A finite data set is consistent with infinitely many alternative theories. Scientific realists recommend that we prefer the simplest one. Anti-realists ask how a fixed simplicity bias could track the truth when the truth might be complex. It is no solution to impose a prior probability distribution biased toward simplicity, for such a distribution merely embodies the bias at issue without explaining its efficacy. In this note, I argue, on the basis of computational learning theory, that a fixed simplicity bias is necessary if inquiry is to converge to the right answer efficiently, whatever the right answer might be. Efficiency is understood in the sense of minimizing the least fixed bound on retractions or errors prior to convergence. Keywords: learning, induction, simplicity, Ockham’s razor, realism, skepticism 1

Many distinct theories are compatible with current experience. Scientific realists recommend that we choose the simplest. Anti-realists object that such appeals to “Ockham’s razor ” cannot be truth-conducive, since they lead us astray in complex worlds. I argue, on behalf of the realist, that always preferring the simplest theory compatible with experience is necessary for efficient convergence to the truth in the long run, even though it may point in the wrong direction in the short run. Efficiency is a matter of minimizing errors or retractions prior to convergence to the truth.

In science, one faces the problem of selecting the true theory from a range of alternative theories. The typical response is to select the simplest theory compatible with available evidence, on the authority of “Ockham’s Razor”. But how can a fixed bias toward simplicity help one find possibly complex truths? A short survey of standard answers to this question reveals them to be either wishful, circular, or irrelevant. A new explanation is presented, based on minimizing the reversals of opinion prior to convergence to the truth. According to this alternative approach, Ockham’s razor does not inform one which theory is true but is, nonetheless, the uniquely most efficient strategy for arriving at the true theory, where efficiency is a matter of minimizing reversals of opinion prior to finding the true theory. 1

We describe a machine-learning system that uses linear vector-space based techniques for inference from observations to extend previous work on model construction for particle physics [11, 10, 6]. The program searches for quantities conserved in all reactions from a given input set; given current data it rediscovers the family conservation laws: baryon#, electron#, muon # and tau#. We show that these families are uniquely determined by the data; they are the only particle families that correspond to a complete set of selection rules. 1

Contents 1 Induction: The Problem and How To Solve It 7 1.1 The Problem of Induction . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Hypothetical Imperatives for Inductive Inference . . . . . . . . . 9 1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.1 Means-Ends Vindications of Traditional Proposals . . . . 11 1.3.2 Novel Solutions to Traditional Problems . . . . . . . . . . 11 1.3.3 New Questions and Answers . . . . . . . . . . . . . . . . . 12 1.3.4 Analysis of Inductive Problems from Scientific Practice . 14 1.3.5 Rational Choice in Games . . . . . . . . . . . . . . . . . . 14 1.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 A Model of Scientific Inquiry 17 2.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 A Model of Scientific Inquiry . . . . . . . . . . . . . . . . . . . . 18 2.3 Examples of Inductive Problems and Scientific Methods . . . . . 26

Particle physics experiments, like the Large Hadron Collider in Geneva, can generate thousands of data points listing detected particle reactions. An important learning task is to analyze the reaction data for evidence of conserved quantities and hidden particles. This task involves latent structure in two ways: first, hypothesizing hidden quantities whose conservation determines which reactions occur, and second, hypothesizing the presence of hidden particles. We model this problem in the classic linear algebra framework of automated scientific discovery due to Valdés-Pérez, ˙Zytkow and Simon, where both reaction data and conservation laws are represented as matrices. We introduce a new criterion for selecting a matrix model for reaction data: find hidden particles and conserved quantities that rule out as many interactions among the nonhidden particles as possible. A polynomial-time algorithm for optimizing this criterion is based on the new theorem that hidden particles are required if and only if the Smith Normal Form of the reaction matrix R contains entries other than 0 or 1. To our knowledge this is the first application of Smith matrix decomposition to a problem in AI. Using data from particle accelerators, we compare our algorithm to the main model of particles in physics, known as the Standard Model: our algorithm discovers conservation laws that are equivalent to those in the Standard Model, and indicates the presence of a hidden particle (the electron antineutrino) in accordance with the Standard Model.