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Actuality from Potentiality

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The Pythagorean World

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Abstract

In this chapter, I am going to explore how actuality emerges from potentiality. In the last chapter I engaged in some speculative thinking about the general process but in this chapter, I want to be more specific and that will entail a retreat to a more manageable problem. I will be considering two paradigms: (1) Leibnizian possible worlds, which is rooted in classical physics, and (2) the consistent histories quantum theory of Griffiths, Gell-Mann, Hartle, and Omnès (I will refer to this as CHQT). Hence, at least initially, I will not be considering all of mathematics but merely that part which is required for these paradigms. It is a moot point how much mathematics that encompasses and I will have more to say about that at the conclusion.

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Notes

  1. 1.

    Drawing on Leibniz’s dream of a universal science, Gödel dreamed of a theory of concepts which would go beyond the theory of sets (i.e. beyond mathematics). Whereas set theory deals with extensions, concept theory would deal with intensions. Note that different concepts can have the same extension whereas, by definition, different sets cannot since they are determined by their extension. For more on this topic, see Wang (1996): Chap. 8.

  2. 2.

    Heinrich Schepers said that “one can view Leibniz’s metaphysics as the work of rationally explicating the different expressions of the polarity of the One and the Many” (Schepers 2000: 171).

  3. 3.

    A Hermitian operator is an operator which is equal to its adjoint, so its eigenvalues are real numbers.

  4. 4.

    See, for example Zeilinger (1999), and Yu and Nikolic (2011).

  5. 5.

    See, for example, Griffiths (2002), Omnès (1994), and Gell-Mann and Hartle (1990).

  6. 6.

    A property of a system is given when one can assert that the value of an observable A is in some given set D of real numbers (Omnès 1992: 342).

  7. 7.

    More generally, an IGUS is a complex adaptive system which is coupled to its environment in some way, able to model its local environment by some form of logical processing, and able to act on the results of its computations. It could be biological, or it could be something like a thermostat or a robot (Hartle 2007b: 3111).

  8. 8.

    Intuitively, if the components of a quantum system are in a superposition, the only way they can decohere is through interaction with an environment, and an environment can only be defined by coarse-graining.

  9. 9.

    A projector is a Hermitian operator on the Hilbert space which is equal to its square, so its eigenvalues are 0 and 1.

  10. 10.

    For example, it is meaningless to ask which slit a photon passed through in Young’s double-slit experiment when an interference pattern is observed.

  11. 11.

    And because of complications such as thermodynamic irreversibility and chaos.

  12. 12.

    Omnès (1994) has applied the methods of quantum information theory to rigorously reconstruct measurement theory from the principles of CHQT—from quantum observables to macroscopic observables.

  13. 13.

    For example, delayed-choice paradoxes, indirect measurement paradoxes, incompatibility paradoxes and the EPR paradox. In each case, when the relevant experiment is clearly defined in a consistent histories framework, the paradox is shown to be the result of comparing, or combining, incompatible frameworks.

  14. 14.

    The table cannot be reproduced here for copyright reasons and the details are not of importance for the current discussion. The interested reader is referred to the original.

  15. 15.

    There are some events which have probability one in a particular framework but it is difficult to identify events which occur with probability one in any fundamental consistent set fine-graining the actual history (Dowker and Kent 1996: 1611–1616).

  16. 16.

    Note that “seemingly contradictory events” contradict our intuitions but do not lead to inconsistencies in the formalism.

  17. 17.

    The physical analogy is a Bose-Einstein condensate.

  18. 18.

    A force that originates in a system with many degrees of freedom by the statistical tendency to increase its entropy is called an entropic force. It has been suggested that gravity is an entropic force (Verlinde 2011).

  19. 19.

    Note that Appendix 2 considers problems which arise from postulating plenitude and offers a defence.

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  1. Philosophy Department Clayton, Victoria, Australia

    Jane McDonnell

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McDonnell, J. (2017). Actuality from Potentiality. In: The Pythagorean World. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-319-40976-4_6

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