The Small Number System

I argue that the human mind includes an innate domain-specific system for representing precise small numerical quantities. This theory contrasts with object-tracking theories and with domain-general theories that only make use of mental models. I argue that there is a good amount of evidence for innate representations of small numerical quantities and that such a domain-specific system has explanatory advantages when infants’ poor working memory is taken into account. I also show that the mental models approach requires previously unnoticed domain-specific structure and consequently that there is no domain-general alternative to an innate domain-specific small number system

Safe Minimum Standards in Dynamic Resource Problems—Conditions for Living on the Edge of Risk

Abstract Safe Minimum Standards (SMSs) have been advocated as a policy rule for environmental problems where uncertainty about risks and consequences are thought to be profound. This paper explores the rationale for such a policy within a dynamic framework and derives conditions for when SMS can be summarily dismissed as a policy choice and for when SMS can be defended as an optimal policy based on standard economic criteria. We have determined that these conditions can be checked with quite limited information about damages and risks. In order to analyze the SMSs in a dynamic setting, we have developed a method for solving optimal control problems where the state space is divided into risky and non-risky subsets.safe minimum standards; optimal control; critical zone; threshold effects; mixed risk spaces

A reassessment of the shift from the classical theory of concepts to prototype theory

Abstract A standard view within psychology is that there have been two important shifts in the study of concepts and that each has led to some improvements. The first shift was from the classical theory of concepts to probabilistic theories, the most popular of which is prototype theory. The second shift was from probabilistic theories to theory-based theories. In this article, I take exception with the view that the first shift has led to any kind of advance. I argue that the main reasons given for preferring prototype theory over the classical theory are flawed and that prototype theory suffers some of the same problems that have been thought to challenge the classical theory

Ash's type II theorem, profinite topology and Malcev products

This paper is concerned with the many deep and far reaching consequences of Ash's positive solution of the type II conjecture for finite monoids. After rewieving the statement and history of the problem, we show how it can be used to decide if a finite monoid is in the variety generated by the Malcev product of a given variety and the variety of groups. Many interesting varieties of finite monoids have such a description including the variety generated by inverse monoids, orthodox monoids and solid monoids. A fascinating case is that of block groups. A block group is a monoid such that every element has at most one semigroup inverse. As a consequence of the cover conjecture - also verified by Ash - it follows that block groups are precisely the divisors of power monoids of finite groups. The proof of this last fact uses earlier results of the authors and the deepest tools and results from global semigroup theory. We next give connections with the profinite group topologies on finitely generated free monoids and free groups. In particular, we show that the type II conjecture is equivalent with two other conjectures on the structure of closed sets (one conjecture for the free monoid and another one for the free group). Now Ash's theorem implies that the two topological conjectures are true and independently, a direct proof of the topological conjecture for the free group has been recently obtained by Ribes and Zalesskii. An important consequence is that a rational subset of a finitely generated free group G is closed in the profinite topology if and only if it is a finite union of sets of the form gH1H2... Hn, where each Hi is a finitely generated subgroup of G. This significantly extends classical results by M. Hall. Finally we return to the roots of this problem and give connections with the complexity theory of finite semigroups. We show that the largest local complexity function in the sense of Rhodes and Tilson is computable

Learning Matters: The Role of Learning in Concept Acquisition

In LOT 2: The Language of Thought Revisited, Jerry Fodor argues that concept learning of any kind—even for complex concepts—is simply impossible. In order to avoid the conclusion that all concepts, primitive and complex, are innate, he argues that concept acquisition depends on purely noncognitive biological processes. In this paper, we show (1) that Fodor fails to establish that concept learning is impossible, (2) that his own biological account of concept acquisition is unworkable, and (3) that there are in fact many promising general models for explaining how concepts are learned