Coherent, well organized text familiarizes readers with complete theory of logical inference and its applications to math and the empirical sciences. Part I deals with formal principles of inference and definition; Part II explores elementary intuitive set theory, with separate chapters on sets, relations, and functions. Last section introduces numerous examples of axiomatically formulated theories in both discussion and exercises. Ideal for undergraduates; no background in math or philosophy required.

Using Sneed''s metatheory an attempt is made to reconstruct Hodgkin and Huxley''s theory of excitation of cell membranes. The structure of this theory is uncovered by defining set-theoretical predicates for the partial potential models, potential models, and models of the theory. The function of permeability is said to be the only theoretical function with respect to this theory. The main underlying assumptions of the theory are briefly outlined.

Imre Lakatos; II—Criticism and the Methodology of Scientific Research Programmes, Proceedings of the Aristotelian Society, Volume 69, Issue 1, 1 June 1969, Page.

SummaryIn this paper we develop a general concept of a theory analogous to that of an empirical theory. It is shown that axioms can be regarded as rules for performing operations. Using this connection we give a definition of an intellectual structure, and it turns out that mathematical theories represent intellectual structures in a natural way.RésuméCet article développe un concept général de théorie analogue à celui de théorie empirique. II montre que les axiomes peuvent être considérés comme des régles pour (...) réaliser des opérations . Cette connexion permet de définir une structure intellectuelle, et il se trouve que les theories mathématiques représentent ces structures intellectuelles de façon naturelle.ZusammenfassungIn dieser Arbeit wird der allgemeine Begriff einer mathematischen Theorie entwickelt, der dem einer empirischen Theorie analog ist. Es wird gezeigt, dass sich Axiome als Regeln zur Durchführung von Operationen auffassen lassen. Unter Benutzung dieses Zusam‐rnenhangs wird eine Definition von intellektueller Struktur gegeben und es zeigt sich, dass mathe‐matische Theorien in natürlicher Weise intellektuelle Strukturen repräsentieren. (shrink)

Imre Lakatos; II—Criticism and the Methodology of Scientific Research Programmes, Proceedings of the Aristotelian Society, Volume 69, Issue 1, 1 June 1969, Page.

Paul Feyerabend's globally acclaimed work, which sparked and continues to stimulate fierce debate, examines the deficiencies of many widespread ideas about scientific progress and the nature of knowledge. Feyerabend argues that scientific advances can only be understood in a historical context. He looks at the way the philosophy of science has consistently overemphasized practice over method, and considers the possibility that anarchism could replace rationalism in the theory of knowledge. -- Amazon.com.

Patrick Suppes' set-theoretical approach to the analysis of theories, and Joseph D. Sneed's metatheory are briefly outlined. The notions of observation, illusion and hallucination are reconstructed according to these approaches. It is argued that the terms ‘perception’ and ‘truth’ are theoretical with respect to observation but nontheoretical with respect to illusion and hallucination. Hallucination is construed as a special kind of illusion.

Using Sneed's metatheory an attempt is made to reconstruct Hodgkin and Huxley's theory of excitation of cell membranes. The structure of this theory is uncovered by defining set-theoretical predicates for the partial potential models, potential models, and models of the theory. The function of permeability is said to be the only theoretical function with respect to this theory. The main underlying assumptions of the theory are briefly outlined.

A theory from the behavioral and social sciences is presented from the structuralist point of view. A more comprehensive theory-net is outlined, some basic terms and core assumptions are formulated, and an expansion of the theory towards two intended applications is given. Finally, some results of a first empirical test of the theory are reported. The aim of the paper is to show that the structuralist account of scientific theories is not confined to mathematical theories from the natural sciences, but (...) can also be applied to relatively informal constructions of the behavioral and social sciences. (shrink)

Patrick Suppes'' set-theoretical approach to the analysis of theories, and Joseph D. Sneed''s metatheory are briefly outlined. The notions of observation, illusion and hallucination are reconstructed according to these approaches. It is argued that the terms perception and truth are theoretical with respect to observation but nontheoretical with respect to illusion and hallucination. Hallucination is construed as a special kind of illusion.

Discussion of the relation between exchange-value and use-value (as defined inCapital I) is clarified by the construction of set-theoretical models of these concepts. Marx argues fallaciously for the independence of exchange-value and use-value. His fallacy is diagnosed as depending upon a mistaken assumption about the impossibility of inferring a certain linear order on a set from a certain (different) partial order on that set.

This is the basis of the first part of the book.

Tasks in Primary Mathematics Teacher Education is intended to advance relevant research and innovative international practices in the preparation and professional development of mathematics teachers. Emerging from discussion at the ICMI study on teacher professional development, this volume, focused on primary and elementary teachers, culls a richness that can only be found by gathering wisdom from varied experiences around the world. The choice of tasks, and the associated pedagogies, is a key aspect of teaching and learning mathematics. Arguing that what (...) students learn is largely defined by the tasks they are given, several major themes are presented. One such major strand, the form, function and focus of tasks, is discussed throughout several chapters, offering analysis, discussion of implementation, and exemplars of a broader category of illustrative techniques for developing critical understanding. (shrink)

A theory from the behavioral and social sciences is presented from the structuralist point of view. A more comprehensive theory-net is outlined, some basic terms and core assumptions are formulated, and an expansion of the theory towards two intended applications is given. Finally, some results of a first empirical test of the theory are reported. The aim of the paper is to show that the structuralist account of scientific theories is not confined to mathematical theories from the natural sciences, but (...) can also be applied to relatively informal constructions of the behavioral and social sciences. (shrink)