Download Steven French and Décio Krause, Identity in Physics: A Historical

BOOK SYMPOSIUM
THE PHYSICS AND METAPHYSICS OF IDENTITY AND INDIVIDUALITY
Steven French and Décio Krause, Identity in Physics: A Historical, Philosophical, and Formal
Analysis. Oxford: Clarendon Press, 2006. Pp. HB
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 nonindividuals, 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.
1. Are Elementary Particles Individuals?
I don’t 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.
2. 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’ favorite 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 they 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
isn’t 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’s the harm in regarding two
indistinguishable bosons as, nonetheless, individuals that ‘jointly and severally’ enjoy all the
rights and responsibilities accorded to systems? That’s 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’ve 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’m 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’m
not. I’m asking not about the individuation of the systems occupying different regions of spacetime, at least not in the first instance. I’m asking, instead, whether those systems possess states of
a certain kind, namely separate, real states that fix, univocally, all joint states. It’s 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’s
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’m 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’m tempted to add, ‘And I don’t care’.
3. 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’s 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’s 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’s hard to dismiss the worry that there is a lack of univocalness in the way in which QFT
constrains its ontology. Moreover, it’s 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 favored 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 Platonist 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; 6365; 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 or not 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 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 antisymmetric 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 ½ P[φ] + ½ P[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
1
For me, Michel Bitbol (2007) drove the last nail into the coffin of my earlier view; see also French and Krause
(192-193, 355-373).
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.
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). Isn’t
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 were 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 made 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.
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.
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 James Ladyman (2008) 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 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,
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.
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 colors. 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 righthand 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 nonindividuals. 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’m 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 nonindividuals 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 at the technical level, in particular. The nonindividuals package requires the development of a suitable formal framework for quantum nonindividuality. 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 nonindividuals 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.
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, (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 wellknown 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 nonindividuals). 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 so 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 with 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 (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 nonindividual 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 ‘nonindividuals’ 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 nonindividual 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 it 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’s
follow French and Krause and use the letter Q to refer to this version of set theory, called quasiset 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’s 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 Matoms (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. Beside 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 nonindividuals. 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 ‘formalframework 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’d 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 two-fold: 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 emphasise 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 emphasises 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
4
We wish to thank Marisa Dalla Chiara and John Stachel for very helpful comments on a draft version.
Muller and Seevinck (2009) have tried to extend this result to bosons but this is contentious; see Ladyman and
Bigaj (2010).
5
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-nonindividuals
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 (363-64) 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 emphasise, 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 ‘pseudoindividuality’ 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
forthcoming). In particular, Domenech et al (2008), (2010), 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 quasicardinality, 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
(2008) 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 nonindividuals 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 & 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, Flum and Thomas 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 can’t 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 can’t 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 postdetection 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-nonindividuals, 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’d like to thank our reviewers again
for helping to indicate the future paths that might be taken.
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