Leibniz's Principle, Physics, and the Language of Physics - Springer Link

Foundations of Physics, Vol. 30, No. 10, 2000

Leibniz's Principle, Physics, and the Language of Physics Elena Castellani 1 and Peter Mittelstaedt 2 Received July 15, 1998 This paper is concerned with the problem of the validity of Leibniz's principle of the identity of indiscernibles in physics. After briefly surveying how the question is currently discussed in recent literature and which is the actual meaning of the principle for what concerns physics, we address the question of the physical validity of Leibniz's principle in terms of the existence of a sufficient number of naming predicates in the formal language of physics. This approach allows us to obtain in a formal way the result that a principle of the identity of indiscernibles can be justified in the domain of classical physics, while this is not the case in the domain of quantum physics.

``Poser deux choses indiscernables, est poser la me^me chose sous deux noms.'' (G. W. Leibniz, GP VII, p. 372)

1. INTRODUCTION In recent philosophy, logic and foundations of physics, a lively debate has arisen on the significance of the notions of identity, individuality, and indistinguishability in physics, especially in the quantum domain. A central role in this discussion is undoubtly played by the question of the validity of Leibniz's principle of the identity of indiscernibles in modern physics. We could even say that, in a sense, the question as to whether Leibniz's principle is violated in the quantum domain has originated the debate in its present form. 3 1

Department of Philosophy, University of Florence, Italy. Institute for Theoretical Physics, University of Cologne, Germany. 3 See the seminal discussion provided by the contributions from Cortes (1976), Barnette (1978), Ginsberg (1981), and Teller (1983) on the validity of Leibniz's principle in quantum physics. 2

1587 0015-9018001000-158718.000 2000 Plenum Publishing Corporation

1588

Castellani and Mittelstaedt

Leibniz's principle provides a relation between the indiscernibility (indistinguishability) of things and their identities. According to this principle, it is not possible for two things to be indiscernibles and yet numerically distinct (not the same thing). 4 The meaning and validity of Leibniz's principle, especially in connection with the problem of defining the notions of identity and individuality, are traditionally debated in the philosophical and logical literature. But why to discus Leibniz's principle with regard to physics, and in particular with regard to quantum physics? The basic motivation for such a discussion has to do with the so-called ``problem of indistinguishable particles.'' 5 The historical roots of this problem are to be found in some physical developments in the 1920's which were of decisive importance for quantum physics: first of all, the appearance of new statistics (the BoseEinstein and FermiDirac statistics) for aggregates of similar physical systems, and the connection between those ``quantum statistics'' and the recognition of permutation invariance as a property of quantum systems of particles of the same kind (the so-called `ìdentical particles''). 6 Permutation invariance, seen as a condition, of physical indistinguishability for `ìdentical particles,'' 7 seems to imply the existence of entities which are indistinguishable and yet ``numerically distinct.'' In this peculiarity of the quantum descriptionnumerically different entities can coincide in all their known physical properties or, in current terminology, there can be `ìndistinguishable'' entities which are not `ìdentical''we can trace the starting point for the discussion of Leibniz's principle with regard to modern physics. 8 In what follows, we first illustrate how the question is currently discussed in recent literature by enucleating two main ways (a ``formal'' and a ``physical'' way in which the subject is usually approached. After a short

4

Regarding the question of tile real meaning of Leibniz's principle, we shall enter into some more detail in Sec. 3 below. 5 Also known, in the literature, as the ``problem of identical particles.'' 6 According to a terminology which is somewhat misleading but current (especially among physicists), particles having the same `ìntrinsic'' or state-independent properties, like mass, spin and charge, are usually referred to as `ìdentical particles.'' For a critical discussion of the connection between quantum statistics and the principle of permutation invariance see in particular van Fraassen (1991), whose Chap. 11 is entirely devoted to the problem of identical particles. 7 From an historical point of view, a useful account of the first developments of the indistinguishability concept in microphysics is provided by Kastler (1983). For a recent critical account of the meaning of the `ìndistinguishability of identical particles'' see e.g., Dicks (1990). 8 A detailed account of the sources, nature and first developments of this debate can be found in van Fraassen (1984).

Leibniz's Principle, Physics, and the Language of Physics

1589

interlude in which we reconsider the question of the physical meaning of the principle by taking into account also its philosophical origin, we turn to the specific subject of our paper: namely, the discussion of the physical validity of Leibniz's principle in terms of the problem of existence of a sufficient number of naming predicates in the formal language of physics.

2. THE CURRENT DISCUSSION: FORMAL AND PHYSICAL APPROACHES As already said, discussing the validity of Leibniz's principle with regard to physics is deeply connected with determining in what the identity and individuality of physical entities do consist. In particular, the question as to whether. Leibniz's principle can be regarded as valid in a given physical domain is inevitably associated with the question as to whether individuals can be defined in that domain: if individuals can be defined, the validity of Leibniz's principle largely depends on how this individuality is obtained; if individuals cannot be defined, Leibniz's principle is simply inaplicable and the problem of its validity dissolves. With regard, to this discussion, the prevalent positions that are to be found in the literature can be grouped into two main categories. According to a first kind of position, it is sustained that a form of individuality may be ascribed to classical particles, but not to quantum particles. Correspondingly, it is also commonly held that a form of the principle of the identity of indiscernibles is valid in the domain of classical physics, while the principle is inapplicable in the quantum case. 9 The positions of the second group maintain, on the contrary, that it is possible to define individuals also in quantum mechanics, under some specific interpretive 9

This is undoubtly the prevailing position in the literature. For discussions of the nonindividuality of quantum particles, see, among others, Reichenbach (1956, Sec. 26), Post (1963), Cortes (1976), Ginsberg (1981), Teller (1983), van Fraassen (1984), van Fraassen (1991), Redhead and Teller (1991), Readhead and Teller (1992). A very illuminating analysis concerning the meaning of the notions of identity, indistinguishability and individuality in both the classical and the quantum case, as well as the two main sorts of positions which can be found in regard, is provided by French (1989a). See also French (1989c), which is especially devoted to examining the status of various forms of Leibniz's principle in classical and quantum physics. A most recent discussion of the two ``metaphysical packages'' supported by the same quantum formalism (one in which particles are regarded as non-individuals, in sortie sense, and another in which particles are regarded as individuals. for which certain sets of states are rendered inaccessible) and their meaning with respect to the discussion of Leibniz's principle can be found in French (1998).

1590


assumptions on the quantum description of particle states. 10 Leibniz's principle may then be applied to quantum particles and whether such `ìndividuals'' do refute the principle depends on the kind of interpretation assumed. 11 It is not of interest, here, to enter into the details of the various positions. It will be sufficient, for the purpose of our paper, to give an outline of how the problem of the physical significance of Leibniz's principle is commonly approached and which are the most frequented themes. In recent literature two main ways of approaching the problem can be distinguished, depending on whether formal aspects or physical features are given a prominent role in the discussion. Let us therefore speak, for the sake of convenience, of ``formal'' and ``physical'' approaches and treat them separately. (a) The formal approaches are essentially grounded on the assumption that a reconstruction of Leibniz's principle within a formal language is a convenient basis for adequately treating the problem. A seminal work in this sense is that of Kuno Lorenz, whose logical reformulation of Leibniz's principle has much contributed to clarify the essential features of the question. 12 As is known, when formally reformulated Leibniz's principle can be stated as a theorem of second-order logic: \ P \ a, b(P(a) W P(b)) Ä a#b

(2)

i.e., two objects a and b are identical if they have all the properties in common, that is if they are indistinguishable. In Lorenz's paper, this formal statement. of Leibniz's principle is proved to be neither trivial nor circular, with the assumption that naming predicates are included into the set of predicates corresponding to the properties of the objects under consideration. 13 10

For a analysis of such a possibility see especially French and Redhead (1988) (discussing the position: individual particles and state accessibility restrictions), and French (1989b) (individual particles and non-supervenient relations holding between them). With regard to the possibility of recovering forms of individuality for quantum particles front the viewpoint of hidden-variables interpretations of quantum mechanics (modal interpretations, Bohmian mechanics), a recent contribution is Huggett (1997). In relation to the position considering fermions as individuals within the framework of a modal interpretation, see in particular van Fraassen (1991) and the critical review of his work in Butterfield (1993). 11 In French and Redhead (1988), for example, the weakest form of Leibniz's principle is shown to be violated for quantum particles (understood as individuals) on the basis of a particular but plausible interpretation of what the state-dependent properties of atomic particles should be taken to be. 12 See Lorenz (1969). 13 This point will be discussed in more detail in Sec. 4 below.


1591

From a formal point of view, it is therefore possible to approach the problem of the validity of a ``formal Leibniz's principle'' in physics by investigating whether a sufficient number of naming predicates is admitted in the language of physics. The last part of our paper will be devoted to this formulation of the problem. Our result will be that the conditions for the validity of a formal Leibniz's principle are fulfilled in classical mechanics, while this is not the case in quantum mechanics: as we shall show, there is not a sufficient number of naming predicates in the formal language of quantum physics. If quantum entities are to be considered non-individuals in some sense, what does this imply from a logico-mathematical point of view? Taking the non-individuality of quantum particles as starting point, some authors concentrate on the logical implications of this result, and especially on the semantical aspects connected with the peculiar features of quantum entities. 14 Contributions in this area are mainly devoted to analysing new semantics for adequately treating objects like the indistinguishable particle of microphysics. The most notable example is the theory of ``quasi-sets'' developed by Dalla Chiara and Toraldo di Francia on the one side, and by Da Costa, Krause and French on the other side, for describing collections of entities that are indistinguishable but nevertheless not the same thing, that is entities to which the concept of identity of classical logic and mathematics apparently does not apply. 15 Leibniz's principle is generally understood to mean that it is not possible for two or more individuals to share exactly the same properties. But what is intended by `ìndividuals'' and what kinds of properties are to be considered? Depending on how individuals are defined and on which properties (formal, non-relational, relational, properties of whatever kind) are taken to be essential for an individual to be distinct from another one, different versions of Leibniz's principle are discussed in the literature. 16 14

See e.g., Dalla Chiara and Toraldo di Francia (1983), Dalla Chiara (1985), and Mittelstaedt (1985). 15 The notion of ``quasi-set'' (or ``quasets'') was first introduced in 1983 by Dalla Chiara and Toraldo di Francia in their `Ìndividuals, Kinds and Names in Physics'' (Dalla Chiara and Toraldo di Francia, 1983). For further developments of this approach, see Dalla Chiara (1985). Concerning the other approach to the theory of quasi-set developed by Da Costa, Krause and French, see e.g., da Costa, French, and Krause (1992), Krause (1992), French and Krause (1995). For a recent survey of the whole subject matter see Dalla Chiara, Giuntini and Krause (1998), where the basic ideas of two approaches to the notion of quasi-set are analysed and compared. 16 For the distinction between ``stronger'' and ``weaker'' versions of Leibniz's principle, depending on what kinds of properties are excluded or allowed to count, see for example Hoy (1984). With regard to physics, a thorough discussion of the different versions of Leibniz's principle and their status in both the classical and quantum domain is provided by French (1989c).

1592


(b) In the physical approaches to the question of the validity of Leibniz's principle, the nature of the individuating and distinguishing factors is usually investigated by focusing on the physical characteristics of the entities under consideration, that is, from the viewpoint of the physical theories involved and their interpretations. As regards in particular the microphysical domain, the discussion of what physical properties or assumptions are relevant to the identity and individuality of the basic entities is closely associated to the debate on the nature of quantum mechanical states. Because of the inc: liminable ``holism'' of quantum theory, for composed quantum systems (like the systems composed of `ìndistinguishable particles'') what states can be attributed to the individual components and, consequently, what properties can be attributed to the individual physical systems that are ``parts'' of the compounds, are very controversial questions. Discussing Leibniz's principle for quantum entities therefore inevitably implies entering into the basic interpretive problems of microphysics. As an illustration, let us mention here three points that are typically discussed in the physical approaches: (b 1 ) Is there a difference between fermions and bosons with respect to Leibniz's principle? More precisely, how relevant to the discussion of Leibniz's principle is the fact that fermions obey Pauli's exclusion principle, in this differing from bosons? The nature of the exclusion principle (no two fermions of the same kind can occupy the same state) suggests a possible connection with Leibniz's principle (no two substances can be completely equal but different in number): on this ground it is sometime claimed, in the literature, that Leibniz's principle is vindicated for fermions. 17 The discussion on this point, starting with Margenau's argument that electrons violate Leibniz's principle, generally focuses on the question of what states can be attributed to the individual components of a system of `ìdentical'' fermions. 18 While it is now commonly accepted that, if quantum-mechanical description is to be considered complete, there is no way of vindicating

17

In Weyl (1949), for example, the exclusion principle is explicitely referred to as ``Leibniz Pauli exclusion principle.'' The connection was proposed also in van Fraassen (1969). For a more recent analysis of this point, see van Fraassen (1984), Sec. 3, and van Fraassen (1991), Chap. 11. An illustration of the view that Pauli's principle offers a vindication of Leibniz's principle is provided, among others, by Schadmi (1978). 18 Cf. Margenau (1944) and (1950), Chap. 10. A discussion of Margenau's argument can be found in van Fraassen (1984), Sec. 3. See also van Fraassen (1991), Chap. 11, Sec. 10.


1593

Leibniz's principle for fermions, 19 it is still matter of discussion whether it is possible, to defend the validity of Leibniz's principle for fermions in the context of a modal interpretation of quantum mechanics (by means of a ``non-orthodox view'' of value attributions allowing to find some observables for which it is possible to assign distinct values to the fermionic components), as first suggested and argued by van Fraassen. 20 (b 2 ) Can history provide an individuation principle for an object? In particular, can a quantum object be individuated by its history? Two `ìdentical particles'' could then be distinguished by their histories. The idea that two things, apparently entirely alike, could nonetheless be distinguished by virtue of their preceding histories can be traced back to Thomas Aquinas, who could in this way generalize the Aristotelean individuation principle (based on the matter) to immaterial entities. 21 With respect to the question of the validity of Leibniz's principle in quantum physics, the re-proposal of the history-argument by van Fraassen (1969) was at the origin of a specific debate. What ``history'' exactly means is quite controversial: is history to be defined in terms of some kind of property, or is history something transcending the set of properties pertaining to the objects considered? If history is to be intended as `ìdentity through time'' or ``genidentity,'' using Reichenbach's terminology, what is the nature of the relations between the different temporal stages belonging to a single persistent object (and therefore constituting its ``history'')? Without entering here in the details of such general and traditional philosophical issues, let us just mention that, in the literature on the physical validity of Leibniz's principle, history is prevalently discussed in relation to the spatio-temporal paths of the particles. On this ground, the possibility of individuating quantum particles is denied in Cortes (1976), because of the ambiguity of the concept of history at space-time ``forks'' of similar entities; such a position, criticized by Barnette (1978) as confusing metaphysics and epistemology, is defended again in Ginsberg (1981) from the 19

Particularly relevant, in this regard, is the impossibility of the so-called `ìgnorance interpretation'' for a certain class of mixed states (the proof of the non-objectification theorem stating that general mixed states do not admit an ignorance interpretation can be found in Busch and Mittelstaedt (1991)). A discussion of the violation of (a weak form of ) Leibniz's principle in the case of fermions can be found, for example, in French and Redhead (1988), Sec. 2. 20 Cf. van Fraassen (1984) and van Fraassen (1991), Chap. 11. For comments on van Fraassen's view regarding the individuability of fermions in the modal interpretation see, for example, Butterfield (1993), Sec. 5. 21 Cf. Thomas Aquinas, De Ente et Essentia V, 5070. On this point and its significance with regard to the discussion of Leibniz's principle in quantum physics, see van Fraassen (1969) and again van Fraassen (1991), Chap. 12, Sec. 4.

1594


viewpoint of quantum field theory, where the creation and annihilation processes seem to constitute a decisive challenge to any concept of history. The whole discussion is finally re-considered in Teller (1983), who generalizes it by calling the attention to the basic question of the nature of the fundamental entities described in quantum field theory. 22 Irrespective of this controversy, one should keep in mind that there is a pure physical history of a quantum system, which contains more information about the system than its present state, even if this one is pure and maximal. 23 The relevance of this historical information for the individualization of a system has not yet be studied in sufficient detail. Finally, let us note that new interesting arguments regarding the history issue can result from the recent ``consistent history approach'' to quantum mechanics. (b 3 ) Is the question of the physical significance of Leibniz's principle meaningless in quantum field theory, where the `òbjects'' are fields or quantized excitations of fields? Objects which can be created and destroyed as well as entities whose existence is only ``virtual'' seem to elude any attempt at a definition of individuality. 24 It is in fact a widespread attitude to hold that the problems related to quantum mechanical objects can be solved only within the framework of quantum field theory, which is the ``true'' theory (or, at least, a more advanced theory). As regards specifically the individuality question, it is accordingly maintained that the idea of individual particles should be simply eliminated from the physical description, by shifting to the Fock space formalism of quantum field theory and so abbandoning the misleading lables of the tensor product Hilbert space formalism. 25 It seems however to be problematic whether genuine problems of non-relativistic quantum mechanics should be solved by a transition to the more extensive and general quantum field theory. We think it more 22

Regarding this basic ontological problem of contemporary physics see especially Teller (1995), entirely devoted to the interpretative problems of quantum field theory. 23 See on this point Aharonov and Albert (1984). 24 As already mentioned, the specific difficulties arising with regard to the individuality question in quantum field theory were first analized in Ginsberg (1981) and Teller (1983). For a more recent treatment of this issue, see Redhead and Teller (1991) and (1992), and Teller (1995). 25 The position that the use of labels suggests that quantum physics is about individuatable entities and that we should instead take quantum entities to be entirely unindividuatable in the way suggested by the unlabelled Fock space description is defended especially in Redhead and Teller (1991) and (1992), and further argued in Teller (1995) arid (1998). This position has been criticized by van Fraassen, claiming that there is a substantial equivalence between the quantum field formalism and the many-particle formalism (van Fraassen (1991), Chap. 12). A critical discussion of van Fraassen's argument is provided in Butterfield (1993), and Teller (1998).


1595

desirable to have the possibility of of determining a ``classical object'' from within the framework of classical mechanics and a (non-relativistic) ``quantum object'' from within the framework of (non-relativistic) quantum mechanics. 26 Not to mention the fact that shifting from a given theory to a more comprehensive one may often create new difficulties rather than help to solve the problems in question.

3. INTERLUDE: RECONSIDERING THE IDENTITY OF INDISCERNIBLES So far we have spoken of Leibniz's principle without any reference to the philosophical context in which the principle was first formulated. A natural question, at this point, is whether the principle as it is currently discussed in relation to physics is really the principle as Leibniz intended it. Answering this question requires considering both the meaning of the principle within the framework of Leibnizian philosophy and the sense of a principle of the identity of indiscernibles with regard to physics. In order to clarify these points, let us re-start the discussion of the principle from its historical origin and then consider the meaning it has currently acquired when referred to the objects of modern physics. As is known, in Leibniz's writings the principle of the identity of indiscernibles is formulated many times, in different contexts and in different ways. There is much discussion, among Leibniz scholars, about Leibniz's own view of the identity of indiscernibles: whether Leibniz intended the principle to be ``logical'' (``necessary'') or ``non-logical'' (``contingent''), whether Leibniz's ideas on the status of the principle changed in time (according to a ``Darwinian approach'' to Leibniz's views), 27 whether the principle was meant to refer only to the ``monads,'' are complicate historical and interpretive questions and it is not our scope to enter into such a debate. 28 Our intention will be, instead, just to call the attention to some of Leibniz's most famous statements regarding the identity of indiscernibles, and stress what is of special interest from the point of view of the physical meaning of the principle. 26

For a ``methodological pluralism'' of this kind in addressing the individuality question in physics see, for example, Beltrametti and Cassinelli (1981), Mittelstaedt (1985), Mittelstaedt (1995), and Castellani (1998). 27 The opposition between a ``Darwinian approach'' and an `Àthenian approach'' to the Leibnizian corpus is due to Castaneda (1975). 28 On such questions relative to Leibniz's views on the identity of indiscernibles see, among others, Frankel (1981), Chernoff (1981), and Brown (1990), Chap. VI.

1596


Let us start with the following statement that can be found in the Monadologie, in connection with the Leibnizian conception of individual substances or ``monads:'' ``Car il n'y a jamais dans la nature deux Etres, qui soyent parfaitement l'un comme l'autre, et ou il ne soit possible de trouver une difference interne, ou fondee sur une denomination intrinseque'' (GP VI, p. 608). 29 The principle of the identity of indiscernibles is here related to a distinction between `ìntrinsic'' and `èxtrinsic'' denominations: extrinsic denominations are not sufficient for distinguishing between two substances. Leibniz's distinction between intrinsic and extrinsic denominations is a subject of much discussion in the literature. 30 Spatial localization and temporal determination, for example, are not intrinsic properties, and hence not sufficient for distinguishing between two things (`ìl faut tousjours qu'outre la difference du temps et du lieu, il y ait un principe interne de distinction'' (GP V, p. 213)). On the other hand, extrinsic determinations are not independent from the intrinsic ones. Here comes into play the Leibnizian conception of an individual substance. For Leibniz, the notion of an individual substance is essentially characterized by its completeness: ``la nature d'une substance individuelle ou d'un e^tre complet est d'avoir une notion si accomplie qu'elle soit suffisante a comprendre et a en faire deduire tous lee predicate du sujet a qui cette notion est attribuee'' (GP IV, p. 433). Such a completeness is also meant in a historical sense, the complete concept of an individual substance involving all its past, present and future predicates: ``la notion d'une substance individuelle enferme une fois pour toutes tout ce qui lui peut jamais arriver'' (GP IV, p. 436). Each individual substance therefore appears to be identified in virtue of its complete concept: in this metaphysical framework, the principle of the identity of indiscernibles seems to be drawn from Leibniz's view of individual substances and their complete concepts. 31 If we accept this interpretation of Leibniz's doctrine, what about the phenomena? If the identity of indiscernibles is to be established only for such individual substances as the monadsin other words, if it precisely the nature of the monads which makes it possible, for Leibniz, to sustain 29

GP=Die Philosophishen Schriften Von Gottfried Wilhelm Leibniz, IVII, C. I. Gerhardt (ed.), Berlin, 18751890, repr. in G. Olms Verlag, Hildesheim, 1960. 30 Also the use of the terms ``denominations,'' `ìntrinsic'' and `èxtrinsic'' in Leibniz is not uncontroversial (see, for example, Ishiguro (1990), Chap. 6). We shall not enter into such interpretive questions, here, and we shall generally speak of ``properties'' also where Leibniz uses the term ``denominations.'' 31 A recent defense of such an interpretation of Leibniz's view of the identity of indiscernibles can be found in Brown (1990). For an idea of the debate concerning the exact nature of the relation between actual individual substances and their complete conceptsa much discussed question among Leibnizian scholarssee for example Frankel (1981).


1597

that `ìl n'est pas vrai que deux substances se ressemblent entierement, et soyent differentes solo numero'' (GP IV, p. 433)what is the scope of referring the principle to physical entities like two drops of water, two eggs or two leaves in a garden, as Leibniz appears to do in some of his most known passages on the principle? Let us quote in detail the most famous Leibnizian example for the identity of indiscernible, illustrating the impossibility of finding two ``perfectly similar'' leaves in the garden of Herrenhausen: `ùne grande Princesse, qui est d'un esprit sublime, dit un jour en se promenant dans son jardin, qu'elle ne croyouait pas, qu'il y avait deux feuilles parfaitement semblables. Un gentilhomme d'esprit, qui etait de la promenade, cru^t qu'il serait facile d'en trouver; mais quoiqu'il en chercha^t beaucoup, il fut coinvaincu par ses yeux, qu'on pouvait toujours y remarquer de la difference'' (GP V, p. 214). 32 A common position in the literature is that such passages should be taken as simply furnishing empirical examples of a metaphysically necessary truth (rather than proving the identity of indiscernibles to be an empirical principle itself ). 33 Phenomena become relevant when shifting from a metaphysical to an epistemological level: the principle can be hard for limited human minds to comprehend, but empirical examples may be of help. As human beings, that is limited `èsprits,'' we cannot have the knowledge of all the infinite attributes which contribute to the complete notion of an individual; this is possible only for God. With our limited capacities, we can never have the complete knowledge for adequately identifying an individual; but empirical examples help us, showing that we cannot find in nature two indiscernible things. Moreover, what empirical examples show us can be inferred as a consequence of the principle of sufficient reason: 34 the supposition of two indiscernibles may indeed be possible in abstract terms, but it is not compatible with the order of things nor with the divine wisdom, by which nothing is admitted without reason. 35 Let us now turn to physics and consider what may be the meaning of the principle of the identity of indiscernibles when referred to the entities which it is the scope of physical theories to describe. In physical terms, objects are generally characterized through two kinds of properties: (a) properties such as, for example, mass, charge, and spin, usually indicated as `ìntrinsic'' (``state-independent'' or, according to a more traditional 32

For the same example, cf. also GP VII, p. 372. See for example Frankel (1981), and Brown (1990), Chap. 6, Sec. 3. 34 It is not of interest, here, to enter in more detail regarding the role played by the principle of sufficient reason in relation to the principle of the identity of indiscernibles (another crucial point of discussion in the literature on Leibniz's doctrine). 35 Cf. GP VII, p. 394. 33

1598


terminology, `èssential'') properties; and (b) `èxtrinsic'' (``state-dependent,'' `àccidental'') properties, like spatio-temporal determinations (position in space, and so on). The above distinction is not uncontroversial, 36 but it is possible to give it a definite meaning within the group-theoretical formalism of physical theories. The application of group-theoretical techniques for exploiting the symmetry properties of physical systems and physical theories allows indeed to identify the `ìntrinsic'' properties as those which can be obtained through given sorts of representations of the fundamental symmetry groups of the theory (for example: the representations of Galilei group in phase space if the theory is classical mechanics, the unitary representations of Galilei group in Hilbert space if the theory is quantum mechanics, and the unitary representations of Poincare group in Hilbert space if the theory is relativistic quantum mechanics). 37 Properties which can be obtained in this way are surely the most relevant ones, sine they are necessary to characterize a physical object as such (a particle, to be identified as an electron, must necessarily have a well defined value for mass, charge, and spin); they are, however, not sufficient to determine an object as an individual (only classes of objects, for example the class of all electrons, can be determined through such properties). 38 Such `ìntrinsic'' physical properties could be seen, at first sight, as the best candidates for the `ìntrinsic'' determinations in Leibnizian sense. In physics, however, we verify a sort of inversion with respect to the Leibnizian case: as already said, the intrinsic properties cannot furnish a principle of individuation (only classes of objects can be determined on the ground of such properties). It is rat most by virtue of extrinsic properties such as spatio-temporal determinations that objects of the same class could be distinguished, provided spatial impenetrability is also presupposed. 39 Of course it is well possible to speak of a principle of the identity of indiscernibles also in a different sense than Leibniz, and discuss it with respect to a given set of (monadic andor relational) properties, according to the specific criteria and framework adopted. As we have seen, this is the usual way of proceeding in the literature: in investigating the question of the physical validity of a ``Leibniz's principle,'' different form of a principle 36

For a recent analysis of this distinction, especially with regard to its application to quantum systems, see Butterfield (1993). 37 For the sake of simplicity, only the fundamental space-time symmetries have been taken into account here. 38 On this point, and in general on the relevance of the group-theoretic formalism to the problem of determining physical objects, see Mittelstaedt (1995) and Castellani (1998). 39 For an illuminating discussion of this view of space-time individuality, grounded on spacetime determinations and on an impenetrability assumption, see in particular French (1989a).


1599

of the identity of indiscernibles are discussed, from a ``weaker'' to a ``stronger'' version, depending on what sorts of properties are taken to be of importance for distinguishing individual physical objects. In what follows, we shall discuss the question of the validity of a formal version of the principle of the identity of indiscernibilitywe shall call it the formal Leibniz's principlein the context of both classical and (nonrelativistic) quantum mechanics. The significant properties will be, for our discussion, `èxtrinsic'' determinations of physical objects.

4. THE FORMAL LEIBNIZ'S PRINCIPLE AND THE LANGUAGE OF PHYSICS Following a ``formal approach'' to the problem of the validity of Leibniz's principle in physics, we shall start our discussion with recalling how a principle of the identity of indiscernibles may be reformulated in a abstract and general way. A formulation of this kind, which makes use of the theory of definite descriptions and which is in accordance with the usual requirements of formal languages, had been first proposed in a different context by Lorenz. This reformulation turns out to be very useful also for the present investigation. Let S=[x, y,...] be a set of subjects and P= [A, B,...] a set of predicates, then we can form propositions A(x), B(x), etc. The indiscernibility of two subjects x and y is then defined by x= y # \ A # P (A(x) W A( y)). On the other hand we can define the identity of x and y by x# y # c(x y), making use of the relation x y (x is different from y) which is introduced by explicite definition. A formal Leibniz's principle can then be formulated in a non-trivial and non-circular way through the theorem: s=t Ä s#t. 40 This theorem can be proved in a purely formal way either in the given form or in the equivalent reformulation st Ä c(s=t). 41 For the proof one must presuppose that for any s # S there exist a naming predicate N s( } ), ``to be an s,'' which is fulfilled by s and which is unique with respect to the relation ``#'' of identity. A naming predicate can then be used for a definite description of the subject s. In a given language of physics with sets S and P respectively, one can now investigate whether the mentioned condition is fulfilled in such a way that the formal Leibniz's principle can be proved. We shall consider here (a) the language S C of classical mechanics and (b) the language S Q of quantum mechanics. 40 41

Cf. Lorenz (1969). See Giuntini and Mittelstaedt (1989).

1600


(a) In the language S C of classical mechanics, the set S is given by mechanical objects of a certain kind, e.g., by mass points, which are determined by some permanent (`èssential'') properties E 1 , E 2 ,... of the objects in question, corresponding to the `èssential'' predicates [E 1 , E 2 ,...]. The (`àccidental'') mechanical predicates P i # P are then given by subsets (Borel-sets) of the phase space. It is well known that these predicates P 1 , P 2 ,... form a lattice L C which is, according to the assumptions mentioned, complete, orthomodular, distributiv and atomic. In the present case, the atoms of the Boolean lattice L C are the points [x i , p i ] of the phase space. Classical mechanics is the theory of the predicates P i as functions of time. The general space-time invariances of this theory, which are given by the 10-parameter Galilei-group G 10 , determine, through the representations of this group, the essential properties E i mentioned above. A naming predicate N s , which is appropriate for the definite description of a subject s, must be fulfilled by s, i.e., N s(s) must hold, and it must be unique with respect to the identity, i.e., the relation \ x # S(N s(x) Ä (x#s)) must be valid. If the set S of subjects is sufficiently large, such that any predicate is fulfilled by at least one subject, then it follows. from the mentioned properties of a naming predicate that N s is an atom in the lattice L C . However, the uniqueness condition requires that in addition a contingent exclusion principle holds, in the present case the spatial impenetrability of the mechanical bodies. Under the additional condition of impenetrability, the points [x k , y k ] in phase space can be used as naming predicates. Hence it follows that the conditions which are sufficient for the validity of the formal Leibniz' principle in classical mechanics are fulfilled. Moreover, in a Boolean lattice L C it follows that an atom N s # L C is also ``complete,'' i.e., for any other predicate A # P we have the relation N s A 6 N s cA. The naming predicates which correspond to the points [x k , y k ] in phase space are thus also complete in the sense of a physical ``complete concept:'' they determine all the other predicates A # P which are fulfilled by s. Consequently, in this particular situation, names correspond at the same time to complete concepts. (b) In the language S Q of quantum mechanics, the set S is given by quantum mechanical systems of a certain class, which is determined by some permanent properties of the respective systems. Examples for such proper ties which characterize a class are rest-mass, charge, spin, etc. The possible predicates P i # P of a subject s are then given by the subspaces of the Hilbert space H s which is used as a state-space of s. These quantum mechanical predicates form a lattice L Q which is again complete, orthomodular and atomic, but, on the contrary to what happens in the case of


1601

the lattice L C of classical mechanics, not distributiv. For the validity of the formal Leibniz's principle, the decisive question is also here, whether there is a sufficient number of naming predicates in L Q . If we start, as in the language S C , with the assumption that the set S of subjects is not restricted and is sufficiently large, then the naming predicates appropriate for definite descriptions are again atoms of the lattice of the predicates, that is, in this case, atoms of the lattice L Q . In the language S Q , however, a naming predicate N s which can be used for definite description must fulfill also another condition: propositions of the form A(s) have a meaning only if the predicate A is compatible with the naming predicate N s used for the definite description of the subject s, i.e., if the relation N s (N s 7 A) 6 (N s 7 cA) holds. Since this requirement must be fulfilled for all the predicates A # P, the name N s must be in the center Z(L Q ) of the lattice L Q . Hence the problem whether there are convenient naming predicates leads to the more formal question of the existence of a sufficiently large set of atoms in the center of L Q . With respect to this question, in quantum mechanics one has to distinguish between two cases, both of which are of physical importance. If the lattice L Q is irreducible, then the center of L Q is trivial, i.e., it consists of the elements 0 and 1. In this case there are no ``superselection rules'' which determine the `èssential'' properties and the answer to the question is quite simple: there are no atoms at all in the center and hence no naming predicates in L Q . A formal Leibniz's principle can not be justified in this case. If the lattice L Q is reducible, we are confronted with a complete different situation. The center is no longer trivial, i.e., there are elements X, Y, Z,... # L Q which are different from 0 and 1 and which are compatible with all the elements of L Q . These predicates correspond to the essential properties mentioned above, which are not subject to the Hilbert space quantum mechanics. In this situation, which is more interesting from the point of view of a formal Leibniz' principle, we have to distinguish again between two cases. There are lattices of the kind mentioned the non-trivial center of which does not contain atoms: such lattices do not have any naming predicates. This result corresponds to the irreducible case. On the other hand there are also lattices of the kind considered here the center of which contains atoms which correspond to naming predicates. These lattices, which at first sight appear to be appropriate canditates for languages structures for which a formal Leibniz's principle holds, must be excluded for another reason. Namely, the set NZ(L Q ) of naming predicates in the center is in no case sufficient for the denomination of all the elements x # S which are possible subjects of a predicate A. Hence it could happen that there is a subject s with the property A such that A(s) is true, but that s

1602


cannot be named by some predicate P i # L Q . Even in this theoretically interesting case, a proof of a formal Leibniz's principle cannot be given, since a sufficient set of naming predicates is not available. Hence we find that the lattice L Q of formal quantum language never contains a sufficient number of elements which could be used as naming predicates. Moreover, this deficiency of convenient predicates cannot be removed by embedding the lattices L Q into a Boolean lattice with a sufficient number of naming predicates. 42 We can therefore conclude that, while a formal principle of the identity of indiscernibles can be justified in the domain of classical mechanics, this is not the case in the domain of (non-relativistic) quantum mechanics.

REFERENCES Y. Aharonov and D. Albert, `Ìs the usual notion of time evolution adequate for quantummechanical systems?'' I-II, Phys. Rev. D 29, I: 223227, II: 228234 (1984). R. L. Barnette, ``Does quantum mechanics disprove the principle of the identity of indiscernibles,'' Phil. Sci. 45, 466470 (1978). E. G. Beltrametti and G. Cassinelli, The Logic of Quantum Mechanics (AddisonWesley, Reading, Massachusetts, 1981). C. Brown, Leibniz and Strawson. A New Essay in Descriptive Metaphysics (Philosophia, Munchen, 1990). P. Busch and P. Mittelstaedt, ``The problem of objectification in quantum mechanics,'' Found. Phys. 21, 889904 (1991). J. Butterfield, `Ìnterpretation and identity in quantum theory,'' Studies in History and Philosophy of Science 24, 443476 (1993). H. Castaneda, ``Leibniz's 1686 views on individual substances, existence, and relations,'' J. Phil. 72, 687690 (1975). E. Castellani, ``Galilean particles: An example of constitution of objects,'' in Interpreting Bodies: Classical and Quantum Objects in Modern Physics, E. Castellani, ed. (Princeton University Press, Princeton, 1998). F. Chernoff, ``Leibniz's principle of the identity of indiscernibles,'' The Philosophical Quartely 31, 126138 (1981). A. Cortes, ``Leibniz' principle of the identity of indiscernibles: A false principle,'' Phil. Sci. 43, 491505 (1976). N. C. A. da Costa, S. French, and D. Krause, ``The Schrodinger problem,'' in Erwin Schrodinger: Philosophy and the Birth of Quantum Mechanics, M. Bitbol and O. Darrigol, eds. (Editions Frontieres, Paris, 1992). M. L. Dalla Chiara and G. Toraldo di Francia, `Ìndividuals, kinds and names in physics,'' in Logica e Filosofia della Scienza oggi, E. Agazzi and M. Mondadori, eds. (Clueb, Bologna, 1983).

42

Within the formal language S Q of quantum physics, the impossibility of a consistent extension of S Q has been demonstrated in particular by Giuntini (1987).


1603

M. L. Dalla Chiara, ``Names and descriptions in quantum logic,'' in Recent Developments in Quantum Logic, P. Mittelstaedt and E. W. Stachow, eds. (Bibliographishes Institut, Mannheim, 1985). M. L. Dalla Chiara, R. Giuntini, and D. Krause, ``Quasiset theories for microobjects: A comparison,'' in Interpreting Bodies, E. Castellani, ed. (Princeton University Press, Princeton, 1998). D. Dieks, ``Quantum statistics, identical particles and correlations,'' Synthese 82, 127155 (1990). L. Frankel, ``Leibniz's principle of identity of indiscernibles,'' Studia leibnitiana 13, 192211 (1981). S. French, `Ìdentity and individuality in classical and quantum physics,'' Australasian J. Phil. 67, 432446 (1989a). S. French, `Ìndividuality, supervenience and Bell's theorem,'' Phil. Studies 55, 122 (1989b). S. French, ``Why the principle of the identity of indiscernibles is not contingently true either,'' Synthese 78, 141166 (1989c). S. French and M. Redhead, ``Quantum physics and the identity of indiscernibles,'' Brit. J. Phil. Sci. 39, 233246 (1988). S. French, `Òn the withering away of physical objects,'' in Interpreting Bodies, E. Castellani, ed. (Princeton University Press, Princeton, 1998). S. French and D. Krause, `À formal framework for quantum non-individuality,'' Synthese 102, 195214 (1995). A. Ginsberg, ``Quantum theory and identity of indiscernibles revisited,'' Phil. Sci. 48, 487491 (1981). R. Giuntini, ``Quantum logic and Lindenbaum property,'' Studia Logica 46, 1735 (1987). R. Giuntini and P. Mittelstaedt, ``The Leibniz principle in quantum logic,'' Internat. J. Theoret. Phys. 28, 159168 (1989). R. C. Hoy, `Ìnquiry, intrinsic properties, and the identity of indiscernibles,'' Synthese 61, 275297 (1984). N. Huggett, `Ìdentity, quantum mechanics and common sense,'' The Monist 80, 118139 (1997). H. Ishiguro, Leibniz's Philosophy of Logic and Language (Cambridge University Press, Cambridge, 1990). D. Krause, `Òn a quasi-set theory,'' Notre Dame Journal of Formal Logic 33, 402411 (1992). K. Lorenz, ``Die Begrundung des principium identitatis indiscernibilium,'' Studia Leibnitiana, Supplementa Vol. III, 149159 (1969). A. Kastler, `Òn the historical development of the indistinguishability concept for microparticles,'' in Old and New Questions in Physics, Cosmology, Philosophy, and Theoretical Biology, A. van der Merwe, ed. (Plenum, New York, 1983). H. Margenau, ``The exclusion principle and its philosophical importance,'' Phil. Sci. 11, 187208 (1944). H. Margenau, The Nature of Physical Reality (McGrawHill, New York, 1950). P. Mittelstaedt, ``Constituting, naming and identity in quantum logic,'' in Recent Developments in Quantum Logic, P. Mittelstaedt and E. W. Stachow, eds. (Bibliographishes Institut, Mannheim, 1985). P. Mittelstaedt, ``Constitution of objects in classical mechanics and in quantum mechanics,'' Internat. J. Theoret. Phys. 34, 16151626 (1995). H. Post, `Ìndividuality and physics,'' The Listener 70, 534537 (1963). M. Redhead and P. Teller, ``Particles, particle labels, and quanta: The toll of unacknowledged metaphysics,'' Found. Phys. 21, 4362 (1991).

1604


M. Readhead and P. Teller, ``Particle labels and the theory of indistinguishable particles in quantum mechanics,'' Brit. J. Phil. Sci. 43, 201218 (1992). H. Reichenbach, ``The genidentity of quantum particles,'' in The Direction of Time (University of California Press, Berkeley, 1956). Y. Shadmi, ``Teaching the exclusion principle with philosophical flavor,'' Am. J. Phys. 48, 844848 (1978). P. Teller, ``Quantum physics, the identity of indiscernibles, and some unanswered questions,'' Phil. Sci. 50, 309319 (1983). P. Teller, An Interpretive Introduction to Quantum Field Theory (Princeton University Press, Princeton, 1995). P. Teller, ``Quantum mechanics and haecceities,'' in Interpreting Bodies, E. Castellani, ed. (Princeton University Press, Princeton, 1998). B. van Fraassen, ``Probabilities and the problem of individuation'' (presented at the American Philosophical Association in 1960), in Probabilities, Problems and Paradoxes, S. Luckenbach, ed. (Dickenson, Encino, California, 1972). B. C. van Fraassen, ``The problem of indistinguishable particles,'' in Science and Reality: Recent Work in the Philosophy of Science, J. T. Cushing, C. F. Delaney, and G. M. Gutting, eds. (University of Notre Dame Press, Notre Dame, 1984). B. C. van Fraassen, Quantum Mechanics: An Empiricist View (Clarendon, Oxford, 1991). H. Weyl, Philosophy of Mathematics and Natural Science (Princeton University Press, Princeton, 1949).