Published at EOLSS Encyclopedia

Quantum (non-)individuality

Décio Krause & Jonas R. Becker Arenhart

Federal University of Santa Catarina

Department of Philosophy

Santa Catarina - Brazil

In this article we review some metaphysical topics concerning individuality and

non-individuality associated with the discussion of the ontology of non-relativistic

quantum mechanics (QM). Our approach will proceed in three distinct fronts. First, we

expose the physical aspects of quantum indiscernibility and how quantum statistics were

understood as giving rise to the view that quantum particles are not individuals. In the

second part, we discuss how the view of quantum particles as non-individuals was

challenged, and how the idea of an individual is to be understood from a metaphysical

point of view. We relate the metaphysical views on individuality with quantum

mechanics. In the third part we focus on some formal aspects of identity and

individuality, so that after presenting the main motivations for distinct metaphysical

ways one may understand the ontology of quantum mechanics, we first argue that

classical logic and set theory may be seen as coping with an ontology of individuals.

Second, we present a formal system which we believe best captures some of the alleged

features of those items when they are interpreted as being non-individuals. As we shall

see, quantum mechanics was taken ever since its early days as being committed with an

ontology of items deprived of individuality, and that view may be put in a sound

mathematical basis. Our emphasis on formal aspects are justified since as we shall see,

both approaches –i.e. individuals and non-individuals- to quantum ontology are

legitimate from a purely quantum mechanical point of view, so that legitimate

metaphysical and formal matters must be called upon when discussing those issues.

Part I – Physical aspects of the problem

1. Quantum indiscernibility and non-individuality

It is often claimed that the quantum revolution has brought a disruption with our

understanding of the world as it was guided by the notions we inherited from common

sense and classical physics. Many concepts relevant for both physicists and

philosophers, such as determinism, causality, measurement and others, were affected by

the development of quantum physics. The notions of individuality and identity, as they

were understood by philosophers since a long time ago were also in the target of the

quantum revolution; or at least that this is so is claimed by some philosophers

investigating the metaphysical nature of particles in the theory.

Just to make clear what is at stake when it comes to identity and individuality,

we begin by considering our understanding of those concepts when dealing with

common objects. This first approach will be informal and intuitive. Let us begin by

considering two similar exemplars of the same book. Suppose one is yours and the other

is mine. Mine has no single scratch, while yours has your name written in the first page,

and some pages are already missing, while others are already torn. Given those

differences, we have no trouble in saying which is mine and which is yours. If I were to

find your exemplar in my bookshelf, in the same place where my exemplar used to be, I

can be sure that someone has exchanged the books. We may ground the reasons to say

that my book is different from yours in terms of the books’ properties already

mentioned. Also, even considering the possibility that your exemplar of the book were

just like mine in every respect, we believe that it makes perfect sense to say that they are

different. We believe, for example, that the very fact that the books are in distinct places

to begin with grants that they are distinct individuals. That is, even granting that it is not

possible for us to distinguish the books in terms of their properties, and even conceding

that we may disagree on whether we could find one such property in case we looked

harder after it, we may concede that they are distinct individuals by the fact that they

have distinct spatial locations.

What goes for the exemplars of a book goes for other common everyday objects

as well (leaving place for disagreement in the case of persons, but we shall not deal with

that here). However, for quantum particles those remarks do not seem to hold at all.

Even supposing I could in some sense have my own electron (for instance) trapped in an

experimental apparatus in my bedroom and you could have yours in your bedroom, it

seems there is no sense in saying they could even in principle have their own features

making it possible for us to tell them apart. If they were exchanged one by the other, no

one would be in a position to say that the exchange really did happen or worse yet, no

one could grant that they both were not exchanged by electrons having nothing to do

with the one we thought to be yours and mine originally trapped. Now, before we go

ahead with the more technical aspect of the matter, consider Roger Penrose, discussing

this particular feature of the nature of quantum particles, who says that “[i]n the

physical world, all particles of the same species appear to be identical. Any two

electrons, for example, are just the same as each other'' (see Penrose (1989), p.297). Just

to make an important terminological point: identical and the same as each other are the

physicists’ way of saying that items are indiscernible, i.e. no property whatever can be

claimed as a distinguishing feature.

Now, someone could protest: the statement of the problem is unfair, since the

case of the books we allowed spatial locations to do the job in grounding the

individuality and numerical difference. Really, if my electron is the one that is located

in my bedroom and your electron is the one located in your bedroom, can’t we simply

distinguish them by these very features, as we did for the books? That is, let us suppose

for a moment that electrons can bear names (not everyone is willing to concede that

they can, of course, but it serves just for the sake of argument). If we call ‘Bob’ the

electron in my room and ‘Rob’ the electron in your room, why can’t we then simply

write |Bob, my room> as an awkward notation to represent the fact that Bob is in my

room and |Rob, your room> for the fact that Rob is the electron in your room, so that the

whole situation could be described (to keep with the sophisticated notation) by the

concatenation of both descriptions |Bob, my room>|Rob, your room>? In that case, it

seems that we may be sure that the “real” situation that obtains is different from the

other one, which would arise from linking together the other two distinct possibilities,

that is, |Rob, my room> and |Bob, your room>, giving us the complex description |Rob,

my room>|Bob, your room>. What is the problem with that kind of protest? That is,

why can’t we employ spatial location as proper a means to distinguish and individuate

quantum entities?

First of all, the problem is that the situation cannot be so simply put in the

quantum mechanical description of states of physical systems. Recall that we are using

quantum mechanics to discover what is going on with the electrons in this case, so that

it dictates to some extent how we should understand their behavior (this explain the

awkward notation, which is the Dirac notation usually employed in the theory).

Quantum mechanics does not allow for descriptions like |Bob, my room>|Rob, your

room>. To deal with a system composed of both electrons, we must (keeping our

terminology) describe the situation employing the following state:

C(|Bob, my room>|Rob, your room> - |Rob, my room>|Bob, your room>).

Here C is a mathematical constant granting some desired feature of the state (unitary

length, as we shall see), the (|Bob, my room>|Rob, your room> - |Rob, my room>|Bob,

your room>) part grants us that all we may know is that one of the electrons is in my

room, the other in your room. Which electron is in which room is a question that really

cannot be answered by quantum mechanical theoretical apparatus. As we shall explain

in a moment, the states in quantum mechanics are described by symmetric and anti-

symmetric vectors, and that means roughly that “things” described by the theory cannot

be distinguished even by spatio-temporal locations. The correct way to describe the

situation is not “Bob here” and “Rob there”, but rather, “one here” and “one there”; no

simple way to say which is which and where is which of them (remember that we are

still working under the assumption that we could reasonably label the items being dealt

with). That situation is seen to amount to a kind of qualitative indiscernibility for

quantum entities.

That issue of indiscernibility has to do with the exotic behavior of quantum

particles when they are aggregated in more than one item. To keep with our comparison

of quantum and classical objects, let us suppose we have two particles, named 1 and 2,

and that those particles share all their state independent properties (sometimes called

their intrinsic properties, but that terminology and its applicability in this case is

contentious). So, 1 and 2 are indiscernible relatively to their state independent

properties. Consider also two states a1 and a

2, which these particles may occupy. We

may write the resulting distribution as follows, putting |n, ai,> for particle n (with n

=1,2) occupying state ai for i = 1,2:

1. |1, a1> and |2, a

1>;

2. |1, a2> and |2, a

2>;

3. |1, a1> and |2, a

2>;

4. |2, a1> and |1, a

2>.

The first situation has both particles in state a1. The second describes both particles in

a2. Third situation describes particle 1 in a

1 and particle 2 in a

2. The fourth situation

reverses the third, that is, particle 2 in a1 and particle 1 in a

2. If 1 and 2 are classical

particles, then the above possibilities describe exactly the way classical physics

describes the possible distribution of two particles over two states. This particular

distribution is known as Maxwell-Boltzmann statistics.

What is it that makes classical statistics go hand in hand with the features of

usual objects as we described before? It is precisely the fact that a permutation of

particles 1 and 2 as described in third and fourth situations above do count as distinct

possibilities. That is, even though 1 and 2 may be indiscernible by their state

independent properties, there is a difference between the cases when it is 1 or 2 that is in

a1 and the other one in a

2. A permutation counts as giving rise to a distinct state. If we

were to concede that all the possibilities are equally probable, then each of the above

cases would have probability ¼ of happening.

Quantum particles, on the other hand, behave very differently. For them, states

described by the third and fourth situations do not count as distinct possibilities. In fact,

to be more precise, the third and fourth situations are not even allowed for quantum

particles. Supposing 1 and 2 are quantum particles of the same kind indiscernible by

their intrinsic properties, then the states for each of them are described in a Hilbert space

H (the same for both), and the states of the system containing both particles are

described in the tensorial product of H with H. Following the above notation, we write

|m, ai>|n, a

j> for a vector in this space, with m, n = 1,2 and i, j = 1,2, and also allowing

linear combinations of such vectors. The allowed situations in quantum mechanics are

describable then as follows:

1. |1, a1>|2, a

1>;

2. |1, a2>|2, a

2>;

3. C(|1, a2>|2, a

2> |1, a

2>|2, a

2>);

Here, once again, C is a constant to ensure that the vector has a desirable feature, viz.

that it has unitary length. Case 3 is the real novelty in the quantum statistics, but there is

a little more to the issue of statistics for quantum particles than only this. First of all, we

must consider that quantum particles come in two distinct kinds: bosons and fermions.

Bosons are the particles obeying possibilities 1-3, when 3 takes the “+” sign. In that

case, assuming once again that each distribution is equally probable, we have that each

case has probability 1/3, and the respective statistics is called Bose-Einstein statistics

(BE). Fermions are the particles obeying only condition 3 when it takes the “–“ sign.

Fermions obey the Pauli Exclusion Principle, which, roughly speaking, forbids that

fermions share all the same quantum numbers. So, any two of them cannot be in the

same state which is a component in the composed state (that is why situation 1 and 2

above are ruled out). In this case, the probability of occurrence of state 3 above is equal

to 1, and the accompanying statistics is called Fermi-Dirac statistics (FD).

As a first point to be noticed that is relevant for our purposes, one should see at

once that what situation 3 above does, in particular, is to render permutations of

particles labeled 1 and 2 innocuous. That is, the labels do not seem to be any more than

a formal device employed by us to keep track of the number of elements being dealt

with. Permuting 1 and 2 in the previous states does not bring us to a distinct state in the

case of bosons, and only changes sign in the case of fermions, something that does not

change the expected value of measurements of any observable carried over a system in

this state.

The idea that quantum particles may be permuted without changing the state

they are in is explained by the fact that they obey the so-called Indistinguishability

Postulate (IP). Before we state IP, we should notice that permutations of particle labels

may be performed inside the Hilbert Space formalism through permutation operators P.

Then, the mentioned fact that particles may be permuted without giving rise to any

observable difference is expressed by saying that the initial state and the final permuted

state do not differ in the predictions they allow us to make for any observable O.

Insisting on the role of observables, the same may be put as follows: for any observable

O and any permutation operator P,

[O,P] = OP – PO = 0. (IP)

That is, operators representing observables and permutation operators commute. For the

expected value version of the Postulate, given a state |a>, an observable O and a

permutation operator, one may put it as follows:

<a |O| a > = <Pa |O| Pa> = <a |P-1

OP| a> (IP).

One should be aware that IP allows for more than only Bose-Einstein and Fermi-

Dirac statistics. That is, it imposes a restriction on observables but this restriction does

not guarantee that only symmetric and anti-symmetric states are allowable (the ones

employed by BE and FD statistics, respectively). Other forms of statistics, known as

Parastatistics, are also allowed, and they bring along their respective particles, the

Paraparticles. Since no Paraparticle has been observed in nature yet, some have

preferred to restrict IP and grant that the only allowable states are the symmetric and

Anti-Symmetric ones. This restriction is known as the Symmetrization Postulate (SP).

Now, it seems that it does not matter how we regard it, in the IP or SP form, we

can be sure that quantum particles are not distinguishable by measurements before or

after permutations. The very fact that permutations do not give rise to distinct physical

situations is one of the main motivations for those claiming that quantum particles are

metaphysically different from their classical cousins. The situation may be put as

follows: classical particles do have something in them that makes the difference when

they are permuted. Quantum particles do not have that something. What is this

mysterious feature accounting for their differences? Some have called it “identity”,

others “individuality” (and others still have employed both concepts interchangeably

without stipulating a distinction between them), but the fact is that both notions always

appear related in those discussions. Even though both types of particles may be deemed

as indiscernible by their intrinsic properties, classical particles are considered to be

individuals, while quantum particles are not. Roughly, this is what came to be known as

the Received View on quantum particles’ non-individuality (see French and Krause

(2006) chap. 4).

Part II – Metaphysical aspects of individuality

2. Metaphysical topics on individuality

But is the foregoing description really all that there is to the discussion

concerning particles’ identity and individuality in quantum mechanics? Not at all. The

supremacy of the Received View among the founding fathers of quantum mechanics has

faced some challenges in recent times. First of all, it is claimed that despite their

indiscernibility, quantum particles may indeed be regarded as individuals, granted that

their individuality is not understood as being grounded in some form of discernibility.

That is, facing the fact that quantum particles are indiscernible and obey IP, one may

propose that there is still hope for those wishing to regard quantum particles as

individuals, provided that what accounts for their individuality be compatible with

indiscernibility through properties (see also French (1989), French and Krause (2006)

chap. 4).

In fact, quantum mechanics does not help us very much in deciding in an

absolute way some issues concerning the nature of the ontology associated with it. One

may hold to the Received View and conceive quantum particles as non-individuals (as

suggested by the previous discussion), or else abandon that option and look for some

individuation principle compatible with it (and indeed there is a bunch such principles

available). None of these options is excluded by the theory alone, and in both cases

extra-theoretical assumptions must be evoked to decide the issue, if it is to be decided.

That fact may sound frustrating for those expecting to extract from quantum mechanics

a definitive answer to the question as to the nature of the entities it deals with, or, to put

it in a more grandiose tone, it is frustrating to discover that quantum mechanics alone

does not provide some of the expected answers as to the ultimate furniture of the world.

It seems that the answer (if there is one), granted that it cannot be given in a definitive

way by quantum mechanics alone, will have to be looked for in metaphysical arguments

mixed with some of the features of the theory. Both options may be developed, the

individuals as well as the non-individuals “packages”, and one may profit from their

study.

The situation described in the previous paragraph was called The

Underdetermination of Metaphysics by the Physics (see French (1998), French and

Krause (2006)). The idea is that physics alone does not determine what kind of ontology

is associated with it. Ontological naturalism, the view that we should read our

ontologies from our best scientific theories, then, must recognize that at least some

metaphysics must be assumed without being grounded in science. As it was put by

French ((1995), p. 466):

“it is erroneous to suggest that physics may be used to uniquely support a particular

piece of metaphysics. This is an instance of a perfectly general point: a physical

theory may support more than one metaphysical package. There is, in effect, a kind

of underdetermination of metaphysics by physics and any apparent inclination of

the latter to support a particular form of the former is a result of some prior tacit

assumptions that will themselves be metaphysical in character. Putting it bluntly,

you get only as much metaphysics out of a physical theory as you put in and

pulling metaphysical rabbits out of physical hats does indeed involve a certain

amount of philosophical sleight of hand.”

In some sense, it was traditionally assumed that this metaphysical

underdetermination poses some challenges for those trying to be scientific realists about

quantum mechanics (see French and Ladyman (2003)). But instead of entering into the

problem of naturalism and its relation to realism, let us proceed with the discussion of

some of the alternatives to the Received View. It would be profitable to analyze some of

the current theories of individuation, since they allow us to better appreciate the nature

of current debate and the proper nature of non-individuals, as we shall understand them.

Really, before we attempt to decide which side to take on that problem, and whether it

is worth to break the underdetermination, some questions remain which are directly

related to this issue: how are we to understand quantum particles as individuals? What

exactly do we mean when we say that they may be seen as non-individuals? These are

notions asking for definitions.

Let us consider what it means for something to be an individual, so that we may

(hopefully) be in a better position to understand what it is for something to be a non-

individual. As is well-known, the problem of individuality is an ancient one, comprising

so many different views that it would be impossible for us to enter into the details here.

Roughly speaking, the problem is generally put in terms of an item’s self-identity and

how it differs numerically from every other thing. A Principle of Individuation should

provide for those aspects of an individual. Some also hold that is must have some kind

of explanatory role; that is, it must explain why an individual is what it is (see also

Lowe (2003)).

Most of the discussion of individuality in quantum mechanics is associated with

looking for Individuation Principles which try to ground individuality in something else.

That is, in general, the problem will be to find some in principle intelligible item or

feature which could be responsible for the individuality of its bearer. There are at least

two options for those intending to reduce individuality to some other notion: the bundle

theories and the transcendental individuality option. Both are viewed generally as

“reductive” theories, that is, they intend to somehow define what it is that makes an

individual once it is given that some further notions are taken for granted. Obviously,

that the discussion is generally framed this way does not forbid us to try to account for

individuality in terms of some primitive kind of principle, grounded in nothing else.

This form of brute individuality was not favored by many in conjunction with the

ontological status of quantum particles, but a recent proposal pursues precisely those

lines (see Dorato and Morganti (2011)).

Let us briefly discuss bundle theories first. Roughly speaking, the bundle theory

defines an individual as a bundle of properties. The idea is that the properties

characterizing the individual are instantiated together, and this “togetherness” means

that they are related by a special relation of compresence that ties the properties in such

a way as to give rise to an individual. The bundle theory, taken by itself, was seen as

unable to grant the uniqueness of the bundle thus defined. Really, how can we make

sure that there are no two bundles instantiating all the same properties? If there were, it

seems, the bundle theory would then be unable to account for the individuation of the

particulars; that is, it would not be able to ground the required uniqueness of the

individual.

To solve that problem, it is usual to adopt a version of the famous Principle of

the Identity of Indiscernibles (PII). According to that principle, if two items share all of

their properties, then they are identical. This would grant all that is necessary for the

bundle theory to work as an individuation theory. If PII is true, none of the problems

concerning multiple instantiation of a single bundle just mentioned seems to arise.

However, before jumping too quickly to that conclusion, one must specify what really is

being meant by “properties” in this context. Three versions of the principle appear

accordingly (the following is really only one among a variety of distinct possible

taxonomies):

PII1) Two items sharing all their intrinsic monadic properties are identical;

PII2) Two items sharing all their intrinsic monadic properties and relations, except for

spatio-temporal relations, are identical;

PII3) Two items sharing all their properties and relations are identical.

It is usually argued that PII3 trivializes the problem: if a and b share all their

properties, then, in particular one could mention the property “being identical to a”.

Since by hypothesis b should also have that property, then it is trivially identical to a.

So, for individuation purposes in a bundle theory, PII3 seems out of question. The

strongest version, PII1, is seen to be false even in classical mechanics (French and

Krause (2006), chap. 2). So, if the bundle theory relies on it, it then it falls prey to many

counterexamples in which we are presented with items sharing all their monadic

properties and not being the same, such as the famous Max Black’s universe containing

only two iron spheres absolutely indiscernible and 5m apart from each other (see Black

(1952)). The only remaining version is PII2. How should we consider a version of the

bundle theory with PII2? Are we allowed to take relations into account when it comes to

individualize objects?

At first sight, it seemed that PII2 had good chances in classical mechanics, but

not in quantum mechanics; that is, quantum particles are such that they share even their

relations (see French and Redhead (1988)). That would be the end on PII in quantum

mechanics and also the end of the chances of bundle theory as an individuation option

in this case. That was taken for granted for some time, until recently this conclusion was

challenged in a series of papers by Saunders, Muller and Seevicnk (see specially

Saunders (2003), (2006), Muller and Saunders (2008), Muller and Seevinck (2009)).

These authors tried to save PII in the context of quantum mechanics by allowing some

special kind of relations to do the job of discerning items.

Conceding that quantum particles share all their monadic attributes, Saunders,

Muller and Seevinck sought to discern particles through relations that could be

furnished exclusively by the formalism of quantum mechanics. Indeed, they discovered

that all of the arguments against the validity of PII in quantum mechanics only took into

account relations stated in terms of conditional probabilities, that is, probabilities stated

thus: if the system has such a property, then it has such another property. Quantum

particles are indiscernible according to that account of relations. However, they argued,

no one had showed that categorical relations (i.e., non-probabilistic relations) could not

be employed to discern quantum particles. One example of a relation that was purported

to do that, in the opinion of those authors, was the relation “having spin in the opposite

direction to”, which holds between fermions in the singlet state. That is, given a

direction in space, one may be sure that fermions a and b related by that relation are

distinguished, since no fermion has spin opposite to itself. So, given a and b

numerically distinct, the idea is that we have discerned them through a relation, and so

PII is vindicated in quantum mechanics.

That kind of result was extended to bosons by Muller and Seevinck (2009), and

so, it seemed, there would be good chances of saving bundle theory in quantum

mechanics. However, the issue is not so simple. Metaphysical considerations were

already adduced against the use of relations in discerning and individuating long before

the existence of quantum mechanics. Roughly, the point is that relations already

presuppose that the relata be distinguished and individuated, so that they may enter into

the relation. In that occasion, relations are not allowed to account for the discernibility

and individuality of the items in question, since they presuppose that discernibility and

numerical diversity (see also Hawley (2009), Ladyman and Bigaj (2010) and Krause

(2010) for further criticism).

So, one may consider that particles are discernible by relations only at the price

of assuming that relations do some kind of metaphysical work not generally expected

from relations. Anyway, the arguments against the use of relations in the individuation

process are very strong ones, and even Muller and Saunders have suggested that

quantum particles are not individuals, they are relationals, entities for which the

individuation principle fails, but which nonetheless are discernible by some kinds of

relations (of course, provided that we accept that relations do discern). Under those

circumstances, one may accept that the bundle theory plus PII seems to fail to account

for the individuality of the particles, so that those wishing to argue that particles may be

individuals must seek for the individuation elsewhere (see also Arenhart (2012)).

One possible option to escape from those difficulties concerns adoption of

bundle theory without PII (see O’Leary-Hawthorne and Cover (1998), and Rodriguez-

Pereyra (2004)). In that case, the possibility of multiple instantiated indiscernible

bundles is allowed, and their numerical distinctness is explained in terms of spatial

separation of instances of the same bundle or in terms of numerically distinct instances

of the same bundle (so that there is a crucial difference between a bundle of universals,

which is taken to be unique, and its instances, which may be multiple). That option

accounts for quantum indiscernibility, but is open to another objection: the contextuality

of quantum mechanics. Really, it seems that the understanding of “property bearing” in

bundle theory and in quantum mechanics are at odds here. According to traditional

property bearing, one may safely say that an entity either has or does not have any given

property. In quantum mechanics, on the other hand, most of the times what properties a

given system has depends on the experimental setup being employed, and incompatible

properties (those represented by non-commuting operators) cannot be attributed to

system simultaneously. So, it seems that there is no easy road for the bundle theorist

here too.

The second option available for the friends of individuality is transcendental

individuality. The idea is that what confers individuality for the individual objects is

something transcending the qualities, not accountable by the use of qualitative

properties. Something may be discernible from other objects by their possessing distinct

properties, but what really confers individuality is the transcendental principle. This

something has been understood either in terms of particular items such as Lockean

substratum, bare particulars, or else in terms of non-qualitative, non-shareable

properties, such as haecceities and primitive thisness (see Moreland (1998), Lowe

(2003) for further discussions). Either way, with transcendental individuality one is

allowed to have indiscernible items which count as numerically distinct (and self-

identical). Properties and relations may do the epistemological role of discerning, if

there is a possibility of discerning at all, but the ontological task of individuation is

performed by something else.

Saying precisely what is that something else is an embarrassing problem. It is

said that it bears the particulars’ properties, but is itself not definable in terms of

properties. Its individuality is primitive, is not grounded in anything else. It is precisely

the mysterious and elusive nature of the substratum (or bare particular) which have

convinced many not to follow through these lines. Indeed, if one was expecting to find

the answer of the individuality question in a scientific theory, as the naturalist did, then,

it seems, one is not very happy with accepting the postulation of a mysterious savior

behind the phenomena. So, even though this option is compatible with quantum

mechanics, it has not received much attention, in the sense of being eagerly defended as

the right option.

Now, what to say about our first option, i.e. non-individuals? Well, we have seen

that individuals have an all-important defining feature: they are self-identical. Also, they

may be said to be numerically distinct from other individuals, even though they may be

indiscernible. So, it has been suggested that non-individuals, the items quantum

mechanics may be seen as dealing with in a plausible interpretation, could be

understood as items without identity. Precisely, identity and difference do not really

make sense for non-individuals. Also, to account for some of their quantum mechanical

features, non-individuals should be indiscernible, that is, they may be indiscernible and

non-identical. As a last defining feature of non-individuals, they may be seen as

aggregating themselves in pluralities with a defined cardinal, but no ordering or

counting of them being possible (see French and Krause (2006), Krause (2002), also

Dalla Chiara (1985)). In fact, counting, as it is usually understood, involves labeling

each of the counted items by a one-to-one correlation of the collection being counted

with an appropriate ordinal number, something we cannot hope to do with quantum

particles.

Those features of a non-individual really conflict with the main features of the

items dealt with in classical logic. So, the usual give and take discussion between

ontological commitments and explanatory power acquired with it is to be performed.

Some have seen this revision in ontology and its accompanying change in logic as too

revolutionary, so that non-individuality really does not have any preference over an

ontology comprising individuals. Keeping in quantum mechanics the same logic as in

other scientific domains is seen as a virtue by some. However that may be (and we shall

discuss this issue in the next section), there is a third option claiming to get away with

the problem from the beginning: structural realism. As is well-known, there are two

kinds of structural realism nowadays, epistemic structural realism (ESR) and

ontological structural realism (OSR) (Ladyman (1998), (2009)). Both are concerned

with solving the challenge posed to the scientific realist of explaining the success of

empirical science and accumulation of knowledge through radical theoretical revisions.

Roughly put, both kinds of structuralism agree that it is structure (in some sense of the

word) that is retained when theories change, not the entities described by them.

Differences appear precisely on what concerns the entities.

According to ESR, the entities dealt with by our best scientific theories are

somehow hidden from our knowledge, so that the best we can know of them is the

relations they enter into (this is very roughly put; for the details see Worrall (1989)).

There is no knowledge of the entities; we only know the relations, but the entities are

there. This account for the fact that in scientific revolutions the ontology of our theories

may change, but our knowledge, as encoded in relations (expressed by equations) are

retained. Since ESR is aimed at epistemology more than ontology, we shall not enter

into the discussion on the nature of quantum entities according to this view. It seems

that it is proposed that we cannot know whether the items in question are individuals or

non-individuals, and such ignorance is no trouble for the success of science.

On the other hand, OSR may come in many distinct varieties, from moderate to

eliminativist. We cannot enter into the details of every one of them here, so we shall

focus on the eliminativist kind, since its proposers derive their motivation precisely

from the dispute concerning individuality in quantum mechanics (French and Ladyman

(2003), Ladyman (1998), (2009)). According to eliminativist OSR objects should be

eliminated from ontology, keeping only relations at their place. Just as in the ESR case,

relations are what is preserved through theory change, but instead of being a hidden

reality, objects can are to be reconceptualized as secondary entities. The real things are

the relations. Really, since in the face of metaphysical underdetermination quantum

mechanics does not decide the matter concerning individuality, we should not sin

against naturalistic scruples and try to decide the matter on other grounds; let us do

without objects. The proposal is to adopt an ontology of relations without the items

related (the relata), seeing individuals and non-individuals as particular instances of a

common structure, in the sense that both cases present enough commonalities, the ones

that characterize the structure of the theory (French (2011)).

It is clear that this is a very sketchy presentation of OSR. This is a growing field

of research nowadays, with papers appearing every day. However, there is still a lack of

rigor in explaining what a structure is, and consequently, in providing an explanation of

what the position amounts to. Really, since relations are primary and objects should be

introduced in terms of relations, the usual accounts of structure will not be useful.

Structures, recall, are mostly seen as some specific entities in some set theory,

characterized by a set of objects and a family of relations (the relations need not be only

between the objects in the domain, but also higher order relations between them. That

happens because in science we need higher order structures). So, much formal work is

still required in the case of OSR.

What is a problem for OSR is already available for non-individuals and for

individuals. We can systematize the features of a non-individual mentioned before in an

appropriate formal system, known as quasi-set theory, with some remarkable

achievements already obtained in the foundations of quantum mechanics (for example,

in Domenech, Holik and Krause (2008)). Before we do that, however, we shall expose

briefly why classical set theory (and classical mathematics in general) is committed to a

view according to which every object is an individual, in the sense of being self-

identical and discernible from every other item. We now expose those points.

Part III – Formal aspects of identity and individuality

3. Classical mathematics and identity

In this section we discuss the relationship between classical logic and

mathematics and the philosophical concept of individuality. It is a very widespread view

among philosophers that logic is “topic neutral”, in the sense that it does not make

assumptions about its subject matter. That is, according to this view, logic deals with

any kind of thing without restrictions. Seen from that point of view, some could claim

that logic does not impose any kind of feature on the items dealt with by it; it deals with

persons and armies as well as quantum particles and molecules indistinctly. However,

we do not agree with that view in its full generality, for we believe that logic may be

seen at least in an indirect way as having ontological import (we mean that logic has an

indirect bearing on ontology because it is possible and reasonable that one may study

any kind of logic with purely mathematical interests). The kind of impact we believe

that classical logic has on ontology comes mainly from its imposition of identity for

every object it deals with. Indeed, some philosophers define the very notion of “object”

as an item obeying the classical logic notion of identity. In that sense, we are claiming

that classical logic deals only with items having identity, which forbids non-individuals

from being in its domain of discourse. Let us discuss what we mean by such statements

as “having identity” and how it happens that classical logic may be seen as imposing

identity on everything.

Let us say that an item has identity if it obeys the conditions of being self-

identical and numerically different from every other item. In that sense, we could claim

that something having identity is an individual in a minimal sense of that notion, and

that even stronger demands on what an individual is presuppose this one. Lowe (2012)

for instance, argues that an individual must obey a unity condition besides self-identity,

with which we agree. However, not everyone puts that demand on individuals, so we

deal with the more permissive idea that an individual must be self-identical. In fact,

some authors take primitive individuality to consist on those simple identity facts

(Dorato and Morganti (2011)).

Classical logic, we argue, commits us with entities satisfying that minimal

requirement, so that we may say that it deals only with individuals. The standard

approach to identity in classical first-order logic consists in assuming that there is a

primitive relation symbol “=” with suitable axioms. As is well-known, the axioms for

identity are the following:

A1) For all x, (x = x);

A2) x=y → (F(x) → F(y)) (Substitution Law, with the usual restrictions).

The law of reflexivity grants us that, considered from a syntactical point of view,

everything is identical to itself. So, one could conclude that, even from a syntactical

point of view, classical logic already commits one to some form of the minimal

condition for individuality mentioned before. If we allow that classical logic is the

underlying logic of every scientific theory, then, everything scientific theories are

committed with are individuals. Really, to appreciate the effect of that remark one must

only consider that classical logic is the underlying logic of classical set theory, the

underlying mathematics of every mathematized scientific theory.

But one could claim that this conclusion is just drawn too fast. Recall that a

formal system may have an intended interpretation, but unless some rigorous

interpretation is really given, all we have is syntactical manipulation of symbols. So, to

grant that the minimal condition for individuality is really being satisfied one must also

make sure that the identity relation symbol is really being interpreted as the relation of

identity among the items in the domain of interpretation, that is, one must make an

incursion into the semantics of first-order languages. The first point to be noticed is that

establishing the universal validity of the reflexivity law of identity is not that

straightforward: the axioms A1 and A2 grant us that the interpretation of identity is real

identity in the domain of interpretation only in some cases, as is well-known. That is,

the relation denoted by the symbol “=” is really a reflexive relation in every model of

the theory we deal with, but that relation may not always be the identity relation. Let us

briefly present the main points.

The intended interpretation of the symbol = is the diagonal of the domain D of

interpretation, that is, the set

Diag(D) = {<x,x>: x is in D}.

The diagonal is the identity from the semantic point of view. To grant that the syntactic

symbol of identity is really identity we should be able to grant that it is always

interpreted in the diagonal of the domain, no matter what the interpretation is (that is,

that the syntax and the semantics match each other). However, since there is no

axiomatization of the relation of identity in first-order languages, we cannot grant that

this is the interpretation of the identity relation in every domain.

The problem may be stated in the following way: in first-order languages, the

interpretation of identity, unless it is fixed in the metalanguage, is conditioned by A1

and A2. However, those axioms are not strong enough to fix the diagonal of the domain

of interpretation as the interpretation of = in every structure. What those axioms grant us

is only that the interpretation of identity is a congruence relation: an equivalence

relation (that is, a reflexive, symmetric and transitive relation) and compatible with

every other relation in the structure. The compatibility condition means that the items

related by a congruence relation may be substituted one by the other without problems.

For example, if C is a congruence relation in a domain D, and R is a binary relation in

D, then if xCy and xRz, we have by the compatibility of C with R also that yRz. Of

course, if we are to have a congruence relation then compatibility must hold for every

relation in the structure. That grants that the law of substitution holds.

But what is the relation between congruences over a structure and the diagonal

of the domain? Well, the diagonal is itself a congruence relation; however, it is not true

that it is always the only one. In fact, there are many structures for which there is more

than one congruence relation, and in those cases we may define an order relation over

the congruences, so that identity may be seen as the weakest one, contained in every

other. In those cases where the diagonal is not alone as the congruence for the structure,

the interpretation of identity will be not on the diagonal but on the strongest congruence

relation. In those cases, the syntactic symbol of identity will denote an indiscernibility

relation by the predicates of the language, not identity as required by the minimal

condition for individuality.

So, how can one claim, as we did, that classical logic commits us with that

requirement? How can we grant that self-identity is true for every item dealt with by

classical logic? First of all, we shall call a structure normal if the relation of identity is

interpreted in the diagonal of the domain (that is, where the diagonal is the only

congruence relation allowed). For normal structures there is the required match between

syntax and semantics. But what about the ones that are not normal? In those cases, it

seems, the identity relation is interpreted in some kind of indiscernibility relation.

One simple solution to those difficulties consists in assuming that identity is a

logical symbol. In that case, its interpretation will be fixed too, just as it happens with

every other logical symbol. Obviously, in that case identity will be interpreted as the

diagonal of the domain of interpretation, in every structure. However, that must be put

in the metalanguage, as a requirement on the interpretation of identity.

There is also a second simple way out of the dilemma concerning identity.

Recall that the semantics is defined inside some classical set theory. In that framework,

one is always able to grant that the indiscernibility we are talking about is only some

form of epistemic indiscernibility: the items dealt with may and indeed discerned. To

see how, let us introduce another possible way of defining indiscernibility. An

automorphism for a structure is a bijection of the structure into itself. We say that items

a and b in the domain of the structure are indiscernible if and only if there is an

automorphism taking a to b. If a structure has no such automorphisms, then its only

automorphism is the trivial one, identity. In those cases, we say that the structure is

rigid. A rigid structure is normal, that is, in a rigid structure the interpretation of identity

is the diagonal. However, not all normal structures are rigid, as for example the field of

complex numbers, which is normal but in which the complex conjugate is an

automorphism distinct from identity (see also Ketland (2006) p. 310).

Structures that are not rigid may be turned into rigid ones. This brings us to the

main point of the thesis that every item dealt with by classical logic may be seen as

satisfying the minimal condition for individuality. By a theorem of set theory, every

structure may be extended to one that is rigid, so, to one that is normal, and as a

consequence, identity in the syntax and in the semantics match. Then, structures that are

not normal may be seen as simply masking the fat that the items in the domain are

individuals. To discover their “real nature”, it is enough to rigidify the structure. An

example of one such process of making a rigid structure, consider the simple structure

of group <I, +>, where I is the set of integers. The automorphism f taking each integer

into its inverse f(x) = -x is an automorphism of the structure distinct from identity, so

that this structure is not rigid. To make it into a rigid structure, however, it is enough

that we add the usual order relation in the structure, so that we have <I, +, < >. In that

case, f is no longer an automorphism, and identity is the only automorphism of the

structure, that is then rigid (and normal).

So, it seems, assuming classical logic as the underlying logic of our theories

commits us with the possibility of making any structure rigid. In that case, any attempt

to propose a failure of self-identity in cases of structures that are not normal is only

apparent, and there is always the possibility that we can extend the structure in question

to a rigid one, so that the minimal condition for individuality is satisfied. Any attempt to

ground a metaphysics of non-individuals, understood as items that fail in the minimal

condition for individuality will be doomed to be only a fake attempt, since individuality

is always able to come at the cheap price of an extension (notice that this means

problem also for ontic structural realism, but we shall not discus that issue here). So,

what should we do to grant that non-individuals are properly treated? We deal with this

topic in the next section.

4. Quasi-set theory Q

The main purpose of quasi-set theory is to be a formal approach to non-

individuality as it may be seen as being suggested by quantum mechanics (i.e., with the

features described by the interpretation provided by the Received View). The idea is

that some of the entities dealt with by the theory should have the following features:

i) They have no identity, that is, identity is a relation that does not hold between those

items;

ii) They must be indiscernible by their properties;

iii) They must be able to form collections with a well-defined cardinality.

Logic, for our present purposes, could be understood as dealing with the most

general concepts of a scientific theory or field of knowledge. Quantum mechanics deals

with concepts such as ‘force’, ‘spin’, ‘mass’, among others which are specific to it. To

have a grasp of them we must study quantum mechanics. However, the study of

quantum mechanics also employs other concepts such as ‘property’, ‘relation’ and

‘object’. Those concepts, and others of the same kind, are more general then the

previous specific ones, and we may call them the categories associated with the theory.

A system of logic, then, may be seen as dealing with some or all of those categories. In

our case, the logic we wish to present will deal with some of the categories quantum

mechanics may be said to have commitment with. The category of non-individuals is

one of those categories, and we shall pay special attention to it. Let us see now how the

formal representation of those features involved in quantum non-individuality is

accomplished in theory Q.

We shall develop Q here without entering into every detail. The theorems are

stated, but not proved (for more details, see French and Krause (2006) chap. 7). The

underlying logic is first-order classical logic without identity, in any standard

formulation. The specific vocabulary of Q is composed of the following collection of

primitive symbols LQ = {m, M, Z, , , qc}.

The terms are the variables and expressions of the form qc(t). We employ t, w, u

as metavariables for terms. The intuitive readings of the specific symbols of the

vocabulary LQ are as follows:

1) M, m and Z are unary predicate symbols. We read m(t) and M(t) as “t is an m-

atom” and “t is an M-atom”, respectively; Z(t) means “t is a set”.

2) Membership and indistinguishability are symbols for binary relations. We read

tw as “t is a member of w” and tw as “t is indistinguishable from w”.

3) The qc is a functional symbol of arity 1 such that qc(t) denotes intuitively the quasi-

cardinal of t, extending the notion of cardinality for arbitrary quasi-sets.

The definition of formula is the usual one. We must only remark once again that

the identity sign “=” does not appear neither as a primitive symbol in the vocabulary of

Q nor as a symbol of the underlying logic, so that formulas of the form “t = w” are not

part of our official language.

In what follows, for convenience in the exposition, we shall divide the specific

postulates in three groups. The first group is composed of postulates laying the

foundations of the theory and the main relations holding among the distinct kinds of

entities dealt with by the theory. In the second group we postulate the existence of some

collections to allow for the introduction of the usual set theoretical operations. The third

group is composed of specific axioms for the notion of quasi-cardinality. As we

mentioned, this division is made only for greater convenience in exposition.

4.1 – Structural postulates

In this section we have three main goals: establish the main properties of the

distinct kinds of atoms, some of the properties of identity and indistinguishability, and

classify distinct kinds of collections according to the kinds of members they have. Our

main inspiration in providing the axioms for Q shall be ZFU (Zermelo-Fraenkel with

Ur-elements). Since we have two kinds of atoms and very different plans for them, we

begin by granting that nothing is both an m-atom and an M-atom. Also, a quasi-set, or

briefly q-set, is something that is not an atom, and atoms do not contain elements.

Q1) x(m(x) M(x)). (Nothing is both an m-atom and an M-atom)

Def. Q(x) =Def (m(x) M(x)). (x is a q-set)

Q2) xy(xy Q(y)). (Atoms have no elements)

Now, we begin the distinction between both kinds of atoms. The intended

interpretation for m-atoms is simple: m-atoms represent quantum particles, taken as the

Received View depicts them, that is, items with neither identity nor individuality. On

the other hand, M-atoms are well-behaved individuals, they have identity conditions and

are not indistinguishable but numerically distinct. To grant that state of affairs, we

define the identity relation as follows:

Def. w =E t =Def {[Q(w) Q(t) z(zw zt)] [M(w) M(t) z(wz

tz)]}. (Extensional identity sign =E)

According to that definition, two q-sets having the same elements are identical,

and two M-atoms belonging to the same q-sets are also identical. The first clause

ensures us that Q is extensional, the second one ensures us that M-atoms have identity

conditions. Nothing is said about m-atoms, so that they are officially items without

identity from now on. When there is no danger of confusion, we drop the index from the

identity sign, writing it simply as =.

To grant that identity has its first-order properties (reflexivity and the

substitution law), we have the following theorem and a postulate:

T1) If Q(x) or M(x), then x = x.

Q3) xy(x = y (A(x) A(y)), with A(x) a formula in which x occurs free, and

A(y) results from the substitution of some of the free occurrences of x by y in A(x),

with y free for x in A

The other properties of identity, such as symmetry and transitivity are derived

from T1 and Q3. As an easy result we also have the following:

T2) If M(x) and x = y, then M(y). If Z(x) and x = y, then Z(y).

Now we deal with the indistinguishability relation. The idea behind the

introduction of a special symbol for this relation is that m-atoms may enter in the

indistinguishability relation without being identical. Since m-atoms do not enter in the

identity relation but may be indiscernible, the distinction between both relations for

them may be kept. The only danger remaining is that indiscernibility is itself an identity

relation. Really, it seems reasonable to impose that indiscernibility should be reflexive,

symmetric and transitive. Given that if it is also compatible with every other relation,

then it will collapse into identity. That is, if it is granted that t w, for any property P, if

P(t) then P(w), then, formally identity and indistinguishability would collapse. We

prevent that from happening by avoiding that indistinguishability be compatible with

membership for m-atoms. That is, given x and y m-atoms, from xy and xz we are not

allowed to infer yz, and also from xy and zx we are not granted that zy. So,

identity and indiscernibility are distinct relations. It seems clear that this demand is not

unreasonable: just because two items are indiscernible we should not be allowed to infer

from the fact that one of them is an element of a q-set that the other one is also there.

However, for objects other than m-atoms, indistinguishability and identity should

coincide. Let us see the postulates:

Q4) The following formulas are axioms of Q:

x(xx) (reflexivity);

xy(xy yx) (symmetry);

xyz((xy yz) xz) (transitivity).

To accomplish our project of making identity and indiscernibility equivalent for

items other than m–atoms, we shall introduce some notation. As we mentioned before,

Z should represent the sets in Q, that is, copies of classical ZFU sets in quasi-set theory.

Together with M-atoms, we shall call items satisfying the predicate Z the “classical

objects” of Q. Following Zermelo, we shall briefly refer to them as the Dinge of our

theory, the classical things:

Def. D(x) =Def M(x) Z(x). (x is a Ding)

Q5) Dxy(xy → x = y).

In Q5 we use relativized quantifiers. That is, we restrict the scope of application

of a quantifier to a given property. In general, given a formula F, if we want to restrict

ourselves to objects having the property designed by that property we write simply

FxB. This expression abbreviates the formula x(F(x) B). The same holds for

existential quantifiers: FxB abbreviates x(F(x) B). So, Q5 plus the theorems that

follow mean that for Dinge identity and indistinguishability are equivalent.

T3) If D(x) and x = y, then xy.

T4) If M(x) and xy, then M(y). If Z(x) and xy, then Z(y).

Now, to prove that indistinguishability is compatible with the predicate Q (for q-

sets) we must adopt also the following postulate:

Q6) xy(m(x) xy m(y)).

T5) If Q(x) and xy, then Q(y).

So, until now we have that the indistinguishability relation is compatible with M,

m, Z and Q. However, indistinguishability is not compatible with membership; hence

we have granted that it is not a disguised relation of identity.

As our third goal, we now establish some basic properties of a very special kind

of collections, those satisfying the predicate Z. Our desire is to reproduce in Q all the

mathematics available in classical set theories. We have already begun to determine a

“classical part” of Q, comprising those items satisfying M and Z. Now, we determine

that the sets in our theory have no m-atoms as elements in their transitive closure, with

this concept understood in the classical way. That is, the elements of a set are classical

things; the elements from the elements are classical things too, and so on, never meeting

m-atoms in the process. First we grant that sets are q-sets.

Q7)x(Z(x) Q(x));

Q8) Qx(y(yx D(y)) Z(x)).

4.2 – Existential postulates

We have not postulated the existence of any particular q-set yet. Obviously, as

we have already established, some q-sets will be sets, and their elements are classical

things. Some q-sets will have both classical things as elements as well as m-atoms and

q-sets that are not sets. A third category will comprise those q-sets having only m-atoms

as elements. Let us establish some general facts in this section.

Our first q-set is the empty one.

Q9) Qxy(yx).

As a theorem, we have:

T6) There is only one empty q-set, and it is in fact a set.

This theorem allows us to introduce the name “” for the empty q-set.

In Q we have a version of the separation scheme: given a formula F(x) with x

free and in which y does not occur free, the following is an axiom schema:

Q10) QzQyx(xy xz F(x)).

We denote the q-set y postulated in Q10 by [xz: F(x)], and when we know it is

a set, we write simply {xz: F(x)}. Our next axiom is the union axiom:

Q11) QxQyz(zy Qw(wx zw)).

This q-set is denoted by x. For the introduction of the power set axiom we

define first the notion of a sub-qset:

Def. x y =Def z(zx zy). (sub-qset)

Q12) QxQyz(zy zx).

We introduce the symbol (x) for the power set of x. Now, to postulate the

existence of unordered pairs we first grant that for any items x and y there is another q-

set having both as elements:

Q13) xyQz(xz yz).

Obviously, in Q13 we do not grant that only x and y belong to z. Applying the

separation scheme to z with F(w): wx wy, we obtain the q-set [wz: wx wy].

We shall denote that q-set by [x, y]z. If x and y are indiscernible, we write simply [x]z.

For classical elements, the usual notation {x,y} is employed. It should be noted that if z

has elements other than x and y but indistinguishable from any one of these two

elements, then the unordered pair may contain more than only two elements. That is, if

for example z contains m-atoms indistinguishable from x, then, the unordered pair of x

and y will have those m-atoms as elements too. That is the reason why we have kept an

index z to the pair: the q-set obtained is a collection of all the indistinguishable from x

or from y relatively to z.

The definition of unordered pair had to follow along the lines discussed in the

previous paragraph for a very simple reason: we cannot simply employ the usual

formula postulating the existence of a collection having as elements the things identical

to x or identical to y. In fact, as it is easy to notice, this definition would forbid the

existence of pairs of m-atoms. So, to keep the treatment as general as is possible, we

have employed indistinguishability in the definition of pair. When we are dealing with

classical objects, since identity is equivalent to indistinguishability, we are back to the

classical case.

The ordered pair is introduced through the notion of unordered pair as follows:

Def. <x, y>z =Def [[x]z, [x,y]z]((z))

The fundamental property of ordered pairs, granting that if <x, y>z = <u, v>z

then x=u and y=v cannot be formulated in Q with full generality. Since we want to

make ordered pairs with m-atoms too, identity is not allowed between the members of

the pairs. To generalize the fundamental property, we will have to show that, granted

the indistinguishability of the pairs <x, y>z and <u, v>z, then xu and yv. We shall

prove that result later (see theorem 18).

We may also introduce the concept of Cartesian product between two q-sets:

Def. u v =Def [<x, y>AB ((AB)): xA yB]

From the notion of ordered pair and Cartesian product one may obtain easily the

notion of a binary relation:

Def. R =Def [<u, v>xy : ux vy] (R is a binary relation between x and y)

A theory of relations could be developed from that definition, dealing with

equivalence relations, order relations among others, but some restrictions should be

observed, mainly in relation to order relations. Notions such as trichotomy, anti-

symmetry and others that employ the notion of identity may not work generally, when

m-atoms are being related. The problem with specifying order for m-atoms comes

already from ordered pairs of m-atoms, since it does not seem to make much sense to

say that one m-atom comes before another one indistinguishable from it. We shall have

that vague idea turned into precise results in a few moments (in corollary to theorem

19).

Now we come to functions. Since we have no identity for m-atoms, we cannot

define with full generality a function as a relation R such that if <x, y>R and <x,

z>R, then y=z. To grant that our definition applies for everything in the domain of Q,

we shall adopt a simple strategy: the idea of a function is to associate members of a

domain uniquely to members of a co-domain. Since some of these members may be

indistinguishable, we shall require that whenever <x, y>R and x is indistinguishable

from w, then if <w, z>R, then y is also indistinguishable from z. That is, we associate

indistinguishable elements of the domain to indistinguishable elements in the co-

domain, and we call the resulting concept a “quasi-function”:

Def. f is a quasi-function from A to B if and only if f is a quasi-relation between A and

B such that for every uA there is a vB such that <u, v>f, and if <w, z>f and uw,

then vz.

If all the items involved are classical then the definition gives us the usual

definition of function, since identity and indistinguishability are equivalent. Further

specifications of the notion of quasi-function, such as an injective and surjective quasi-

function shall be defined only after we have introduced the notion of quasi-cardinal. The

usual definition of injection, for example, will not work for q-sets having m-atoms.

Really, suppose we have a q-set with 3 elements, taken intuitively, such that two of

them are indiscernible among themselves, but discernible from the third, and another q-

set with only two elements, discernible from each other. A quasi-function could be

defined coordinating the two indistinguishable elements of the first q-set to one of the

elements of the other q-set, and pairing the remaining element of the first q-set with the

remaining element of the second q-set. In this case, there is some kind of one-to-one

attribution, but it would not be possible to say we are mapping the first q-set into the

second one, since it has more elements. To remedy those cases, the concept of quasi-

cardinal is employed in the definition. But that will have to wait for the next section.

We finish the exposition of our official axioms in this section with an axiom of

infinity and the axiom of regularity:

Q14) Qx(x y(yx y[y]xx))

Q15) Qx(x yz(zx (zy))

As a matter of completeness, we should also add the axiom of choice and the

axiom scheme of replacement. We do not include them officially now, but we announce

both as follows:

(AC) For any q-set A, for every non-empty q-set B such that B A, there exists a

choice quasi-function f such that f(B) A.

The axiom of replacement must be assumed for the translation in the next

section to be adequately made. That is, in the next section we shall estipulate a

translation from the language of ZFU to the language of Q so that every axiom of ZFU

should be a theorem of Q. Since the scheme of replacement is an axiom of ZFU, when

translated to the language of Q we shall have the required version of the axiom in Q in

order to prove the corresponding theorem. To introduce replacement, we begin with a

formula A(x,y) with x and y free variables. This formula expresses a y-functional

condition over a q-set t if the following holds:

w(w t → sA(w,s) ˄ wu (w t ˄ u t → ss'(A(w, s) ˄ A(u, s') ˄ wu →

ss'))).

We abbreviate this formula by x!yA(x,y). The axiom of replacement is given by the

following scheme:

(Rep) x!yA(x,y) → Q u Q v (z(z v → w( w u ˄ A(w,z)))).

4.3 – Cardinals and quasi-cardinals

Now we begin our journey to determine the postulates for the symbol qc, a

generalization of the notion of cardinal for a quasi-set. As we mentioned before, non-

individuals are being understood as entities for which identity and difference do not

make sense. That idea is already captured by the fact that identity is not defined for m-

atoms. Non-individuals may also be indiscernible, a feature represented in Q by the

adoption of a primitive relation of indistinguishability. The final main characteristic

possessed by non-individuals is that they may be aggregated in collections having a

well-defined cardinal. We shall achieve that in this section through the concept of quasi-

cardinal.

The main reason for us to introduce a primitive notion qc in our language and

provide specific axioms for it is directly related to the fact that some of the collections

may contain m-atoms. In that case, as we mentioned before, the definition of some

kinds of order relations is no longer possible as it is usually done for classical sets, in

particular, well-ordered relations are not definable. In that case, one cannot attribute an

ordinal to any q-set containing m-atoms as elements. As a consequence, the usual

definition of cardinal numbers as a specific ordinal is no longer available for us.

One may look for alternative definitions, in particular, for one not stated in terms

of the concepts of identity and of bijection (another concept still not available for us).

For finite collections, Domenech and Holik have furnished one alternative (see their

(2007)). The idea, briefly stated, is that one may begin with a q-set and, through a

process of elimination of its elements, try to make it empty in a finite number of steps.

The “elimination” of the elements is performed in such a way that only operations not

presupposing the very notion of cardinal are employed, and also, at an intuitive level it

is clear that only one element is being taken at each time. If after a finite number of

steps the q-set is empty, then, the number of times we repeated the procedure of

eliminating its elements is the number of elements the q-set had in the beginning.

This formal procedure is built in such a way as to mirror an analogous procedure

which may (at least in principle) be performed in a laboratory. Suppose we want to

count how many electrons a Helium atom has. Following the idea described in the last

paragraph, we may put the atom in a cloud chamber and use radiation to ionize it. The

result is the track of an electron and the track of an ion. The track of the electron

represents a system of only one element. If we ionize the atom once again, we shall see

the track of one electron and the track of an ion, this time with charge 2e. Since we can

extract no more electrons from the atom, we know it had only two electrons. The idea is

that this process does not rely on the identity of the electrons, that is, we do not need to

have a well-defined identity relation holding among them to conduct the mentioned

process. Nowhere in the procedure described were we required to determine which

electron was eliminated first, which was eliminated second.

Besides this alternative account, one may follow the route we are taking here,

and assume that cardinality is a primitive notion. Our purpose is the following: through

a convenient translation of the language of ZFU in the language of Q we shall show that

every axiom of ZFU is itself a theorem of Q. With that in hand, we may develop inside

quasi-set theory a “copy” of ZFU, so that the notion of a cardinal number may be

defined in the usual way inside this copy. Then, through the notion of qc we shall

attribute cardinals to every quasi-set, so that classical sets will have as quasi-cardinals

their cardinals defined in the classical part of Q, and the other q-sets will have cardinals

attributed according to some reasonable postulates. Let us see.

Suppose ZFU has as usual first-order logic with identity as its underlying logic,

a binary membership relation and a monadic predicate C, such that C(w) means that w

is a set. We now define the desired translation from the formulas of ZFU to the formulas

of Q through a function t given by the following clauses (see French e Krause (2006),

chap. 7):

1) If A is C(x), then t(A) is Z(x);

2) If A is x = y, then t(A) is ((M(x) M(y)) (Z(x) Z(y)) x =E y);

3) If A is xy, then t(A) is ((D(x) Z(y)) xy);

4) If A is B, then t(A) is t(B);

5) If A is B C, then t(A) is t(B) t(C);

6) If A is xB, then t(A) is x((D(x) t(B)).

The desired theorem follows:

T7) If A is an axiom of ZFU, t(A) is the translation described above, then, if A is an

axiom from ZFU, t(A) is a theorem in Q.

One proves this theorem by applying t to the axioms of ZFU and showing that

they are theorems of Q. With that result in hands, we are then able to introduce some

further concepts by definition: i) card(x) is the cardinal of a set x (i.e., an object

satisfying Z); ii) Cd(x) means that x is a cardinal; iii) Fin(x) means that x is a finite

collection. Notice that all the classical mathematics developed in ZFU is now directly

available also in Q.

4.4 – Specific postulates for qc

Now we begin the presentation of the postulates for qc. As we mentioned, every

quasi-cardinal is a cardinal, and quasi-cardinal attribution respects the classical

definition of cardinality, in the sense that for classical objects, cardinality and quasi-

cardinality coincide. Our first postulate states precisely that:

Q16) x!y(Cd(y) qc(x) = y (Z(x) y = card(x))).

An immediate theorem follows:

T10) qc() = 0.

As a result of Q16, one should notice that atoms also have a quasi-cardinal. To

make sure that this is compatible with our view that only q-sets have elements, then, we

grant that atoms have quasi-cardinal zero:

Q17) x(Q(x) qc(x) = 0).

Also, we shall require that a proper sub-qset x of a q-set y should have a smaller

quasi-cardinal than y:

Q18) Q xQy(x y qc(x) qc(y)).

As a theorem, we have the following:

T11)Q x(x E qc(x) E 0).

Notice that the theorem is relative to q-sets. It does not grant in general that if

something has quasi-cardinal 0 then it is the empty q-set (for atoms are also empty).

Other simple theorems follow:

T12) If x = y, then qc(x) = qc(y).

T13) Q xQ y(x y qc(x) qc(y)).

T14) If x y and y x, then qc(x) = qc(y).

T15) Q xQ y(Fin(x) x y qc(x) qc(y)).

Now, we grant that for any cardinal k smaller than the cardinal of a q-set x there

exists a sub-qset y of x having k as its quasi-cardinal. Here, and denote ordinal

numbers:

Q19) Q x(qc(x) = ( Q y(y x qc(y) = ))).

Now, we generalize to quasi-cardinals two simple facts of cardinal arithmetic:

Q20) Q xQ y(w(w x w y) qc(xy) = qc(x) + qc(y)).

Q21) Q x(qc((x)) = 2qc(x)

)

Now, before we proceed, since we already have the notion of quasi-cardinal in

hands and we know that it has inherited from the classical part of Q an order relation

among quasi-cardinals, we introduce definitions of injective, surjective and bijective

quasi-functions:

Def. A quasi-function f is injective if the following condition is satisfied:

xx’yy’(<x, y> f <x’, y’> f y y’ → x x’) qc(dom(f)) qc(Ran(f)).

That is, indiscernible items in the range must be associated to indiscernible items

in the domain, and the domain must be at most as big as the range. This captures the

idea that the elements of the domain are being taken into the elements of the co-domain.

Def. A quasi-function f between D and C is surjective if the following condition is

satisfied:

y(y C → x(x D <x, y> f)) qc(C) qc(D).

This illustrates the idea that every element in the co-domain has its respective

correspondent in the domain. The last clause on quasi-cardinals is there to ensure that

every element of the co-domain will be mapped by something in the domain.

A bijection is a quasi-function that is an injection and a surjection. In that case,

the cardinals of both domain and co-domain are equal. That fact, however, is a

consequence of the definition, which itself employs the quasi-cardinal concept and

cannot be used to define cardinality as it does in the classical approach.

4.4.1 – Weak extensionality

Now we are in a position to state one of the most important axioms of Q, which

formalizes one of the main properties of the indistinguishability relation: the weak

extensionality axiom. Intuitively, it says that two q-sets are indistinguishable when they

have exactly the same kinds of elements in the same quantity. Before writing that

statement in the language of Q, we shall state two previous auxiliary definitions: two q-

sets x and y are similar if all their elements are indistinguishable among themselves.

Further, x and y are quasi-similar if besides being similar, they have the same quasi-

cardinal. Formally, we have the following:

Def. Sim(x, y) =Def zw((z x w y) z w). (x and y are similar)

Def. Qsim(x, y) =Def Sim(x, y) qc(x) = qc(y). (x and y are quasi-similar)

In the following statement, x\ denotes an equivalence class of x by the

indistinguishability relation. Weak extensionality is the following axiom:

Q22) QxQy(z(z x\ w(w y\ qsim(z, w))) w(w y\ z(z x\

qsim(w, z))) x y).

As we mentioned, the axiom grants that q-sets x and y are indistinguishable if

they have the same quantity of indistinguishable elements. The following theorems are

then derivable from Q22:

T16) If x y and y x, then x y.

T17) (i) Q xQ y(Sim(x,y) qc(x) =E qc(y) x y).

(ii) xy qc([x]z) =E qc([y]w) [x]z [y]w.

Theorem 17 helps us proving the generalized property of ordered pairs and other

interesting results:

T18) If <x, y>z <u, v>w then xu and yv.

T19) Given x, y, z, w belonging to A, if xz and yw, then <x,y>A<z,w>A.

Corollary to T19) Given x, y A, if xy then <x,y>A<y,x>A.

This last corollary is a formal version of the informal remark made before in

which we stated that m-atoms could not be ordered. The idea is that if items are

indistinguishable, one cannot distinguish which of them comes first and which comes

second.

Our next goal is to state in Q a theorem to the effect that indistinguishable items

are permutable without giving rise to any kind of distinction. As we mentioned in the

introduction, this is one of the characteristics that has been taken as most relevant in the

proposal of a metaphysics of non-individuals. Before that, we need to define the concept

of a strong singleton of an item. Given any object x, we can obtain in Q a q-set

containing only one element such that this element is indistinguishable from x. In the

case of an m-atom, it is impossible to prove that this element belonging to the strong

singleton is really x; if x is a classical element, however, then the strong singleton

reduces to the classical singleton {x}.

Def. <x> =Def y such that y [x] qc(y) =E 1. (<x> is the Strong singleton of x)

The motivation behind the next theorem, as we mentioned, is the idea that non-

individuals may be permuted without changing the state of the system. In Q this state of

affairs is obtained by permuting one element in a q-set by another indistinguishable

from it; the resulting q-set, as expected, is indistinguishable from the original

unpermuted one.

T20) [Permutations are not observable] Let x be a q-set, z a m-atom such that z x.

Given w such that w x and wz, then (x\<z>)<w>x.

5. Further Developments

So, after the development of quasi-set theory, what else can we do to give the

metaphysics of non-individuals as conceived here further support? One line in which

the research may follow consists in providing a modification of Q to account for

collections without a well-determined cardinal. Systems with those features appear in

quantum field theory, and it would be interesting to try to formalize them for rigorous

study (see further discussions and suggestions in Domenech and Holik (2007)). That

would involve deep modifications in the structure of the theory, since even the weak

extensionality axiom is formulated employing the concept of quasi-cardinal in an

essential way.

Another kind of development already undertaken consists in assuming non-

individuality as formalized in quasi-set theory and, with some further resources of Q, try

to develop quantum mechanics making essential use of non-individuality. That

approach has been followed by Domenech, Holik and Krause (2008). The idea is that

one can, by using collections of m-atoms, build inside Q a mathematical structure

analogous to Fock spaces, in which quantum mechanics may be developed. In their

construction, the authors have not made use of labels or other features which could be

associated to individuality. So, this construction lies essentially on the non-individuality

as formalized by m-atoms, and we have a quantum mechanical treatment of non-

individuals.

One could go on and explore even further the formalization of the notion of non-

individuality. Our proposal here was not to impose one particular ontological

interpretation of quantum mechanics as right, but rather to show that it makes perfect

sense and may be dealt with in a rigorous fashion. Just as many other novelties of

quantum mechanics have brought revisions in many areas of our traditional knowledge

(just remember the discussions on determinism, locality, Bell’s theorem…), we believe

that ontology may also benefit from that theory and have new areas of exploration

waiting for the philosopher.

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