In this collection of previously published papers on mathematical probability theory, Zabell addresses the idea of symmetry from philosophical, historical, and mathematical points of view. Arguments such as the following are considered:
[I]f future outcomes are viewed as exchangeable, i.e., no pattern is viewed as any more or less likely than any other (with the same number of successes), then when an event occurs with a certain frequency in an initial segment of the future, we must, if we are to be consistent, think it likely that event will occur with approximately the same frequency in later trials. Conversely, if we do not accept this, it means that we must have – prospectively — thought certain patterns more likely than others. (p. 6)
This approach, Zabell notes, is a key departure (“a chasm … had to be crossed” (p. 16)) from influential Greek thought in which an inability to place the likelihood of occurrence of one event over that of another could not be used to forecast the future. Breaking from this tradition helped set the stage for the emergence of the key idea of a fundamental probability set. Zabell writes that this is
a partition of the space of possible outcomes into equiprobable outcomes. The recognition and use of such sets to compute numerical probabilities for complex events was a key step in the birth of mathematical probability. Once the ability to calculate probabilities in this simple case had been mastered, the outlines of the mathematical theory discerned, and its practical utility recognized, all else followed. (p. 13)
Symmetry assumptions have provoked much theoretical discussion. Zabell remarks after referring to an observation by the philosopher Goodman — “A rule is amended if it yields an inference we are unwilling to accept; an inference is rejected if it amends a rule we are unwilling to amend” (p. 64):
Symmetry assumptions must therefore be tested in terms of the particular inferences they give rise to. But — and this is the rub — particular inferences can only be reasonably judged in terms of particular situations, whereas symmetry assumptions are often proposed in abstract and theoretical settings devoid of concrete specifics. (p. 29)
This and the rest of the short discussion (in which Zabell notes symmetry assumptions were in practice modified) coming after Goodman’s observation comprise one of the book’s many insightful passages.
Goodman, N. (1979). Fact, Fiction, and Forecast, 3rd edition. Indianapolis, Hackett.
Dennis Lomas (firstname.lastname@example.org) has studied computer science (MSc), mathematics (half dozen, or so, graduate courses), and philosophy (PhD). He resides in Prince Edward Island (Canada).
Part I. Probability: 1. Symmetry and its discontents; 2. The rule of succession; 3. Buffon, Price, and Laplace: scientific attribution in the eighteenth century; 4. W. E. Johnson’s sufficientness postulate. Part II. Personalities: 5 Abraham De Moivre and the birth of the Central Limit Theorem; 6 Ramsey, truth, and probability; 7. R. A. Fisher on the history of inverse probability; 8. R. A. Fisher and the fiducial argument; 9. Alan Turing and the Central Limit Theorem; Part III. Prediction: 10. Predicting the unpredictable; 11. The continuum of inductive methods revised.