On Occam's Razor

\Occam's Razor" is an important metatheoretical principle in science. Mathematical methods from ergodic theory, information theory, and the theory of algorithms and stochastic processes can be used to help clarify the meaning of Occam's Razor and address the philosophical problems it raises. 1 1 \Pluralitas non est ponenda sine nec-cesitate." Thus, William of Occam articulated the famous principle ontological conservatism which bears his name. Although Occam himself was interested in metaphysical questions about the existence of God or Platonic ideals, his maxim has become an important principle of sci-entiic metatheory, usually formulated: \The simplest theory that ts the data is the best one." If we are to take this principle seriously, however, we must answer two questions: 1. What does it mean for one scientiic explanation to be simpler than another? 2. Why is a simpler theory better? There are at least ve diierent responses to the rst question: Ontological Conservatism: The \simplest" theory is the one which postulates the existence of the fewest number of theoretical entities. Predictive Conservatism: The \simplest" theory is the one which claims the fewest possible correlations between events, and thus, advances the fewest possible predictions which could be empirically falsiied. Descriptive Simplicity: The \simplest" theory is the one which can be described most concisely. Computational Simplicity: The \simplest" theory is the one which requires the least computational eeort to explain and predict phenomena. Structural Simplicity: The \simplest" theory is the one possessing the \simplest" mathematical structure, where we characterize the simplicity of a structure in terms of mathematical properties like symmetry, approximability, or dimension. There are two responses to the second question: Pragmatic: A simpler theory is better for purely practical reasons: simpler theories make prediction tasks more straightforward , make easier the communication of models and interpretations between scientists, involve less error-prone computations, etc. 2 Epistemological: A simpler theory is more likely to be true, by virtue of its simplicity. The Pragmatic justiication seems quite plausible and unproblem-atic. The Epistemological justiication, however, raises a diicult questions. How can we evaluate claims about the relative \likelihood" of scientiic theories? If the empirical evidence gives equal support to two theories, why should one be deemed more likely on a priori grounds? Nonetheless, it is the second, stronger interpretation which seems ubiquitous within the scientiic community. We need to construct a philosophical framework wherein we can at least make sense of this interpretation, and hopefully even …