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**Contextual semantics in quantum mechanics from a categorical point of view.**
*(English)*
Zbl 1396.81014

The technical core of this paper is a categorical adjunction between the sheaves of variable local Boolean frames and the category of quantum event algebras. The aspect which is analysed is the subobject classifier in the former category, which is a Grothendieck topos, that provides via a suitable adjunction a subobject classifier in the latter category, thus equipping it with a notion of non-classical truth allowing to interpret propositions in quantum mechanics, which is the ultimate purpose of the article.

The article appeared in Synthese, a leading journal in philosophy, and thus the focus of the work is to discuss the philosophical implications of the previous construction. Mathematically, the construction follows a standard path, and although it is by far non-elementary, the result is not surprising in a purely mathematical sense. In the physical sense, the result is, as far as the reviewer can understand, significant and of interest beyond the purely mathematical content, since it provides the conceptual framework to assemble observations in a coherent way, considering that the act of observing influences the outcome of an experiment. Of course, this is an over-simplification, adopted in this review to illustrate the content of the paper; however the same idea is discussed in depth in the first part of the work, in its full complexity, and with a particular emphasis on the physical and philosophical implications, e.g., its relation with the well-known Kochen-Specker’s theorem.

The paper starts providing the general framework which explains how the physical phenomena are linked to propositions, thus showing what a notion of truth means in the interpretation of a physical system in terms of its events. While in classical mechanics such an interpretation boils down to classical logic, as the semantics of propositions reduces to the lattice of subspaces of the phase space, which is Boolean (Section 1), the same interpretation in quantum Physics does not reduce to classical logic, as the structure of the phase space is not that of a Boolean lattice (Section 2). Studying the structure of the quantum algebra of events (Section 3), it becomes natural to interpret quantum event algebras in Category Theory (Section 4). This study is abstract, claiming for a concrete representation, which is pursued in Section 5 by means of pointed Boolean frames, and deepened in Section 6 by Boolean localization functors. The mathematical instruments developed in these sections allow to explicitly characterise the subobject classifier in the lattice of quantum event algebras (Section 7), and to provide an explicit interpretation of truth of physical propositions (Section 8) by means of the so-synthesised quantum truth-values algebras. Then, in Section 9, the contextual semantics valuation is analysed by its conceptual and philosophical implications. It turns out that elementary propositions associated with the behaviour of a quantum system receive a locally classical meaning with respect to an observational framework, while the proposition itself possesses a non-classical truth value which is linked to the classical truth-values it gets in all the possible observational framework by the corresponding Boolean localization functors. It is this link between the global quantum semantics and the local classical semantics that is the core of the paper and the discussed issue here.

The article is relevant to whoever is interested in the epistemology and ontology of quantum mechanics, as it provides not only a general discussion or an ideal framework, but rather it studies the fundamental mathematical tools to model the semantics of physical propositions along the proposed line of philosophical investigation. Mathematically, the paper provides a non-obvious use of the notion of subobject classifiers in a framework that requires a non-classical notion of truth to be synthesised, showing the power of topos-theoretic tools in this respect. Thus, the paper may have an independent interest for category theorists interested in the application of topos theory outside the strict domain of Mathematics.

The article appeared in Synthese, a leading journal in philosophy, and thus the focus of the work is to discuss the philosophical implications of the previous construction. Mathematically, the construction follows a standard path, and although it is by far non-elementary, the result is not surprising in a purely mathematical sense. In the physical sense, the result is, as far as the reviewer can understand, significant and of interest beyond the purely mathematical content, since it provides the conceptual framework to assemble observations in a coherent way, considering that the act of observing influences the outcome of an experiment. Of course, this is an over-simplification, adopted in this review to illustrate the content of the paper; however the same idea is discussed in depth in the first part of the work, in its full complexity, and with a particular emphasis on the physical and philosophical implications, e.g., its relation with the well-known Kochen-Specker’s theorem.

The paper starts providing the general framework which explains how the physical phenomena are linked to propositions, thus showing what a notion of truth means in the interpretation of a physical system in terms of its events. While in classical mechanics such an interpretation boils down to classical logic, as the semantics of propositions reduces to the lattice of subspaces of the phase space, which is Boolean (Section 1), the same interpretation in quantum Physics does not reduce to classical logic, as the structure of the phase space is not that of a Boolean lattice (Section 2). Studying the structure of the quantum algebra of events (Section 3), it becomes natural to interpret quantum event algebras in Category Theory (Section 4). This study is abstract, claiming for a concrete representation, which is pursued in Section 5 by means of pointed Boolean frames, and deepened in Section 6 by Boolean localization functors. The mathematical instruments developed in these sections allow to explicitly characterise the subobject classifier in the lattice of quantum event algebras (Section 7), and to provide an explicit interpretation of truth of physical propositions (Section 8) by means of the so-synthesised quantum truth-values algebras. Then, in Section 9, the contextual semantics valuation is analysed by its conceptual and philosophical implications. It turns out that elementary propositions associated with the behaviour of a quantum system receive a locally classical meaning with respect to an observational framework, while the proposition itself possesses a non-classical truth value which is linked to the classical truth-values it gets in all the possible observational framework by the corresponding Boolean localization functors. It is this link between the global quantum semantics and the local classical semantics that is the core of the paper and the discussed issue here.

The article is relevant to whoever is interested in the epistemology and ontology of quantum mechanics, as it provides not only a general discussion or an ideal framework, but rather it studies the fundamental mathematical tools to model the semantics of physical propositions along the proposed line of philosophical investigation. Mathematically, the paper provides a non-obvious use of the notion of subobject classifiers in a framework that requires a non-classical notion of truth to be synthesised, showing the power of topos-theoretic tools in this respect. Thus, the paper may have an independent interest for category theorists interested in the application of topos theory outside the strict domain of Mathematics.

Reviewer: Marco Benini (Buccinasco)

### MSC:

81P13 | Contextuality in quantum theory |

81P05 | General and philosophical questions in quantum theory |

03G30 | Categorical logic, topoi |

18B25 | Topoi |

81P10 | Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) |

### Keywords:

Kochen-Specker theorem; category theory; categorical adjunction; subobject classifier; correspondence truth; contextual semantics
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\textit{V. Karakostas} and \textit{E. Zafiris}, Synthese 194, No. 3, 847--886 (2017; Zbl 1396.81014)

### References:

[1] | Abramsky, S., & Brandenburger, A. (2011). The sheaf-theoretic structure of non-locality and contextuality. New Journal of Physics, 13, 113036. · Zbl 1448.81028 |

[2] | Awodey, S. (2010). Category theory (2nd ed.). Oxford: Oxford University Press. · Zbl 1194.18001 |

[3] | Bell, J. L. (1988/2008). Toposes and local set theories. Dover: New York. · Zbl 0649.18004 |

[4] | Blackburn, S., & Simmons, K. (1999). Truth (Oxford readings in philosophy). Oxford: Oxford University Press. |

[5] | Burgess, A., & Burgess, J. P. (2011). Truth (Princeton Foundations of Contemporary Philosophy). Princeton: Princeton University Press. |

[6] | Coecke, B. (2010). Quantum picturalism. Contemporary Physics, 51(1), 59-83. |

[7] | Dalla Chiara, M., Giuntini, R., & Greechie, R. (2004). Reasoning in quantum theory. Dordrecht: Kluwer. · Zbl 1059.81003 |

[8] | Davis, M. (1977). A relativity principle in quantum mechanics. International Journal of Theoretical Physics, 16, 867-874. · Zbl 0392.03040 |

[9] | Devitt, M.; Lynch, M. (ed.), The metaphysics of truth, 579-611 (2001), Cambridge MA |

[10] | Dirac, P. A. M. (1958). Quantum mechanics (4th ed.). Oxford: Clarendon Press. · Zbl 0080.22005 |

[11] | Domenech, G., & Freytes, H. (2005). Contextual logic for quantum systems. Journal of Mathematical Physics, 46, 012102. · Zbl 1076.81003 |

[12] | Döring, A., & Isham, C. J. (2011). “What is a thing?”: Topos theory in the foundations of physics. In B. Coecke (Ed.), New structures for physics. Lecture notes in physics (Vol. 813, pp. 753-937). Berlin: Springer. · Zbl 1253.81011 |

[13] | Epperson, M., & Zafiris, E. (2013). Foundations of relational realism: A topological approach to quantum mechanics and the philosophy of nature. Book series: Contemporary Whitehead Studies. New York: Lexington Books. |

[14] | Goldblatt, R. (1984/2006). Topoi: The categorial analysis of logic (rev. 2nd ed.). New York: Dover. · Zbl 0528.03039 |

[15] | Heunen, C.; Landsman, NP; Spitters, B.; Halvorson, H. (ed.), Bohrification, 271-313 (2011), New York · Zbl 1234.81025 |

[16] | Isham, C. J., & Butterfield, J. (1998). A topos perspective on the Kochen-Specker theorem: I. Quantum states as generalized valuations. International Journal of Theoretical Physics, 37, 2669-2733. · Zbl 0979.81018 |

[17] | Karakostas, V. (2007). Nonseparability, potentiality, and the context-dependence of quantum objects. Journal for General Philosophy of Science, 38, 279-297. |

[18] | Karakostas, V. (2012). Realism and objectivism in quantum mechanics. Journal for General Philosophy of Science, 43, 45-65. |

[19] | Karakostas, V. (2014). Correspondence truth and quantum mechanics. Axiomathes, 24, 343-358. |

[20] | Kochen, S., & Specker, E. P. (1967). The problem of hidden variables in quantum mechanics. Journal of Mathematics and Mechanics, 17, 59-87. · Zbl 0156.23302 |

[21] | Lawvere, W., & Schanuel, S. (2009). Conceptual mathematics (2nd ed.). Cambridge: Cambridge University Press. · Zbl 1179.18001 |

[22] | MacFarlane, J. (2005). Making sense of relative truth. Proceedings of the Aristotelian Society, 105, 321-339. |

[23] | MacLane, S., & Moerdijk, I. (1992). Sheaves in geometry and logic. New York: Springer. |

[24] | Malpas, J. (Ed.). (2003). From Kant to Davidson: Philosophy and the idea of the transcendental. London: Routledge. |

[25] | Rédei, M. (1998). Quantum logic in algebraic approach. Dordrecht: Kluwer. · Zbl 0910.03038 |

[26] | Svozil, K. (1998). Quantum logic. Singapore: Springer. · Zbl 0922.03084 |

[27] | Takeuti, G. (1978). Two applications of logic to mathematics. Part I: Boolean valued analysis. Publications of mathematical society of Japan 13. Tokyo and Princeton: Iwanami and Princeton University Press. · Zbl 0393.03027 |

[28] | Tarski, A. (1935/1956). The concept of truth in formalized languages. In Logic, semantics, metamathematics: Papers from 1923 to 1938 (pp. 152-278). Oxford: Clarendon Press. · Zbl 0013.28903 |

[29] | van den Berg, B., & Heunen, C. (2012). Noncommutativity as a colimit. Applied Categorical Structures, 20(4), 393-414. · Zbl 1261.46051 |

[30] | Von Neumann, J. (1955). Mathematical foundations of quantum mechanics. Princeton: Princeton University Press. · Zbl 0064.21503 |

[31] | Zafiris, E. (2004). Boolean coverings of quantum observable structure: A setting for an abstract differential geometric mechanism. Journal of Geometry and Physics, 50, 99-114. · Zbl 1068.18013 |

[32] | Zafiris, E. (2006a). Generalized topological covering systems on quantum events structures. Journal of Physics A: Mathematical and General, 39, 1485-1505. · Zbl 1100.81002 |

[33] | Zafiris, E. (2006b). Sheaf-theoretic representation of quantum measure algebras. Journal of Mathematical Physics, 47, 092103. · Zbl 1112.81007 |

[34] | Zafiris, E., & Karakostas, V. (2013). A categorial semantic representation of quantum event structures. Foundations of Physics, 43, 1090-1123. · Zbl 1282.81017 |

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