Abstract
The paper address the question of whether quantum mechanics (QM) favors Priority Monism, the view according to which the Universe is the only fundamental object. It develops formal frameworks to frame rigorously the question of fundamental mereology and its answers, namely (Priority) Pluralism and Monism. It then reconstructs the quantum mechanical argument in favor of the latter and provides a detailed and thorough criticism of it that sheds furthermore new light on the relation between parthood, composition and fundamentality in QM.
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Notes
These axioms are familiar. Therefore I will not give a formal rendition. From now on, formulas, unless otherwise specified, are intended to be universally closed. First order logic with identity is used throughout.
The literature regarding this principle is enormous. The interested reader can start from Markosian (1998) for it provides a clearly written overview of the issue.
This is but one definition of mereological sum that is found in the literature. There is another, stronger definition, i.e.:
\(Sum^{*}(z,\varphi (x))=_{df} (\varphi (x)\rightarrow x\prec z)\wedge (\forall y)(y\prec z\rightarrow (\exists x)(\varphi (x)\wedge O(x,y))\).
To see that this definition is stronger note that only the right to left direction of the following biconditional, \((Sum(z,\varphi (x))\leftrightarrow Sum^{*}(z,\varphi (x))\) can be proven using only the partial ordering axioms. This argument is from Hovda (2009).
It can be proven that \(U\) is unique. The argument will go roughly as follows. Suppose it is not, i.e. there exists \(U^{*}\ne U\). Then there will be an entity, namely \(U^*\) that is not part of \(U\). But this contradicts the assumption that \(U\) is the universal element.
He explicitly admits that this assumption is controversial, for it could be maintained that such relations, if there are any at all, hold for example between properties rather than objects. I will not pursue this line of argument in the paper and grant Schaffer this assumption. Moreover I will simply assume that the dependence relation is primitive.
It will be clear in a moment why the definition does not require that \(x\ne y\). This is because the relation of dependence is irreflexive.
Schaffer points out some examples of dependence relations holding among entities that belong to distinct ontological categories. He claims for example that the truth of the sentence “Socrates exists”, that he takes to be a fact, depends on Socrates, a concrete object, and not the other way round (Schaffer 2010, p. 35).
Note that basic entities are concrete by (1.6) so that there is no need to add a concreteness clause. I thank an anonymous referee for this journal for having pointed this out.
Priority Monism is distinct from Ontological Monism for the latter holds that there exists only one object whereas the second maintains that there exist more than one object but only one is metaphysically fundamental. Clearly Ontological Monism (plus Well-Foundedness) entails Priority Monism. The same applies, mutatis mutandis, for Pluralism. In this case however it is Priority Pluralism that entails Ontological Pluralism. For an argument in favor of Ontological Monism inspired by QM see Esfeld (1999). This paper deals only with Priority Monism. I will thus omit this specification from now on.
I will use \(D\) for density operators. No confusion can arise between them and the relation of metaphysical dependence of § 1.2.
A sum of projection operators is again a projection operator. See Hughes (1992, p. 51).
In the terminology of § 1.1 we have that \(S_{12} =Sum(z,\varphi (x))\) where \(\varphi (x)=(x=S_1 \vee x=S_2 )\).
The following notations are all equivalent: \(|\varphi \rangle \otimes |\psi \rangle =\varphi \otimes \psi =|\varphi \rangle |\psi \rangle =|\varphi \psi \rangle \).
Equation (3.2) is a form of the Schmidt decomposition theorem, where \(\dim H_i \) stands for the dimension of Hilbert space \(H_i \).
It can be proven that projection operators that project onto one-dimensional subspaces are density operators and thus represent states of the system, in particular pure states. It also follows that pure states can be represented by vectors that span that one dimensional subspace (Beltrametti and Cassinelli 1981, p. 7).
Naturally the same holds for \(A_2, I_1 \otimes A_2 \) defined over \(H_2\) and \(H_{12}\) respectively.
Provided \(D_{12}\) is a pure state.
It can be proven that projector operators that project onto one-dimensional subspaces represent pure states. It follows that \(D_1, D_2\) are mixed states.
Whereas \(D_{12}\) is a mixed state.
I am being deliberately vague here. I do not intend “fact” in any technical sense. The supervenience basis might comprise concrete objects, properties, relations, facts and so on.
Schaffer’s argument is a little more elaborate than that. However I will not be focusing too much on EU, so I will rest content on this reconstruction.
The collapse postulate ensures that if a measurement is made on a quantum system that is in a superposition of two pure states \(c_1 |\varphi _1 \rangle +c_2 |\varphi _2 \rangle \) the state of the system collapses in one of the terms of the superposition with probability \(|c_i |^{2}\). This in turn entails that a measurement turns an entangled state into a product state.
I should add that pluralists would know what entanglement relations to add to the supervenience basis only after composition has occurred, so to speak. Consider two quantum composite systems \(S_{12}, {S_{12}}^{*}\) that happen to be in eigenstates of the following projection operators: \(P_{like} =(|\!\!\uparrow _1 \rangle \otimes |\!\!\uparrow _2 \rangle +|\!\!\downarrow _1 \rangle \otimes |\!\!\downarrow _2 \rangle ); P_{opposite} =(|\!\!\uparrow _1 \rangle \otimes |\!\!\downarrow _2 \rangle +|\!\!\downarrow _1 \rangle \otimes |\!\!\uparrow _2 \rangle )\) (Darby 2012, p. 779). They will instantiate the relation of having correlated or opposite (anti-correlated) spin respectively. Yet the reduced states of \(S_{1(2)}, {S_{1(2)}}^{*}\) will be the same. Pluralists will only be able to choose which entanglement relation to add to the supervenience basis, correlation or anti-correlation, only after they know whether the composite system is in an eigenstate of \(P_{like}\) or \(P_{opposite}\). The argument in this section would still go through. The argument is not about whether pluralists should know in advance what kind of entanglement relations they should add, but rather that adding the correct ones will provide them an adequate supervenience basis. And this is exactly the case. I thank an anonymous referee for having pointed this out.
The first problem Schaffer raises is that this move necessarily requires a “particle” ontology that will be difficult to maintain when passing to quantum field theories. I cannot do justice to such a claim here. I will simply point out that it seems hazardous to draw conclusive metaphysical consequences from two different theories (Morganti 2009, p. 277).
It could be argued that the Unrestricted Composition axiom is not strictly speaking needed. The argument entails that \(S_{12}\) is in an entangled state. This is enough to guarantee that \(S_{12}\) is a whole without the invocation of mereological principles. I find this line of argument correct. However nowhere in this paper I have presented an argument in favor of the claim (which I regard as true) that being in an entangled state is at least a sufficient condition for composition. I owe these considerations to an anonymous referee.
Their argument is quite technical and it is not possible to enter into any details here. I will only sketch it in broad outline as follows. Suppose you have a system \(S_{123}\) in a pure entangled state \(D_{123}\) such that this state does not admit a 2-particle entanglement. Then, calculate the reduced states \(D_{12}, D_3\). These are mixed states. To each mixed state we can associate an entropy that is proportional to the logarithm of their dimension. It will follow that the entropy of \(S_{12}\) is greater than that of \(S_3\) which contradicts a fundamental theorem by Araki and Lieb.
It might be argued for example that the entanglement relation holding between \(S_1, S_2\) on the one hand and the one holding between \(S_{12}, S_3 \) on the other are not exactly the same relation. This should be checked case by case. Consider system \(S_{1234} \) in state (4.1). The entanglement relation holding between \(S_1, S_2 \) is that of anticorrelation. System \(S_{12}\) is in an eigenstate of the projection operator \(P_{opposite}\) in Darby (2012), which I mentioned in footnote 24. The same goes for \(S_{34} \). System \(S_{123}\) would not be in such eigenstate and so systems \(S_1, S_2 \) and systems \(S_{12}, S_3 \) would not stand in the same entanglement relation. But in these cases pluralists will be free to analyze the entanglement relation holding between \(S_1, S_2, S_3, S_4\) in terms of the same 2-place entanglement relation, namely the anti-correlation relation, holding between \(S_1, S_2\) on the one hand and between \(S_3, S_4\) on the other. They would not need to resort to an entanglement relation between \(S_{12}, S_3\) in the first place. Moreover, as I note immediately afterwards, pluralists would in any case only need to invoke one entanglement relation whose adicity depends on the number of subsystems considered.
Though he uses it for a different reason, this is the same suggestion Darby (2012, p. 786) makes. I thank an anonymous referee for having pointed this paper out to me.
As he actually once did in a private conversation.
Or \(S_{12}, S_{34} \) in the case of state (4.1).
Actually it should also be checked whether the argument for WMF would carry over unaltered.
This was pointed out to me by Mauro Dorato.
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Acknowledgments
For their comments and discussion I would like to thank J. Schaffer, A. Varzi, M. Morganti, C. Hoefer, M. Dorato, V. Fano, G. Torrengo, J. Diez, A. Solè and D. Dieks. I am especially grateful to two anonymous referees for this journal for their insightful, thorough and helpful suggestions.
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Calosi, C. Quantum mechanics and Priority Monism. Synthese 191, 915–928 (2014). https://doi.org/10.1007/s11229-013-0300-6
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DOI: https://doi.org/10.1007/s11229-013-0300-6