19th Brazilian Logic Conference: Book of Abstracts

This is the book of abstracts of the 19th Brazilian Logic Conferences. The Brazilian Logic Conferences (EBL) is one of the most traditional logic conferences in South America. Organized by the Brazilian Logic Society (SBL), its main goal is to promote the dissemination of research in logic in a broad sense. It has been occurring since 1979, congregating logicians of different fields — mostly philosophy, mathematics and computer science — and with different backgrounds — from undergraduate students to senior researchers. The meeting is an important moment for the Brazilian and South American logical community to join together and discuss recent developments of the field. The areas of logic covered in the conference spread over foundations and philosophy of science, analytic philosophy, philosophy and history of logic, mathematics, computer science, informatics, linguistics and artificial intelligence. Previous editions of the EBL have been a great success, attracting researchers from all over Latin America and elsewhere. The 19th edition of EBL takes place from May 6-10, 2019, in the beautiful city of João Pessoa, at the northeast coast of Brazil. It is conjointly organized by Federal University of Paraíba (UFPB), whose main campus is located in João Pessoa, Federal University of Campina Grande (UFCG), whose main campus is located in the nearby city of Campina Grande (the second-largest city in Paraíba state) and SBL. It is sponsored by UFPB, UFCG, the Brazilian Council for Scientific and Technological Development (CNPq) and the State Ministry of Education, Science and Technology of Paraíba. It takes place at Hotel Luxxor Nord Tambaú, privileged located right in front Tambaú beach, one of João Pessoa’s most famous beaches

Building and testing a cognitive approach to the calculus using interactive computer graphics

This thesis consists of a theoretical building of a cognitive approach to the calculus and an empirical testing of the theory in the classroom. A cognitive approach to the teaching of a knowledge domain is defined to be one that aims to make the material potentially meaningful at every stage (in the sense of Ausubel). As a resource in such an approach, the notion of a generic organiser is introduced (after Dienes), which is an environment enabling the learner to explore examples of mathematical processes and concepts, providing cognitive experience to assist in the abstraction of higher order concepts embodied by the organiser. This allows the learner to build and test concepts in a mode 1 environment (in the sense of Skemp) rather than the more abstract modes of thinking typical in higher mathematics. The major hypothesis of the thesis is that appropriately designed generic organisers, supported by an appropriate learning environment, are able to provide students with global gestalts for mathematical processes and concepts at an earlier stage than occurs with current teaching methods. The building of the theory involves an in-depth study of cognitive development, of the cultural growth and theoretical content of the mathematics, followed by the design and programming of appropriate organisers for the teaching of the calculus. Generic organisers were designed for differentiation (gradient of a graph), integration (area), and differential equations, to be coherent ends in themselves as well as laying foundations for the formal theories of both standard and non-standard analysis. The testing is concerned with the program GRADIENT, which is designed to give a global gestalt of the dynamic concept of the gradient of a graph. Three experimental classes (one taught by the researcher in conjunction with the regular class teacher) used the software as an adjunct to the normal study of the calculus and five other classes acted as controls. Matched pairs were selected on a pre-test for the purpose of statistical comparison of performance on the post-test. Data was also collected from a third school where the organisers functioned less well, and from university mathematics students who had not used a computer. The generic organiser GRADIENT, supported by appropriate teaching, enabled the experimental students to gain a global gestalt of the gradient concept. They were able to sketch derivatives. for given graphs significantly better than the controls on the post-test, at a level comparable with more able students reading mathematics at university. Their conceptualizations of gradient and tangent transferred to a new situation involving functions given by different formulae on either side of the point in question, performing significantly better than the control students and at least as well, or better, than those at university

Banishing Ultrafilters from Our Consciousness

The way in which Martin Davis conceived the first chapter of his book \u201cApplied nonstandard analysis \u201d is a brilliant example of information hiding as a guiding principle for the design of widely applicable constructions and methods of proof. We discuss here a common trait that we see between that book and another writing of the year 1977, \u201cMetamathematical extensibility for theorem provers and proof-checkers\u201d, which Martin coauthored with Jacob T. Schwartz . To tie the said part of Martin\u2019s study on nonstandard analysis to proof technology, we undertake a verification, by means of a proof-checker based on set theory, of key results of the non-standard approach to analysis