Recent studies on neural pathways in a broad spectrum of vertebrates suggest that, in addition to migration and an increase in the number of certain select neurons, a significant aspect of neural evolution is a “parcellation” (segregation-isolation) process that involves the loss of selected connections by the new aggregates. A similar process occurs during ontogenetic development. These findings suggest that in many neuronal systems axons do not invade unknown territories during evolutionary or ontogenetic development but follow in their ancestors' paths (...) to their ancestral targets; if the connection is later lost, it reflects the specialization of the circuitry.The pattern of interspecific variability suggests (1) that overlap of circuits is a more common feature in primitive (generalized) than in specialized brain organizations and (2) that most projections, such as the retinal, thalamotelencephalic, corticotectal, and tectal efferent ones, were bilateral in the primitive condition. Specialization of these systems in some vertebrate groups has involved the selective loss of connections, resulting in greater isolation of functions. The parcellation process may also play an important role in cell diversification.The parcellation process as described here is thought to be one of several underlying mechanisms of evolutionary and ontogenetic differentiation. (shrink)

On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, and that (...) standard models of formal deduction are not well-equipped to support such evaluations. I discuss a model of proof that is used in the automated deduction community, and show that this model does better in that respect. (shrink)

We present a formal system, E, which provides a faithful model of the proofs in Euclid's Elements, including the use of diagrammatic reasoning.

Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned areas.

Many mathematicians and philosophers of mathematics believe some proofs contain elements extraneous to what is being proved. In this paper I discuss extraneousness generally, and then consider a specific proposal for measuring extraneousness syntactically. This specific proposal uses Gentzen's cut-elimination theorem. I argue that the proposal fails, and that we should be skeptical about the usefulness of syntactic extraneousness measures.

Since the early days of physics, space has called for means to represent, experiment, and reason about it. Apart from physicists, the concept of space has intrigued also philosophers, mathematicians and, more recently, computer scientists. This longstanding interest has left us with a plethora of mathematical tools developed to represent and work with space. Here we take a special look at this evolution by considering the perspective of Logic. From the initial axiomatic efforts of Euclid, we revisit the major milestones (...) in the logical representation of space and investigate current trends. In doing so, we do not only consider classical logic, but we indulge ourselves with modal logics. These present themselves naturally by providing simple axiomatizations of different geometries, topologies, space-time causality, and vector spaces. (shrink)