Elementary Particles, Hidden Variables, And Hidden Predicates

ADONAI S. SANT’ANNA

ELEMENTARY PARTICLES, HIDDEN VARIABLES, AND HIDDENPREDICATES∗

ABSTRACT. We recently showed that it is possible to deal with collections of indistin-guishable elementary particles (in the context of quantum mechanics) in a set-theoreticalframework, by using hidden variables. We propose in the present paper another axiomaticsfor collections of indiscernibles without hidden variables, where hidden predicates areimplicitly assumed. We also discuss the possibility of a quasi-set theoretical picture forquantum theory. Quasi-set theory, based on Zermelo-Fraenkel set theory, was developedfor dealing with collections of indistinguishable, but, not identical objects.

1. INTRODUCTION

We begin by considering that it is necessary to settle some philosophicalterms in order to avoid confusion. When we say thata andb areidenticals,we mean that they are the verysameindividual, that is, there are no ‘two’individuals at all, but only one which can be named indifferently by eithera or b. By indistinguishabilitywe simply mean agreement with respect toattributes. We recognize that this is not a rigorous definition. Neverthelesssuch an intuition is better clarified in the next section.

In physics, elementary particles that share the same set of state-independent (intrinsic) properties are usually referred to asindistinguish-able. Although ‘classical particles’ can share all their intrinsic properties,there is a sense in saying that they ‘have’ some kind ofquid which makesthem individuals, since we are able to follow the trajectories of classicalparticles, at least in principle. That allows us to identify them. In quantumphysics that is not possible, i.e., it is not possible,a priori, to keep track ofindividual particles in order to distinguish among them when they share thesame intrinsic properties. In other words, it is not possible to label quantumparticles.

The problems regarding individuality of quantum particles have beendiscussed in recent literature by several authors. A few of them areDalla Chiara and di Francia (1993), Krause (1992, 1996), Mandel (1991),Sakurai (1994), Sant’Anna and Krause (1997), Schrödinger (1952), vanFraassen (1991), Redhead and Teller (1991). Many intricate puzzles on the

Synthese125: 233–245, 2000.© 2000Kluwer Academic Publishers. Printed in the Netherlands.

234 ADONAI S. SANT’ANNA

logical and philosophical foundations of quantum theory have been raisedfrom these questions. For instance, there is the possibility that the collec-tions of such entities may be not considered as sets in the usual sense. Yu.Manin (1976) proposed the search for axioms which should allow to dealwith collections of indistinguishable elementary particles. Other authors(Dalla Chiara and di Francia 1993; Krause 1996), have also consideredthat standard set theories are not adequate to cope with some questionsregarding microphysical phenomena. These authors have emphasized thatthe ontology of microphysics apparently does not reduce to that one ofusual sets, due to the fact thatsetsare collections of distinct objects.

Quasi-set theory, based on Zermelo-Fraenkel set theory, was developedfor dealing with collections of indistinguishable but, not identical ob-jects (Krause 1996). Hence, quasi-set theory provides a mathematicalbackground for dealing with collections of indistinguishable elementaryparticles as it has been shown in (Krause 1996). In that paper, it hasbeen shown how to obtain the quantum statistics into the scope of thisnon-standard approach.

Nevertheless, it has been recently proposed that standard set theoryis strong enough to deal with collections ofphysically indistinguishablequantum particles (Sant’Anna 1997), if we use some sort of hidden vari-able formalism. In the present paper we make a comparison between thehidden variables formalism proposed in (Sant’Anna 1997) and quasi-settheory.

Section 2 presents the hidden variable picture. In Section 3 the hiddenpredicates formalism for elementary particles is introduced. In Section 4we discuss very briefly some points about quasi-set theory. In the nextsection we show how to derive quantum statistics in the hidden predicateformalism. Finally, in the last section, we discuss some related lines ofwork.

2. THE HIDDEN VARIABLES FORMALISM

Here we intend to show that it is possible to distinguish, at least in prin-ciple, among particles that are ‘physically indistinguishable’, where by‘physically indistinguishable’ particles we mean, roughly speaking, thoseparticles which share the same set of measurement values for their in-trinsic properties.1 In a previous work (Sant’Anna 1997), we assumedthat ‘physicallyindistinguishable particles’ are those particles which sharethe same set of measurement values for a correspondent complete set ofobservables. It seems clear that such a modification simplifies our concep-tual framework and it is still related to the usual understanding about the

ELEMENTARY PARTICLES, HIDDEN VARIABLES, AND HIDDEN PREDICATES 235

meaning of (physical) indistinguishability. A kind of distinction is possibleif we associate to each particle an ordered pair whose first element is thementioned set of measurement values of the intrinsic properties and thesecond element is a hidden property (a hidden variable) which intuitivelycorresponds to something which was not yet measured in laboratory. This‘hidden property’ does assume different values for each individual particlein a manner that it allows us to distinguish those particles which are inprinciple ‘physically’ indistinguishable. Obviously, such a hidden propertyseems to have a metaphysical nature. Indeed, our proposed hidden variablehypothesis does have a metaphysical status. This is the kind of metaphysicsthat we advocate. The ‘reasonable’ metaphysics should be that one whichcould provide a hope for a future new physics. This future new physicsmay correspond to more extended physical systems that are not, until now,measured in laboratories.

As remarked above, our concern here is only with the process oflabeling physically indistinguishable particles. So, although we are notinterested in describing here an axiomatic framework for quantum physics,quantum mechanics or even mechanics, we expect that our approach canbe extended in order to encompass them. All that follows is performed ina standard set theory like Zermelo-Fraenkel withUrelemente(ZFU).

Our picture for describing indistinguishability issues in quantum phys-ics is a set-theoretical predicate, following P. Suppes’ ideas about axiomat-ization of physical theories (Suppes 1967).

Hence, our system has five primitive notions:λ, X, P , m, andM. λ isa functionλ : N → <, whereN is the set{1,2,3 . . . , n}, n is a positiveinteger, and< is the set of real numbers;X andP are finite sets;m andM are unary predicates defined on elements ofP . Intuitively, the imagesλi of the functionλ, wherei ∈ N , correspond to our hidden variables. Wedenote by3N the set of allλi, whereI ∈ N . X is a set whose elementsshould be intuitively interpreted as sets associated to measurement val-ues of the intrinsic or state-independent properties like rest mass, electriccharge, absolute value of spin, etc. The elements ofX are denoted byx,y, etc.P is to be interpreted as a set of particles. One of the axioms givenbelow states that the elements ofP are ordered pairs whose first elementsbelong toX and whose second elements belong to3N . When there isno risk of confusion we denote the elements ofP asp, q, r, etc.m(p),wherep ∈ P , means thatp is a microscopic particle, or a micro-object.M(p) means thatp ∈ P is a macroscopic particle, or a macro-object.Actually, the distinction between microscopic and macroscopic objects, asmentioned here, does not reflect, at least in principle, the great problem ofexplaining the distinguishability among macroscopic objects, since these


are composed of physically indistinguishable things. As it is well known,Schrödinger explained that in terms of aGestalt(Schrödinger 1952). Nev-ertheless, this still remains as an open problem from the foundational(axiomatic) point of view. The following is a set-theoretical predicate fora system of ontologically distinguishable particles. We use the symbol ‘=’for the standard ‘equality’, and the symbols ‘⇒’, ‘∧’, ‘∨’, and ‘¬’ for thelogical connectives ‘if . . . then’, ‘and’, ‘or’, and ‘not’, respectively.

DEFINITION 2.1.DHV = 〈λ,X,P,m,M〉 is a system ofontologicallydistinguishable particles, abbreviated asDHV -system, if and only if thefollowing six axioms are satisfied:

λ : N → < is an injective function, whose set of images isdenoted by3N .

(HV1)

P ⊂ X×3N . The elements ofP are denoted by〈x, λi〉, wherex ∈ X andλi ∈ 3N .

(HV2)

DEFINITION 2.2.〈x, λi〉 .= 〈y, λj 〉; if, and only if; x = y.

DEFINITION 2.3. If p ∈ P andq ∈ P , we say thatp is ontologicallyindistinguishablefrom q if, and only if, p = q, where= is the usualequality between ordered pairs.

The usual equality among ordered pairsp = 〈x, λi〉 ∈ P is a binaryrelation which corresponds to our ontological indistinguishability betweenparticles, while

.= is another binary relation which corresponds to thephysical indistinguishability between particles.

Two particles are ontologically indistinguishable if and only if theyshare the same set of measurement values for their intrinsic physical prop-erties and the same value for their hidden variables. Definition 2.2 says thattwo particles are physically indistinguishable if, and only if, they share thesame set of measurement values for their intrinsic (physical) properties.

(∀x, y ∈ X)(∀λi ∈ 3N)((〈x, λi〉 ∈ P ∧ 〈y, λi〉 ∈ P) ⇒ x =y).

(HV3)

(∀p, q ∈ P)(M(p) ∧M(q)⇒ (p.= q ⇒ p = q)).(HV4)

(∀p, q ∈ P)(p .= q ∧ ¬(p = q)⇒ m(p) ∧m(q)).(HV5)

(∀p ∈ P)((m(p)∨M(p)) ∧ ¬(m(p)∧M(p)).(HV6)


Axiom (HV1) allows us to deduce that the cardinality of3N coin-cides with the cardinality ofN (#3N = #N). Axiom (HV2) just saysthat particles are represented by ordered pairs,2 where the first elementintuitively corresponds to measurement values of all the intrinsic phys-ical properties, while the second element corresponds to the hidden innerproperty that allows us to distinguish particles at an ontological level.Yet, axioms (HV2) and (HV3) guarantee that two particles that share thesame values for their hidden variable are the very same particle, since ourstructure is set-theoretical and the equality= is the classical one. Axiom(HV4) says that macroscopic objects that are physically indistinguishable,are necessarily identicals. Axiom (HV5) says that two particles physicallyindistinguishable that are not ontologically indistinguishable (they are on-tologically distinguishable) are both microscopic particles. Axiom (HV6)means that a particle is either microscopic or macroscopic, but not both.

Axiom (HV4) deserves further explanation. It entails that (ontologic-ally) distinct macro-objects are always distinguished by a measurementvalue; if two particles are macro-objects, then there exists a value for ameasurement which distinguish them. Then, macro-objects, in particular,obey Leibniz’s Principle of the Identity of Indiscernibles and we may saythat (according to our axiomatics) classical logic holds with respect tothem while micro-objects may be physically indistinguishable without thenecessity of being ‘the same’ object.

In (Sant’Anna and Krause 1997) the axiomatic framework for a systemof ontologically distinguishable particles is a bit different from the presentformulation. The main difference is on axiom (D5), which does not existin (Sant’Anna and Krause op. cit.). Such an axiom is necessary to provesome results which may be found in (Sant’Anna 1997).

We discuss in (Sant’Anna and Krause 1997) how our approach is outof the range of the well known proofs on the impossibility of hidden vari-ables in the quantum theory, like von Neumann’s theorem, Gleason’s work,Kochen and Specker results, Bell’s inequalities or other works where itis sustained that no distribution of hidden variables can account for thestatistical predictions of the quantum theory.

3. HIDDEN PREDICATES

The ontological interpretation for hidden variables given above suggeststhat indistinguishability in quantum theory is just relative to certain prop-erties usually referred to as intrinsic. Thus, in this section we show that itis possible to deal with collections of indistinguishable particles withoutany explicit reference to hidden variables or hidden properties.


Our system has four primitive notions:P , X, m, andM. P is a finiteset whose elements areUrelemente. Intuitively, P corresponds to the setof particles.X is a family of unary predicates{xi}i∈I , whereI is a non-empty set. Such predicates are defined on elements ofP . When we saythat an elementp of P satisfies the predicatexi , i.e., xi(p), we meanby that that particlep has the intrinsic propertyxi . By intrinsic proper-ties we mean the state-independent properties of particles, like rest mass,charge, absolute value of spin, etc.m andM are unary predicates definedon elements ofP .m(p) (M(p)) corresponds to say thatp is a microscopic(macroscopic) particle, or a micro-object (macro-object). The followingis a set-theoretical predicate for a system of ontologically distinguish-able particles. We use the symbol ‘=’ for ‘equality’ and the same logicalnotation which appears in the previous section.

DEFINITION 3.1. DHP = 〈P,X,m,M〉 is a system ofontologicallydistinguishable particles, abbreviated asDHP -system, if and only if thefollowing eight axioms are satisfied:

P is a finite non-empty set whose elements areUrelemente. Weinterpret suchUrelementeas elementary particles.

(HP1)

X = {xi}i∈I is a family of unary predicates defined on elementsof P , whereI is a non-empty set. The intuitive meaning ofxi(p) is: particlep does have an intrinsic property (predicate)xi .

(HP2)

m andM are unary predicates defined on elements ofP , whichdo not belong to the familyX.

(HP3)

Every particlep ∈ P satisfies at least one intrinsic property.(HP4)

(∀p∃i)((p ∈ P ∧ i ∈ I )⇒ xi(p)).

DEFINITION 3.2. We say that particlep is ontologically indistinguishablefrom particleq if p = q.

DEFINITION 3.3. We say that particlesp and q are physically indis-tinguishable, and denote it byp

.= q, if there existsi ∈ I such thatxi(p) ∧ xi(q).

Definition 3.3 justifies our previous reference to hidden predicates. Ourmathematical framework allows the existence of two particles inP which


are indistinguishable with respect to a setX of unary predicates. Neverthe-less, since they aretwoparticles, then there should exist a hidden predicate(in the sense that it does not belong toX) which allows to distinguish bothparticles.

Each particle satisfies just one predicate of the familyX ofintrinsic properties.

(HP5)

(∀i∀j∀p)((i ∈ I ∧ j ∈ I ∧ p ∈ P ∧ xi(p) ∧ xj (p))⇒ i = j).

Macroscopic particles that are physically indistinguishable, arenecessarily identicals.

(HP6)

(∀p∀q)((p ∈ P ∧ q ∈ P ∧M(p) ∧M(q))⇒ (p

.= q ⇒ p = q))).Two particles physically indistinguishable that are ontologic-ally distinguishable (i.e., they are not ontologically indistin-guishable) are both microscopic particles.

(HP7)

(∀p∀q)((p ∈ P ∧ q ∈ P ∧ p .= q ∧ ¬(p = q))⇒ m(p) ∧m(q)).

Any particle is either microscopic or macroscopic, but not both.(HP8)

(∀p)(p ∈ P ⇒ (m(p)∨M(p)) ∧ ¬(m(p)∧M(p)).Axiom (HP2) deserves some additional explanations. Our axiomatics

does make sense (intuitively) when we conveniently interpret the unarypredicates from the familyX. For instance, since each particle shouldsatisfy at least one predicate fromX (axiom (HP4) and just one (axiom(HP5)), it is not convenient to say thatxi(p) means that electronp doeshave an electric charge 1e. It is rather convenient to say that electronp

does have an electric charge 1e, a rest mass 0.5110 MeV and spin12, since

muons, for example, share the same intrinsic properties of electrons, exceptthe rest mass. Usually it is considered that the only intrinsic propertiesof an electron are its rest mass and quantum numbers. Here we allowthe possibility of existence of some hidden predicate which permits theidentification of elementary particles (axiom (HP7)) but which does notbelong to the setX.


Axiom (HP6) entails that (ontologically) distinct macro-objects arealways distinguished by intrinsic properties; if two particles are macro-objects, then there exists an intrinsic property which distinguishes them.Then, macro-objects, in particular, obey Leibniz’s Principle of the Iden-tity of Indiscernibles and we may say that (according to our axiomatics)classical logic holds with respect to them while micro-objects may bephysically indistinguishable without the necessity of being ‘the same’object.

The main advantage of this picture, if we compare it to that one presen-ted in the section above, is that it does not make any explicit reference tohidden variables. If particles are physically indistinguishable in the senseof Definition 3.3, we mean by that that they are indistinguishable withrespect to a given predicatexi . Since we are talking about a set-theoreticalpredicateDHP , it is quite clear that two physically indistinguishableparticles should be individualized with respect to a ‘hidden’ predicatewhich does not belong toX. Nevertheless, we say that if a given predicatebelongs toX, then it does have a physical meaning, otherwise it does not.

Another advantage of our picture is that it recognizes, in an explicitmanner, the limits of physics. Physicists cannot claim that they knowallintrinsic properties of elementary particles. The predicates that belong toX just refer to those intrinsic properties that physicists know about.

One logical consequence from these axioms is that every particle phys-ically indistinguishable from a micro-object is also a micro-object, as itcan be seen in the following theorem:

THEOREM 3.1.(∀p∀q)(p ∈ P ∧ q ∈ P ∧m(p) ∧ p .= q)⇒ m(q).Proof. If p = q, the proof is trivial. For the case wherep 6= q, suppose

that¬m(q). Then, according to axiom (HP8),M(q), sinceq ∈ P . Butp.= q andp 6= q. Thus, according to axiom (HP7),m(p) ∧ m(q), which

contradicts the hypothesis¬m(q). Hence,m(q). 2THEOREM 3.2. IfX is a unitary set andP has more than one element (thecardinality ofP is greater than 1) then(∀p)(p ∈ P(m(p))).

Proof. If x1 is the only element ofX, then according to axiom (HP4),(∀p)(p ∈ P ⇒ x1(p)). Thus, from the Definition 3.3, we infer that(∀p∀q)((p ∈ P ∧ q ∈ P)⇒ p

.= q). Hence, if¬(p = q),m(p)∧m(q),according to axiomHP7. 2


4. QUASI-SET THEORY AND QUANTUM THEORY

Quasi-set theory is based on Zermelo-Fraenkel axioms and permits to copewith collections of indistinguishable objects by allowing the presence oftwo sorts of atoms (Urelemente), termedm-atoms andM-atoms (Krause1992, 1996). A binary relation of indistinguishability betweenm-atoms(denoted by the symbol≡), is used instead of identity, and it is postu-lated that≡ has the properties of an equivalence relation. The predicate ofequality cannot be applied to them-atoms, since no expression of the formx = y is a formula ifx ory denotem-atoms. Hence, there is a precise sensein saying thatm-atoms can be indistinguishable without being identical.

The universe of quasi-sets is composed bym-atoms,M-atoms andquasi-sets. The axiomatics is adapted from that of ZFU (Zermelo-Fraenkelwith Urelemente), and when we restrict the theory to the case whichdoes not considerm-atoms, quasi-set theory is essentially equivalent toZFU, and the corresponding quasi-sets can then be termed ‘ZFU-sets’. TheM-atoms play the role of theUrelementein the sense of ZFU.

In order to preserve the concept of identity for the ‘well-behaved’ ob-jects, anExtensional Equality(=E) is introduced for those entities whichare notm-atoms on the following grounds: for allx and y, if they arenot m-atoms, thenx =E y corresponds to say that∀z(z ∈ x ↔ z ∈y) ∨ (M(x) ∧M(y) ∧ x ≡ y). It is possible to prove that=E has all theproperties of classical identity.

In (Krause et al. 1999) a quasi-set theory for bosons and fermions ispresented. The authors obtain the quantum distribution functions (Fermi–Dirac and Bose–Einstein) and discuss the Helium atom. To use quasi-settheory for dealing with collections of quantum particles means that weconsider non-individuality right at the start and that is an assumption thatwe intend to avoid in this paper. As we remarked in the beginning of ourtext, our intention is to work on the possibility that elementary particlesmay be considered as individuals of some sort.

Nevertheless, even in quasi-set theory non-individuality may be natur-ally interpreted as an individuality which is somehow ‘veiled’.

5. QUANTUM STATISTICS

In order to obtain the quantum distribution functions in the usual manner,it is necessary to assume that quantum particles may be indistinguishable.In the case of fermions, it is also assumedPauli’s Exclusion Principle. Asit is well known, bosons do not satisfy such a principle.


Roughly speaking, Pauli’s Principle states that two or more fermionscannot occupy the same state. This happens because a state like| k′〉 | k′〉is necessarily symmetrical, which is not possible for fermions. States alsocannot be used to label fermions, since a fermion can change its state.In the case of bosons, the situation is more dramatic, since we may haveseveral bosons occupying the same single state. Hence, even if we takea collection of physically indistinguishable bosons or physically indis-tinguishable fermions, is it possible to find a reasonable way to expressthis into the framework of a standard set theory? In our mathematicalframework, the answer is positive.

We consider the fermion case as our first example to illustrate our ideas.To cope with a collection of fermions, we consider, as a first assumption, aDHP -system withX as a unitary set. According to Theorem 3.2, ifX is aunitary set in aDHP -system with more than one particle, then all particlesare microscopic. So, fermions are microscopic particles because they arephysically indistinguishable objects. Our second assumption is Pauli’s Ex-clusion Principle. But before that, we need to establish the meaning of thesymmetrical and the antisymmetrical states.

For the sake of simplicity, but with no loss of generality, we considera system of two physically indistinguishable particles, whereP = {p, q}.The state of particlep (q), in the Hilbert space formalism, is| k′, p〉 (|k′′, q〉). Hence, the state ket for the two particles system is

| k′, p〉 | k′′, q〉.(1)

If a measurement is performed on this system, it may be obtainedk′for one particle andk′′ for the another. But, in practice, it is not possible toknow if the state ket of the system is| k′, p〉 | k′′, q〉, | k′′, p〉 | k′, q〉 or anylinear combinationc1 | k′, p〉 | k′′, q〉 + c2 | k′′, p〉 | k′, q〉. This is calledthe exchange degeneracy, which means that to determine the eigenvalue ofa complete set of observables is not sufficient for uniquely specifying thestate ket.

Using a notation similar to Sakurai’s (?, 11)e define the permutationoperatorPpq by

Ppq | k′, p〉 | k′′, q〉 =| k′′, p〉 | k′, q〉.(2)

It is obvious thatPpq = Pqp and thatP2pq = 1. In the case of fermions:

Ppq | k′, p〉 | k′′, q〉 = − | k′, p〉 | k′′, q〉,(3)

or, in the more general situations:

Pij | n physically indistinguishable fermions〉(4)


= − | n physically indistinguishable fermions〉,wherePij is the permutation operator that interchanges the particlepi withthe particlepj , wherei andj are arbitrary but distinct elements of a finiteset.

In our picture, it is possible to count fermions, since we are working ina set-theoretical framework. So, we may deal with collections of fermionsas sets. It is also clear what it is meant by saying that a system of fermionsis totally antisymmetrical under the interchange of any pair, since now themeaning of the word ‘interchange’, according to Eq. (2) was made clear.Thus, we observe that, by Eq. (2),

Ppq | k′, p〉 | k′, q〉 =| k′, p〉 | k′, q〉,(5)

which contradicts Eq. (4). Hence, as expected, fermions cannot occupy thesame physical state, which is the exclusion principle in our language ofhidden predicates.

Since we have characterized the permutation operator, symmetrical andantisymmetrical states, Pauli’s Exclusion Principle and thelabeling ofindistinguishable quantum particles, we can easily deduce the quantumdistribution functions3 in a manner which does not differ from the usualone, from now on.

6. SOME RELATED QUESTIONS

The main point we want to make in this paper is that quasi-sets arenot necessary to describe the world of micro-phenomena, that is, theworld usually described by quantum physics. If collections of element-ary particles may be described by means of quasi-set theory, that doesnot mean that the quantum world is completely different from classical(macroscopic) world, at least from the mathematical point of view. Wemay describe the same physical phenomena by using a set-theoreticalframework.

Hence, we suggest the possibility of a complete classical picture formicroscopic phenomena. Some authors have proposed something like that.Bohmiam mechanics is a well known example of a semi-classical picturefor quantum mechanics (Bohm and Hilley 1994). Suppes and collabor-ators have also developed a particular description for some microscopicphenomena usually described by quantum physics (Suppes et al. 1996a,b).

One of the main points that makes quantum physics quite different fromclassical physics is the presence of nonlocal phenomena in the first one.


Nevertheless, it is usually considered that the interference produced bytwo light beams, which is a nonlocal phenomenon, is determined by boththeir mutual coherence and the indistinguishability of the quantum particlepaths. Mandel (1991), e.g., has proposed a quantitative link between thewave and the particle descriptions by using an adequate decomposition ofthe density operator. So, perhaps an adequate theory for the problem ofindistinguishability between elementary particle trajectories, in terms ofhidden variables, may allow us a classical picture for interference.

On the other hand, there is another nonlocal phenomenon, namely,Einstein–Podolsky–Rosen (EPR) experiment, which entails some fascin-ating results like teleportation (Watson 1997). If we take very seriouslythe non individuality of quantum particles, space-time coordinates cannotbe used to label elementary particles. Nevertheless, at the moment, thereis no satisfactory answer to the question if there is a relation betweenindistinguishability and nonlocality in the sense of EPR correlations.

ACKNOWLEDGMENTS

We acknowledge with thanks the suggestions and criticisms made byDécio Krause.

NOTES

∗ I am very pleased to dedicate this work to Prof. Newton da Costa, on the occasion of his70th birthday. Prof. Da Costa is the antithesis of the scientist as a professional who achievesjust a high – and ever increasing – level of specialization. His outstanding achievementsrather point to the great and ever increasinginterweavingof formerly separated disciplinessuch as mathematics, philosophy and physics. One of my personal endeavours is to try tofollow his example.1 By measurement values of intrinsic properties of a given particle we mean real numberstimes an adequate unit of rest mass, charge, spin, etc., associated to the respective restmass, charge, spin, etc. of this particle.2 In (da Costa and Krause 1994) the authors discuss the possible representation ofquantum particles by means of ordered pairs〈E,L〉, whereE corresponds to a predicatewhich in some way characterizes the particle in terms, e.g., of its rest mass, its charge, andso on, whileL denotes an appropriate label, which could be, for example, the location ofthe particle in space-time. Then, even in the case that the particles (in a system) have thesameE, they might be distinguished by their labels. But if the particles have the same label,the tools of classical mathematics cannot be applied, since the pairs should be identified. Inorder to provide a mathematical distinction between particles with the sameE andL, theseauthors use quasi-set theory. In the present picture, according to axioms (HV1)–(HV3), itis prohibited the case where two particles have the same (ontological) label.


3 These arguments work for the usual quantum statistics as well as to parastatistics.

REFERENCES

Bohm, D. and B. J. Hilley: 1994,The Undivided Universe, Routledge, London.da Costa, N. C. A. and D. Krause, 1994, ‘Schrödinger logics’,Studia Logica53, 533–550.Dalla Chiara, M. L. and G. Toraldo di Francia: 1993, ‘Individuals, Kinds and Names in

Physics’, in G. Corsi et al. (eds.),Bridging the Gap: Philosophy, Mathematics, Physics,261–283, Kluwer Academic, Dordrecht. Reprint fromVersus40, 29-50 (1985).

Krause, D.: 1992, ‘On a Quasi-Set Theory’,Notre Dame Journal of Formal Logic33402–411.

Krause, D.: 1996, ‘Axioms for Collections of Indistinguishable Objects’,Logique etAnalyse153–15469–93.

Krause, D., A. S. Sant’Anna, and A. G. Volkov: 1999, ‘Quasi-set Theory for Bosons andFermions: Quantum Distributions’,Found. Phys. Lett.12, 51–66.

Mandel, L.: 1991, ‘Coherence and Indistinguishability’,Optics Letters161882–1883.Manin, Yu. I.: 1976, ‘Problems of Present Day Mathematics: I (Foundations)’, in F. E.

Browder (ed.),Proceedings of Symposia in Pure Mathematics28, American Mathemat-ical Society, Providence, 36.

Redhead, M. and P. Teller: 1991, ‘Particles, Particle Labels, and Quanta: The Toll ofUnacknowledged Metaphysics’,Foundations of Physics21, 43–62.

Sakurai, J. J.: 1994,Modern Quantum Mechanics, Addison-Wesley, Reading.Sant’Anna, A. S.: 1997, ‘Some Remarks about Individuality and Quantum Particles’,

Logique et Analyse157, 45–66.Sant’Anna, A. S. and D. Krause: 1997, ‘Indistinguishable Particles and Hidden Variables’,

Found. Phys. Lett.10409–426.Schrödinger, E.: 1952,Science and Humanism, Cambridge University Press, Cambridge.Suppes, P.: 1967,Set-Theoretical Structures in Science, mimeo. Stanford University,

Stanford.Suppes, P., A. S. Sant’Anna, and J. A. de Barros: 1996a, ‘A Particle Theory of the Casimir

Effect’, Found. Phys. Lett.9, 213–223.Suppes, P., J. A. de Barros, and A. S. Sant’Anna: 1996b, ‘Violation of Bell’s Inequalities

with a Local Theory of Photons’,Found. Phys. Lett.9, 551–560.van Fraassen, B.: 1991,Quantum Mechanics: an Empiricist View, Clarendon, Oxford.Watson, A.: 1997, ‘Teleportation Beams Up a Photon’s State’,Science278, 1881–1882.

Department of MathematicsFederal University of ParanáCuritibaBrazil

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Elementary Particles, Hidden Variables, And Hidden Predicates