BOOK SYMPOSIUM

The physics and metaphysics of identityand individuality

Steven French and Decio Krause: Identity in physics: A historical,philosophical, and formal analysis. Oxford: Clarendon Press,2006, 440 pp, £68.00 HB

Don Howard • Bas C. van Fraassen • Otavio Bueno •

Elena Castellani • Laura Crosilla • Steven French •

Decio KrausePublished online: 3 November 2010

� Springer Science+Business Media B.V. 2010

Don Howard

Steven French and Decio Krause have written what bids fair to be, for years to

come, the definitive philosophical treatment of the problem of the individuality of

elementary particles in quantum mechanics (QM) and quantum-field theory (QFT).

The book begins with a long and dense argument for the view that elementary

particles are most helpfully regarded as non-individuals, and it concludes with an

earnest attempt to develop a formal apparatus for describing such non-individual

entities better suited to the task than our customary set theory.

D. Howard (&)

Department of Philosophy and Graduate Program in History and Philosophy of Science,

University of Notre Dame, Notre Dame, IN 46556, USA

e-mail: dhoward1@nd.edu

B. C. van Fraassen

Philosophy Department, San Francisco State University, 1600 Holloway Avenue,

San Francisco, CA 94132, USA

e-mail: fraassen@sfsu.edu

O. Bueno

Department of Philosophy, University of Miami, Coral Gables, FL 33124, USA

e-mail: otaviobueno@mac.com

E. Castellani

Department of Philosophy, University of Florence, Via Bolognese 52,

50139 Florence, Italy

e-mail: elena.castellani@unifi.it

L. Crosilla

Department of Pure Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, UK

e-mail: matmlc@leeds.ac.uk

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DOI 10.1007/s11016-010-9463-7

Are elementary particles individuals?

I do not know. It depends on what one means by ‘individuals’. This much is

certain—elementary particles as described by QM and QFT are not individuals inthe same sense in which classical mechanical systems, the molecules constituting a

Boltzmann gas, or Daltonian atoms are individuals. The elementary particles of QM

obey either bosonic or fermionic statistics. The molecules of a Boltzmann gas do

not. An individual elementary particle can find itself in a state that is a superposition

of eigenstates of some observable. Dalton’s atoms cannot. On the standard

interpretation of QM, interacting quantum systems are described by an entangled

joint state that is not fixed completely by any possible separate states of those

systems. Not so the interacting systems described by Newtonian mechanics and

gravitation theory. In QFT, there might be states of indefinite or indeterminate

particle number. In classical physics, a given region of space during a given time

always contains a determinate number of atoms. In QFT, particle number is frame

dependent, mutually accelerated observers feeling or seeing different numbers of

photons in one and the same region of space. In classical physics, particle number is

frame independent. You and I must detect the same number of atoms or molecules

regardless of our relative states of motion.

So, are elementary particles individuals? The philosopher’s customary way of

approaching the question is via the Principle of the Identity of Indiscernibles (PII),

which asserts the identity—hence the lack of individuality—of two things that share

all of the same properties. The strength of PII depends upon what one takes to be the

relevant individuating properties. French and Krause distinguish three versions, in

order of increasing strength. The weakest, PII(1), includes all properties. Slightly

stronger is PII(2), which excludes spatio-temporal properties. Stronger still is PII(3),

which excludes all relational properties.

Some version of PII seems to be relevant in the case of at least some kinds of

elementary particles. Consider the more straightforward case of bosons. Photons are

massless, spin-1 systems obeying Bose–Einstein statistics. Hence, on the standard

interpretation, QM does not regard as physically different two configurations that

one might think describable as involving merely the switch of physical location of

two otherwise identical photons. Not even a difference in spatial situation suffices to

endow two such photons with discernible or distinct identities of a kind sufficient to

mark the two configurations as physically different. Two such photons being, thus,

indiscernible, in spite of a difference in spatial situation, one is tempted to conclude

by way of PII(2), that they are identical, and, for that reason, not individuals.

S. French

Department of Philosophy, University of Leeds, Leeds, UK

e-mail: S.R.D.French@leeds.ac.uk

D. Krause

Department of Philosophy, Federal University of Santa Catarina,

88040-900 Campus Trindade, Florianopolis, SC, Brazil

e-mail: deciokrause@gmail.com

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This metaphysical state of affairs has definite, testable, physical consequences.

There being only one way of arraying two bosons in two different locations, that

configuration receives a statistical weight equal to that attaching to each of the two

different ways of arraying the two bosons in one or another of the same locations.

When, then, we ask what is the probability of the two bosons being together in the

same place, the answer is 2/3, significantly higher than the ‘classical’ value of 1/2,

which was computed by applying equiprobability to what were, classically, four

different configurations. As noted, this difference between quantum and classical

statistics has testable consequences, most famously in the form of Einstein’s 1925

prediction of the phenomenon of Bose–Einstein condensation, wherein, at

sufficiently low temperatures the atoms of a bosonic gas all condense into the

same lowest quantum state.

So, again, one is tempted to conclude that two (or more) indiscernible bosons are

not individuals. This conclusion can be resisted in various ways. Thus, one might

choose a non-standard interpretation of QM, such as the Bohmian interpretation,

which ascribes determinate trajectories to all elementary particles, endowing those

particles with in-principle distinguishing identities, and locating all of the quantum

weirdness elsewhere, mainly in the quantum potential. Or one can assume that, in

spite of their seeming indiscernibility, two such indiscernible bosons are neverthe-

less rendered metaphysically distinct by virtue of their each possessing some form

of transcendental individuality, primitive ‘thisness’, or, in the language of the

Scholastics, haecceitas. One recovers the standard quantum statistics by applying

the appropriate probabilities—1/6—to what are, on this view, distinct configurations

differing only by particle exchange.

A price must be paid for one’s resisting the conclusion that bosons are not

individuals. Going the Bohmian route requires lots of extra apparatus, including the

quantum potential, quantum ergodicity, and relativistic non-localities at the level of

the hidden variables. Choose the route of haecceitas and one requires a seemingly ad

hoc weighting of configurations, a weighting for which there is no plausible physical

explanation beyond the ex post facto argument that those are the weightings needed

to get the predictions to turn out right. But if one is willing to pay some such price,

neither logic nor empirical evidence can debar one’s taking one of these routes.

Are we asking the right question?

While it is clear, as I said above, that elementary particles as described by QM are

not individuals in the same sense as the molecules of a Boltzmann gas, what it

means positively to say that they are non-individuals is less clear. The second major

goal of French and Krause’s book is to develop formal tools in the form of quasi-set

theory to make easier our speaking and thinking clearly about non-individuals. But

space is wanting in this essay for a proper appreciation of that effort. Instead here,

like French and Krause—but for slightly different reasons—I want to suggest that

some of the preliminary unclarity about the sense in which elementary particles are

non-individuals derives from our having chosen to approach the question in the first

instance via PII. That PII is the philosophers’ favourite tool does not imply that it is

the most helpful tool for the physicist to employ.

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Two considerations suggest that there is something odd about deploying PII in

assaying the individuality of bosons. Firstly, even if one accepts the conclusion that,

owing to their indiscernibility, two bosons are not individuals, in spite of their

difference in spatial situation, this lack of individuality does not come in the form

of the bosons’ being numerically identical. That is to say, that this variety of

indiscernibility does not imply that we have just one boson, rather than two. One

cannot tell which boson is which, perhaps not just because of a lack of epistemic

access to that which might constitute the ‘whichness’ of the bosons, but because

they lack such ‘whichness’ from the start. But that there are two bosons, not one, is a

fact.

Secondly, in thinking about Bose–Einstein statistics, what are more helpfully

regarded as being indiscernible in the sense relevant for the applicability of PII are

not the bosons themselves, but rather the configurations classically distinguished by

particle exchange. This is because the indiscernibility of the configurations does, as

noted, imply the strict numerical identity of those configurations. In bosonic

quantum statistics, there is, literally, only one configuration in which there is exactly

one boson in each of two cells of phase space.

The point towards which I mean now to be tending in so querying the helpfulness

of PII is that, in making all of this out to be an argument about the individuality of

elementary particles, we might be asking the wrong question or a question that is not

well posed. French and Krause agree in querying the helpfulness of PII in assessing

the individuality of elementary particles. Indeed, they go so far as to assert the

contingent falsity of the principle in this context. However, their alternative is not to

drop the question about the individuality of elementary particles but to recast the

question of individuality in other terms.

French and Krause have a reason for wanting the question about elementary

particles to be a question about individuality. It is because arguments for the non-

individuality of elementary particles are supposed to be arguments for some version

of structuralism, the idea being that one wants a structuralist characterization of

elementary particles as being not, themselves, ontologically primitive, but

ontologically derivative from relations of some kind. And yet, no version of

structuralism known to me stands or falls on a claim about the non-individuality of

elementary particles. Structuralism would not be refuted if bosons were distin-

guishable particles, and the very same physics that encourages a view of elementary

particles as non-individuals entails other problems for structuralism, in the form of

the necessary existence of non-unitarily equivalent representations of algebraic

QFT, an issue to which I will return below.

If PII proves unhelpful in assaying the individuality of elementary particles, in

what other way are we to pose the individuality question? The answer is that French

and Krause concur in what they term the ‘Received View of particle (non-)

individuality’. On this view, indistinguishability, alone, establishes the non-

individuality of elementary particles.

So what kind of individuality or non-individuality is intended here? French and

Krause stress the importance of noting a distinction between individuality and

distinguishability, so what is assumed cannot be a definition of individuality as

distinguishability. Instead, what seems to be assumed is something that one might

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dub the ‘Principle of the Non-Individuality of Indistinguishables’. Well and good.

We have here a criterion. But, again, what kind of individuality or non-individuality

is at issue here? Two bosons are indistinguishable. Should that suffice to make them

non-individuals? After all, there are still two of them, not one, even if one cannot

tell which is which, or even if there is no ‘whichness’ in the first place. So if

determinate cardinality is one manifestation of the individuality of the members of a

set, as surely it is, then is not there at least something ‘individual’-like about bosons?

Think about it this way. Are two identical, hence, indistinguishable human twins to

be regarded as non-individuals simply because they are indistinguishable? Assume

for the sake of argument that human twins could also not be distinguished even

by their spatio-temporal situations, that continuity of trajectory, as it were, would

also not suffice to tell us which is which. Would one think them, therefore, devoid of

the kinds of legal rights and responsibilities that attach to individuals? No. Indeed,

our legal tradition has a concept ready-made for such situations, for in certain

circumstances we regard the members of a group as ‘jointly and severally

responsible’ for this or that, and for the purpose of deploying this notion of

responsibility the ‘whichnesses’ of the members of the group are irrelevant. In such

cases, the law does not care who is who, but we do not, therefore, regard the

members of the group as non-individuals. So where is the harm in regarding two

indistinguishable bosons as, nonetheless, individuals that ‘jointly and severally’

enjoy all the rights and responsibilities accorded to systems? That is surely not the

kind of individuality evinced by classical physical systems, but perhaps the lesson is

that progress in science has forced us to revise our notion of individuality rather than

denying its applicability to elementary particles.

My own view is that all of the complications now accumulating around the notion

of individuality suggest that, in making the question one about individuality or non-

individuality, we have been asking an unhelpful and perhaps ill-posed question.

But if individuality—whatever it might mean—is not the most helpful issue to be

raising about elementary particles, what is? From the point of view of the physicist,

the relevant question is simply this: Does difference of spatial (or spatio-temporal)

situation suffice to endow physical systems always and everywhere with separate,

real, physical states of such a kind that determine, univocally, the real physical

states of any aggregation of such systems? In classical physics as well as in

Einstein’s general relativity, the answer is, clearly, ‘yes’. In QM and QFT, on the

standard interpretation, the answer is, clearly, ‘no’, because of entanglement and

such special cases of entanglement as those represented by bosonic and fermionic

statistics.

One might think that I am suggesting here just a different way of approaching the

individuality question, that I am asking whether, in the quantum domain, what

French and Krause term the Principle of Space–time Individuation (STI) is true.

One might ask that question, but I am not. I am asking not about the individuation of

the systems occupying different regions of space–time, at least not in the first

instance. I am asking, instead, whether those systems possess states of a certain

kind, namely separate, real states that fix, univocally, all joint states. It is a question

about the states, not the systems. Impressed by the ubiquity of quantum

entanglement, one might then also ask what kinds of things are the systems that

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can be the bearers of entangled joint states and that might lead one back into the

thicket of questions about individuality. But that is not the question I asked. The

question I asked is simply whether quantum mechanical states are always separable.

And that is a clean, well-posed question, the answer to which is, in general, ‘no’.

Non-separability or entanglement is everywhere.

We live, or so we think at the moment, in a quantum universe. Does the ubiquity

of entanglement entail that there are no individual elementary particles? Yet again,

it depends in large measure on what one means by ‘individuality’. But in no case

does entanglement in any form entail that two indistinguishable particles become,

by virtue of that indistinguishability, just one particle. Nor does it mean, in many

cases, that two entangled particles lose their trackable identities, their ‘whichness’.

They do if they are ‘identical’ particles, such as two photons in a laser beam. They

do not if they are not ‘identical’, in the sense of sharing all intrinsic properties, such

as the members of an entangled particle–antiparticle pair resulting from a pair-

creation event, or better still, an elementary particle of any kind and the detector

apparatus with which it becomes entangled in a measurement interaction.

If quantum entanglement is the important novelty in the quantum realm, then

asking whether entangled quantum systems are individuals is not likely to help us

much in understanding how an entangled quantum world differs from a classical

one. As has happened before with progress in the sciences, we find science giving us

here not new answers to old metaphysical questions, but, instead, new questions.

Answering those new questions might well be helped by the development of new

kinds of formal apparatus, as has so often been the case in the history of physics,

from Newton and Leibniz’s invention of the calculus to Einstein and Grossmann’s

having prompted the development of modern differential geometry. And such might

well prove to be the case with French and Krause’s elaboration of quasi-set theory.

But I am not sure that I see how we are helped by trying to assimilate the new

question about entanglement to old and confused questions about individuality.

So if, now, one asks again, ‘Are elementary particles individuals?’, I answer, as

before, ‘I don’t know’. It depends on what one means by ‘individuals’. But now,

speaking from a physicist’s point of view, I am tempted to add, ‘And I don’t care’.

A different question: structuralism and QFT

If QM and QFT render problematical the view of elementary particles as

individuals, they lend support to a structuralist way of characterizing elementary

particles, not as individuals that are bearers of relations but as nodes implicitly

defined by a presumed structure of relations. The problem first appears in non-

relativistic QM, with bosonic and fermionic quantum statistics. It becomes only

more acute in relativistic QM and QFT, where one encounters states of

indeterminate particle number, and where particle number becomes a frame-

dependent quantity. So the quantum theory appears to afford a sustained and

compelling argument for a structuralist quantum ontology.

But not so fast. Let us pause and think about the ontology of QFT. Ask, first, what

formalism is in question. The structuralist likes algebraic formulations of theory. In

non-relativistic QM, in the QM of systems with finite numbers of degrees of

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freedom, it is the algebraic point of view developed by Wigner, Weyl, von

Neumann, and others that made clear the way in which the seeming difference

between such radically different ontologies as those assumed by Schrodinger’s wave

mechanics and Heisenberg’s matrix mechanics represented no relevant difference at

all. For, in the context of algebraic QM, the Stone-von Neumann theorem tells us

that wave mechanics and matrix mechanics are just different representations of one

and the same algebraic structure, and that, as with all representations of that algebra,

they are, of necessity, unitarily equivalent. Well and good. A triumph of the

structuralist point of view.

But the non-relativisitic QM of systems with a finite number of degrees of

freedom is not our best current theory. Our best current theory is QFT. It is a

relativistic theory (in the sense of special, not general relativity), and it is a theory of

systems with an infinite number of degrees of freedom. As such, in its most natural

algebraic form, it can be shown to possess representations that are, of necessity,

unitarily inequivalent. This is the algebraist’s way of saying that the theory is not

categorical, that it does not constrain the class of its models up to the point of

isomorphism. In other words, if the algebraic point of view that elsewhere serves the

structuralist so well is here, too, the right point of view, then there is a prima facie

problem for the structuralist, inasmuch as the theory does not and, so it would seem,

cannot privilege a single structural representation of quantum reality.

The work-a-day particle physicist replies that the framework of algebraic QFT

represents nothing but a philosopher’s fetishizing of axiomatic simplicity and that

one can only calculate in a specific representation. But absent an argument that

picks out a single correct representation, such as a Fock space or occupation-number

representation—and I know of no such argument—it is hard to dismiss the worry

that there is a lack of univocalness in the way in which QFT constrains its ontology.

Moreover, it is not just that there happen to be a variety of alternative ontological

pictures among which theory is impotent to choose. No, in this case there exist, ofnecessity, a variety of structurally inequivalent representations of physical reality.

If the structuralist is committed to the view that theory affords a unique structural

representation of nature, and if the algebraic point of view is the right point of view

in QFT, the structuralist has a problem. My own view is that structuralism should

never have committed itself to a uniqueness claim in the first place. But then, I am

not a structuralist. So I have no right to an opinion.

Bas C. van Fraassen

This admirable book, after the initial clarification of the problematique surrounding

the Principle of Identity of Indiscernibles, presents us with a cornucopia of technical

tours de force in response, but also with a distinct thesis. While insisting on the

under-determination of metaphysics by physics, the authors present a telling case for

adopting one of two metaphysical packages, that of quantum particles as non-

individuals. When a metaphysical position is spelled out in such detail, with such

technical virtuosity, it does indeed become formidable. How daunting for the

a-metaphysical!

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1. The quasi-set theory through which this favoured view is presented here is

distinctive. It is not the identity sign that appears as a primitive, but an equivalence

relation that can be read as ‘are indiscernible’. The universe contains non-sets,

which are of two sorts: m-atoms and M-atoms. The former may or may not be

discernible but can sometimes be counted, so that intuitively at least we think of

some of them as distinct from one another. For the latter, the M-atoms that we can

think of as macroscopic objects, identity can be defined (‘belonging to the same

qsets’), and they are identical if and only if indiscernible.

In response, I will here suggest an understanding of science that locates the sorts

of things that are distinct but indiscernible solely inside the mathematical

structures—inside the models—physics presents to represent physical phenomena.

At least as far as a feasible understanding of the sciences is concerned, I will submit,

one could add that everything in nature is distinct only if discernible. I will also

suggest that this view follows, more or less, Leibniz’s own position, to the effect

that violations of PII are to be met with only among ‘ideal’ objects.

2. Discernibility is by unshared properties. But what makes for distinctness? That

is the problem of individuation and has two competing answers: (A) by identifying

properties (which implies PII), (B) by something ‘transcendent’: substratum,

haecceity, ‘thisness’. Each faces its difficulties. If (A), what would guarantee that all

distinct entities are mutually discernible? What is the logical status of that

Principle? If (B), what sense can we make of such notions as haecceity?

Both options and their variants presuppose that individuation is by something that

‘confers individuality’. This ‘something’ seemed to be needed in the Aristotelian or

Thomist tradition (hylemorphism: neither matter nor form can exist except together) to

answer: What makes for the difference between a particular and a universal? There can

be many teachers of Plato, but not many Socrates – why not? (Aristotle Metaphysics VII,

1034a, 5–8; for discussion of the history see van Fraassen and Peschard 2008) That

question does not arise for nominalists, nor for extreme Platonists who take universals to

be (abstract) individuals. What could be the problem for us today?

We do have to answer the question: is it possible for there to be distinct entitiesthat are indiscernible? But it is not at all obvious that a yes would require

postulating special features that ‘confer individuality’. Secondly, we do have to

respect the well-known problem of identical particles in physics. But right from the

outset we should recognize, as French and Krause emphasize (83, 189–192), that thephysics underdetermines the metaphysics.

3. With quantum physics, the classical candidates for discernibility disappear,

and the ‘Received View’—of which French and Krause defend a sophisticated

variant—is that quantum particles are ‘non-individuals’. As they mention, in the

past I was intent on exploring interpretations in which PII is not violated.

Individuating properties or transcendent individuation can be postulated consis-

tently—but they are empirically superfluous, and as such offend empiricist scruples.

Hence, I did not advocate believing that the theory is true on such an interpretation;

the only virtue I would claim is that understanding is furthered by all consistent

answers to the question of how the world could possibly be the way that the theory

says it is. But with all that behind us, I would like to place these questions about

identity and individuation in a larger context.

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4. Leibniz’s Disputatio metaphysica de principio individui was, at age 15, still

scholastic. Aquinas and Scotus had both taken the position that a material entity is

individuated by something partial, so to speak—in Aquinas’ case, ‘designated’

matter, and in Scotus’ haecceity. Leibniz’s opposing view here is that nothing

partial—whether a part or a characteristic—is sufficient to individuate; only the

whole entity will do.

Aquinas had taken that position for the case of immaterial substances, the angels,

but not for material entities, in which he distinguished the individuating ‘designated

matter’ from the form. Hence, for Aquinas it was possible—in fact actual—to have

several material beings of the same infima species and not ruled out that they should

have all qualities (accidents) in common as well, though that was not possible for

beings such as angels which have no matter to individuate them. Leibniz insisted on

an account that covers all cases in the same way.

This we can see as the initial step to his mature view, that ‘‘[i]t is not true that two

substances may be alike and differ only numerically … and … what St. Thomas

says on this point regarding angels and intelligences … is true of all substances’’

(Discourse on Metaphysics IX).

This is the position I became convinced of initially (van Fraassen 1969, Ch. III

Sect. 1b; 63–65; van Fraassen 1972.) Apparent counterexamples, such as Max

Black’s universe consisting of two spheres completely alike, I took to include

implicitly assumed distinguishing characteristics to individuate, such as counter-

factual properties. If counterfactual statements do not have objectively determined

truth-values, this two sphere universe has the same status as the round square,

whatever that status may be. Since then, I vacillated between several views. (1) The

metaphysical alternatives make literal sense but are additions to the physics. (2) The

metaphysical alternatives do not make any literal sense. (3) Whether those

alternatives make sense, there was no problem in the first place, for which they

could be solutions: the problem dissolves when made explicit.

5. A simple, classical example to support (3): what is the difference between

There are many cows and Cowhood is multiply instantiated? The latter appears to

have the additional implication that cowhood exists. But it can be parsed so that the

implication disappears (Quine’s virtual class theory, combinatory logic). So it’s just

‘book-keeping’!

Two problems have dissuaded me. The first is a logical point: can’t every property

be multiply instantiated? Leibniz answers: the complete individual concept can be at

most singly instantiated. Suppose that it is part of the complete concept of Caesar that

he is the greatest general of all time. There can’t be more than one! The complete

concept of an individual in effect identifies the entire world it inhabits, with all its

relations to all its parts; and all the relations among those. But even so, what could

guarantee that no complete concept of an individual can be multiply instantiated?

Only PII could. The second pertains to physics. Adapting that scheme to QM requires

at least the argument that QFT could be understood as a theory of particles as well.

I thought this was supportable, but I am now convinced that I was wrong.1

1 For me, Bitbol (2007) drove the last nail into the coffin of my earlier view; see also French and Krause

(192–193, 355–373).

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Though in some vague sense the difference is mere book-keeping, the two forms

of language are not inter-translatable without significant loss.

6. A lesson from ‘incongruous counterparts’? If God had created a universe

containing just one thing, a hand, would it have to have been either a left hand or a

right hand? Three possible responses: (1) It has the same status as the round square;

(2) there is handedness, which comes in left and right, and without which something

is not a hand; (3) it would be a hand but neither left hand nor right hand.

PII is satisfied in all three cases, if handedness is a property; but it is hard to make

sense of any of them. The real challenge this example poses is that we can

consistently offer descriptions that imply distinctness without implying discernabil-

ity. The predicate ‘‘… exists in a universe containing only one other object’’ can

only be multiply instantiated, but does not bring along anything to differentiate its

instances.2 So PII is not a logical truth.

7. What about a Kantian perspective? If we can advance that for the troubling

examples in quantum physics, perhaps we can take it as definitive. And here there is

definite hope. Remember Margenau’s contention: Pauli’s exclusion principle, which

Weyl had associated with PII, actually provides a counter-example to it. If a pair of

identical fermions are in the anti-symmetric tensor product state (1/H2)[(uw - wu)],

where u, w are orthogonal spin states, states can be ascribed by ‘reduction of the

density matrix’. But each particle is assigned the same mixed state �P[u] ? �P[w].

As far as this representation goes, these particles are indiscernible.

Can interpretation, on the level of semantics rather than pragmatics, save PII?

Yes, but only by empirically superfluous parameters. I developed such an account

(see French and Krause’s critique, 164–167) but since those parameters are either an

idle addition or intrusive metaphysics, PII seems at this point to be totally dubious

from an empiricist point of view.

So here the transcendentalist can place a wedge (vide van Fraassen and Peschard

2008). Is not there still, in such an entangled state, a possibility of discernment? In

Aspect’s famous experiment to detect violations of the Bell inequalities, a pair of

entangled photons is in the pure state (1/H2)[(uu ? ww)]. Results of polarization

measurements with two polarizers aligned in parallel are 100% correlated. Each

photon may be found randomly either in channel ? or - of the corresponding

polarizer, but in this case, when one photon is found positively polarized, then so is

its twin companion. Notice now that if the experiment was interrupted early on with

a ‘number’ measurement on the entangled pair, the outcome would have been ‘2’.

But that we can also say in the context of a quantum-field or Fock space

representation, in which that ‘2’ is an occupation number, in a model in which there

are no particles, only a field. So this measurement is a literal ‘counting of distinct

2 Quine’s and Simon Saunders’ points about relations are similar. If an asymmetric relational predicate,

or a symmetric irreflexive one, is instantiated at all then there are at least two things. Saunders’ term

‘‘weakly discernible’’ does not seem apt: we have here a way to show that logically, things can be distinct

without being discernible, by an argument that does not appeal to PII but to its innocuous converse. Hence

I do not consider the PII saved or satisfied even in the case of fermions by reflections on ‘weak

discernability’, contrary to Muller and Saunders (2008), despite the impressively nuanced distinctions

they make.

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entities’ operation for the experimenter who thinks in terms of the first, elementary

quantum mechanical model, not for the quantum-field theorist. We do not have here

an empirically given distinctness without discernibility!

Now suppose that this interruption does not happen—instead the Aspect type of

experiment is allowed to run its course. We get two clicks or none. These two clicks

happen in counters set some distance apart from each other, and now the quantum–

mechanical model has the two photons localized, hence clearly discerned.

Measurements taken immediately after the polarization filters predictably reveal

photons in those specific places. But something more is true of this situation, as

represented in the model: the photons are not, at that point, in an entangled state. We

have two definitely positively polarized photons there, for example. Thus, the

individual photons could be discerned, but the discernment could only be done withviolence.

That is a crucial point for the transcendentalist. Kant responds to Leibniz’s

conviction—that all distinctions must be capturable conceptually—by insisting

that there are intuitions that cut across conceptual lines. In the complete concept

of the solitary hand, there is nothing to specify right- or left-handedness. But theconditions of knowledge of such a hand entail that if we did have acquaintance

with it via perception or measurement, we would discern it either as definitely

right-handed or as definitely left-handed, for we could only know it by

acquaintance in spatial relation to ourselves. So the very conditions under which

alone it could be directly known are ones in which, necessarily, it is discernible.

No implication beyond that, beyond the conditions under which such a finite,

natural being as we are must be, comes along with this. Mutatis mutandis for

particles.

Thus, if we accept that the limits set by conditions of knowledge also set the

limits for any reasonable epistemic or doxastic attitude, then we cannot hold the

belief that there is distinctness in nature without discernibility—despite recognition

of the purely logical leeway. For an empiricist that ‘if’ is still a big ‘if’, however.

8. So now we come to a final further reaction to the problems of identity and

individuation in physics. Counter-examples to PII are either so fanciful that we can

refuse to see them as realizable, or else they are of abstract objects. Leibniz

explicitly exempted the latter from PII!

People … commonly use only incomplete and abstract concepts, … Such

concepts men can easily imagine to be diverse without diversity – for

example, two equal parts of a straight line, since the straight line is something

incomplete and abstract, which needs to be considered only in theory. But in

nature every straight line is distinguishable by its contents from every other.3

We can understand why PII violations should be rampant in mathematics, in two

ways.

3 In correspondence with de Volker; I am relying here on Jauernig (2004), and am additionally in her

debt for correspondence and for partial drafts of her book in progress, from all of which I have greatly

benefitted.

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The first concerns abstraction, in two senses. On one traditional view,

mathematical entities are formed by abstraction in the sense of removal of features

from consideration—the removed features could be the differentiae, thus leaving

indiscernibles. There is a newer sense of abstraction too: introduction of an entity

that is, or corresponds to, an equivalence class in a previously given range of

entities. Even if the original admitted of no non-trivial automorphism, the result

may, because the reduction ‘erases’ differences.

The second way Ladyman (2007) presented as ‘lessons from graph theory’.

Graphs have nodes and edges; the edges may be directed or directionless, an edge

has beginning and end nodes which may be the same (loops). If a graph has non-

trivial automorphisms, it does have more than one node, but equally, some of these

nodes may have all the same properties, even relational properties. The moral is not

evaded by complexity. For the real number continuum, to identify numbers modulo

translations would leave us with just a single one.

So, when it comes to understanding mathematics, we had best be completely

sanguine about violations of PII! The real question for us is only whether we can

relegate all apparent such violations to the realm of mathematics.

9. Since learning a great deal from Ladyman and French about structuralism in

philosophy of science, I have been working for some time now on an empiricist

structuralism, which is not a view about the world but a view of science:

I. Science represents the empirical phenomena solely as embeddable in certain

abstract structures (theoretical models).

II. Those abstract structures are describable only up to structural isomorphism.

Given the above discussion of mathematics and abstraction, it would on this view

be truly a mystery if we found PII everywhere satisfied in the sciences’ theoretical

models. An embedding is itself an one–one map that preserves relevant features; it

is not perfect copying, and distinct elements of nature may not be distinguishable in

terms of such embeddings.

There is a striking moral to be drawn from QM: the phenomena cannot all be

embedded in structures in which all distinct elements are mutually discernible,

unless those structures include empirically superfluous hidden parameters. That is

remarkable and revolutionary new empirical knowledge, concerning what the

observable phenomena are like. However, the indiscernibility pertains to properparts of the embedding images in the models, and nothing implies that it pertains as

well to the phenomena that are thus represented. Admittedly, there are empirical

criteria that are to be satisfied also by models in which there is much that may not

correspond to anything real. All parameters, including those that pertain to the

indiscernibles, ought to be empirically grounded, but that means only that relative tothe theory, there must be conditions, realizable in principle, in which certain

operations count as measurements whose results imply specific values for those

parameters. That empirical criterion does not change the point.

With the violations of PII, theoretical postulation has unmistakably shown its true

colours. The mutually indiscernible entities ‘postulated as real’ are, I submit, best

understood as located solely in the useful but abstract representations that the

sciences offer us.

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Otavio Bueno

Identity is arguably a fundamental concept. It is typically presupposed by (almost)

all conceptual schemes, and it does not seem that it can be properly defined in terms

of more basic concepts (see, e.g., McGinn 2000). Even the attempt at defining

identity in second-order logic—for instance, by noting that two objects are identical

if and only if they have exactly the same properties—clearly presupposes identity.

After all, the same properties (i.e., those that are identical to one another) are

invoked. The point goes through even if the alleged definition is formulated

explicitly. Consider: x = y if and only if for every P (Px if and only if Py). In order

for this statement to make sense, it is crucial that the same variables (x, y, and P) are

in the right-hand side and in the left-hand side of the relevant bi-conditionals. The

identity of the variables is required for the proper formulation of the statement. But,

in this case, the concept of identity is presupposed by the statement that was

intended to define it. Given how fundamental identity is, it is not surprising that this

concept is also crucial to physics, physical theorizing, and the philosophy of

physics. In Identity in Physics, French and Krause present the most thorough,

systematic, and insightful treatment of identity in physics available, covering

classical physics, QM, and QFT.

A central feature of French and Krause’s proposal is to emphasize the

unavoidable underdetermination of the metaphysics by the physics, in the sense

that metaphysically irreconcilable views are compatible with the same physical

theory (24, 83, 159, and 354). In particular, French and Krause argue, QM is

compatible with two radically distinct metaphysical packages: one which empha-

sizes that quantum objects are bona fide individuals and thus have well-defined

identity conditions (the individuals package), whereas the other stresses that

quantum objects are non-individuals and thus lack properly characterized identity

conditions (the non-individuals package). Each of these packages has two key

components: (a) a particular philosophical view and (b) a technical development. It

is by articulating these components that the resulting conception emerges.

The individuals package assumes a fairly straightforward metaphysical under-

standing of individuals, as objects that do have well-defined identity conditions.

These objects can be systematically identified, distinguished, and re-identified; they

are, by and large, well behaved. This standard philosophical outlook is accompanied

by a particular technical development, which consists in the need for finding a

suitable reinterpretation of the so-called indistinguishability postulate in QM.

Roughly speaking, this postulate entails that no measurement of particle permu-

tations can result in a discernible difference between permuted (final) and non-

permuted (initial) states.

In contrast, the non-individuals package advances a far more radical view about

the metaphysics of quantum objects and emphasizes the need for taking the latter as

non-individuals. According to this proposal, which was defended, in different ways,

by Schrodinger and Weyl, among others, quantum objects are not the kind of thing

to which identity can be properly applied. Thus, such objects lack precisely the

identity conditions that are so important to classical objects. (I am using ‘object’

here as a neutral expression that, as opposed to ‘entity’, does not presuppose

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well-specified identity conditions for the items it is applied to.) One of the main

motivations for this view emerges from the behaviour that quantum objects seem to

display, at least assuming a certain interpretation of QM. For instance, if two

electrons swap places, the quantum states they are in remain the same. No

individual features of particular electrons contribute to the characterization of those

states. Only the kind of objects that is involved—in this case, electrons—rather than

the individual members of that kind, is relevant. As Weyl (1931) once noted, there is

no alibi for an electron. Not surprisingly, the non-individuals package offers a fairly

straightforward interpretation of the indistinguishability postulate. If it does not

matter which particular electron is permuted, the lack of well-defined identity

conditions for such objects easily explains why the postulate holds.

The key challenges for this view surface, in particular, at the technical level. The

non-individuals package requires the development of a suitable formal framework

for quantum non-individuality. Two interrelated issues need to be addressed: How

exactly can we describe such non-individuals mathematically? And what exactly is

their ontological status? One of the strengths of French and Krause’s book is the

way they engage—thoughtfully, provocatively, and ingeniously—with these issues.

In particular, among the most significant technical innovations of their work is the

careful development of a formal framework for quantum non-individuality in terms

of quasi-set theory. This is a set theory that, as opposed to classical set theories, does

not presuppose the identity of the objects it quantifies over. In one sense, quasi-set

theory provides a generalization of classical set theories—in particular, the

Zermelo-Fraenkel set theory with Urelemente (ZFU), which is a set theory that

includes objects that are not sets. After all, the system of quasi-set theory developed

in French and Krause’s volume contains a ‘copy’ of such classical ZFU sets as part

of its own set theoretical hierarchy (285). In another sense, however, quasi-set

theory provides a revision of classical set theories, since, as opposed to the latter,

there is no requirement that only objects that have well-defined identity conditions

are quantified over.

As French and Krause do not fail to note, the ontological picture associated with

non-individuals requires some careful rethinking of Quine’s criterion of ontological

commitment. But one issue that the non-individuals package faces is the

intelligibility of quantification without the presupposition of the identity of the

objects that are quantified over. The two Quinean slogans, ‘to be is to be the value of

a variable’ and ‘no entity without identity’, are of course closely connected. In order

for something to be taken as the value of a variable—in order for something to be

intelligibly quantified over—it is crucial that it be distinguished from other objects,

whether objects of the same or of different kinds. It is crucial, that is, that the objects

in question be entities, thus having well-defined identity conditions. In fact, such

conditions need be in place in order for universal quantification to behave

classically. Consider, for instance, the equivalence that holds in classical logic

between ‘Every x is Fx’ and ‘Each x is Fx’. On the one hand, if every x is Fx, then

being F is a property that holds universally, for every object in the domain of

interpretation. In this case, the property then holds for each object in particular. For

if it failed for one, it would not hold for all. On the other hand, if each object in the

domain is F, that is, if each x is Fx, then F is a property that holds for all objects in

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the domain. For if it did not hold for all, it would fail for one, and this would violate

the assumption that it holds for each. Thus, in this case, every x is Fx. The Quinean

way is that of the classical logician.

It turns out, however, that (classical) universal quantification presupposes the

identity of the objects that are quantified over. In order for us to validly infer from

each x is Fx that every x is Fx, it is crucial that the quantification over each object in

the domain covers distinct objects. If that quantification ranges successively over

exactly the same object in the domain, again and again, it will never cover that

whole domain. But in this case, it need not follow that every object in the domain

has the relevant property: distinct objects need to be quantified over in each case.

So, the identity of the objects in question is clearly assumed. Similarly, in order for

us to infer from every x is Fx that each x is Fx, it is not enough that all objects in the

domain of quantification instantiate the property F; that property also needs to be

held by each object individually. But that requires the distinguishability of each of

these objects, which in turn presupposes that identity can be applied to them. Once

again, identity is crucial for (classical) quantification. As a result, with the

introduction of quasi-set theory, the issue arises as to how to make sense of

universal quantification without presupposing the identity of the objects in the range

of that quantifier.

Furthermore, without the extensional equivalence between ‘for each’ and ‘for

all’, how can the universal quantifier in quasi-set theory be distinguished from the

existential one? After all, the existential quantifier (according to classical logic) is

such that:

(E) If a is F, then something is F.

Consider now the corresponding inference for the (classical) universal quantifier:

(U) If a is F, then every object is F.

Clearly, as opposed to (E), (U) is not generally valid. It only holds if a is arbitrary,

that is, if it does not matter which object in the domain of quantification is

considered. But to show that (U) is violated, we need to establish that although a is

F, some object distinct from a is not F. The condition in italics is crucial. In classical

logic, objects are typically taken to be consistent, in the sense that it is not the case

that (classical, well-behaved) objects both have and do not have a given property.

Thus, in order to show the violation of (U), we need to establish that at least some

object—distinct from the relevant object a—does not have the property F. If the

object a both had and did not have F, it would provide a counterexample to the

inference. But it would also violate another principle of classical logic, namely,

the law of non-contradiction. The underlying logic of quasi-set theory, at least in the

version elegantly developed by French and Krause, is not paraconsistent: it is not

supposed to be taken as the underlying logic of inconsistent but non-trivial theories

(that is, theories that, despite being inconsistent, do not entail every sentence in the

language in which they are formulated). Hence, the violation of the inference (U)—

in a non-paraconsistent context—seems to require that identity be applicable to the

objects involved. Without the notion of identity in place, it is unclear how to prevent

that an inference that is generally valid only for the existential quantifier, namely,

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(E), also becomes valid for the universal quantifier, i.e., (U). Thus, a significant

distinction between the two quantifiers becomes curiously blurred.

A related concern also emerges at this point. Consider a mixed statement in

which both individuals and non-individuals are quantified over, such as: ‘Some

electrons were detected inside the apparatus last night’. In quasi-set theory, we can

quantify over both individuals (for whom identity can be applied) and non-

individuals (for whom it does not). Suppose that we extend the language of quasi-set

theory in such a way that we can express the content of the mixed statement above.

How can the same existential quantifier pick out objects for which identity applies

and objects for which it does not? In quasi-set theory, quantification is always

restricted. The quantifiers range over both individuals and non-individuals, and the

predicates in the sentence sort them out: non-individuals are specified as micro-

objects, and individuals are specified as macro-objects. The point, however,

remains: the range of the quantifiers includes both sorts of objects.

But if this is the case, what exactly are the ontological commitments of quasi-set

theory? Is it committed to both types of objects (individuals and non-individuals), to

only one of these types, or to none of them at all? The answer, of course, depends on

the interpretation of those quantifiers. I think there are basically four options here.

(a) If we adopt an indispensability argument—insisting that we should be

ontologically committed to all and only those entities that are indispensable to

our best theories of the world (Colyvan 2001)—and advance an interpretation of

QM that favours non-individuals, we can claim that the quantifiers only provide

ontological commitment to such non-individuals. After all, the latter are the

fundamental components of the ontology according to our best theory of the

quantum world on that interpretation. However, in this case, the non-individualspackage is clearly being favoured, something that French and Krause do not intend

to do, given that they correctly support the underdetermination of the two packages

(individuals and non-individuals). (b) If we still adopt an indispensability argument

but reject the non-individuals interpretation of QM in favour of the individuals

package, we will conclude that the quantifiers provide ontological commitment only

to such individuals. But, as a result, the individuals package is then advocated,

which, once again, is in tension with the defence of the underdetermination of the

two packages. (c) If we continue to adopt an indispensability argument and realize,

perhaps based on the considerations that supported alternatives (a) and (b), that both

individuals and non-individuals are needed, we then conclude that the theory is

ontologically committed to both types of objects. But, as a result, yet again we need

to abandon the underdetermination argument. (d) Finally, if we reject the

indispensability argument and insist that the quantifiers need not be read in an

ontologically loaded way, the ontological commitments of the theory are left

entirely open. However, in this case, unless we add by hand a specification of the

conditions under which we are entitled to claim that certain objects, properties, or

structures exist (see, e.g., Azzouni 2004), we will not have any form of realism left.

This option, however, conflicts with French and Krause’s well-known preference for

a realist stance.

There is no doubt that, among the various forms of realism, French and Krause

favour a structural realist view (118). In fact, structural realism is supposedly

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supported by the underdetermination between the two metaphysical packages

(individuals versus non-individuals). We may be unable to settle the ultimate nature

of the ontology of objects in the quantum world. But, for the structural realist, far

from being a problem, this indicates that only the commitment to the overall

structures that are described by quantum theory should be ontologically serious.

However, it is unclear that the underdetermination of the two packages ultimately

supports structural realism. After all, the same underdetermination can also be used

in defence of views as diverse as constructive empiricism (van Fraassen 1991) and,

to some extent, semi-realism (Chakravartty 2007).

Which alternative is left then? Perhaps the argument in French and Krause’s

book is better understood as supporting the disjunction of (a) and (b) above. In this

way, they may be able to keep the underdetermination argument. But in order to

establish such a disjunction, they need to establish at least one of the disjuncts first.

However, if one of the disjuncts is settled, we seem to have reason to challenge the

very project of establishing the underdetermination in the first place—at least in this

way.

In the end, perhaps the best alternative here, given the style of argument

developed by French and Krause, would be simply to give up on realism. As van

Fraassen (1991) would say, each metaphysical package may offer us some

understanding of how the world could be; but we are in no position to establish their

truth (or even their approximate truth). It would be truly a pleasure to have French

and Krause on the empiricist camp. In fact, I would be the first to welcome them to

the club!

Elena Castellani and Laura Crosilla

French and Krause’s Identity in Physics is a fundamental, long-awaited, and unique

reference in the field. The discussion of the impact of current physical theories on

our understanding of the identity and individuality of objects has a long history and

a various literature, but this is the first monograph entirely devoted to the issue and

in such a detailed and complete way. The book stands out in many respects: in

particular, by carefully combining the philosophical, historical, and formal

dimensions in its analyses. The authors thus succeed in offering an excellent and

exhaustive map of the landscape of these issues.

In fact, the material covered is rich enough for three books at least! A book on

statistics and individuality, consisting of the very detailed historical and

philosophical analyses of the developments of classical statistical mechanics and

quantum statistics provided in chapter 2 and chapter 3, respectively. Another book

on quantum physics and individuality, centred on the philosophical discussion of the

individuality issue in the context of QM outlined in chapter 4. This discussion,

complemented by French and Krause’s reflections on the notions of identity,

individuality, and distinguishability and on the related role of Leibniz’s famous

Principle of Identity of Indiscernibles (PII) (Chapters 1–3), forms the very core of

their philosophical analysis. Finally, a book on logic and individuality resulting

from the formal analysis provided in the chapters 5–9 and representing the logical

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counterpart of what French and Krause maintain from a philosophical point of view

in the former chapters.

Summarized briefly the central argument of the book proceeds as follows. The

position that ‘classical particles are individuals, but quantum particles are not’,

described by the authors as the ‘Received View’ (RV) of particle (non)

individuality, is not forced upon us by the physics. The QM formalism ‘can be

taken to support two very different metaphysical positions, one in which the

particles are regarded as ‘non-individuals’ in some sense and another in which they

are regarded as (philosophically) classical individuals for which certain sets of states

are rendered inaccessible’ (189). The difference crucially depends on how we

understand the ‘Indistinguishability Postulate’ (IP), expressing the fact that, in an

aggregate of particles of the same kind (the so-called identical or indistinguishable

particles), the arrangements obtained by particle permutations do not count as

distinct. RV explains this reduction in statistical weight by the non-classical

metaphysical nature of the particles as non-individuals. But another perspective is

possible, from which the ontology is not directly tied to the statistics: One can

continue to regard the particles as individuals, but for these particles certain states

are inaccessible (i.e., possible, but never actually realized). From this alternative

view, IP is understood as an extra requirement concerning state accessibility. In

conclusion, we have a kind of underdetermination of the metaphysics by the physics

(two possible metaphysical packages—individual or non-individual objects—for the

same physics). The lesson is that we cannot simply read our metaphysics off the

relevant physics. One is thus justified in maintaining that quantum objects are

individuals if one wishes. But ‘if one were to adopt RV, it should be in the context

of an appropriate logico-mathematical framework’ (169). In fact, while theories of

individuals are well accounted for by our mathematical theories, a genuine theory of

non-individuals is still lacking proper formal treatment: ‘it is not clear how the idea

of non-individuality is even expressible in formal terms’ (192).

In essence, French and Krause (1) argue for the underdetermination thesis and (2)

claim that there should be a correspondence between the individuals/non-

individuals metaphysics adopted and the logico-mathematical framework in which

the theory is formulated. Motivated by this second claim (let us call it the ‘formal-

framework claim’), French and Krause devote the third part of the volume to

proposing a formal counterpart to the ‘non-individuals package’ (since, they

maintain, an adequate one is still missing).

French and Krause’s underdetermination argument constitutes a precious

contribution to the debate on the so-called identical particles. By carefully

unveiling the often implicit assumptions on which RV is grounded, the authors offer

an invaluable lesson to both philosophers and physicists. One should not attribute to

physics things that have not to do with it. At the same time, the subtleties of the

philosophical analysis can show how apparently natural options concerning the

objects of the ‘relevant physics’ are not the only one available. We cannot but

strongly appreciate the work of the authors in this part. Nonetheless, we also have an

initial worry here, regarding a central feature of both this and the following ‘formal’

part of the volume: namely, the authors’ concept of ‘non-individual’ particles.

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For French and Krause, quantum particles are non-individuals in the sense that

they do not have to satisfy the reflexive law of identity (for all a, a = a), that is, they

do not need to possess ‘self-identity’. This definition of non-individuality is at the

basis of the whole logical construction developed by the authors. It therefore plays a

crucial role; in particular, it marks (according to the authors) the difference with

respect to another formal approach that has been proposed to deal with the

individuality question raised by indistinguishable quantum particles, namely, Dalla

Chiara and Toraldo di Francia’s quaset theory. This approach is discussed in

connection with the question of names in physics tackled in chapter 5. Here, the

authors describe quaset theory as ‘the appropriate formal counterpart’ of ‘a

metaphysical framework in which the particles can be considered as named

individuals for which distinguishing descriptions cannot be given’ (237). (Although,

as the authors acknowledge, this does not necessarily correspond to Dalla Chiara

and Toraldo di Francia’s own understanding of their work). In other words, quaset

theory is seen as a theory of indistinguishable individuals rather than as a theory of

‘non-individuals’ like the one put forward by the authors. In fact, quaset theory is an

intensional version of the Zermelo-Fraenkel (ZF) set theory, in which self-identity

holds, but for which both extensionality and PII may fail.

While an intensional approach may allow to stress the epistemological difficulty

in accessing objects rather than their presupposed objective non-individuality,

French and Krause stress that ‘the problem is not epistemological but ontological’;

that is, they are ‘supposing [there is] a strong relationship between individuality and

identity’ (248). A quantum particle is a non-individual in the sense that the relation

of self-identity does not make sense for it. Our first worry regards precisely this

‘ontological’ reading of the indistiguishability issue raised by the so-called identical

particles. Think about the following situation: there is a system consisting of just

one hydrogen atom, with just one electron; hence, there is no problem of

distinguishing the electron from other similar particles. Why should we think that

this particle has, ontologically, no self-identity? In general, we can easily imagine

physical systems in which a quantum particle can be recognized as an individual,

whatever individuality theory we choose. If what counts as an individual or not may

depend on the physical situation considered, this seems to suggest a contextual

account of the individuality issue—as, for example, the one proposed by Stachel

(2005)—rather than an ontological reading in terms of self-identity.

Our second worry concerns the very motivation behind French and Krause’s

formal account; that is, what we have called their ‘formal-framework claim’. We

think that, although the ‘relevant physics’, when interpreted according to RV, can

play an important heuristic function in suggesting new routes to the mathematical

and logical investigations, this does not imply that a formal framework without a

notion of self-identity is more adequate for formulating the physical theory (in fact,

the task of re-formulating the whole theory in such a formalism seems quite hard).

In what follows we focus on this point, starting with an account of French and

Krause’s formal framework.

In chapter 6, the authors put forward their motivation for their variant of ZF set

theory. Let us follow French and Krause and use the letter Q to refer to this version

of set theory, called quasi-set theory by the authors. In the opening of the chapter,

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the authors briefly hint at mathematical theories (other than Q) which have been

proposed to deal with sets of quantum objects. However, the description of these

theories is rather concise. We take it that one of the aims of Q is to enhance the

philosophical analysis; in this case, a comparison of Q with these alternative

proposals would have been useful to assess the benefits of each theory. This would

have been very much in harmony with the in-depth analyses in the previous chapters

of the book.

Before going into a discussion of the motivation for quasi-set theory, let us

briefly recall what this theory is. Q is an extension of ZFU—the Zermelo-Fraenkel

set theory with Urelements—in which two kinds of urelements feature: the m-atoms

(thought of as micro-objects) and the M-atoms (the macro-objects). Quasi-sets are

then constructed from the urelements by applying the usual set-theoretic operations

of ZF (appropriately formulated). The M-atoms behave as standard urelements, and

there are also standard fully extensional sets in Q. In other words, the theory Q is an

extension of ZFU which includes, besides the usual set-theoretic objects, also

urelements representing elementary particles. Besides the standard notion of

equality, the theory also postulates an equivalence relation of indistinguishability. A

form of weak extensionality holds of quasi-sets, based on the predicate of

indistinguishability.

The main claim which makes its way through chapter 6 is that standard set theory

encapsulates a notion of identity which makes it unsuitable for the treatment of

theories of non-individuals. In particular, French and Krause recall ‘Weyl’s

strategy’ of using sets endowed with an equivalence relation for the purpose of

mimicking indistinguishability. The authors seem to hold that this approach, though

satisfactory for the mathematical description of physics, is not appropriate

‘philosophically’ (263). We have already mentioned French and Krause’s

‘formal-framework claim’, according to which there should be a correspondence

between the entities postulated by a metaphysical package and the mathematical

entities used to describe them. In the case of the non-individual package, the theory

Q provides a direct correspondence between the non-individuals and the m-atoms.

In the case of Weyl’s strategy, instead, the correspondence is between the non-

individuals and the elements of a mathematical structure. The latter, when

represented in ZF, can be pictured as a set endowed with an equivalence relation.

The correspondence in this case can thus be seen as between a non-individual and an

equivalence class. In standard set theory, sets satisfy the identity relation (as

elements of the set-theoretic universe) and thus can be seen as individuals with

respect to the set-theoretic universe. French and Krause seem to argue that the

individuality conferred by the identity relation, which holds on the set-theoretic

universe as a whole, somehow prevents us from seeing the elements of a set

endowed with an equivalence relation as non-individuals.

We think this latter conclusion can be resisted, as the individuality of the

elements of the structure seems more appropriately described solely in terms of the

equivalence relation which is postulated within the structure itself and the role they

play within the structure. The ‘external’ set-theoretic representation could be

considered as extrinsic to the mathematical structure. In this respect, we wish also to

mention that in the context of Bishop’s style constructive mathematics (that is,

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mathematics based on intuitionistic logic), a concept of set occurs that best suits the

views we here express. According to Bishop, a set is defined by postulating which

elements belong to it and also when two of its elements are equal. That is, a

set always comes equipped with its identity relation (which in general can be non-

extensional).

Even though we are not totally persuaded that there is a real need to abandon

standard set theory for quasi-set theory as a formal account for the non-individual

metaphysical package, we do acknowledge a genuine purely logical interest in a

theory of sets which challenges one of the strongest bases of classical logic: its

identity predicate. The theory Q appears to be also motivated as a semantic tool in

relation to other logical theories presented in subsequent chapters.

Let us now turn our attention to some more technical points. Although at first

sight the theory Q could seem a simple enough modification of ZFU, it does have a

number of non-trivial outcomes. In fact, we have to admit that there are a number of

technical aspects of the theory which are still not totally clear to us. The most

prominent is the notion of quasi-cardinal of a quasi-set. Quasi-cardinals appear as a

way of accounting for the talk of size of a set of indistinguishable elements. This is a

fascinating concept and so is the discussion clarifying the difficulty involved in

assigning an ordinal to a quasi-set. However, as we are used to thinking of cardinals

as ordinals, we would need more information on how a quasi-cardinal is assigned to

a quasi-set (for example, in the case of finite quasi-sets). More generally, it would be

extremely useful to have more examples showing how the theory actually works.

Another concern is that it is not clear to us how the modifications of ZF which give

rise to Q could affect the development of the mathematics which is needed for the

physics. However, it might be premature to judge Q on this issue, as it is a very

novel theory, and the authors themselves clearly state that more work is needed to

clarify various aspects of it.4

Authors’ response: Steven French & Decio Krause

We would like to thank all five reviewers for their generous comments and

thoughtful criticisms. They have encouraged us to reflect further on both the aims of

our book and the details of our constructions. Our aims, we recall, were twofold:

first to consider the so-called Received View that takes particles to be non-

individuals and render it formally respectable by setting it within the framework of

quasi-set theory; secondly, however, to emphasize that the physics is also

compatible with an alternative view that takes particles to be individuals, subject

to certain constraints, and spell out the details of such an account. As a result, the

book presents two alternative metaphysical packages, amounting to a form of

underdetermination of the metaphysics by the physics.

Thus, we agree with Howard when he emphasizes that the claim that quantum

particles are non-individuals can be resisted in various ways. However, we differ

4 We wish to thank Marisa Dalla Chiara and John Stachel for very helpful comments on a draft version.

Laura Crosilla’s research supported by EPSRC grant EP/G029520/1.

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when it comes to his suggestion that there is a lack of clarity in these discussions

that results from approaching the issue of particle individuality from the perspective

of PII. For us, PII is just one means of grounding individuality, albeit perhaps the

most obvious one to deploy in the case of physical objects. As Howard notes, we

canvas various versions of PII only to conclude that it fails in the quantum context.

Thus, it might appear that the particles-as-individuals package requires the

introduction of haecceity or something similar.

However, Saunders has argued that a form of PII can be maintained in the

quantum realm, at least for fermions (see Muller and Saunders 2008).5 If successful,

this would be all to the good as it would further reinforce the particle-as-individuals

option (at least for fermions). Nevertheless, the claim that fermions are relationally

discernible (via an irreflexive but symmetric relation such as ‘has opposite spin to’)

but not absolutely discernible (via a monadic property) is problematic given the

underlying framework of first-order Zermelo-Fraenkel set theory (ZF), at least in the

case of finitely many fermions. This is because, given two fermions, say, we can

enlarge the language of ZF with two additional constants a and b (since in ZF, every

structure can be extended to a rigid structure) and define the monadic properties ‘‘to

be identical with a (b)’’, which distinguish the electrons absolutely. These properties

are not relations in disguise, for once we have named the fermions—and we can do

it within extended ZF in the case of finitely many—these definitions can be regarded

as presenting truly monadic properties. Thus, the chosen formal framework may not

be appropriate for the philosophical aim here.

This, of course, is our central contention when it comes to the particles-as-non-

individuals option. Here, PII gets no purchase, as there is no individuality for it to

‘ground’. Interestingly, van Fraassen suggests a reading of Leibniz according to

which PII does not apply to ideal or abstract objects. If quantum particles are

regarded as ideal in this sense, then the non-applicability of PII would mean the

question as to its status in this context is obviated. Thus, from the perspective of the

constructive empiricist, the apparent ontology of distinct but indiscernible objects is

just like modality in being merely a feature of our models.

This is an interesting move. However, it removes the point of PII as a metaphysical

principle; or at least, its current point since under this view it would apply only to

observable objects. Even if our metaphysical packages only correspond to ways the

world could be, according to the constructive empiricist, there is still value to exploring

these metaphysical options, else how could we understand the way the world could be,

otherwise? That is, it is not enough for the constructive empiricist to withdraw into the

models—if any sense is to be made of her stance on interpretations, some articulation of

the relevant metaphysics and underlying formalism is still required.

Returning to the non-individuals package, it is not the case that, as Howard

suggests, ‘arguments for the non-individuality of elementary particles are supposed

to be arguments for some version of structuralism’. Rather, the suggestion is that the

very underdetermination that Howard also seems to support, between particles-

as-individuals and particles-as-non-individuals, motivates structuralism, by pressing

5 Muller and Seevinck (2009) have tried to extend this result to bosons but this is contentious; see

Ladyman and Bigaj (2010).

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the removal of the notion of object from a metaphysics appropriate to the quantum

domain. This is where modern forms of structuralism differ from those of Cassirer

and Eddington whose versions of this position stand or fall on the Received View

(French 2003). Bueno is certainly right that van Fraassen also accepts this

underdetermination, and hence that it does not univocally support realism, but the

point is that if one wants to resist the temptations of constructive empiricism and

remain a realist in light of this result, one must abandon an object-oriented

framework and adopt a structuralist metaphysics.

Howard certainly raises a significant problem here, concerning the existence of

inequivalent representations in QFT. As we note (363–364) it is this that lies behind

Haag’s Theorem, which raises problems for the particle concept in this context.

Nevertheless, there are options available. One would be to adopt so-called naıve

Lagrangian QFT (Wallace 2006), in which the inequivalent representations encode

inaccessible information and hence can be dismissed. Alternatively, one could focus

on the underlying Weyl algebra, adapt Ruetsche’s ‘Swiss Army Knife’ approach

(2002) and modify one’s structural realism accordingly (see French forthcoming).

However, addressing these issues would take us far beyond the scope of the book

which, we must emphasize, was not intended to offer a comprehensive defence of

structural realism itself.

Castellani and Crosilla also raise concerns about the non-individuals package.

First, they suggest that there may exist situations for which adopting this package

would be inappropriate. Thus, they invite us to consider a system consisting of a

single hydrogen atom. In this case, they suggest, where there are no other electrons,

say, from which the sole electron of the atom will be indiscernible, there is no need

to regard it as a non-individual. In general, we can imagine systems in which the

particles can be regarded as individuals, and thus they advocate a form of contextual

individuality.

However, just as Leibniz, in his debate with Clarke, questioned whether the

latter’s imaginings were chimerical, so here we have to examine carefully the basis

for the above situation. There would seem to be two options. First, we could imagine

the hydrogen atom as the sole existent, thereby delineating a possible world distant

from this one. In this case, the physics would be very different since there would be

no quantum statistics incorporating permutation invariance, or at least not of the kind

that underpins the Received View. It might well be that we would then have no

grounds for attributing non-individuality and bringing in quasi-set theory but that is

obviously not the world we are dealing with! Alternatively, one could insist that we

remain within this, the actual world, but consider the hydrogen atom as an isolated

system. In that case, drastic idealizations must be introduced and what we would be

philosophically reflecting upon is not QM per se, but a particular model, applicable to

that situation. In this case, the contextual individuality that might be introduced is

akin to Toraldo di Francia’s ‘pseudo-individuality’ that is introduced post-

measurement and which we discuss in the book. In the case of the hydrogen model,

the contextual individuality would be metaphysically fleeting, as it were, since as

soon as we relax the idealization we have to deal with the full consequences of

quantum statistics. And it is those consequences, as the foundational level, that we

were concerned with and that we think can be captured by quasi-set theory.

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Secondly, Castellani and Crosilla suggest that although the relevant physics can

provide the heuristic basis for our formal framework, concerns might remain as to

the framework’s overall adequacy. Of course, we agree that more than one

framework may be available—indeed, as we have said, the particles-as-individuals

package is just as natural and of course is underpinned by our standard set

theoretical understandings. However, Castellani and Crosilla argue that even within

the non-individuals approach we need not be committed to quasi-set theory. They

suggest taking what we have called Weyl’s strategy and allying it with Bishop’s

intuitionistic formulation of set theory. This is certainly an interesting suggestion,

yet we would need to have more information about the way Bishop’s constructive

mathematics could help here before we could comment in detail. Setting aside

formal issues that might arise, the difference has to do with how seriously—

metaphysically speaking—we take the notion of non-individuality. Weyl’s strategy

attempts to accommodate the Received View by retaining standard set theory and

forming the relevant equivalence classes. However, as we say in the book, this

effectively ‘masks’ what is going on (given the metaphysics of non-individuals)

since we retain discernible elements of the set but ensure that we can never know

which are which. Quasi-set theory plus a metaphysics of non-individuals offers the

possibility of bringing metaphysics into closer harmony with epistemology.

Castellani and Crosilla also raise more technical issues about the manner in

which our framework applies to other examples, and the form of mathematics that

quasi-set theory would support. These are all on-going matters of investigation and

we, and others in the field, fully intend to continue developing and exploring this

formal option (see French and Krause 2010). In particular, Domenech et al. (2008,

forthcoming), present the first steps in the development of a ‘non-reflexive quantum

mechanics’, in the direction suggested in our book.

With regard to the distinction between cardinality and ordinality, in particular,

this obviously bears on how we understand numerical diversity, one of the crucial

issues in this context, as Howard notes. Thus, we agree that in a two-boson state, for

example, there will be two of them, not one, even if one cannot tell which is which,

and that is why we introduce the notion of quasi-cardinality, taken as primitive in

our theory. Within the quasi-set framework, this permits us to introduce a sense in

which the bosons can be ‘counted’; that is, we assign a cardinal to a collection of

them, without having an associated ordinal. But this does not make them

‘individual-like’.

Now, a quasi-cardinal is a cardinal extended to quasi-sets containing m-objects as

elements, and it is a cardinal (in the standard sense) when these objects are not

considered (the theory keeps standard ZF intact when m-objects are not being taken

into account). Domenech and Holik (2007) have shown that for finite quasi-sets

(those quasi-sets whose quasi-cardinal is finite), the concept of quasi-cardinal can be

defined, but even this move does not prevent us from pursuing the way quasi-

cardinals are ascribed to quasi-sets, as Castellani and Crosilla claim. From the

mathematical point of view, it does not matter how the quasi-cardinal is attributed to

a quasi-set. What is important is that we can speak (within the theory) of collections

(quasi-sets) of non-individuals having a cardinal (its quasi-cardinal), but with no

associated ordinal.

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Bueno likewise presents a number of important issues with regard to our formal

framework. In particular, he raises concerns about the intelligibility of quantification

in this context. Thus, he insists that in order for the usual quantifiers to apply, the

objects must have well-defined identity conditions. This is of course the case

involving the usual objectual interpretation. We maintain that quasi-set theory

shows that this is not the case in general and in Arenhart and Krause (2009), an

account is given of how quantifiers apply in quasi-set theory, by reproducing (and

adapting) what we find in standard ZF (cf. Ebbinghaus et al. 1994).

In particular, Bueno argues that in order to validly infer from each x is Fx that

every x is Fx, quantification over each object in the domain must cover distinctobjects. The alternative is for quantification to range successively over exactly the

same object in the domain, again and again, but then it will never cover the whole

domain, and it would not necessarily follow that every object would have the

requisite property—a disastrous consequence. However, quasi-set theory offers a

third choice, according to which we have elements that are indistinguishable, and

hence not distinct in having well-defined identity conditions but yet are not ‘the

same’ in the above sense. Since we cannot in general distinguish between the

m-objects, our talk of ‘many’ is given in terms of cardinals: when we have a quasi-

set with, say n [ 0 indistinguishable m-objects, we have grounds for saying that we

have ‘more than one’ (exactly n) objects. The afore-mentioned account of

quantification then allows us to make the above inference.

Furthermore, Bueno suggests that we lose an important distinction between the

existential and universal quantifiers, expressed in terms of the inference ‘If a is F,

then every object is F’ which should not be valid. However, he argues, for this to be

so, there must exist an object b, distinct from a such that b is not an F, and this

requires identity. We maintain that we can adopt the above distinction between

quantifiers within quasi-set theory, where we may have an object b that is not

indistinguishable from a which is F. Furthermore, we can prove the following

results: define Fx as x : a where a is some fixed qset (it may be an m- or an

M-atom). It follows that Fa is true, for the relation : of indiscernibility is

reflexive, but from this we cannot infer that Fb, even if b : a. And we don’t need

identity to do that.

As for the possibility of mixed statements involving both m- and M-atoms, within

our framework, such statements which could confuse the use of quantifiers do not

arise, for the atoms are distinguished by their predicates, and the quantifiers are

relativised to them. Of course, one might wonder how we would deal with the claim

that macroscopic objects, such as a piece of apparatus, are composed (in part) by

electrons. As we noted in the book, quasi-set theory could be extended to embrace

appropriate mereological axioms. However, there are at least two obstacles that

would need to be overcome: first, such axioms would have to allow for a discernible

M-object to be composed of indiscernible m-objects; secondly, they would have to

accommodate the holistic nature of QM in some form or other, possibly by altering

the concept of ‘sum’ by which the whole is viewed classically as the sum of its

parts. Although steps have been taken to explore such an extension, considerable

work remains to be done.

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There might be the further concern, related to Castellani and Crosilla’s above,

that statements about the detection of electrons, say, involve implicit reference to

pre- and post-detection electrons, described by m- and M-objects respectively, thus

leading to further confusion regarding the use of quantifiers. Again, we would

suggest that one should conceive of the post-detection situation in terms of pseudo-

individuals and no confusion should arise. Of course, one could eschew quasi-set

theory and the view of particles as non-individuals entirely and adopt the particles-

as-individuals package (with its attendant constraints), in which case standard set

theory would apply throughout.

Similarly, with regard to the indispensability argument, if one accepts its terms

one must decide on the metaphysical package to be adopted and hence the

appropriate formal framework, prior to applying the argument. Bueno’s concerns

then evaporate, since (at the fundamental, micro-level) not only are his (a) and (b)

the only legitimate options, they are disjunctive. But of course, the indispensability

argument is supposed to be a mechanism that supports realism. Given the

underdetermination that holds between these metaphysical packages, the structural

realist would argue that the relevant entities that fall within the terms of the

argument are neither objects-as-individuals nor objects-as-non-individuals, but the

relevant structures that are indispensable for our physics.

These concerns and our attempts to assuage them have given us considerable

food for thought. It is hugely gratifying to see this level of technical and

philosophical engagement with our book. There is, as always, further work to be

done and we would like to thank our reviewers again for helping to indicate the

future paths that might be taken.

References

Arenhart, J.B.B., and D. Krause. 2009. Quantifiers and the foundations of quasi-set theory. Principia 13:

251–268.

Azzouni, J. 2004. Deflating existential consequence. New York: Oxford University Press.

Bitbol, M. 2007. Materialism, stances, and open-mindedness. In ed. Monton, 229–270.

Chakravartty, A. 2007. A metaphysics for scientific realism: Knowing the unobservable. Cambridge:

Cambridge University Press.

Colyvan, M. 2001. The indispensability of mathematics. New York: Oxford University Press.

Domenech, G., and F. Holik. 2007. A discussion on particle number and quantum indistinguishability.

Foundations of Physics 37: 855–878.

Domenech, G., F. Holik, and D. Krause. 2008. Q-spaces and the foundations of quantum mechanics.

Foundations of Physics 38: 969–994.

Domenech, G., F. Holik, L. Kniznik, and D. Krause. forthcoming. No labeling quantum mechanics of

indiscernible particles. International Journal of Theoretical Physics.Ebbinghaus, H.D., J. Flum, and W. Thomas. 1994. Mathematical logic. New York: Springer.

French, S. 2003. Scribbling on the blank sheet: Eddington’s structuralist conception of objects. Studies inHistory and Philosophy of Modern Physics 34: 227–259.

French, S. forthcoming. Unitary inequivalence as a problem for structural realism.

French, S., and D. Krause. 2010. Some remarks on the Theory of Quasi-sets. Studia Logica 95: 101–124.

Jauernig, A. 2004. Leibniz freed from every flaw: A Kantian reads Leibnizian metaphysics. Dissertation:

Princeton University.

250 Metascience (2011) 20:225–251

123

Ladyman, J. 2007. Scientific structuralism: On the identity and diversity of objects in a structure.

Proceedings of the Aristotelian Society Supplementary 81: 23–43.

Ladyman, J., and T. Bigaj. 2010. The principle of the identity of indiscernibles and quantum mechanics.

Philosophy of Science 77: 117–136.

Luckenbach, Sidney. (ed.). 1972. Probabilities, problems and paradoxes. Encino: Dickenson.

McGinn, C. 2000. Logical properties. Oxford: Clarendon Press.

Monton, B. (ed.). 2007. Images of empiricism: Essays on science and stances, with a reply from Bas C.van Fraassen. Oxford: Oxford University Press.

Muller, F.A., and S.W. Saunders. 2008. Discerning fermions. British Journal Philosophy of Science 59:

499–548.

Muller, F., and M. Seevinck. 2009. Discerning elementary particles. Philosophy of Science 76: 179–200.

Ruetsche, L. 2002. Interpreting quantum field theory. Philosophy of Science 69: 348–378.

Stachel, J. 2005. Structural realism and contextual individuality. In Hilary Putnam, ed. Y. Ben-Menahem.

Cambridge: Cambridge University Press.

van Fraassen, B. 1969. An introduction to the philosophy of time and space. New York: Random House.

van Fraassen, B. 1972. Probabilities and the problem of individuation. In ed. Luckenbach, 121–138.

van Fraassen, B. 1991. Quantum mechanics: An empiricist view. Oxford: Clarendon Press.

van Fraassen, B., and I. Peschard. 2008. Identity over time: objectively, subjectively. PhilosophicalQuarterly 58: 15–35.

Wallace, D. 2006. In defence of Naivete: on the conceptual status of Lagrangian quantum field theory.

Synthese 151: 33–80.

Weyl, H. 1931. The theory of groups and quantum mechanics. (Translated from the second, revised

German edition by H. P. Robertson. The first edition was published in 1928.) New York: Dover.

Metascience (2011) 20:225–251 251

123