Today, the Stanford Encyclopedia of Philosophy (SEP) published a substantive revision of its article “Interpretations of Probability,” by Alan Hájek (originally published in 2002). I hope the article would be of interest to folks visiting my blog for its content on probability—especially of the sort that doesn’t appear philosophically fraught on its surface. The SEP entry provides a nice introduction to the relationship between the basic axioms of probability and the major competing schools of interpretation, which Hájek designates as follows, at least for the sake of guiding the discussion:
1. An epistemological concept, which is meant to measure objective evidential support relations. For example, “in light of the relevant seismological and geological data, California will probably experience a major earthquake this decade.”
2. The concept of an agent’s degree of confidence, a graded belief. For example, “I am not sure that it will rain in Canberra this week, but it probably will.”
3. A physical concept that applies to various systems in the world, independently of what anyone thinks. For example, “a particular radium atom will probably decay within 10,000 years.”
Hájak of course acknowledges that the lines between these are “somewhat permeable” and that there is internal disagreement among proponents of each category. These and other such observations point to what an immensely difficult topic probability is, at its roots. As its put in the article’s opening quotation:
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means. –Bertrand Russell, 1929 Lecture1
The article also includes a brief mention of interpretative issues surrounding physical probabilities being viewed as fundamental within certain interpretations of quantum mechanics. What interests me most in that context is how to make sense of probability under an Everettian—or many-worlds—interpretation, in which every outcome of an experiment that can occur does occur in some world; or, more simply, everything that can happen does happen. (I touch on some of my puzzlement over this my post, “Some Naive Questions About Many-Worlds Quantum Mechanics.”)
Hájak doesn’t touch on this subject, but does link to a 2016 article called “Philosophical Issues in Quantum Theory,” which in turn links to a more focused article called “Many-Worlds Interpretation of Quantum Mechanics,” which has a section on probability that describes the problem as follows:
The difficulty with the concept of probability in a deterministic theory, such as the [many-worlds interpretation], is that the only possible meaning for probability is an ignorance probability, but there is no relevant information that an observer who is going to perform a quantum experiment is ignorant about. The quantum state of the Universe at one time specifies the quantum state at all times. If I am going to perform a quantum experiment with two possible outcomes such that standard quantum mechanics predicts probability 1/3 for outcome A and 2/3 for outcome B, then, according to the MWI, both the world with outcome A and the world with outcome B will exist. It is senseless to ask: “What is the probability that I will get A instead of B?” because I will correspond to both “Lev”s: the one who observes A and the other one who observes B.
We are then given a synopsis to some interesting solutions to the problem, with suggestions for further reading.2
I’d love to see an entire SEP article dedicated to the question of probability in the many-worlds context, particularly as my impression is that more and more physicists of (cultural) influence seem to not only be on board with the idea, but are aiming to convince the rest of us to get on board with it as well. (See, for example, Sean Carroll’s upcoming book Something Deeply Hidden: Quantum Worlds and the Emergence of Spacetime, which I’ve already preordered.)
Ok, enough about all that (for now).
For more introductory reading on the philosophy of probability, I recommend Aidan Lyon’s colorful 2010 article, “Philosophy of Probability” (from the 2010 volume Philosophies of the Sciences: A Guide, Allhoff (ed.), Wiley-Blackwell, pp. 92–127), available on Lyon’s website. The article covers similar ground as that of the above SEP entry—the two authors cite each other in their respective articles, and Lyon thanks Hájek, in footnote number 24, for “many discussions and feedback on earlier drafts”—but it’s nice to read both to note differences in style as well as in how the concepts are carved up and for what a given author chooses to emphasize (
Lyon one-ups compare Hájek’s 0.999 credence in a sentence ruling as rationally questionable with Lyon’s example of assigning 0.9999999 to the Earth being flat).
Lyon also includes one of my favorite quotes on probability, this one from Pierre-Simon Laplace’s 1814 book, A Philosophical Essay on Probabilities: “the most important questions of life, are indeed for the most part only problems of probability.” (Read it for free at Google Books.)
To go deeper into the topic, I recommend the 2011 anthology, Philosophy of Probability: Contemporary Readings (Eagle (ed.), Routledge). It is, according to the back cover, “the first anthology to collect essential readings in this important area of philosophy. Featuring the work of leading philosophers in the field such as Carnap, Hájek, Jeffrey, Joyce, Lewis, Loewer, Popper, Ramsey, van Fraassen, von Mises, and many others…” I’ve gotten a lot out of this collection. I’m sure there are other good ones out there, like The Oxford Handbook of Probability and Philosophy (2016, Hájek & Hitchcock (eds.), Oxford), but I haven’t read that one as it’s out of my price range and one can afford so many of these expensive books.
Finally, for going deeper into the topic by way of its historical dimensions, I’ll enthusiastically recommend a book (referenced in Hájek’s SEP article) that I recently finished, in which James Franklin persuasively argues, among other things, that probability as we much think of and rely on it today—namely, as a largely qualitative evaluation of uncertain evidence and risk—pre-existed the quantitative turn of the 17th century: The Science of Conjecture: Evidence and Probability before Pascal (2015 edition, Johns Hopkins University Press).
I’ll resist the temptation to list more. What works in the philosophy of probability do you recommend?
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- Cited on page 587 of E.T. Bell’s 1945 book The Development of Mathematics (2nd edition, McGraw-Hill Book Company/Dover).
- They are:
Albert, D. and Loewer, B., 1988, ‘Interpreting the Many Worlds Interpretation‘, Synthese, 77: 195–213.
Saunders, S., 2010, ‘Chance in the Everett Interpretation’, in S. Saunders, J. Barrett, A. Kent and D. Wallace (eds.), Many Worlds? Everett, Quantum Theory, & Reality, Oxford and New York: Oxford University Press, pp. 181–205. (Note that this one also includes a chapter by David Albert.)
Wilson, A., 2013, ‘Objective Probability in Everettian Quantum Mechanics’, British Journal for the Philosophy of Science, 64: 709–737.
Kent, A., 2010, ‘One World Versus Many: The Inadequacy of Everettian Accounts of Evolution, Probability, and Scientific Confirmation’, in S. Saunders, J. Barrett, A. Kent and D. Wallace (eds.), Many Worlds? Everett, Quantum Theory, & Reality, Oxford and New York: Oxford University Press, pp. 307–354.