## Abstract

We examine two quite different threads in Pitowsky’s approach to the measurement problem that are sometimes associated with his writings. One thread is an attempt to understand quantum mechanics as a probability theory of physical reality. This thread appears in almost all of Pitowsky’s papers (see for example 2003, 2007). We focus here on the ideas he developed jointly with Jeffrey Bub in their paper ‘Two Dogmas About Quantum Mechanics’ (2010) (See also: Bub (1977, 2007, 2016, 2020); Pitowsky (2003, 2007)). In this paper they propose an interpretation in which the quantum probabilities are objective chances determined by the *physics of a genuinely indeterministic* universe. The other thread is sometimes associated with Pitowsky’s earlier writings on quantum mechanics as a Bayesian theory of quantum probability (Pitowsky 2003) in which the quantum state seems to be a credence function tracking the experience of agents betting on the outcomes of measurements. An extreme form of this thread is the so-called Bayesian approach to quantum mechanics. We argue that in both threads the measurement problem is solved by implicitly adding structure to Hilbert space. In the Bub-Pitowsky approach we show that the claim that decoherence gives rise to an effective Boolean probability space requires adding structure to Hilbert space. With respect to the Bayesian approach to quantum mechanics, we show that it too requires adding structure to Hilbert space, and (moreover) it leads to an extreme form of idealism.

## Access this chapter

Tax calculation will be finalised at checkout

Purchases are for personal use only

### Similar content being viewed by others

## Notes

- 1.
Everett’s (1957) interpretation of quantum mechanics has a similar attempt in this respect.

- 2.
- 3.
- 4.
The ‘no signaling’ principle asserts, roughly, that the marginal probabilities of events associated with a quantum system are independent of the particular set of mutually exclusive and collectively exhaustive events associated with any

*other*system. - 5.
Or Einstein’s principle of relativity and the light postulate.

- 6.
- 7.
For decoherence theory, and references, see e.g., Joos et al. (2003).

- 8.
Interactions never begin, so state (1) (which is a product state) should be replaced with a state in which Alice + electron are (weakly) entangled; otherwise the dynamics would not be reversible.

- 9.
In discrete models, both | ψ

_{↑↓}⟩ and |*E*_{±}(*t*)⟩ are states of the form: \( \prod_{i=1}^N\mid \left.{\mu}_i\right\rangle \) defined in the corresponding multi-dimensional Hilbert spaces. - 10.
In statistical mechanics, here is an analogous scenario: the phase space of the universe can be partitioned into infinitely many sets corresponding to infinitely many

*macrovariables*. Some of these partitions exhibit certain regularities, for example, the partition to the thermodynamic sets which human observers are sensitive to, exhibit the thermodynamic regularities; while other partitions to which human observers are not sensitive, although they are*equally real*exhibit other regularities and may even be anomalous (that is, exhibit no regularities at all). We call this scenario in statistical mechanics*Ludwig’s problem*(see Hemmo and Shenker 2012, 2016). But one*can*explain by straightforward physical facts why (presumably) human observers experience the thermodynamic sets and regularities but not the equally existing other sets (although it might be that in our experience some other sets also appear). This case too is dis-analogous to the basis-symmetry in quantum mechanics, from which the problem of*no*preferred basis follows. - 11.
- 12.
- 13.
For the role of decoherence in the Bayesian approach and its relation to Dutch-book consistency, see Fuchs and Scack (2012).

- 14.
- 15.
- 16.
These two roles are carried out by the mind at one shot (as it were), but analytically they are different.

## References

Bell, J. S. (1987a). Are there quantum jumps? In J. Bell (Ed.),

*Speakable and unspeakable in quantum mechanics*(pp. 201–212). Cambridge: Cambridge University Press.Bell, J. S. (1987b). How to teach special relativity. In J. Bell (Ed.),

*Speakable and unspeakable in quantum mechanics*(pp. 67–80). Cambridge: Cambridge University Press.Brown, H. (2005).

*Physical relativity: Spacetime structure from a dynamical perspective*. Oxford: Oxford University Press.Brown, H. R., & Timpson, C. G. (2007). Why special relativity should not be a template for a fundamental reformulation of quantum mechanics. In Pitowsky (Ed.),

*Physical theory and its interpretation: Essays in honor of Jeffrey Bub, W. Demopoulos and I*. Berlin: Springer.Bub, J. (1977). Von Neumann’s projection postulate as a probability conditionalization rule in quantum mechanics.

*Journal of Philosophical Logic, 6*, 381–390.Bub, J. (2007). Quantum probabilities as degrees of belief.

*Studies in History and Philosophy of Modern Physics, 38*, 232–254.Bub, J. (2016).

*Bananaworld: Quantum mechanics for primates*. Oxford: Oxford University Press.Bub, J. (2020). ‘Two Dogmas’ Redux. In Hemmo, M., Shenker, O. (eds.)

*Quantum, probability, logic: Itamar Pitowsky’s work and influence*. Cham: Springer.Bub, J., & Pitowsky, I. (2010). Two dogmas about quantum mechanics. In S. Saunders, J. Barrett, A. Kent, & D. Wallace (Eds.),

*Many worlds? Everett, quantum theory, and reality*(pp. 431–456). Oxford: Oxford University Press.Bohm, D. (1952). A suggested interpretation of the quantum theory in terms of “hidden” variables.

*Physical Review, 85*(166–179), 180–193.Caves, C. M., Fuchs, C. A., & Schack, R. (2002a). Unknown quantum states: The quantum de Finetti representation.

*Journal of Mathematical Physics, 43*(9), 4537–4559.Caves, C. M., Fuchs, C. A., & Schack, R. (2002b). Quantum probabilities as Bayesian probabilities.

*Physical Review A, 65*, 022305.Caves, C. M., Fuchs, C. A., & Schack, R. (2007). Subjective probability and quantum certainty.

*Studies in History and Philosophy of Modern Physics, 38*, 255–274.de Finetti, B. (1970).

*Theory of probability*. New York: Wiley.Everett, H. (1957). ‘Relative state’ formulation of quantum mechanics.

*Reviews of Modern Physics, 29*, 454–462.Fuchs C. A. & Schack, R. (2012). Bayesian conditioning, the reflection principle, and quantum decoherence. In Ben-Menahem, Y., & Hemmo, M. (eds.),

*Probability in physics*(pp. 233–247). The Frontiers Collection. Berlin/Heidelberg: Springer.Fuchs, C. A., & Schack, R. (2013). Quantum Bayesian coherence.

*Reviews of Modern Physics, 85*, 1693–1715.Fuchs, C. A., Mermin, N. D., & Schack, R. (2014). An introduction to QBism with an application to the locality of quantum mechanics.

*American Journal of Physics, 82*, 749–754.Fuchs, C. A., & Stacey, B. (2019). QBism: Quantum theory as a hero’s handbook. In E. M. Rasel, W. P. Schleich, & S. Wölk (Eds.),

*Foundations of quantum theory: Proceedings of the International School of Physics “Enrico Fermi” course 197*(pp. 133–202). Amsterdam: IOS Press.Ghirardi, G., Rimini, A., & Weber, T. (1986). Unified dynamics for microscopic and macroscopic systems.

*Physical Review, D, 34*, 470–479.Hagar, A. (2003). A philosopher looks at quantum information theory.

*Philosophy of Science, 70*(4), 752–775.Hagar, A., & Hemmo, M. (2006). Explaining the unobserved – Why quantum mechanics ain’t only about information.

*Foundations of Physics, 36*(9), 1295–1324.Hemmo, M., & Shenker, O. (2012).

*The road to Maxwell’s demon*. Cambridge: Cambridge University Press.Hemmo, M., & Shenker, O. (2016).

*Maxwell’s demon*. Oxford University Press: Oxford Handbooks Online.Hemmo, M. & Shenker, O. (2019).

*Why quantum mechanics is not enough to set the framework for its interpretation*, forthcoming.Joos, E., Zeh, H. D., Giulini, D., Kiefer, C., Kupsch, J., & Stamatescu, I. O. (2003).

*Decoherence and the appearance of a classical world in quantum theory*. Heidelberg: Springer.Pitowsky, I. (2003). Betting on the outcomes of measurements: A Bayesian theory of quantum probability.

*Studies in History and Philosophy of Modern Physics, 34*, 395–414.Kolmogorov, A. N. (1933).

*Foundations of the Theory of Probability*. New York: Chelsea Publishing Company, English translation 1956.Pitowsky, I. (2007). Quantum mechanics as a theory of probability. In W. Demopoulos & I. Pitowsky (Eds.),

*Physical theory and its interpretation: Essays in honor of Jeffrey Bub*(pp. 213–240). Berlin: Springer.Ramsey, F. P. (1990).

*Truth and Probability*. (1926); reprinted in Mellor, D.H. (ed.), F. P. Ramsey: Philosophical Papers. Cambridge: Cambridge University Press.Tumulka, R. (2006). A relativistic version of the Ghirardi–Rimini–Weber model.

*Journal of Statistical Physics, 125*, 821–840.von Neumann, J. (2001). Unsolved problems in mathematics. In M. Redei & M. Stoltzner (Eds.),

*John von Neumann and the foundations of quantum physics*(pp. 231–245). Dordrecht: Kluwer Academic Publishers.

## Acknowledgement

We thank Guy Hetzroni and Cristoph Lehner for comments on an earlier draft of this paper. This research was supported by the Israel Science Foundation (ISF), grant number 1114/18.

## Author information

### Authors and Affiliations

### Corresponding author

## Editor information

### Editors and Affiliations

## Rights and permissions

## Copyright information

© 2020 Springer Nature Switzerland AG

## About this chapter

### Cite this chapter

Hemmo, M., Shenker, O. (2020). Quantum Mechanics as a Theory of Probability. In: Hemmo, M., Shenker, O. (eds) Quantum, Probability, Logic. Jerusalem Studies in Philosophy and History of Science. Springer, Cham. https://doi.org/10.1007/978-3-030-34316-3_15

### Download citation

DOI: https://doi.org/10.1007/978-3-030-34316-3_15

Published:

Publisher Name: Springer, Cham

Print ISBN: 978-3-030-34315-6

Online ISBN: 978-3-030-34316-3

eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)