Originally published in 1988. This text gives a lucid account of the most distinctive and influential responses by twentieth century philosophers to the problem of the unity of the proposition. The problem first became central to twentieth-century philosophy as a result of the depsychoiogising of logic brought about by Bradley and Frege who, responding to the 'Psychologism' of Mill and Hume, drew a sharp distinction between the province of psychology and the province of logic. This author argues that while Russell, Ryle and Davidson, each in different ways, attempted a theoretical solution, Frege and Wittgenstein (both in the Tractatus and the Investigations) rightly maintained that no theoretical solution is possible. It is this which explains the importance Wittgenstein attached in his later work to the idea of agreement in judgments. The two final chapters illustrate the way in which a response to the problem affects the way in which we think about the nature of the mind. They contain a discussion of Strawson's concept of a person and provide a striking critique of the philosophical claims made by devotees of artificial intelligence, in particular those made by Daniel Dennett.

Originally published in 1941. Professor Ushenko treats of current problems in technical Logic, involving Symbolic Logic to a marked extent. He deprecates the tendency, in influential quarters, to regard Logic as a branch of Mathematics and advances the intuitionalist theory of Logic. This involves criticism of Carnap, Russell,Wittgenstein, Broad and Whitehead, with additional discussions on Kant and Hegel. The author believes that the union of Philosophy and Logic is a natural one, and that an exclusively mathematical treatment cannot give an adequate account of Logic. A fundamental characteristic of Logic is comprehensiveness, which brings out the affinity between logic and philosophy, for to be comprehensive is the aim of philosophical ambition.

Originally published in 1964. This book is concerned with general arguments, by which is meant broadly arguments that rely for their force on the ideas expressed by all, every, any, some, none and other kindred words or phrases. A main object of quantificational logic is to provide methods for evaluating general arguments. To evaluate a general argument by these methods we must first express it in a standard form. Quantificational form is dealt with in chapter one and in part of chapter three; in the remainder of the book an account is given of methods by which arguments when formulated quantificationally may be tested for validity or invalidity. Some attention is also paid to the logic of identity and of definite descriptions. Throughout the book an attempt has been made to give a clear explanation of the concepts involved and the symbols used; in particular a step-by-step and partly mechanical method is developed for translating complicated statements of ordinary discourse into the appropriate quantificational formulae. Some elementary knowledge of truth-functional logic is presupposed.

Originally published in 1962. This book gives an account of the concepts and methods of a basic part of logic. In chapter I elementary ideas, including those of truth-functional argument and truth-functional validity, are explained. Chapter II begins with a more comprehensive account of truth-functionality; the leading characteristics of the most important monadic and dyadic truth-functions are described, and the different notations in use are set forth. The main part of the book describes and explains three different methods of testing truth-functional aguments and agument forms for validity: the truthtable method, the deductive method and the method of normal forms; for the benefit mainly of readers who have not acquired in one way or another a general facility in the manipulation of symbols some of the procedures have been described in rather more detail than is common in texts of this kind. In the final chapter the author discusses and rejects the view, based largely on the so called paradoxes of material implication, that truth-functional logic is not applicable in any really important way to arguments of ordinary discourse.

Originally published in 1966. Professor Rescher's aim is to develop a "logic of commands" in exactly the same general way which standard logic has already developed a "logic of truth-functional statement compounds" or a "logic of quantifiers". The object is to present a tolerably accurate and precise account of the logically relevant facets of a command, to study the nature of "inference" in reasonings involving commands, and above all to establish a viable concept of validity in command inference, so that the logical relationships among commands can be studied with something of the rigour to which one is accustomed in other branches of logic.

This volume tells the story of the legacy and impact of the great German polymath Gottfried Wilhelm Leibniz (1646-1716). Leibniz made significant contributions to many areas, including philosophy, mathematics, political and social theory, theology, and various sciences. The essays in this volume explores the effects of Leibniz's profound insights on subsequent generations of thinkers by tracing the ways in which his ideas have been defended and developed in the three centuries since his death. Each of the 11 essays is concerned with Leibniz's legacy and impact in a particular area, and between them they show not just the depth of Leibniz's talents but also the extent to which he shaped the various domains to which he contributed, and in some cases continues to shape them today. With essays written by experts such as Nicholas Jolley, Pauline Phemister, and Philip Beeley, this volume is essential reading not just for students of Leibniz but also for those who wish to understand the game-changing impact made by one of history's true universal geniuses.

Originally published in 1923 Chance and Error examines the vagaries of chance, and how this is the result of the interference of yes and no. The book basis its examination of chance on the idea of a two-sided coin. The book stipulates that contradictories are head and tail, or yes and no. When the coin is flipped in the air yes normally wins half of the trials, but this includes half of the half that normally go to no. Thus, normally in one quarter of the trials there is an interference of yes and no. From this the chance of any number of heads or tails can be easily calculated, and all results that are attained by more difficult mathematics are secured. The book uses this idea to examine interference of yes and no in everyday life and argues that this causes the variations in everything that goes on around us in nature and in our daily life. This book will be of interest to philosophers of logic, as well as mathematicians.

A New World of Geometry Awaits Your Discovery! The last stone falls out ... a rush of ancient air ... the glint of gold ... the tingle of discovery ... When explorers first opened the tombs of the ancient pharaohs, they knew that they had discovered something wonderful. That feeling, that same passionate sense of discovery, is one of the most powerful educational tools a text can deliver. Geometry by Discovery is an exciting new approach to geometry. This ground-breaking text taps the pedagogical value of discovery to help students stretch their geometric perspective and hone their geometric intuition. It actively engages students in solving mathematical problems, and empowers them to be successful problem-solvers and discoverers of mathematical ideas.

Spherical Geometry and Its Applications introduces spherical geometry and its practical applications in a mathematically rigorous form. The text can serve as a course in spherical geometry for mathematics majors. Readers from various academic backgrounds can comprehend various approaches to the subject. The book introduces an axiomatic system for spherical geometry and uses it to prove the main theorems of the subject. It also provides an alternate approach using quaternions. The author illustrates how a traditional axiomatic system for plane geometry can be modified to produce a different geometric world - but a geometric world that is no less real than the geometric world of the plane. Features: A well-rounded introduction to spherical geometry Provides several proofs of some theorems to appeal to larger audiences Presents principal applications: the study of the surface of the earth, the study of stars and planets in the sky, the study of three- and four-dimensional polyhedra, mappings of the sphere, and crystallography Many problems are based on propositions from the ancient text Sphaerica of Menelaus

Originally published in 1994, The Incommensurability Thesis is a critical study of the Incommensurability Thesis of Thomas Kuhn and Paul Feyerabend. The book examines the theory that different scientific theories may be incommensurable because of conceptual variance. The book presents a critique of the thesis and examines and discusses the arguments for the theory, acknowledging and debating the opposing views of other theorists. The book provides a comprehensive and detailed discussion of the incommensurability thesis.

Contents:
Introduction. 1. Philosophy of logic 2. Philosophy of mathematics in the 20th century. 3. Frege 4. Wittgenstein's Tractatus 5. Logical postivism 6. The philosophy of physics 7. The philosophy of science 8. Chance, cause and conduct: probability theory and the explanation of human action 9. Cybernetics 10. Descartes' legacy: the mechanist/vitalist debates.

During the first few decades of the twentieth century, philosophers and mathematicians mounted a sustained effort to clarify the nature of mathematics. This led to considerable discord, even enmity, and yielded fascinating and fruitful work of both a mathematical and a philosophical nature. It was one of the most exhilarating intellectual adventures of the century, pursued at an extraordinarily high level of acuity and imagination. Its legacy principally consists of three original and finely articulated programs that seek to view mathematics in the proper light: logicism, intuitionism, and finitism. Each is notable for its symbiotic melding together of philosophical vision and mathematical work: the philosophical ideas are given their substance by specific mathematical developments, which are in turn given their point by philosophical reflection.

This book provides an accessible, critical introduction to these three projects as it describes and investigates both their philosophical and their mathematical components.

Solutions manual is available upon request.

This edited collection is the first of its kind to explore the view called perspectivism in philosophy of science. The book brings together an array of essays that reflect on the methodological promises and scientific challenges of perspectivism in a variety of fields such as physics, biology, cognitive neuroscience, and cancer research, just as a few examples. What are the advantages of using a plurality of perspectives in a given scientific field and for interdisciplinary research? Can different perspectives be integrated? What is the relation between perspectivism, pluralism, and pragmatism? These ten new essays by top scholars in the field offer a polyphonic journey towards understanding the view called 'perspectivism' and its relevance to science.

One of the only volumes that brings the humanities, social sciences and even the natural sciences under one remit to look at how they can be researched in an integrated and useful way, with policy and real world implications in terms of how we relate in and to the world. Interdisciplinarity and Transdisciplinarity have been around for a long time, but as as we move through a digital age they are becoming more and more important and interesting to the scholarly community and beyond. There is nothing on the market that pulls all of these subjects across disciplines together and works out a framework to construct the analysis in a way that asks and answers useful questions.

Philosophy of science studies the methods, theories, and concepts used by scientists. It mainly developed as a field in its own right during the twentieth century and is now a diversified and lively research area. This book surveys the current state of the discipline by focusing on central themes like confirmation of scientific hypotheses, scientific explanation, causality, the relationship between science and metaphysics, scientific change, the relationship between philosophy of science and science studies, the role of theories and models, unity of science. These themes define general philosophy of science. The book also presents sub-disciplines in the philosophy of science dealing with the main sciences: logic, mathematics, physics, biology, medicine, cognitive science, linguistics, social sciences, and economics. While it is common to address the specific philosophical problems raised by physics and biology in such a book, the place assigned to the philosophy of special sciences is much more unusual. Most authors collaborate on a regular basis in their research or teaching and share a common vision of philosophy of science and its place within philosophy and academia in general. The chapters have been written in close accordance with the three editors, thus achieving strong unity of style and tone.

The Twentieth Century has seen a dramatic rise in the use of probability and statistics in almost all fields of research. This has stimulated many new philosophical ideas on probability.
Philosophical Theories of Probability is the first book to present a clear, comprehensive and systematic account of these various theories and to explain how they relate to one another. Gillies also offers a distinctive version of the propensity theory of probability, and the intersubjective interpretation, which develops the subjective theory.

Reissuing five works originally published between 1937 and 1991, this collection contains books addressing the subject of time, from a mostly philosophic point of view but also of interest to those in the science and mathematics worlds. These texts are brought back into print in this small set of works addressing how we think about time, the history of the philosophy of time, the measurement of time, theories of relativity and discussions of the wider thinking about time and space, among other aspects. One volume is a thorough bibliography collating references on the subject of time across many disciplines.

Originally published in 1976. This comprehensive study discusses in detail the philosophical, mathematical, physical, logical and theological aspects of our understanding of time and space. The text examines first the many different definitions of time that have been offered, beginning with some of the puzzles arising from our awareness of the passage of time and shows how time can be understood as the concomitant of consciousness. In considering time as the dimension of change, the author obtains a transcendental derivation of the concept of space, and shows why there has to be only one dimension of time and three of space, and why Kant was not altogether misguided in believing the space of our ordinary experience to be Euclidean. The concept of space-time is then discussed, including Lorentz transformations, and in an examination of the applications of tense logic the author discusses the traditional difficulties encountered in arguments for fatalism. In the final sections he discusses eternity and the beginning and end of the universe. The book includes sections on the continuity of space and time, on the directedness of time, on the differences between classical mechanics and the Special and General theories of relativity, on the measurement of time, on the apparent slowing down of moving clocks, and on time and probability.

Originally published in 1980. What is time? How is its structure determined? The enduring controversy about the nature and structure of time has traditionally been a diametrical argument between those who see time as a container into which events are placed, and those for whom time cannot exist without events. This controversy between the absolutist and the relativist theories of time is a central theme of this study. The author's impressive arguments provide grounds for rejecting both these theories, firstly by establishing that 'empty' time is possible, and secondly by showing, through a discussion of the structure of time which involves considering whether time might be cyclical, branching, beginning or non-beginning, that the absolutist theory of time is untenable. This book then advances two new theories, and succeeds in shifting the traditional debate about time to a consideration of time as a theoretical structure and as a theoretical framework.

First published in the most ambitious international philosophy project for a generation; the Routledge Encyclopedia of Philosophy.
Logic from A to Z is a unique glossary of terms used in formal logic and the philosophy of mathematics.
Over 500 entries include key terms found in the study of:
* Logic: Argument, Turing Machine, Variable
* Set and model theory: Isomorphism, Function
* Computability theory: Algorithm, Turing Machine
* Plus a table of logical symbols.
Extensively cross-referenced to help comprehension and add detail, Logic from A to Z provides an indispensable reference source for students of all branches of logic.

Brilliant introduction to the philosophy of mathematics, from the question 'what is a number?' up to the concept of infinity, descriptions, classes and axioms Russell deploys all his skills and brilliant prose to write an introductory book - a real gem by one of the 20th century's most celebrated philosophers New foreword by Michael Potter to the Routledge Classics edition places the book in helpful context and explains why it's a classic

How we reason with mathematical ideas continues to be a fascinating and challenging topic of research--particularly with the rapid and diverse developments in the field of cognitive science that have taken place in recent years. Because it draws on multiple disciplines, including psychology, philosophy, computer science, linguistics, and anthropology, cognitive science provides rich scope for addressing issues that are at the core of mathematical learning.
Drawing upon the interdisciplinary nature of cognitive science, this book presents a broadened perspective on mathematics and mathematical reasoning. It represents a move away from the traditional notion of reasoning as "abstract" and "disembodied," to the contemporary view that it is "embodied" and "imaginative." From this perspective, mathematical reasoning involves reasoning with structures that emerge from our bodily experiences as we interact with the environment; these structures extend beyond finitary propositional representations. Mathematical reasoning is imaginative in the sense that it utilizes a number of powerful, illuminating devices that structure these concrete experiences and transform them into models for abstract thought. These "thinking tools"--analogy, metaphor, metonymy, and imagery--play an important role in mathematical reasoning, as the chapters in this book demonstrate, yet their potential for enhancing learning in the domain has received little recognition.
This book is an attempt to fill this void. Drawing upon backgrounds in mathematics education, educational psychology, philosophy, linguistics, and cognitive science, the chapter authors provide a rich and comprehensive analysis of mathematical reasoning. New and exciting perspectives are presented on the nature of mathematics (e.g., "mind-based mathematics"), on the array of powerful cognitive tools for reasoning (e.g., "analogy and metaphor"), and on the different ways these tools can facilitate mathematical reasoning. Examples are drawn from the reasoning of the preschool child to that of the adult learner.

How we reason with mathematical ideas continues to be a fascinating and challenging topic of research--particularly with the rapid and diverse developments in the field of cognitive science that have taken place in recent years. Because it draws on multiple disciplines, including psychology, philosophy, computer science, linguistics, and anthropology, cognitive science provides rich scope for addressing issues that are at the core of mathematical learning.
Drawing upon the interdisciplinary nature of cognitive science, this book presents a broadened perspective on mathematics and mathematical reasoning. It represents a move away from the traditional notion of reasoning as "abstract" and "disembodied," to the contemporary view that it is "embodied" and "imaginative." From this perspective, mathematical reasoning involves reasoning with structures that emerge from our bodily experiences as we interact with the environment; these structures extend beyond finitary propositional representations. Mathematical reasoning is imaginative in the sense that it utilizes a number of powerful, illuminating devices that structure these concrete experiences and transform them into models for abstract thought. These "thinking tools"--analogy, metaphor, metonymy, and imagery--play an important role in mathematical reasoning, as the chapters in this book demonstrate, yet their potential for enhancing learning in the domain has received little recognition.
This book is an attempt to fill this void. Drawing upon backgrounds in mathematics education, educational psychology, philosophy, linguistics, and cognitive science, the chapter authors provide a rich and comprehensive analysis of mathematical reasoning. New and exciting perspectives are presented on the nature of mathematics (e.g., "mind-based mathematics"), on the array of powerful cognitive tools for reasoning (e.g., "analogy and metaphor"), and on the different ways these tools can facilitate mathematical reasoning. Examples are drawn from the reasoning of the preschool child to that of the adult learner.

This book contributes towards the literature in the field of mathematics education, specifically on aspects of empowering learners of mathematics. The book, comprising eighteen chapters, written by renowned researchers in mathematics education, provides readers with approaches and applicable classroom strategies to empower learners of mathematics.The chapters in the book can be classified into four sections. The four sections focus on how learners could be empowered in their learning, cognitive and affective processes, through mathematical content, purposefully designed mathematical tasks, whilst developing 21st century competencies.

Chemists, working with only mortars and pestles, could not get very far unless they had mathematical models to explain what was happening "inside" of their elements of experience -- an example of what could be termed mathematical learning.
This volume contains the proceedings of Work Group 4: Theories of Mathematics, a subgroup of the Seventh International Congress on Mathematical Education held at Universite Laval in Quebec. Bringing together multiple perspectives on mathematical thinking, this volume presents elaborations on principles reflecting the progress made in the field over the past 20 years and represents starting points for understanding mathematical learning today. This volume will be of importance to educational researchers, math educators, graduate students of mathematical learning, and anyone interested in the enterprise of improving mathematical learning worldwide.