1st Edition

Bertrand’s Paradox and the Principle of Indifference

By Nicholas Shackel Copyright 2024
    390 Pages 27 B/W Illustrations
    by Routledge

    Events between which we have no epistemic reason to discriminate have equal epistemic probabilities. Bertrand’s chord paradox, however, appears to show this to be false, and thereby poses a general threat to probabilities for continuum sized state spaces. Articulating the nature of such spaces involves some deep mathematics and that is perhaps why the recent literature on Bertrand’s Paradox has been almost entirely from mathematicians and physicists, who have often deployed elegant mathematics of considerable sophistication. At the same time, the philosophy of probability has been left out. In particular, left out entirely are the philosophical ground of the principle of indifference, the nature of the principle itself, the stringent constraint this places on the mathematical representation of the principle needed for its application to continuum sized event spaces, and what these entail for rigour in developing the paradox itself. This book puts the philosophy and its entailments back in and in so doing casts a new light on the paradox, giving original analyses of the paradox, its possible solutions, the source of the paradox, the philosophical errors we make in attempting to solve it and what the paradox proves for the philosophy of probability. The book finishes with the author’s proposed solution—a solution in the spirit of Bertrand’s, indeed—in which an epistemic principle more general than the principle of indifference offers a principled restriction of the domain of the principle of indifference.

    Bertrand's Paradox and the Principle of Indifference will appeal to scholars and advanced students working in the philosophy of mathematics, epistemology, philosophy of science, probability theory and mathematical physics.

    1. The Principle of Indifference

    2. The Principle of Indifference for Sets

    3. Bertrand’s Paradoxes

    4. The Threat to the Principle and Four Kinds of Solution

    5. The Distinction Strategy

    6. The Well‑posing Strategy

    7. The Irrelevance Strategy

    8. The Maximum Entropy Principle

    9. The Universal Average

    10. Meta‑indifference

    11. Permissivism

    12. Uniqueness a Criterion of Identity

    13. Symmetry: The Forlorn Hope

    14. Unearthing the Root

    15. Bertrand’s Temptations

    16. Rational Strength


    Nicholas Shackel is Professor of Philosophy, Cardiff University and Distinguished Research Fellow at the Oxford Uehiro Centre for Practical Ethics, Oxford University. His research is mainly on paradoxes and rationality. He has published numerous articles in books and leading journals including Journal of Philosophy, Mind, and Philosophy and Phenomenological Research.

    “This is a very useful resource for graduate students and researchers interested in one of the most challenging puzzles in the theory of probability.”

    Hykel Hosni, University of Milan, Italy

    “This is essential reading for anyone seriously interested in Bertrand’s chord paradox or the broader epistemic issue of the status of the principle of indifference (and the maximum entropy principle.”

    Darrell P. Rowbottom, Lingnan University, Hong Kong