This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from leading Brazilian logicians and their Latin-American and European colleagues. All papers were selected by a careful refereeing processs and were revised and (...) updated by their authors for publication in this volume. There are three sections: Advances in Logic, Advances in Theoretical Computer Science, and Advances in Philosophical Logic. Well-known specialists present original research on several aspects of model theory, proof theory, algebraic logic, category theory, connections between logic and computer science, and topics of philosophical logic of current interest. Topics interweave proof-theoretical, semantical, foundational, and philosophical aspects with algorithmic and algebraic views, offering lively high-level research results. (shrink)

In this article I intend to show that certain aspects of the axiomatical structure of mathematical theories can be, by a phenomenologically motivated approach, reduced to two distinct types of idealization, the first-level idealization associated with the concrete intuition of the objects of mathematical theories as discrete, finite sign-configurations and the second-level idealization associated with the intuition of infinite mathematical objects as extensions over constituted temporality. This is the main standpoint from which I review Cantor’s conception of (...) infinite cardinalities and also the metatheoretical content of some later well-known theorems of mathematical foundations. These are, the Skolem-Löwenheim Theorem which, except for its importance as such, it is also chosen for an interpretation of the associated metatheoretical paradox (Skolem Paradox), and Gödel’s (first) incompleteness result which, notwithstanding its obvious influence in the mathematical foundations, is still open to philosophical inquiry. On the phenomenological level, first-level and second-level idealizations, as above, are associated respectively with intentional acts carried out in actual present and with certain modes of a temporal constitution process. (shrink)

Degrees of belief; Dempster's rule of combination; Simple and separable support functions; The weights of evidence; Compatible frames of discernment; Support functions; The discernment of evidence; Quasi support functions; Consonance; Statistical evidence; The dual nature of probable reasoning.

In this paper we present an informal review of our recent work whose goal is to develop a mathematical theory of the physical phenomenon of emission and absorption of radiation by systems of nonrelativistic matter such as atoms and molecules.

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

The Quine/Putnam indispensability approach to the confirmation of mathematical theories in recent times has been the subject of significant criticism. In this paper I explore an alternative to the Quine/Putnam indispensability approach. I begin with a van Fraassen-like distinction between accepting the adequacy of a mathematical theory and believing in the truth of a mathematical theory. Finally, I consider the problem of moving from the adequacy of a mathematical theory to its truth. I (...) argue that the prospects for justifying this move are qualitatively worse in mathematics than they are in science. (shrink)

In our previous writings we have outlined a mathematical theory of biological phenomena. In our systematic construction of “mathematical biology,” similar in its aims to mathematical physics, we have started with the fundamental unit,—the living cell. After having established a physico-mathematical theory of the fundamental properties of the cell, we have studied the interaction of several cells. This led us into two different fields. On the one hand we studied such interactions of cells, which (...) determine the form of cellular aggregates, constituting multicellular organisms. On the other hand we studied different types of functional interactions, which determine the reactions of the aggregate as a whole to different environmental changes. While the first field leads us eventually to the theory of organic form, the second brings us to the physico-mathematical theory of behavior. (shrink)

Auf Grund der Vorstellung, dass die Erregungsleitung auf einer Wiedererregung der benachbarten Gewebebezirke durch lokale bioelektrische Ströme beruht, wurde vorher eine mathematische Theorie der Fortpflanzung der Erregung im Nerv entwickelt, welche einige Tatsachen befriedigend darstellt. In der vorliegenden Arbeit wird die Theorie auf den Fall angewandt, dass die Erregung von einem Gewebe auf ein anderes übertragen wird, wobei die beiden Gewebe verschiedene elektrische Eigenschaften haben. Es zeigt sich, dass dabei gewisse Bedingungen für die Möglichkeit der Übertragung der Erregung erfüllt sein (...) müssen, welche an den vonL. Lapique geforderten Isochronismus erinnern.Dans deux mémoires précédents nous avons développé une théorie mathématique de la propagation de l'excitation nerveuse, basée sur l'hypothèse, que cette propagation est due à une réexcitation par les courants bioélectriques. Dans le mémoire présent nous étudions le cas de deux tissus adjacents, différents dans leurs constantes électriques, du point de vue du passage de l'excitation d'un tissu sur l'autre. Il se trouve, que pour que ce passage put avoir lieu, certaines relations doivent être satisfaites, des relations qui rappellent l'isochronisme postulé parL. Lapique. (shrink)

The aim of this paper is to describe and analyze the epistemological justification of a proposal initially made by the biomathematician Robert Rosen in 1958. In this theoretical proposal, Rosen suggests using the mathematical concept of “category” and the correlative concept of “natural equivalence” in mathematical modeling applied to living beings. Our questions are the following: According to Rosen, to what extent does the mathematical notion of category give access to more “natural” formalisms in the modeling of (...) living beings? Is the so -called “naturalness” of some kinds of equivalences (which the mathematical notion of category makes it possible to generalize and to put at the forefront) analogous to the naturalness of living systems? Rosen appears to answer “yes” and to ground this transfer of the concept of “natural equivalence” in biology on such an analogy. But this hypothesis, although fertile, remains debatable. Finally, this paper makes a brief account of the later evolution of Rosen’s arguments about this topic. In particular, it sheds light on the new role played by the notion of “category” in his more recent objections to the computational models that have pervaded almost every domain of biology since the 1990s. (shrink)

A theory of embolism based on an optimization model of blood flow is proposed and used to explain the topographic distribution of emboli in arterial trees.

Evidence Theory is a branch of mathematics that concerns combination of empirical evidence in an individual’s mind in order to construct a coherent picture of reality. Designed to deal with unexpected empirical evidence suggesting new possibilities, evidence theory is compatible with Shackle’s idea of decision-making as a creative act. This essay investigates this connection in detail, pointing to the usefulness of evidence theory to formalise and extend Shackle’s decision theory. In order to ease a proper framing (...) of the issues involved, evidence theory is compared with sub-additive probability theory and Ewens’s infinite alleles model. Furthermore, the original version of evidence theory is presented along with its most recent developments. (shrink)

For parents wanting their children to get a head start in reading, it can be a challenge to find something that will maintain their attention. Now, learning to read can become a fun and enter- taining thing to do with the help of an extraordinary cat. Join Cleo-cat-tra as she brings reading to life in the charming picture book Rhymes and Times with Cleo-cat-tra by Lucy T. Geringer and illustrated by Bernardita Cox Kollock. Rhymes and Times of Cleo-cat-tra is a (...) picture book about a special cat named Cleo-cat-tra. She is not an ordinary cat, as she cleverly rhymes about animals, people, and things she encounters inside and outside her house. The rhymes are silly which make learning a fun and creative pro- cess for kids and for parents to share. This delightful story makes use of colorful illustrations to heighten a child's imagination. Its simple text and brief sentences make reading easy for the early reader. It is also a great book for children who are learning English and can be enjoyed both at home and in school through lively interactive discussions between students and teachers. It is a book for all ages. (shrink)

Self-taught mathematician and father of Boolean algebra, George Boole (1815-1864) published An Investigation of the Laws of Thought in 1854. In this highly original investigation of the fundamental laws of human reasoning, a sequel to ideas he had explored in earlier writings, Boole uses the symbolic language of mathematics to establish a method to examine the nature of the human mind using logic and the theory of probabilities. Boole considers language not just as a mode of expression, but as (...) a system one can use to understand the human mind. In the first 12 chapters, he sets down the rules necessary to represent logic in this unique way. Then he analyses a variety of arguments and propositions of various writers from Aristotle to Spinoza. One of history's most insightful mathematicians, Boole is compelling reading for today's student of intellectual history and the science of the mind. (shrink)

Though there are holes in the theory of how children move through stages of numerical competence, the current approach offers the most promising avenue for characterizing changes in competence as children confront new mathematical concepts. Like the science of mathematics, children's discovery of number is rooted in intuitions about sets, and not purely in analytic truths.

This paper is on Aristotle's conception of the continuum. It is argued that although Aristotle did not have the modern conception of real numbers, his account of the continuum does mirror the topology of the real number continuum in modern mathematics especially as seen in the work of Georg Cantor. Some differences are noted, particularly as regards Aristotle's conception of number and the modern conception of real numbers. The issue of whether Aristotle had the notion of open versus closed intervals (...) is discussed. Finally, it is suggested that one reason there is a common structure between Aristotle's account of the continuum and that found in Cantor's definition of the real number continuum is that our intuitions about the continuum have their source in the experience of the real spatiotemporal world. A plea is made to consider Aristotle's abstractionist philosophy of mathematics anew. (shrink)

It is a well-known fact of mathematical logic, by now developed in considerable detail, that formalized mathematical theories can be ordered by relative interpretability, and the "strength" of a theory is indicated by where it stands in this ordering. Mutual interpretability is an equivalence relation, and what I call an ordering is a partial ordering modulo this equivalence. Of the theories that have been studied, the natural theories belong to a linearly ordered subset of this ordering.

In this article the problem of unification of mathematical theories is discussed. We argue, that specific problems arise here, which are quite different than the problems in the case of empirical sciences. In particular, the notion of unification depends on the philosophical standpoint. We give an analysis of the notion of unification from the point of view of formalism, Gödel's platonism and Quine's realism. In particular we show, that the concept of “having the same object of study” should be (...) made precise in the case of mathematical theories. In the appendix we give a working proposal of a certain understanding of this notion. (shrink)

This paper discusses a deterministic model of the spread of an infectious disease in a closed population that was proposed byKermack &McKendrick . The mathematical assumptions on which the model is based are listed and criticized. The ‘threshold theorem’ according to which an epidemic develops if, and only if, the initial population density exceeds a certain value determined by the parameters of the model, is discussed. It is shown that the theorem is not true. A weaker result is stated (...) and proved.A new simplified version of the model is discussed. In this model it is assumed that an infected individual becomes infectious after a constant time and that it will removed after a constant time. Numerical simulations of this model show that the form of the fluctuations in the number of diseased individuals depends on the values of the parameters. The number of infectious individuals may oscillate up and down under certain circumstances. Necessary conditions for the number of infectious individuals to rise above previous levels are stated and proved. (shrink)

This book introduces a set of methods and techniques for studying mathematical theories and relating them to each other through the use of Grothendieck toposes.

Mathematical concepts are explications, in Carnap's sense, of vague or otherwise non-clear concepts; mathematical theories have an empirical and a deductive component. From this perspective, I argue that the empirical component of a mathematical theory may be tested together with the fruitfulness of its explications. Using these ideas, I furthermore give an argument for mathematical realism, based on the indispensability argument combined with a weakened version of confirmational holism.

This monograph aims to mathematically codify the notion of "moral systems" and define a sensible distance between them. It consists of three parts, aimed at an audience with varying interests and mathematical backgrounds. The first part steers philosophical, formally defining moral systems and several related concepts. The second part studies black box algorithms, including questions of inference and metric construction. The third part explores the technical construction of metrics amongst conditional probability distributions.

Topics covered in Discrete Mathematics have become essential tools in many areas of studies in recent years. This is primarily due to the revolution in technology, communications, and cyber security. The book treats major themes in a typical introductory modern Discrete Mathematics course: Propositional and predicate logic, proof techniques, set theory (including Boolean algebra, functions and relations), introduction to number theory, combinatorics and graph theory. An accessible, precise, and comprehensive approach is adopted in the treatment of each (...) topic. The ability of abstract thinking and the art of writing valid arguments are emphasized through detailed proof of (almost) every result. Developing the ability to think abstractly and roguishly is key in any areas of science, information technology and engineering. Every result presented in the book is followed by examples and applications to consolidate its comprehension. The hope is that the reader ends up developing both the abstract reasoning as well as acquiring practical skills. All efforts are made to write the book at a level accessible to first-year students and to present each topic in a way that facilitates self-directed learning. Each chapter starts with basic concepts of the subject at hand and progresses gradually to cover more ground on the subject. Chapters are divided into sections and subsections to facilitate readings. Each section ends with its own carefully chosen set of practice exercises to reenforce comprehension and to challenge and stimulate readers. As an introduction to Discrete Mathematics, the book is written with the smallest set of prerequisites possible. Familiarity with basic mathematical concepts (usually acquired in high school) is sufficient for most chapters. However, some mathematical maturity comes in handy to grasp some harder concepts presented in the book. (shrink)