This book intends to show that radical naturalism (or physicalism), nominalism and strict finitism account for the applications of classical mathematics in current scientific theories. The applied mathematical theories developed in the book include the basics of calculus, metric space theory, complex analysis, Lebesgue integration, Hilbert spaces, and semi-Riemann geometry (sufficient for the applications in classical quantum mechanics and general relativity). The fact that so much applied mathematics can be developed within such a weak, strictly finitistic system, is surprising in itself. It also shows that the applications of those classical theories to the finite physical world can be translated into the applications of strict finitism, which demonstrates the applicability of those classical theories without assuming the literal truth of those theories or the reality of infinity. Both professional researchers and students of philosophy of mathematics will benefit greatly from reading this book.

Mathieu Marion offers a careful, historically informed study of Wittgenstein's philosophy of mathematics. This area of his work has frequently been undervalued by Wittgenstein specialists and by philosophers of mathematics alike; but the surprising fact that he wrote more on this subject than on any other indicates its centrality in his thought. Marion traces the development of Wittgenstein's thinking in the context of the mathematical and philosophical work of the times, to make coherent sense of ideas that have too often been misunderstood because they have been presented in a disjointed and incomplete way. In particular, he illuminates the work of the neglected 'transitional period' between the Tractatus and the Investigations. Marion shows that study of Wittgenstein's writings on mathematics is essential to a proper understanding of his philosophy; and he also demonstrates that it has much to contribute to current debates about the foundations of mathematics.

This book presents diverse topics in mathematical logic such as proof theory, meta-mathematics, and applications of logic to mathematical structures. The collection spans the first 100 years of modern logic and is dedicated to the memory of Irving Anellis, founder of the journal 'Modern Logic', whose academic work was essential in promoting the algebraic tradition of logic, as represented by Charles Sanders Peirce. Anellis’s association with the Russian logic community introduced their school of logic to a wider audience in the USA, Canada and Western Europe. In addition, the collection takes a historical perspective on proof theory and the development of logic and mathematics in Eastern Logic, the Soviet Union and Russia. The book will be of interest to historians and philosophers in logic and mathematics, and the more specialized papers will also appeal to mathematicians and logicians.

The proceedings publishes research results on the following disciplines: philosophy and epistemology, history and philosophy of science, mathematics, logic and computer science, social sciences, linguistics, cognitive sciences, artificial intelligence, general systems and operational research.

This book is an exploration of philosophical questions about infinity. Graham Oppy examines how the infinite lurks everywhere, both in science and in our ordinary thoughts about the world. He also analyses the many puzzles and paradoxes that follow in the train of the infinite. Even simple notions, such as counting, adding and maximising present serious difficulties. Other topics examined include the nature of space and time, infinities in physical science, infinities in theories of probability and decision, the nature of part/whole relations, mathematical theories of the infinite, and infinite regression and principles of sufficient reason.

This is a collection of outstanding contributed papers presented at the 11th International Congress of Logic, Methodology, and Philosophy of Science (Kraków, 1999). The articles address current issues in logic, metamathematics, philosophy of language, philosophy of science, and cognitive science, as well as philosophical problems of biology, chemistry and physics. The volume will be of interest to philosophers, logicians and scientists interested in foundational problems.

This title was first published in 2000: What was Wittgenstein's relation to "theory of meaning" in his post-1937 writings and what was his characterization of "philosophy" ? How does "philosophy" in Wittgenstein's later writings differ from what is usually accepted in modern academic 'analytic' philosophy? This book discusses problems encountered in looking at Wittgenstein's texts after-1937, focusing particularly on whether the problem of philosophy amounts to a systematic or a theoretical activity. Arguing that philosophy can be characterized as a form of conceptual investigation, Gefwert aims to demonstrate that a theoretical view does not correspond to Wittgenstein's conception of philosophy. For example, philosophy is not transcendental as he thought it was before 1929. Neither is philosophical language universal as Wittgenstein 1929-1936 thought it was. Proposing that a philosophical conceptual investigation is analogous to a psychotherapeutical session of Freud, with the common aim to dissolve the conceptual problems in language that haunt us in our everyday life, Gefwert's examination of the post-1937 writings of Wittgenstein concludes that "philosophical investigation" is a very different activity than that assumed by the Logical Positives and others adhering to a theoretical view.

Michael Dummett stands out among his generation as the only British philosopher of language to rival in stature the Americans, Davidson and Quine. In conjunction with them he has been responsible for much of the framework within which questions concerning meaning and understanding are raised and answered in the late twentieth-century Anglo-American tradition. Dummett's output has been prolific and highly influential, but not always as accessible as it deserves to be. This book sets out to rectify this situation. Karen Green offers the first comprehensive introduction to Dummett's philosophy of language, providing an overview and summary of his most important arguments. She argues that Dummett should not be understood as a determined advocate of anti-realism, but that his greatest contribution to the philosophy of language is to have set out the strengths and weaknesses of the three most influential positions within contemporary theory of meaning - realism, as epitomised by Frege, the holism to be found in Wittgenstein, Quine and Davidson and the constructivism which can be extracted from Brouwer. It demonstrates that analytic philosophy as Dummett practices it, is by no means an outmoded approach to thinking about language, but that it is relevant both to cognitive science and to phenomenology.

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