# The physics and metaphysics of identity and individuality

## Authors

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DOI: 10.1007/s11016-010-9463-7

- Cite this article as:
- Howard, D., van Fraassen, B.C., Bueno, O. et al. Metascience (2011) 20: 225. doi:10.1007/s11016-010-9463-7

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## Don Howard

Steven French and Décio 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.

### 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 *in the 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.

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 nevertheless 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.

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 distinguishable 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 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 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 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 Schrödinger’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, *of necessity*, 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 *problématique* 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!

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 entities that 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 *the physics 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 consistently—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.

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 counterfactual 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}

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* discernability. 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/√2)[(φψ − ψφ)], where φ, ψ are orthogonal spin states, states can be ascribed by ‘reduction of the density matrix’. But each particle is assigned the same mixed state ½P[φ] + ½P[ψ]. 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/√2)[(φφ + ψψ)]. 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 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 with violence*.

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 *the conditions 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.

*abstract*objects. Leibniz explicitly exempted the latter from PII!

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

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.

- 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.

*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 *proper**parts* 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 to the 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.

## Otávio 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 emphasizes 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 understanding 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 permutations 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 Schrödinger 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 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 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.

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

aisF, then something isF.

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

aisF, then every object isF.

*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, (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*-*individuals* package 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 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 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.

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, 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, 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 & Décio 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 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 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 naïve 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.

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.

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 *distinct* objects. 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.

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.

For me, Bitbol (2007) drove the last nail into the coffin of my earlier view; see also French and Krause (192–193, 355–373)*.*

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.

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.

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.