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The present paper has three closely related aims. We first argue that Agazzi's scientific realism about Quantum Mechanics is in line with Selleri's and Tarozzi's proposal of Quantum Waves. We then go on to formulate rigorously different

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Found Phys (2011) 41:1740–1755DOI 10.1007/s10701-011-9590-z
Quantum Ontology and Extensional Mereology
Claudio Calosi
·
Vincenzo Fano
·
Gino Tarozzi
Received: 1 April 2011 / Accepted: 27 July 2011 / Published online: 2 September 2011© Springer Science+Business Media, LLC 2011
Abstract
The present paper has three closely related aims. We ﬁrst argue thatAgazzi’s scientiﬁc realism about Quantum Mechanics is in line with Selleri’s andTarozzi’s proposal of Quantum Waves. We then go on to formulate rigorously dif-ferent metaphysical principles such as property compositional determinateness andmereological extensionalism. We argue that, contrary to widespread agreement, re-alism about Quantum Mechanics actually refutes only the former. Indeed we evenformulate a new quantum mechanical argument in favor of extensionalism. We con-cludebynotingthat,giventheresultsofthework,Agazzi’sparticularattitudetowardsQuantum Mechanics is still one of the most promising theoretical perspectives.
Keywords
Quantum realism
·
Properties of composite systems
·
Mereologicalextensionalism
1 Introduction
Evandro Agazzi has clearly underlined that the neo-positivistic refusal of an au-tonomous philosophical analysis that goes beyond formal clariﬁcation of scientiﬁcdiscourse is unjustiﬁed. For example, philosophical questions that, on the one handlie at the foundations of physical theories, and on the other are implied by them, canbe addressed in non metaphysical terms. This can be done by assuming those veryneo-positivistic criteria of meaning that, even if useless in deﬁning scientiﬁc proposi-tions, are perfectly able to guarantee a reformulation of some metaphysical theses interms of philosophical principles endowed with empirical meaning. In this paper wewill in fact address some metaphysical principles such as mereological extensional-ism and property compositional determinateness, to name just two. We will provide
C. Calosi (
)
·
V. Fano
·
G. TarozziDepartment of Foundations of Science (Scienze di base e Fondamenti), University of Urbino,Palazzo Albani, Via Timoteo Viti 10, 61029 Urbino, Italye-mail: claudio.calosi@uniurb.it
Found Phys (2011) 41:1740–1755 1741
their rigorous formulation and we will attribute them empirical meaning by compar-ing them with some ontological intimations that come from a realistic interpretationof Quantum Mechanics.
1
Agazzi maintains that recent developments in Quantum Mechanics do not offer adeﬁnitive or exhaustive solution to the philosophical problem of the nature of micro-objects. He underlines, with extraordinary intuition, the need to introduce new con-cepts to analyze the quantum mechanical realm. This novelty should not be just theresult of a combination of classical concepts, but rather, the introduction of an srci-nal set of concepts that could be able, at least in principle, to replace the old ones, aspointed out clearly in Agazzi [2]:“Only by inventing some new concept, that is new in this fundamental sense,could we possibly overcome the present uneasy state of affairs, which is notrelated to the regret of losing the old concepts but to the lack of new conceptscapable of adequately replacing them”.In the very same year that Agazzi was advocating the introduction of a new con-cept to solve the wave-particle duality, Franco Selleri proposed a realistic interpreta-tion of QM based on such a new concept in Selleri [21]. It is the concept of
empty or quantum wave
. This can be thought of as the synthesis of three different conceptionsabout the wave-particle duality held by the fathers of quantum theory.Coherently with his general perspective, Agazzi [2] observes that the importantconceptual novelty of the quantum wave hypothesis is constituted by the refusal of the symmetric nature of wave-particle duality. In fact Selleri endorses neither thoseinterpretations that insist on a unique corpuscular nature, nor those complementaryinterpretationsthatmaintainthattheadoptionofauniqueinterpretationovertheotheris,intheend,contradictory.Selleriacceptsbotharealisticinterpretationofthedualityand some sort of ontological priority of particles over waves:“The essential novelty of this concept is represented by the acceptance of thede Broglie realist interpretation of wave-particle duality, but not of symmetricalnature of the dualism. In Selleri’s approach both particles and waves are simul-taneously real, but the latter can be characterized only with relational proper-ties with particles: that is the observables properties of producing interferenceand stimulated emission. Such a possibility would imply an ontological prior-ity of particles over waves, which would therefore belong to a weaker level of physical reality, containing objects which are sensible carriers of exclusivelyrelational predicates”.
2
Different experiments have been advanced stemming from this new realistic inter-pretation of the wave-function. The interest of these experiments, as Agazzi pointedout, is twofold:“The interest of these experiments is due to the fact that they would allow onenot only to test this new realist interpretation vs. the Copenhagen one—to dis-criminateexperimentallybetweentwodifferent philosophicalinterpretationsof
1
QM from now on.
2
See Agazzi [2], p. 73.
1742 Found Phys (2011) 41:1740–1755
a physical theory—but also the well-known axiom of the reduction of the wavefunction
(...)
”.
3
Agazzi was referring to the fact that the properties of quantum waves can be usedto reconstruct photons’ trajectories within an interferometer while at the same timeregistering the interference pattern, contrary to the prediction based on the collapseof the wave-function. This would also establish an important connection between thewave particle-duality on the one hand, and the fundamental quantum measurementproblem on the other. So far, as Hardy [10] points out, no experiment has been suc-cessful in either revealing such quantum waves’ properties or in refuting the collapsepostulate.However a new thought experiment that could be easily realized has been pro-posed, for example in Auletta and Tarozzi [4, 5], to the point that quantum waves
could produce EPR style correlations. This would actually allow a new and unsus-pected relation to be established between two different realistic interpretations of QM: Agazzi’s realism about theoretical entities on the one hand and the EPR’s prin-ciple of local realism on the other.Here is the plan of the paper. In the next section we will discuss broadly Agazzi’sscientiﬁc realism. In Sect. 3 we will address ontological issues regarding entangledstates in the tensor product space for composite systems and in Sect. 4 we will givea rigorous formulation of such ontological problems in terms of modern mereology,i.e. the formal theory of parthood relations. This will allow us to discuss in Sect. 5different conceptual relations between various ontological thesis, such as mereologi-cal extensionalism and property compositional determinateness. The closing Sect. 6will be devoted to the formulation of possible future research developments.
2 Agazzi’s Scientiﬁc Realism
Agazzi’s conception was elaborated in close connection with the philosophical prob-lems raised by QM, in particular the status and the meaning of the complementarityprinciple and the realistic interpretation of the wave-function.His realistic demand stems from the problem of the epistemological value of sci-entiﬁc theories. The main point is trying to establish whether scientiﬁc theories pro-vide any ground for objectivity. Agazzi underlines, in Agazzi [1], three fundamentalsenses of such a term: “objectivity as inter-subjectivity, as invariance and as corre-spondence to objects”.
4
Through an effective and detailed analysis he argues thatthese three senses can be identiﬁed. The presuppositions grounding the possibility of such an identiﬁcation, from an epistemological point of view, are essentially three:(i) The ﬁrst one is the operational ground of scientiﬁc concepts along with thefact that, this grounding notwithstanding, those concepts cannot be reduced to apurely operational dimension.(ii) The realization that the meaning of scientiﬁc terms is context-dependent.
3
See Agazzi [2], p. 73.
4
Original in Italian. Our translation.
Found Phys (2011) 41:1740–1755 1743
(iii) The fact that scientiﬁc objects are constituted by properties established objec-tively through a set of operations and yet they do not only constitute a simplebundle of such properties but rather a well deﬁned structure of relations betweenthem.We will see that these three aspects are even more inter-related when applied tothose scientiﬁc concepts that are expressed through the so-called theoretical terms,i.e. those terms whose denotations are not immediately observable. Let’s considerbrieﬂy each point.Scientiﬁc theories are constructed on the basis of theoretical terms, but their aimis to account for those immediate facts of experience describable using empiricaland observational terms. This raises the infamous problem of how to guarantee thatsuch theoretical terms maintain a link with empirical terms. A theoretical conceptlike “electron” is, we read in Agazzi [1], “a theoretical construction around whichwe gather many properties operationally deﬁned”.
5
Actually the operational aspectallows theoretical terms to keep in touch with experience, so to acquire a physicalmeaning. Those theoretical terms cannot however be simply reduced to operationalterms, as Agazzi [1] explicitly claims:“
[
...
]
we do not even dream of saying that theoretical concepts can be reducedto operational concepts: those who wanted to do that would do the same thingas someone wishing to reduce his house to the bricks that constitute it”.
6
Various combinations of empirical (observational) terms actually form construc-tions (the theoretical terms) that are not themselves directly operational.From all this, it follows that the meaning of theoretical terms is always contextual,as we underlined in our second point. Agazzi [1] notes that this“does not mean however that [the meaning of theoretical terms] comes from theobservational terms through a context
[
...
]
, but it comes from the context itself.In this case observational terms are present, but they are not alone in it, for thecontext is actually realized by all those logical and mathematical connectionsthat link together each and every concept, observational and not”.
7
The context within which the theoretical terms acquire their meaning is the phys-ical theory of which they are part and that they help to construct. Only the physicaltheory as a whole can be empirically interpreted and thus can be related to possibleobservations. This point is of decisive importance for a realistic interpretation of thequantum wave. According to Agazzi many interpretative problems of QM stem fromthe attempt to apply classical concepts to quantum objects. The solution cannot sim-ply consist however of a new combination of classical notions of wave and particle,exactly because the meaning of such theoretical terms is context-dependent. This isclearly recognized in Agazzi [1] again:
5
Original in Italian. Our translation.
6
Original in Italian. Our translation. This last remark is of particular importance for the rest of the paper.
7
Original in Italian. Our translation.
1744 Found Phys (2011) 41:1740–1755
“not only
can
we say, but we
must
say that it is not the same particle, it is notthe same wave as the particle and wave we talk about in classical and quantummechanics. This is because the contexts are different”.
8
This raises the need to ﬁnd truly srcinal concepts in order to escape the traditionalproblemsrelatedtoarealisticinterpretationofQM.Andthistakesustoourlastpoint.We have already pointed out that scientiﬁc objects, in particular those that aredenoted by theoretical terms, present themselves as relational structures of propertiesdeﬁned operationally. And we have also pointed out that they cannot be reduced tosuch properties. This last point is of fundamental importance in Agazzi’s perspective.And it is deeply related to the contextual character of theoretical terms:“The object is always a structure, a relational structure. Those relations can be,for the most part, the result of some operations but the ‘holding together’ of such relations cannot be justiﬁed in terms of any of those operations”.
9
The attempt to reconstruct such a structure is the primary goal of scientiﬁc theoriesfor“structure is not just what lies beneath experimental determinations and objec-tive characteristics, but is what is constituted by them: it is the object”.
10
It is this structure that makes the world what it is and that can deem our theorieswrong. For our theories presuppose a structure that is not the structure of the worldand yet they are an attempt to reconstruct such a world structure.We subscribe to Agazzi’s opinion whereby theoretical terms acquire their empiri-cal meaning through the whole theory in which they are embedded. Nevertheless thisdoes not mean that whenever one endorses a scientiﬁc realistic thesis about some the-oretical entities, one is also compelled to accept that the world is exactly as the theorystates. In what follows we adopt a mild form of entity scientiﬁc realism, as developedin a semantic approach to theories described for example in Fano and Macchia [7],which is committed to the reality of theoretical entities only in relation to the investi-gated system and to the theoretical framework most suitable for that system.
3 Quantum Mechanics and Properties of Composite Systems
Let
x
be a quantum particle;
x
is a model for Quantum Mechanics (QM) in Sup-pes’ [22] sense. Suppose, for the sake of argument, that
x
is completely describableby the observable
O
11
with two eigenfunctions
|↑
x
and
|↓
x
, with eigenvalues 1and
−
1 respectively. Its state
x
is represented by a normalized vector in the two-dimensional Hilbert space
H
x
spanned by
|↑
x
and
|↓
x
. This Hilbert space contains
8
Original in Italian. Our translation.
9
See Agazzi [1]. Original in Italian. Our translation.
10
See Agazzi [1]. Original in Italian. Our translation.
11
Normally a description of a quantum system is given by at least two observables, but in this context wecan neglect one of them and focus only on the other.

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