Within the framework of the Bar-Hillel Colloquium for the History, Philosophy and Sociology of Science 2015-2016
The classical interpretation of probability together with the Principle of Indifference are formulated in terms of probability measure spaces in which the probability is given by the Haar measure. A notion called Labeling Invariance is defined in the category of Haar probability spaces: it is shown that Labeling Invariance is violated and Bertrand's Paradox is interpreted as the very proof of violation of Labeling Invariance. It is argued that, under the suggested interpretation of Bertrand's Paradox, the paradox does not undermine either the Principle of Indifference or the classical interpretation, and is in complete harmony with how mathematical probability theory is used in the sciences to model phenomena. It is shown in particular that violation of Labeling Invariance does not entail that the labeling of random events affects the probabilities of random events. It is also argued, however, that the content of the Principle of Indifference cannot be specified in such a way that it can establish the classical interpretation of probability as descriptively accurate or predictively successful.
The talk is based on the paper: Z. Gyenis and M. Rédei: "Defusing Bertrand's Paradox", The British Journal for the Philosophy of Science, online, 2015.