Abstract
It is shown that by realizing the isomorphism features of the frequency and geometric interpretations of probability, Reichenbach comes very close to the idea of identifying mathematical probability theory with measure theory in his 1949 work on foundations of probability. Some general features of Reichenbach’s axiomatization of probability theory are pointed out as likely obstacles that prevented him making this conceptual move. The role of isomorphisms of Kolmogorovian probability measure spaces is specified in what we call the “Maxim of Probabilism”, which states that a necessary condition for a concept to be probabilistic is its invariance with respect to measure-theoretic isomorphisms. The functioning of the Maxim of Probabilism is illustrated by the example of conditioning via conditional expectations.
Highlights
Probability theory featured prominently in Reichenbach’s thinking and work throughout his whole career, both as a conceptual tool and as a subject of philosophical investigation: Already in his doctoral thesis (1915) (published in English translation in 2008 Reichenbach (1915), see Padovani (2011) for a compact review of this translation) probability takes center stage in the form of a “principle of lawful distribution”
The conditional probability values provided by the conditional expectation lacks the ambiguity involved in versions. This can be expressed in terms of the Maxim of Probabilism: Using the fact that a mod0 isomorphism generates an isomorphism of the spaces of equivalence classes of integrable random variables, one can show (Gyenis and Rédei (2020)) that the conditional expectation is invariant with respect to mod0 isomorphisms – and so are the conditional probabilities defined by it in the manner of (12)
The Maxim of Probabilism only gives a necessary condition to be satisfied by a concept in order to qualify as probabilistic
Summary
Probability theory featured prominently in Reichenbach’s thinking and work throughout his whole career, both as a conceptual tool and as a subject of philosophical investigation: Already in his doctoral thesis (1915) (published in English translation in 2008 Reichenbach (1915), see Padovani (2011) for a compact review of this translation) probability takes center stage in the form of a “principle of lawful distribution” This principle states, roughly, that the empirical relative frequencies of occurrences of events converge to a true limit to be understood as probability. The principle is transcendental because it is not empirically testable – rather it forms the basis of empirical science (Glymour and Eberhardt (2016)) In his subsequent works, Reichenbach both applied probability theory in the analysis of specific philosophical problems and investigated the foundations of probability theory itself.
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