Abstract
This paper is concerned with the nature of probability in physics, and in quantum mechanics in particular. It starts with a brief discussion of the evolution of Itamar Pitowsky’s thinking about probability in quantum theory from 1994 to 2008, and the role of Gleason’s 1957 theorem in his derivation of the Born Rule. Pitowsky’s defence of probability therein as a logic of partial belief leads us into a broader discussion of probability in physics, in which the existence of objective “chances” is questioned, and the status of David Lewis influential Principal Principle is critically examined. This is followed by a sketch of the work by David Deutsch and David Wallace which resulted in the Deutsch-Wallace (DW) theorem in Everettian quantum mechanics. It is noteworthy that the authors of this important decision-theoretic derivation of the Born Rule have different views concerning the meaning of probability. The theorem, which was the subject of a 2007 critique by Meir Hemmo and Pitowsky, is critically examined, along with recent related work by John Earman. Here our main argument is that the DW theorem does not provide a justification of the Principal Principle, contrary to the claims by Wallace and Simon Saunders. A final section analyses recent claims to the effect that the DW theorem is redundant, a conclusion that seems to be reinforced by consideration of probabilities in “deviant” branches of the Everettian multiverse.
Chance, when strictly examined, is a mere negative word, and means not any real power which has anywhere a being in nature. David Hume (Hume 2008)
[The Deutsch-Wallace theorem] permits what philosophy would hitherto have regarded as a formal impossibility, akin to deriving an ought from an is, namely deriving a probability statement from a factual statement. This could be called deriving a tends to from a does. David Deutsch (Deutsch 1999)
[The Deutsch-Wallace theorem] is a landmark in decision theory. Nothing comparable has been achieved in any chance theory. …[It] is little short of a philosophical sensation …it shows why credences should conform to [quantum chances]. Simon Saunders (Saunders 2020)
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20 May 2020
In the original version of the book, there was a spelling error in the chapter title wherein “Everettian” was incorrectly spelt as “Eerettian”.
Notes
- 1.
Pitowsky (1994, p. 108).
- 2.
Pitowsky (2006, p. 4).
- 3.
Ibid.
- 4.
We will not discuss here Pitowsky’s 2006 resolution of the familiar measurement problem, other than to say that it relies heavily on his view that the quantum state is nothing more than a device for the bookkeeping of probabilities, and that it implies that “we cannot consistently maintain that the proposition ’the [Schrödinger] cat is alive’ has a truth value. Op. cit. p. 28. For a recent defence of the view that the quantum state is ontic, and not just a bookkeeping device, see Brown (2019).
- 5.
The identity is
$$\displaystyle \begin{aligned} \bigcup_{i = 1}^{k} \left( \{B=b_j\}\cap \{ A=a_i\} \right) = \{ B=b_j\} = \bigcup_{i = 1}^{l} \left( \{B=b_j\}\cap \{ C=c_i\} \right) \end{aligned} $$(7.1)where A, B and C measurements have possible outcomes a 1, a 2, …, a k; b 1, b 2, …, b r and c 1, c 2, …, c l, respectively.
- 6.
Op. cit. Footnote 2.
- 7.
For reasons to be skeptical ab initio about non-contextualism in hidden variable theories, see Bell (1966).
- 8.
- 9.
See for example Brown and Svetlichny (1990).
- 10.
- 11.
The fact that it is the same parameter in each case makes the phenomenon even more remarkable.
- 12.
There are good reasons, however, for considering the source of the unpredictability in some if not all classical chance processes as having a quantum origin too; see Albrecht and Phillips (2014). But whether strict indeterminism is actually in play in the quantum realm is far from clear, as we see below.
- 13.
Gillies (2000), pp. 177, 178.
- 14.
Gillies (1972), pp. 150, 151.
- 15.
Maudlin (2007).
- 16.
Wallace (2012), p. 137.
- 17.
Op. cit., p. 138. It should be noted, however, that although Wallace claims that the certain probabilities in statistical mechanics must be objective, he accepts that such a claim is hard to make sense of without bringing in quantum mechanics (see Wallace 2014). For further discussion of Wallace’s views on probability in physics, see (Brown, 2017).
- 18.
- 19.
This last approach is due to David Lewis (1980); an interesting analysis within the Everettian picture of a major difficulty in the best system approach, namely “undermining”, is found in Saunders (2020). It should also be recognised that the best system approach involves a judicious almagam of criteria we impose on our description of the world, such as simplicity and strength; the physical laws and probabilities in them are not strict representations of the world but rather our systematization of it. Be that as it may, Lewis stressed that whatever physical chance is, it must satisfy his Principal Principle linking it to subjective probability; see below.
- 20.
Relatively few philosophers follow Hume and doubt the existence of chances in the physical world; one is Toby Handfield, whose lucid 2012 book on the subject (Handfield 2012) started out as a defence and turned into a rejection of chances. Another skeptic is Ismael (1996); her recent work (Ismael 2019) puts emphasis on the kind of weak objectivity mentioned at the start of this section. See also in this connection recent work of Bacciagaluppi, whose view on the meaning of probability is essentially the same as that defended in this paper: (Bacciagaluppi, 2020), Sect. 10.
- 21.
The half-life is not, pace Google, the amount of time needed for a given sample to decay by one half! (What if the number of nuclei is odd?) It is the amount of time needed for any of the nuclei in the sample to have decayed with a probability of 0.5. Of course this latter definition will be expected to closely approximate the former when the sample is large, given the law of large numbers (see below).
- 22.
In 2011, (Maris et al. 2011) showed for the first time, with the aid of a supercomputer, that allowing for nucleon-nucleon-nucleon interactions in a dynamical no-core shell model of beta decay explains the long lifetime of 14C.
- 23.
de Finetti (1964), p. 117. de Finetti’s argument, as Gillies (2000) stresses on pages 103 and 159, was made in the context of the falsifiability of probabilistic predictions, even in the case of subjective probabilities. But the argument also holds in the case of inferring “objective” probabilities by way of frequencies.
- 24.
Saunders puts the point succinctly:
Chance is measured by statistics, and perhaps, among observable quantities, only by statistics, but only with high chance. (Saunders 2010)
This point is repeated in Saunders (2020, Sect. 7.2). To repeat, this “high chance” translates into subjective confidence; Saunders, like many others, believes that chances and subjective probabilities are linked by way of the “Principal Principle” (see below). Wallace’s 2012 version of the law of large numbers explicitly involves both “personal” and objective probabilities. His personal prior probabilities Pr(⋅|C) are conditional on the proposition expressing all of his current beliefs, denoted by C. X p denotes the hypothesis that the objective probability of E n is p, where as usual it is assumed that p is an iid. Then, Wallace claims, the Principal Principle ensures that his personal probability of the proposition Y i, that heads occurs in the ith repetition of the random process, is p: symbolically Pr(Y i|X p&C) = p. Given the iid assumption for p and the usual updating rule for personal probabilities, it follows from combinatorics that the personal probability Pr(K M|X p&C) is very small unless p = M∕N, where K M is the proposition that the experiment results in M heads out of N. Wallace concludes that
…as more and more experiments are carried out, any agent conforming to the Principal Principle will become more and more confident that the objective probability is close to the observed relativity frequency. (Wallace 2012, p. 141.)
Note that there are versions of the “objective” position that avoid the typicality assumption mentioned above, such as that defended in Gillies (2000), Chap. 7, and (Greaves and Myrvold 2010). For more discussion of the latter, see Sect. 2.3 (ii) below.
- 25.
This is not to say that the law of large numbers allows for probabilities to be defined in terms of frequencies, a claim justly criticised in Gillies (1973), pp. 112–116.
- 26.
Feynman et al. (1965), Sect. 6.1.
- 27.
- 28.
Feynman et al. (1965), Sect. 6.3. We are grateful to Jeremy Steeger for bringing this passage to our attention. For his careful defence of a more objective variant of Feynman’s notion of probability, see (Steeger, ’The Sufficient Coherence of Quantum States’ [unpublished manuscript]).
- 29.
See Page (1995). Page has also defended the view that truly testable probabilities are limited to conditional probabilities associated with events defined at the same time (and perhaps the same place); see Page (1994). This view is motivated by the fact that strictly speaking we have no direct access to the past or future, but only present records of the past, including of course memories. The point is well-taken, but radical skepticism about the veracity of our records/memories would make science impossible, and as for probabilities concerning future events conditional on the past (or rather our present memories/records of the past) they will be testable when the time comes, given the survival of the relevant records/memories. So it is unclear whether recognition of Page’s concern should make much difference to standard practice.
- 30.
- 31.
Greaves and Myrvold (2010), Sect. 3.3.
- 32.
It is noteworthy that in Footnote 4 (op. cit.), Greaves and Myrvold entertain the view that “it is via [the Principal Principle] that we ascribe beliefs about chances to the agent”. This Principle is the topic of the next section of our paper, in which we report Myrvold’s more recent claim that it is not about chances at all.
- 33.
Greaves and Wallace also take issue (p. 22) with the claim made in the final sentence of Sect. 2.2 above that objectivists need to introduce the substantive additional assumption that the statistical data is “typical”. As far as we can see, this is again because the notion of chance they are considering is ontologically neutral.
- 34.
See de Finetti (1964, pp. 146, 147).
- 35.
Gillies (2000, pp. 74, 83, 84) and Wallace (2012, p. 138), point out that de Finetti’s scheme of updating by Bayesian conditionalisation yields reasonable results only if the prior probability function is appropriate. If it isn’t, no amount of future evidence can yield adequate predictions based on Bayesian updating. This a serious concern, which will not be dealt with here (see in this connection Greaves and Myrvold 2010). Note that it seems to be quite different from the reason raised by Wallace (see Sect. 2.1 above) as to why the half-life of a radioactive isotope, for example, must be an objective probability. See also Sect. 7.5.1 below.
- 36.
Myrvold, W. C., Beyond Chance and Credence [unpublished manuscript], Chap. 5.
- 37.
- 38.
- 39.
See Gillies (2000, chapter 4).
- 40.
This version is essentially the same as that in Wallace p. 140. Note that Wallace here (i) does not strictly equate the PP with the decision-theoretic link and (ii) sees PP as underpinning not just the decision-theoretic link but also the inferential link (see Footnote 24 above). We return to point (i) below; we are also overlooking here the usual subtleties involved with characterising the background information as “accessible”. For a more careful discussion of the PP, see (Bacciagaluppi 2020).
- 41.
Hoefer (2019), Sect. 1.3.4.
- 42.
Wallace (2012), Sects. 4.10, 4.12.
- 43.
- 44.
Myrvold, ‘Beyond Chance and Credence’ [unpublished manuscript], Sect. 2.5.
- 45.
See Brown (2011). In this paper it is briefly argued that the related philosophical “problem of induction” should be seen as a pseudo-problem.
- 46.
See Gillies (2000, chapter 2) for a useful introduction to the classical theory.
- 47.
See Wallace (2012, section 4.11).
- 48.
A lucid discussion is found in Gillies (2000, pp. 37–48), which contains an insightful critique of Jaynes’ defence of the principle of indifference in physics – and conceding that the principle has been successfully applied in a number of cases in physics.
- 49.
- 50.
This is the problem of understanding how probabilities can come about when everything that can happen does happen. See Wallace (2012, pp. 40, 41).
- 51.
- 52.
Similar qualms were voiced by Hemmo and Pitowsky (Hemmo and Pitowsky 2007).
- 53.
Wallace (2012), p. 196.
- 54.
Op. cit. p. 196.
- 55.
Op. cit. p. 214.
- 56.
Op. cit. p. 197.
- 57.
Ibid.
- 58.
Ibid. Note that a different, recent approach to proving that credences should be non-contextual in quantum mechanics is urged by Steeger (Steeger, J., ’The Sufficient Coherence of Quantum States’ [unpublished manuscript]).
- 59.
It was mentioned in Sect. 7.2 above that contextual probabilities may lead to the possibility of superluminal signalling. But this does not imply that contextualism is irrational. Indeed, violation of no-signalling is bound to happen in some “deviant” branches in the Everett multiverse; see Sect. 7.6 below.
- 60.
Perhaps this should be understood as an ‘emancipated’ PP, given that Lewis himself did not believe in the existence of chances in a deterministic universe.
- 61.
Wallace (2012), Footnote 26, pp. 150–151.
- 62.
Saunders (2010), p. 184.
- 63.
Wallace (2012), p. 237.
- 64.
Wallace (2012), pp. 208–210 and Appendix D.
- 65.
Wallace (2002), Sect. 2.7.
- 66.
In his 2012 book, Wallace does acknowledge this point; see Wallace (2012, p. 138).
- 67.
Wallace (2012), p. 141.
- 68.
Saunders (2010), p. 184.
- 69.
See Saunders (2020); this paper also contains a rebuttal of the claim that the process of decoherence, so essential to the meaning of branches in EQM, itself depends on probability assumptions.
- 70.
Saunders (2010).
- 71.
Wallace (2012), p. 144.
- 72.
Op. cit. Chap. 6.
- 73.
Op. cit. p. 234.
- 74.
Op. cit. p. 229.
- 75.
Ibid.
- 76.
Saunders (2010), p. 182.
- 77.
Simon Saunders, private communication.
- 78.
Wallace (2012), p. 144. Compare this with the views expressed in Sect. 2.1 above.
- 79.
Wallace 2012, p. 249.
- 80.
Wallace, ibid, compares branch weights with physical money, in our view a very apt analogy. A dollar note, e.g., has no more intrinsic status as money than branch weights have the status of probability, until humans confer this status on it. This point is nicely brought out by Harari, who writes that the development of money “involved the creation of a new inter-subjective reality that exists solely in people’s shared imagination.” See Harari 2014, chapter 10.
- 81.
Wallace describes the Everettian program as interpreting the “bare quantum formalism” – which itself makes no reference to probability (Wallace 2012, p. 16) – in a “straightforwardly realist way” without modifying quantum mechanics (op. cit., p. 36).
- 82.
- 83.
Our approach to probability in EQM is very similar to the “pragmatic” view of Bacciagaluppi and Ismael, in their thoughtful 2015 review of Wallace’s 2012 book (of which more in Sect. 7.6 below):
According to such a view, the ontological content of the theory makes no use of probabilities. There is a story that relates the ontology to the evolution of observables along a family of decoherent histories, and probability is something that plays a role in the cognitive life of an agent whose experience is confined to sampling observables along such a history. In so doing, one would still be doing EQM …(Bacciagaluppi and Ismael 2015, Sect. 3.2).
However, we do not follow these authors in their attempt to define chances consonant with such probabilities and the PP. (We are grateful to David Wallace for drawing to our attention this review paper.)
- 84.
Op. Cit. p. 16.
- 85.
Recall that in the typical case of the infinite dimensional, separable Hilbert space \(\mathcal {H}\), a condition of Gleason’s theorem is that the probability function must be countably additive; see Sect. 7.2 above.
- 86.
See, for instance, (Wallace 2012, section 4.10).
- 87.
But recall the complication referred to in footnote 64 above.
- 88.
See Gillies (2000, chapter 7).
- 89.
In his 1999 paper, Deutsch assumed that agents involved in his argument were initially adherents of the non-probabilistic part of Everettian theory. Wallace was influenced by the work of Greaves and Myrvold (Greaves and Myrvold 2010) who developed a confirmation theory suitable for branching as well as non-branching universes, but he went on to develop his own confirmation theory as part of what he called a “unified approach” to probabilities in EQM (see Sect. 7.4.1(ii) above). However, Deutsch, in his 2016 paper, follows Popperian philosophy and rejects any notion of theory confirmation, thereby explicitly sidelining the work of Greaves and Myrvold, despite the fact that it contains a falsifiability element as well as confirmation.
- 90.
- 91.
Read (2018), p. 138.
- 92.
Wallace (2012), p. 226.
- 93.
However, Hemmo and Pitowsky also argue that
…in the many worlds theory, not only Born’s rule but any probability rule is meaningless. The only way to solve the problem …is by adding to the many worlds theory some stochastic element. (Hemmo and Pitowsky 2007, p. 334)
This follows from the claim that the shared reality of all of the multiple branches in EQM entails “one cannot claim that the quantum probabilities might be inferred (say, as an empirical conjecture) from the observed frequencies.” (p. 337). This is a variant of the “incoherence” objection to probabilities in EQM (see footnote 45 above), but in our opinion it is not obviously incoherent to bet on quantum games in a branching scenario (see Wallace 2010).
- 94.
Although in their above-mentioned review, Bacciagaluppi and Ismael do not go quite this far, they do not regard the DW-theorem as necessary for “establishing the intelligibility of probabilities” in EQM. By appealing to standard inductive practice based on past frequencies, they argue:
- 95.
Read (2018), p. 140. Note that Read, like Dawid and Thébault, rests the rationality of the statistics-driven route on the Bayesian confirmation analysis given by Greaves and Myrvold op. cit..
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Acknowledgements
The authors are grateful to the editors for the invitation to contribute to this volume. HRB would like to acknowledge the support of the Notre Dame Institute for Advanced Study; this project was started while he was a Residential Fellow during the spring semester of 2018. Stimulating discussions over many years with Simon Saunders and David Wallace are acknowledged. We also thank David Deutsch, Chiara Marletto, James Read and Christopher Timpson for discussions, and Guido Bacciagaluppi, Donald Gillies and Simon Saunders for helpful critical comments on the first draft of this paper. We dedicate this article to the memory of Itamar Pitowsky, who over many years was known to and admired by HRB, and to Donald Gillies, teacher to HRB and guide to the philosophy of probability.
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Brown, H.R., Porath, G.B. (2020). Everettian Probabilities, The Deutsch-Wallace Theorem and the Principal Principle. 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_7
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