M1 O

» Œ ( )Œk K kO MO


© 2005 The Author. Journal compilation © 2005 Editorial Board of dialectica

Definition 2a) Let (Y,O(Y)) and (M,O(M)) be two topological spaces. A topological bundleover (M,O(M)) is a continuous surjective mapping8 (to keep the notation tight,the topologies are suppressed)

Associated with the notion of a bundle is the notion of a cross-section or of afield that is itself a continuous mapping.

b) Let (Y,O(Y)) and (M,O(M)) be two topological spaces and p a bundle over(M,O(M)). A cross-section or a field with respect to p is a continous mapping s,

with p(s(m)) = m for all m Œ M. The set p-1(m), the inverse image of m with respectto p, is called a ‘fiber’ over m.9

For the present ontological discussion of bundles, no further qualification isrequired for fibers and cross-sections or fields. It should be kept in mind that twotopological spaces are required, but the main ‘entity’ is a mapping that is, bydefinition, dependent on the spaces. The spaces (Y,O(Y)) and (M,O(M)) maycontribute to different bundles (by different mappings p¢), and any one bundle pgenerally has several cross-sections s¢.

Next to be considered, before turning to the reinterpretation of the relevantmathematics, are ‘fields as tropes’, as proposed by Campbell 1990 and vonWachter 2000b.

3. Tropes and fields

From its very beginning (Williams 1953), classical trope theory understands itselfas an alternative to substance ontology.10 Like substance ontology, it sees its

in the present context. In a different context, White 1988 presents philosophical comments ontopologies helpful for interpretational purposes.

8 For a map to be continuous means, roughly speaking, that any two points y1, y2 of Y, which are ‘near’ to each other with respect to the topology O(Y), are mapped into two pointsp(y1), p(y2) of M that are ‘near’ to each other with respect to the topology O(M). The notion ofcontinuity presupposes the notion of topology. More formally:

Definition 3Let (Y,O(Y)) and (M, O(M)) be two topological spaces a mapping f,

is continuous iff for all U Œ O(Y) f-1(U) Œ O(M).9 More generally: Let p : Y Æ M be a bundle over M and U Ã Y an open set of (M,O(M)),

then a mapping s : U Æ Y with is a cross-section with respect to U that can becalled a ‘local cross-section’.

10 The term ‘classical trope theory’ indicates a trope theory that does not conceive oftropes as fields.

f M Y: Æ

p Y M: Æ

s M Y: Æ

p sU idU∞ =



theoretical task as that of explaining what could be called ‘the world of middle-sized objects’: the world as it presents itself through the senses and as it isintellectually captured in a rather naive manner. In contrast to substance ontolo-gists, classical trope theorists regard as theoretically untenable such putativeentities as substrata, bare particulars, primitive particularizers.11 Classical tropetheories still take what are traditionally termed ‘properties’ (sensible qualities area favorite paradigm) to require theoretical exposition, but they interpret these‘properties’ as tropes or abstract particulars. They do not conceive of them as‘universals’. Instead, tropes are taken to be particulars. Moreover, tropes are takento be ‘abstract’ in the sense of not existing in isolation (i.e. without other tropes).As Campbell puts it: ‘The colour of this pea, the temperature of that wire, thesolidity of this bell, are abstract in this sense only: that they (ordinarily) occur inconjunction with many other instances of qualities (all the other features of thepea, the piece of wire or the bell), and that, therefore, they can be brought beforethe mind only by a process of selection, of systematic setting aside, of these otherqualities of which we are aware’ (Campbell 1990, 2, his emphasis). Trope is theonly category within trope theory. All other entities – all ‘individuals’, for example– must figure as sets or ‘heaps’ (‘bundles’ is the technical term, that must notconfounded with the mathematical term ‘bundle’) of tropes.

In that respect the notion ‘trope’ involves two different ontological aspects, amaterial one – conceiving of tropes under the paradigm of particular properties –and an architectonic one – ‘trope’ is the only category of the ontological frame-work. This is not the only use of the word ‘trope’. For Loux 1998b, 81, e.g.‘tropes’ are ‘attributes understood as particulars’, irrespectively of whether theyare embedded within a one- or a two-category ontology. Simons 1994, 578, usesthe word to indicate a ‘kind of dependent concrete particular’, without relatingthis particular to any specific ontological framework.12 In what follows, the word‘trope’ is used to express the material as well as the architectonic aspect. Elaboratetheories of tropes are put forward, e.g. by Williams 1953, Campbell 1990 andBacon 1995. Because of Campbell’s further claim that tropes may be regarded astropes as fields, the focus here is on his theory.

By abstracting from sensible qualities, Campbell’s classical brand of tropetheory postulates a layer of ultimate tropes, out of which all other (more complex)tropes, up to ‘concrete individuals’, are built. These ultimate tropes are interpretedas simple qualities, having positions in space and time and tending to be compre-sent in space and time. Campbell characterizes the resulting interplay as follows:

11 Arguing against bare particulars and the like is not a topic germaine to trope theorists.M. Loux, e.g. a philosopher who decidedly favours a substance-ontology argues against baresubstrata as well as against bundle theories. (C.f. Loux 1998b, 117 ff and Loux 1998a).

12 Simons’s own theory takes ‘tropes’ in this sense as constituents of his two-sortalnucleus theory of tropes.



At the core of trope ontology is the thought: the basic items, the ‘alphabet of being’in William’s phrase, are cases of kinds. They are entities that are particulars, but notbare particulars. Each of them, if truly basic, has a simple nature. (There are complexderivative tropes. But the basic ones are single in character.) [. . .] The differentmutually independent properties of the same (complex) thing, are on the tropescheme distinct items, each a trope in its own right, each a particular, each with itsown nature. (Campbell 1990, 20)13

Campbell himself critizes the atomistic approach to trope theory on several points(cf. Campbell 1990, 142 ff). Although he agrees that a problemfree trope ontologyshould regard tropes as partless, changeless, and with unambiguous boundaries,he argues that the atomistic approach fails in that it is inherently ‘granular’ (ibid.,143). Campbell ultimately conceives of tropes under the paradigm of fields – morespecifically of the tensor-fields used in mathematical physics, especially in Gen-eral Relativity Theory.14

Campbell searches for items that he characterized as follows: ‘What we need,therefore, is a set of items, different from atoms, which are nevertheless like thempartless, changeless and without boundary problems’ (Campbell 1990, 145)15. Hisfields are such items.

The ontologically interesting intuitions behind Campbell’s proposal do not liein the philosophical (re)interpretation of fields as tropes, but in those mathematicalstructures of fields that Campbell does not recognize or (re)interpret. To repeat, afield is a cross-section with respect to a topological bundle. A fortiori it is acontinuous mapping from one topological space into another:

Campbell should have interpreted the relevant formalism in the following manner:any given field is a trope and for each point m of the topological space (M,O(M)),there is one definite value, s(m) Œ Y, of the field at this point. The trope, as amapping, ‘connects’ two items, M and Y. But what are these items? Campbellvirtually ignores them. He appears to ‘reduce’ the fields or tropes to their respec-tive ‘variegated’ values (cf. Campbell 1990, 128, 132, 148). This means that hereduces mappings to their ranges.

Moreover, he seems to locate ‘fields’ ‘in’ space-time: ‘All basic tropes arespace-filling fields, each of them distributes some quantity, in perhaps varyingintensities, across all of space-time’ (ibid., p. 146). Here again, Campbell identi-

13 Clearly, one important and problematic explicandum within this framework is whatmay pretheoretically be termed ‘object’ or ‘individual’: something extended, internally varie-gated, with a sort of crisp border, clearly different from cognates, etc. Some such ‘objects’ or‘individuals’ are persistent through time, i.e. they ‘remain the very same things, while changingsome of their characteristics (pretheoretically: some of their properties)’.

14 This is clearly indicated in chapter 6, 135 ff.15 Whether or not there are boundary problems for field theories of tropes is not

considered in the present article.

s M M Y Y: , ,O O( )( )Æ ( )( )



fies tropes not with fields, but with the values of the fields. But if values – ortropes/fields, in Campbell’s sense – are to be located anywhere at all, the relevant‘where’ must be specified.16 Nevertheless, Campbell himself at least once refersto space-time as ‘the most ethereal field’ (ibid., 147). It is thus clear that Campbellcannot be interpreted as conceiving of M as space-time, since M is not a field.

The same position can be attributed to von Wachter, since he makes explicituse of ‘points of a space’ (von Wachter 2000b, 211), which he presents as in-dispensable for defining the ‘intensity’ of a field at a locus. Be these specifics asthey may, however, any trope theorist endorsing a field approach should flesh outthe ontological status of the formal items involved.

Regarding tropes as fields and taking this notion as a mathematical notion, oneis committed to several items: the topological spaces (M,O(M)) and (Y,O(Y)), thebundle (mapping) p, and the field/cross-section s associated with the bundle. Atleast those four items should be interpreted.

The space (M,O(M)) looks, ontologically speaking, much like a heap of primi-tive ‘particularizers’. The only role of M is that values can be attached to itselements in a ‘smooth’ manner. The elements of M ‘bear’ these values. In anycase, (M,O(M)) is not a field in any non-trivial sense and not a trope/field in thesense put forward by Campbell. Nor is it a set of tropes. Here we have an instanceof offending the trope-theoretical unicategoriality and thus of displaying incoher-ence. Therefore, a field ontology should seek to avoid this incoherence.

The more interesting question is that of how to interpret the ‘rest of s(M)’,Y\s(M). This amounts to asking how to interpret the other possible fields (cross-sections) s¢ with respect to the bundle p.17 Campbell does not address this topic.If more – or all – cross-sections are taken into account, field ontology is able toaccount for non-deterministic frameworks.

This may be visualised by the following picture. Campbell regards the surfacethat is indicated by the bold curves as the only field, neglecting the surfaceindicated by the thin curve. The presence of this second surface is due to p and

16 For the trope theorist, no conceptual problem necessarily arises from the fact thattropes are located at space-time points, but to regard tropes as fields in a mathematically morerigorous manner, one must also regard ‘space-time’, especially the metric (tensor) field, as itselfa trope rather than as an entity over and above the ‘world of tropes’. The only dimensions ofthis framework within which the tropes could be located are the spaces (M,O(M)) and (Y,O(Y)).The first space, even if taken as a differential manifold, has a structure sparser than that requiredby space-time, and the metaphor ‘in’ does not work for it, because (M,O(M)) is the domain ofthe fields. Fields are, loosely speaking, attached to M, not ‘embedded’ in M.

The second space works much better for the metaphor ‘in’ in that it encloses the rangeof the field, but it is the part of the structure in question that should capture the ‘qualitative’aspect – the particular natures, which should be, by Campbell’s intuition, the trope in question.

17 Generally, these other cross-sections of the bundle p cannot be conceived of as othertropes. Other tropes would instead be cross-sections (or values thereof) of other bundles overM and not of the same bundle over M.



the fact that generally p allows for more than one cross-section. Each surface inthe picture below indicates the range of a cross-section. These cross-sections aredifferent, but their ranges overlap (viz., the ‘flat’ region).

The interpretation of (Y,O(Y)), the bundle p, the cross-sections s, and the fibersmake up the core of the kind of field ontology put forward in this article. Thereare, however, three aspects that should be preserved: 1.) Intuitively speaking,(Y,O(Y)) is not a space of generic aspects, but a single generic aspect, henceforth,a generic aspect that includes all its possible particularizations. This gives rise tothe task of explaining how to combine several generic aspects and their associatedparticular aspects. When fields are combined, the result should also be a field. 2.)A set of primitive particularizers, (M,O(M)), is abandoned altogether. The mainmotivation for this abandonment is the avoidance of incoherencies. 3.) A mappingp : Y Æ Y (which is generally not onto!), together with crosssections, i.e. fields,will flesh out how to understand ‘possible’ in this context. This understandingresults from recognizing and taking seriously the fibers so far neglected.

4. An approach to field ontology

As has been indicated, field ontology is an ontology that dispenses altogether with‘substrata’ and kindred concepts. In this very vague sense it is akin to severalontological bundle theories whose primordial constituents are, in pretheoreticalterms, ‘properties’ (‘universals’ or ‘tropes’).

‘Property’, as usually understood, points to a static entity, such as a colour.The ontology’s generic aspects, the fields, are meant here to capture dynamic aswell as static aspects, e.g. runnings and readings and the like. To some degree –as the examples below will show –, dynamic aspects are the more perspicuousparadigms.

The primordial entities of the approach here to be presented are not, how-ever, conceived of as mathematical fields, i.e. cross-sections of a bundle (Y, p),

Figure 1.



but as bundles (Y, p) (over p(Y)) together with an associated family of cross-sections, S. It is this formal move that makes possible the regarding of ‘theworld’ as a regulated coordination of its different possible histories; this is thekey to making indeterministic intuitions precise. This is mainly due to the factthat generally one bundle allows for different cross-sections. Intuitively speak-ing, Y may be regarded as a generic aspect that includes all its possible particu-larizations or as a combination of several generic aspects with their respectiveparticularizations, the ‘world’ being the most complex case of them. But theassociated cross-sections also play important roles. In what follows, (Y, p, S) iscalled a ‘field-space’.18 A field space completely fits neither the intuition of atrope as a field (in Campbell’s sense) ‘spread in the world’ nor the intuition of a(possibly) realized ‘universal’(since there are no particularizers); instead, itshares some aspects of both. Field spaces will be of various degrees of com-pleteness and complexity.

4.1 The field spaceBefore interpretational issues are considered, the modified formal background ispresented. The field space, the main entity, is 1.) a continuous mapping p with

, from a topological space (Y,O(Y)) into itself (the topology is notation-ally omitted):

This mapping is usually not onto; p(Y) is usually a proper subset of Y. (Y, p) is abundle over p(Y). 2.) Associated with (Y, p), there is the set of continuous cross-sections S : = {sj} associated with p. This set is not necessarily countable and eachsj gives rise to another associated bundle (Y, pj) over Yj : = pj(Y), with .

From now on Y = �Yj is presupposed. (Therefore and for convenience ‘S’ isnotationally omitted.)19 The Yj are joined by homeomorphisms20

18 It would be more correct to call (Y, p) a ‘possibility bundle’, but this would be inconflict with ontological bundle theories, which have a completely different structure.

19 From a mathematical perspective, this assumption formulates a restriction. There aresituations that do not allow for a (global) cross-section. But, for ontological purposes, asformulated here, this would amount to having a world that would ‘jump’ when evolving. Nordoes the restriction mean that all (global) cross-sections that are possible should be taken intoaccount. There may well be ontological persuasions to allow only for some of the cross-sectionsto making up a world. This topic would be relevant for discussing possible worlds, a topic notdealt with in this article.

20 Homeomorphisms are mappings that are one-to-one and onto, as well as beingcontinuous, as are their respective inverse mappings.

p p p∞ =

p Y Y: Æ

p s pj j= ∞

s Y Yj k j k, : Æ



with . Especially, sj,j = id.21 Therefore, one pj is enough to getall the fibers. Moreover, fibers cross each Yj at one point only. Y is partitioned byfibers.22

For a first interpretation of this modified formalism, it is best to regard theelements y Œ Y as instantaneous possible individuals that are configurations ofitems of lesser complexity. They are unpacked and analysed below.23 The rangesof the cross-sections or fields – the Yj – are the sets of possibly coexistinginstantaneous individuals; the Yj are termed surfaces of compossibility. Any sur-face of compossibility can be regarded as one complete possible world history.The fibers may be regarded as an ‘instantaneous individual’ that includes all itsinstantaneously possible realizations, thus all its complete possible states, each ofwhich can – but only one of which will – be realized at a given instant. So, eachelement of a fiber stands for an individual as it is or as it could have been at onedefinite instant of its history. This captures the intuition, shared by the author, thatthe identity of an ‘individual’ is generally not exhausted by how it actually is (orwas), but also includes both all the ways it could be or could have been and allthe ways, in its future, it can possibly be.

This may sound strange, but a concrete example may decrease the strangeness.Imagine a human individual who, at one instant in his life, commits a crime. Toassume that the possibility of his not having committed this crime at that veryinstant of his history did not contribute to the identity of that human individualwould make any later punishment irrational.

It is important to note that there are not two individuals, a possible one thatdecides not to commit the crime and who then leads a quiet future life, and anactual one that commits the crime, and then lives in fear to be punished. There isonly one individual. This individual includes these two possibilities (and perhapsmore) – the possibilities contribute to its identity –; at the relevant instant in hishistory he makes one possibility actual – he commits the crime. If he did notinclude both possibilities, this individual would be subject to punishment for beingwhat he was. This would be like punishing a stone for falling on someone’s head.Of course, there should be room for entities that are completely exhausted by one

21 The clause p � p = p ensures that there is at least one (global) cross-section.22 Formally, the bundle (Y, p) plays no distinctive role; one could begin equivalently

with a family (Y, pj), together with some regularity and maximality conditions, and combiningmappings sj,k as above. This would lead to the same result. The definitional procedure used hereis chosen simply for convenience. In either case, one ends up with a family Yj of homeomorphictopological spaces with an associated family of homeomorphisms sj,k.

23 ‘Instantaneous’ points to the fact that at this stage neither change nor time is explicitlyproblematized. A whole individual, however, will include its changes. The word ‘possible’ pointsto the feature that not all of the instantaneous possibilities of individuals are actualized. Onlyone possibility will be.

p y p s yj k j k- -( ) = ( )( )1 1

,



possibility (stones may be such entities). The mapping p can help to make rigorousthis intuition.

One of David Lewis’s examples may provide additional clarification. Lewisdistinguishes cases of ‘divergence of worlds’ from those of the ‘branching ofworlds’:

In branching, instead of duplicate segments, one and the same segment is alleg-edly shared as a common part by two overlapping worlds. Branching is problem-atic in ways that divergence is not. First, because an inhabitant of the sharedsegment cannot speak unequivocally of the world he lives in. What if he saysthere will be a sea fight tomorrow, meaning of course to speak of the future of hisown world, and one of the two worlds he lives in has a sea fight the next day andthe other doesn’t? Second, because overlap of worlds interferes with the mostsalient principle of demarcation for worlds, viz. that two possible individuals arepart of the same world iff they are linked by some chain of external relations, e.g.of spatiotemporal relations. (I know of no other example.) (Lewis 1998, 176; hisemphasis)

This citation reveals that Lewis conceives of two branching worlds as two dif-ferent worlds with something in common, in his example a past up to the daybefore the sea fight will or will not take place. In contrast to Lewis, within fieldontology there is only one world, a world that includes the actual past up to theday before the sea fight will or will not take place (together with past possibili-ties that have not been actualized) and the two possible branches, neither ofwhich is actualized so far, but one of which will be actual the next day. On theday before, which of the two possible futures that belong to the one world willbe actualized and which one will not remains open. As is indicated above, thesame holds mutatis mutandis for the individual faced with the question ofwhether he will or will not witness a sea fight tomorrow. He also is one individ-ual including his different possibilities, only one of which will be actual thefollowing day.

The surfaces of compossibility express that not all possible states of an instan-taneous individual can coexist with all possible states of other instantantaneousindividuals. Some possible states, however, do of course coexist.

This holds both synchronically and diachronically. Without a book, a humanindividual cannot read a book. Nor is there a reading without something to beread. There is no reading without the requisite biological and intellectual capac-ities of the reader. This feature points to the intrinsic complexity of the elementsof the fibers, an issue addressed below.

Human individuals also tend to change smoothly. A given human individualmay be four feet tall at one instant and five feet tall at another, but there will bemany intervening instants at which the individual is of intervening heights. Thus,surfaces of compossibility initially regulate what is compossible, both diachroni-cally and synchronically.



There is a further intuition. Instantaneously, any individual can actualize onlyone of its possible states. A human being, for example, cannot at once be simul-taneously running and sitting. Formally, this means: each fiber intersects eachsurface of compossibility at one point only.

A sitting may, however, coincide with a reading, and this, too, reflects theintrinsic complexity of the elements of the fibers. There are sittings withoutreadings and vice versa, so there may be (and usually are) different elements ofthe fiber, one combining sitting and reading (along with various other character-istics), one including only sitting but not reading, a third including only readingbut not sitting, and a fourth including neither.

To indicate its status within field ontology, a modified expression is used forwhat in the preceding paragraphs is dubbed ‘(instantaneous) individual’: aninstantaneous proto-individual is an entity comprising all its instantaneouslypossible states. Each possible state is internally complex (as is shown below).Formally, an instantaneous proto-individual is a fiber p-1(y), with y Œ p(Y). Sincep-1(y) = (s � p)-1(y) for any cross-section s, it is independent of any specific Yj;therefore, no hidden particularizers are at work.

‘Possibility’ points to a certain openness concerning realization: a part of asurface of compossibility being actualized may be open to be regularly completedin several different ways. For example, if the notions of present, future, and pastare meaningful, at least locally, then the past actualized make-up of the world willnot completely determine the future make-up, but neither will every imaginablefuture be possible. This holds for the whole actual or present world. Its past –insofar as it is actualized – is unalterable, but concerning its future, there shouldat least be some regions whose futures are not completely determined by thepresent and the past.

Framing one’s ontology by (Y, p) and the associated cross-sections providesmeans to make this intuition rigorous.24 The relevance of different possible futuresmay be visualised as follows:

Figure 2.



The picture above shows a cube enclosing four smooth surfaces, one a slopedescending from the left upper edge (u1–u3) to the right lower edge (l2–l4), one aslope ascending from the left lower (l1–l3) to the right upper (u2–u4) edge, the thirda trough, including both upper edges (u1–u3 and u2–u4) and the region where allsurfaces overlap, and the fourth a ridge, including the lower edges (l1–l3 and l2–l4) and the overlap. Each surface symbolizes a surface of compossibility that canbe interpreted as a ‘possible world’. ‘Before’ (left of) and ‘after’ (right of) theoverlap, the surfaces branch, that is the overlap does not determine which of thetwo possibilities is to be taken; both branches are possible. Any past may haveseveral possible futures, but any past will make some futures impossible. Possiblefutures are limited to those that smoothly or continuously extend from the relevantpresent and past. The region of overlap symbolizes a region where the near pastcompletely determines the near future. It is a region of determinism. At the borderof this region, branching worlds are possible.

The bold line within the cube symbolizes one instantaneous proto-individual.Formally, it is a fiber. The points – the intersection of the instantaneous proto-individual with the surfaces of compossibility – mark the instantaneous proto-individual’s possible actualizations with respect to these surfaces (or ‘possibleworlds’).

Thus, (Y, p) determines not only which surfaces of compossibility there are,but also which parts of these surfaces may complete each other (and which‘futures’ are possible for actual (present) instantaneous protoindividuals), i.e.which futures can complete a given actual world and its past (and instantaneousproto-individuals). This is the aspect that gives rise to the addendum ‘possible’.

Concerning the modes of existence of several possible worlds, there is, first,a region – better, a border – of actuality (the right borderline of the overlap in thepicture above). Second, there is the past, so far actualized, that is unalterable withrespect to this border (the slope descending from the upper left edge, plus theoverlap up to but excluding the right border). Both together form the world as ithas been actualized up to some actual instant, up to some (included) actual present.The past as it has been actualized so far and the actual present are parts of oneand only one surface of possibility. All surfaces of possibility that neither overlapwith the actualized past up to the border of actuality nor are continuous extensionsthereof have no ontological import with respect to the actual state of the world.Surfaces of possibility that overlap the actualized past are ways the world couldhave been. The surfaces that smoothly extend the border of actuality are thepossible futures. As the world evolves, possible futures continue to arise, andextensions that had been possible become impossible.

24 Here, local cross-sections may play a role, since they help to interpret how anactualized past (i.e. a local cross-section) may be completed to a global cross-sections (surfaceof compossibility).



4.2 Some remarks on changeAt the end of the preceding section, reference is made to ‘past’, ‘present’, and‘future’, and thus to time. The task now is to sketch informally how ‘changes’ fitinto the framework just delineated. This sketch provides the basis for introducingtime into the framework, in a precise manner, although this article does notdevelop such an introduction.25

The main intuition underlying the following construction is that change shouldbe regular and suffciently smooth with respect to surfaces of compossibility, andthus with respect to instantaneous proto-individuals, their respective individualpasts, and their actualized instants. A proto-individual now is a changing entitywhose ‘instantaneous states’ are instantaneous proto-individuals. So, each proto-individual has its own evolution, but the different evolutions are coordinated.There is at least a common directedness of evolutions.

The mathematical entity that can correspond to these intuitions is a so-calledaction or dynamic system. Associated with an action are the so-called orbits ofthe action. Orbits can help to express proto-individuals as changing entities. Apicture may be more helpful than would be a presentation of the relevant math-ematics. It is, however, important to note that an action at once coordinates allproto-individuals with respect to their possibilities and their changing. The orbitsrepresent two different single proto-individuals, including their possible and actualstates (including states formerly actualized), and the action prevents actual andpossible histories of proto-individuals from overlapping. Never in their lives anytwo proto-individuals share any possible or actual variants of any of their charac-teristics – they are never identical with respect either to their actualities or theirpossibilities. They may, however, share generic aspects.

25 ‘Change’ must not be confused with ‘time’. Proper consideration of time or spacetimerequires the formulation of a much more refined structure – better: a geometry – than can beincluded in this aricle. For a more precise presentation of change and time, in a formally similarcontext, see Schneider 2001.

Figure 3.



The picture above is the same as that in subsection 4.1, but with some additions.On a surface of compossibility, there are two bold lines that represent the historiesof two proto-individuals up to one instant in their ‘lives’. The surface along whichthe lines run may be regarded as the ‘realized possible world’, as it was up to thisinstant. At the right edge of the horizontal region of overlap with other surfaces,the lines branch. At this (former) instant, there were multiple ways in which theproto-individuals and the corresponding world could have further developed. Thebranches ‘going up’ from there belong to a possible future that, along with therest of the relevant surface of compossibility, could have been actualized, butwhich in fact was not actualized (one branch is, perhaps, the uncommitted crimeof the example above, and the world as it then would have been). Despite its nothaving been actualized, this possible history remains among the possibilities ofthat changing and evolving proto-individual. Because the past is given, along withthe present, the only futures that are possible are those that extend smoothly fromthem (not shooting John F. Kennedy, at the instant when he did, was a possibilityfor Lee Harvey Oswald; shooting Richard Nixon at that instant was not a possi-bility for him, because Oswald was in Dallas and Nixon was not). As the pictureindicates, the two protoindividuals share past and future, but without overlapping.It may be important to note that this picture simplifies matters, but it nonethelessshows that the completion of the actual is holistic in the sense that all individualcompletions must fit within it.26

Up to this point, surfaces of compossibility, along with proto-individualscomprising their changes, have been modelled as simple entities. Even within thishighly abstract framework, however, proto-individuals – instantaneous and chang-ing – are to be conceived of as being more complex. They must be unpacked, asmust the surfaces of compossibility.

Before turning to this task, some interpretational comments concerning theoverall coordination of the framework may be helpful. The world as it presentsitself so far is coordinated by means of three main ‘dimensions’: the dimensionof possibility, expressed by the fibers, the dimension of change, depicted by thebold branching lines, and the dimension of coactuality, depicted by the straightlines where the branchings take place. Instantaneous proto-individuals on such aline can constitute one ‘possible world’ at an ‘instant’. Instantaneous proto-individuals on such a line may be interpreted, with all caution, as being coordi-nated in a ‘space-like’ manner.27 The topology at once and inclusively expresses

26 Formally, this completion could be refined by the addition of a distinction betweencertain ‘dominant’ proto-individuals, whose choices for their futures are relatively independent,and the rest of the proto-individuals, who must adapt their choices to those of their ‘dominant’cognates in ways that lead to a single possible (near) future. Other refinements are possible, butthey would require a more elaborate mathematical presentation.

27 Mutatis mutandis, what holds for ‘space’ or ‘space-like’ holds as well for ‘time’: Thestructure presented here is too weak to include them properly.



three features: that of smooth succession (a child cannot gain ten inches in heightwithin ten minutes), that of being co-actual in ‘small’ ‘space-like’ regions (withinten suqare-inches, a lake cannot both be calm and have high waves), and that ofthe ways a proto-individual can possibly be with respect to similarity (remotepoints on a fiber indicate rather different ways of possibly being, neighboringpoints indicate rather similar ways of possibly being; changing one’s hair colorfrom black to brown makes less difference with respect to similarity than doeschanging it from black to blonde).

4.3 Unpacking proto-individualsThe intuition here points to the categorial framework of the ontology in question:all proto-individuals, instantaneous or changing, as well as every whole possibleworld history, should be understood as configurations of other field spaces that,conceived of in isolation, are too sparse and too poor to be proto-individuals orpossible worlds. Both in analysis and in configuration, there should be field spacesall the way up and all the way down. This has some similarity to one-categoryontologies. The structures of higher complexitiy are configurations of field spaces,but in a more subtle way than pure bundling can account for. As a constituent,each field space can be interpreted as a generic aspect or ‘property’, and insuperposition constituents give rise to increasingly complex generic aspects. Agiven (instantaneous or changing) proto-individual can then be seen as a saturatedcombination of generic aspects, an entity to which no more generic aspects canbe added and from which none can be subtracted. Moreover, each generic aspectincludes all its possible particularizations, all its diverse ways of possible actual-ization. Diverse particularizations of generic aspects are co-actualized, and suc-ceed one another in regulated manners. This regularity of co-actualization andsuccession, together with the mode of co-constituting proto-individuals, gives riseto the regularity and mutual adaptations of the proto-individuals discussed in thepreceding sections.

Formally, the generic aspects that co-constitute surfaces of possibility andproto-individuals are a (not necessarily finite or countable) family (Yj, pj) of fieldspaces, along with their associated cross-sections. The way of constituting genericaspects of higher complexity and ultimately of protoindividuals is regulated by afamily of associated continuous mappings.

The simplest case of such a combination is sketched in what follows. Let ((Yj,pj), j Œ J) be a family of field spaces (each of which may be regarded as acharacteristic with all its possible variants), such that for each pair i, j Œ J, i £ jor j £ i (this points to the simplification mentioned above). Further, there is acontinuous one-to-one mapping for each pair (i, k), i £ k:

s i ki kY Y, : Æ



with si,i = id and sk,l � si,k = si,l for i £ k £ l. These mappings are generally not onto.The ‘field space structure’ is determined by the following condition:

This condition helps to make sure that surfaces of compossibility and fibrescorrespond. It determines which different generic aspects may go together, i.e.can form generic aspects of higher complexity, and which of the respectiveparticularisations thereof are combinable. This means that combinations ofgeneric aspects lead to proto-individuals that are built up from relevant particu-larizations. Here the intuition of ‘bundling’ (in the philosophical reading of theword) shows up, but in a refined manner. As it stands, possible co-actualizationsare regulated only for pairs of characteristics. But ‘pairwise togetherness’ is notenough; more than two generic aspects (but not neccessarily all of them) across(Yj, pj) should be combinable. Formally, the ‘overall’ combination searched for isthe inductive limit of (Yj), , with respect to the mappings si,k. The inductive limitis a topological space, , together with a family of continuous mappings si : Yi

Æ with si = sj � si,j.28 Because of the condition introduced above, has anatural mapping p – inherited from the pj – that makes ( , p) a bundle. The fibersof ( , p) are instantaneous proto-individuals; p and the associated cross-sectionsdefine the surfaces of compossibility. Each instantaneous proto-individual can beregarded as a ‘bundle’ (in the colloquial sense of the word) of fibers of all andonly those characteristics that may be successively lined up by the mappings si,k.

Note that for all k ≥ i, there is one and only one particularization of a genericaspect yk Œ Yk that can be coactualized with a particularization yi of Yi, but,conversely, for l < i, a particularization yi Œ Yi may not be in the image of sl,i.This means that some particularizations of Yi cannot be co-actualized with anyparticularization of Yl. This can be interpreted as follows: although some generic

28 More elaborately:

Definition 4Let Xi, i Œ I be a family of topological spaces, ‘£’ a transitive, reflexive and anti-symmetricrelation on I. For i, j Œ I there exists an l with i £ l and j £ l. Let, further, (si,j) be a family ofcontinuous mappings

A topological space X is the inductive limit of Xi with respect to (si,j) if the following conditionshold:

1. There exists a family of continuous mappings (si : i Œ I), si : Xi Æ X with si = sj � si,j (ifsi,j is defined)

2. X is endowed with the finest topology with respect to them.Note: 1.) Inductive limits exist for topological spaces. 2.) It is not presupposed that the family(si,j) includes a continuous mapping si,j or sj,i for each pair i, j (as was presupposed in the simpleexample above).

s si ki k

i kp p, ,∞ = ∞

YY

Y Y

s i j i jX X, : .Æ

s s sj k i j i k i j k, , , , .∞ = £ £

YY



aspect generically co-constitutes a given proto-individual, there are possible actu-alizations of the proto-individual that are not co-constituted by any particulariza-tion of this generic aspect. Other possible actualizations of the proto-individualmay be co-constituted by particularizations of this generic aspect. This points tothe pretheoretic intuition that there are ‘properties’ that can be either present orabsent. The simple example sketched above displays this feature. The intuitionthat there should be some generic aspects whose particularizations must not beleft out in co-constituting a possible actualization of any proto-individual (prethe-oretically: ‘essential properties’) may be captured by refined orderrelations on Jand refined conditions on the mappings si,k.

The inductive limit exists, mathematically speaking, in rather general contexts.For ontological purposes, these contexts can and must be narrowed down tocapture specific ontological intuitions. Making these various lines of thoughtrigorous is more a technical exercise than a fundamental obstacle.

In order to keep the presentation short and simple, it suffces to note thatchanging proto-individuals may be defined by changes (continuous actions andorbits) of the constituting field spaces.

Two very simple examples may help to clarify the role of inductive limits inthe present context. The first example refers to the simplification mentioned above,and the second example shows that a move beyond this simplification must andcan be made.

1) Imagine an impoverised world with only three field spaces: ‘thinking (t)’,‘living (l)’ and ‘being physical (p)’. There is no particularization of thinkingwithout living, t £ l, but there may be particularizations of living without thinking,so the mapping st,l is not onto. Further, there is no particularization of livingwithout a way of being physical, l £ p, but not every way of being physical is alsoa particularization of living, sl,p is not onto. But, there is also no way of thinkingwithout a way of being physical, t £ p, and each particularization of thinking istied to a particularization of being physical by being a way of living, st,p = sl,p �st,l. A possible actualization of an instantaneous proto-individual is now the set(equivalence class) {x, st,l(x), st,p(x)}. A particularization of thinking, togetherwith its coordinated particularization of living and its coordinated way of beingphysical.

2.) Imagine another sparse world, one with five field spaces: ‘reading (r)’,‘being intelligent (i)’, ‘sitting (s)’, ‘lying (l)’, and ‘being in a physiological state(p)’. There is no particularization of reading without a way of being intelligent,r £ i and sr,i(x) is not onto, since there are particularizations of being intelligentwithout reading. Neither reading nor being intelligent must go together withsitting or lying, but each must go together with some particularization of being ina physiological state, therefore neither r nor i is comparable with s or l, respec-tively, (¬(r £ s ⁄ s £ r) and ¬(r £ l ⁄ l £ r)); and the same holds for ‘being



intelligent’, but r £ p and i £ p and sr,p = si,p � sr,i(x). But because there is neithera sitting nor a lying without a way of being in a physiological state, and not everyphysiological state is conjoined with one of them, s £ p, l £ p and ss,p and sl,p arenot onto. Further, there is no ‘simultaneous’ sitting and lying, therefore ss,p andsl,p have no values in common. This example shows that the set {r, i, s, l, p} isnot linearily ordered and there is a restriction concerning the values of the map-pings ss,p and sl,p. A possible actualization of an instantaneous proto-individualthat is reading while sitting is the equivalence class {x, sr,i(x), sr,p(x), y, ss,p(y)}with ss,p(y) = sr,p(x). Here, x is a particularization of reading. An actually readingproto-individual cannot be ‘simultaneously’ sitting and lying, since then sr,p(x) =ss,p(y) = sl,p(z) would hold for a particularization y of sitting and a particularizationz of lying. But this cannot be the case since ss,p and sl,p have no values incommon. There may of course be a reading while lying and an instantaneousproto-individual may possibly actualize this way of reading, say w for that par-ticular reading, {w, sr,i(w), sr,p(w), z, sl,p(z)} with sl,p(z) = si,p(w).

4.4 Are field spaces universals or particulars?Having now unpacked proto-individuals, and thus, as Campbell puts it, gottenfield spaces ‘before the mind by an act of abstraction’, one may finally posethe question whether field spaces are ‘universal’ or ‘particular’ entities.According to Campbell, ‘[T]he opposite of particular is universal’ (Campbell1997, 477); this suggests that field spaces should fit into one or the other ofthe two meta-categories. The issue, however, is not simple. This is not theplace to consider refined theories of universals and particulars, but some of theimportant characteristics of these meta-categories must be introduced to clarifythis problem.

1.) For present purposes, a universal or a universal entity is an entity servingas a theoretical counterpart to the pretheoretical notion of ‘property’, ‘feature’,‘characteristic’, or the like (understood not only as ‘static’, but also as ‘dynamic’).2.) Within ontological theories recognizing such entities, a given universal thatenters the constitution (or the make-up) of a more complex entity does so as awhole and it can so enter into the constitutions (in whatever sense) of differententities of higher complexity (i.e. universals are multiply instantiable). 3.) Thereare no universals that are different solely numerically (since there is no need forthis).

The first and the third of these characteristics hold for field spaces. The secondcharacteristic, the most important one, does not. No field space enters as a wholeinto the constitution of any more complex entity (any proto-individual). Conse-quently, no field space as a whole can enter into the constitution of different (i.e.multiple) entities. To be sure, as a whole, every field space contributes to theconstitution of the whole world, and this world includes not only the actual ways



it was, is, and will be, but also the ways it possibly could have been, could be,and will become.

Turning to the question whether a field space is a particular entity, it must bekept in mind that field spaces are the theoretical counterparts of ‘properties’, notof individuals (in a robust sense). Therefore, the question is whether field spacesare to be regarded as particular entities in the way that, for many philosophers,are ‘tropes’ or ‘individual accidents’ or ‘moments’. Restricted to this paradigm,a particular entity is an entity that displays three features. 1.) It contributes to theconstitution or make-up of a more complex entity. 2.) A particular entity, sounderstood, can contribute to one and only one such complex entity (particularsare not multiply instantiable). 3.) Particulars in this sense are mutually different,but may be different solo numero or may resemble each other exactly.29

The first characteristic, of course, holds for field spaces, but the second andthird do not. Concerning the third feature, field spaces are mutually different, butmay not be different solo numero. With respect to the second characteristic, eachfield space as a whole contributes (as is said above) to the world as a whole, butnot to more complex entities of that world, as e.g. proto-individuals. Thereforefield spaces are not particular entities in the relevant sense. Every field spaceincludes its own different possible concretizations. These, however, can beregarded as being particular in the relevant sense. Is a field space therefore a set,a collection, or a ‘heap of particulars’? It is a collection of its possible concreti-zations, but it is much more than this. It does more than simply collect itsconcretizations. The field space, by the mapping p and the associated cross-sections, regulates which concretizations may contribute to one surface of com-possibility and which to another. By doing so, the field space regulates whichconcretizations can make up one proto-individual and which another (aided bythe action, instantaneously and evolving).

Field spaces are neither universals nor particulars, in the senses put forwardhere. Field spaces are more than a collection of particulars. They are entities suigeneris, internally complex, that capture general as well as particular aspects. Theparticular aspects are captured by a field space’s concretizations, and the generalones by the coordinating and holistic role the field space plays.

5. Perspectives

The field approach presented so far may be conceived of as at best a skeleton ofa field ontology; to be more than a skeleton, it would have to be fleshed out inseveral ways. Some of those ways are considered in this final section.

29 Concerning the topic of ‘exact resemblance’ cf. Bacon 1995, Mormann 1995 orSimons 1994.



The framework, so far, accommodates only (changing) proto-individuals.Proto-individuals are entities that are closer to Whitehead’s actual entities orLeibniz’s monads than to the entities usually dubbed ‘individuals’; in comparisonwith the latter, proto-individuals are rather abstract. Yet however ‘abstract’ fieldontology may be, it has its theoretical and ontological virtues. Ontological fieldtheory makes sense of possibility and is not wedded to deterministic presupposi-tions: a given actual present and its past may lead to several futures, but not tojust any future. One may, however, be inclined to accept ‘deterministic’ ontologicalregions, or proto-individuals displaying deterministic behavior, and to combinethem with indeterministic proto-individuals within one evolving possible worldthat is not deterministic ‘everywhere’. This world has some regions that are openwith respect to their futures and others that are not. Moreover, one could be inclinedto allow for some non-deterministic proto-individuals that affect their nearbyneighborhoods when they choose their (near) futures, while leaving remote regionsunaffected, be those regions deterministic or not. This can be formulated withina field ontology. Although constructing such an ontology is, formally, a technicalexercise, it is not at all trivial philosophically, in that it presupposes variousrevisions concerning the concept of causality. Causality, however, is an extensivetopic that deserves separate treatment. But it is important to note that the frame-work presented in this article is in no way restricted to any physicalist world view.

Field ontology can accommodate entities corresponding to those understood pre-theoretically as, respectively, (a) non-living, (b) alive, and (c) endowed with spon-taneity, volutional and intellectual capacities (may those entities be ‘physical’ or‘outside space and time’). Its accommodation of this variety of entities avoids theproblems that beset dualistic frameworks in the Cartesian tradition as well as the(counterintuitive) consequences of frameworks, like those of Leibniz and Whitehead,that take all entities to be somehow ‘alive’. Here, this cannot be further pursued.

A field approach to ontology has a sort of ‘holistic character’ and, thus, is aliento the usual ontological/conceptual parsings of the universe of discourse by meansof (philosophical) categories and notions such as ‘universal’ and ‘particular’,‘substance’ and ‘accident’, ‘exemplification’, ‘instantiation’, and the like. In lightof the success of mathematical field theory within our best scientific theories, andbecause discontinuities between those theories and ontology should be minimized,it appears worthwhile to tackle the formidable ontological and meta-ontologicaltask of rethinking the paradigms of and intuitions about ontological categoriesand kindred concepts that have, traditionally, guided ontological inquiry. Fieldontology may be a point in case.*

* The author wishes to express her heartfelt thanks for the generous labor of ProfessorAlan White (Williams College, Massachussetts), who transformed the English of the originaltext into elegant, accurate prose – and whose probing philosophical comments helped to improvemuch more than the style.



References

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pp. 477–488.Earman, J. 1986, A Primer on Determinism, Dordrecht, Boston: D. Reidel Publishing Company.Lewis, D. 1998, “New Work for a Theory of Universals”, in S. Laurence and C. Macdonald. Con-

temporary Readings in the Foundations of Metaphysics, Oxford: Blackwell, pp. 163–197.Loux, M. 1998a, “Beyond Substrata and Bundles”, in S. Laurence and C. Macdonald. Contemporary

Readings in the Foundations of Metaphysics, Oxford: Blackwell, pp. 233–247.Loux, M. 1998b, Metaphysics. A contemporary introduction, London: Routledge.Mormann, T. 1995, “Trope Sheaves: A Topological Ontology of Tropes”, Logic and Logical Philos-

ophy 3, pp. 129–150.Schneider, C. 2001, Leibniz’ Metaphysik. Ein formaler Zugang, München: Philosophia Verlag.Simons, P. 1994, “Particulars in Particular Clothing: Three Trope theories of Substance”, Philosophy

and Phenomenological Research LIV, pp. 553–575.von Wachter, D. 2000a, “A World of Fields”, in M. Urchs, U Scheffer and J. Faye eds, Things, Facts,

and Events, Amsterdam: Rodolpi, pp. 305–326.von Wachter, D. 2000b, Dinge und Eigenschaften, Versuch zur Ontologie, Dettelbach: J. H. Röll.White, M. J. 1988, “On Continuity: Aristotle vs. Topology?”, History and Philosophy of Logic 9.Williams, D. C. 1953, “On the Elements of Being”, Review of Metapysics 7, pp. 171–192.

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