18 Probability and confirmation* The story of deductive logic is well known. Until the beginning of the nineteenth century, deductive logic as a subject was ...

This major publication is a history of the semantic tradition in philosophy from the early nineteenth century through its incarnation in the work of the Vienna Circle, the group of logical positivists that emerged in the years 1925–1935 in Vienna who were characterised by a strong commitment to empiricism, a high regard for science, and a conviction that modern logic is the primary tool of analytic philosophy. In the first part of the book, Alberto Coffa traces the roots of logical (...) positivism in a semantic tradition that arose in opposition to Kant's theory that a priori knowledge is based on pure intuition and the constitutive powers of the mind. In Part II, Coffa chronicles the development of this tradition by members and associates of the Vienna Circle. Much of Coffa's analysis draws on the unpublished notes and correspondence of many philosophers. The book, however, is not merely a history of the semantic tradition from Kant 'to the Vienna Station'. Coffa also critically reassesses the role of semantic notions in understanding the ground of a priori knowledge and its relation to empirical knowledge and questions the turn the tradition has taken since Vienna. (shrink)

The language of mathematics speaks of objects. This is a rather trivial statement; it is not certain that we can conceive any developed language that does not. What is of interest is that, taken at face value, mathematical language speaks of objects distinctively mathematical in character: numbers, functions, sets, geometric figures, and the like. To begin with they are distinctive in being abstract.

In this book Terence Parsons revives the older tradition of taking such objects at face value. Using various modern techniques from logic and the philosophy of language, he formulates a metaphysical theory of nonexistent objects. The theory is given a formalization in symbolism rich enough to contain definite descriptions, modal operators, and epistemic contexts, and the book includes a discussion which relates the formalized theory explicitly to English.

Aristotelian Philosophy, as far as its subject matter is Being, conceives of it as “ousia,” substance. This actual, self-sufficient Being contrasts with the dependent Being of the “symbekóta,” accidents, nonsubstantial attributes. Real Being, existence was of no interest for Antique Philosophy. It was as late as in Scholastic Philosophy that the concept of “existere” gained interest in connection with the problem of universals and the proofs of God’s existence. Albert the Great uses this concept in connection with the concept of (...) a person. (shrink)

In challenging the prevailing paradigm for understanding how the human mind works, Patterns, Thinking, and Cognition is certain to stimulate fruitful debate.

I argue on the basis of an example, Fourier theory applied to the problem of vibration, that Field's program for nominalizing science is unlikely to succeed generally, since no nominalistic variant will provide us with the kind of physical insight into the phenomena that the standard theory supplies. Consideration of the same example also shows, I argue, that some of the motivation for mathematical fictionalism, particularly the alleged problem of cognitive access, is more apparent than real.

In this paper, we develop an alternative strategy, Platonized Naturalism, for reconciling naturalism and Platonism and to account for our knowledge of mathematical objects and properties. A systematic (Principled) Platonism based on a comprehension principle that asserts the existence of a plenitude of abstract objects is not just consistent with, but required (on transcendental grounds) for naturalism. Such a comprehension principle is synthetic, and it is known a priori. Its synthetic a priori character is grounded in the fact that it (...) is an essential part of the logic in which any scientific theory will be formulated and so underlies (our understanding of) the meaningfulness of any such theory (this is why it is required for naturalism). Moreover, the comprehension principle satisfies naturalist standards of reference, knowledge, and ontological parsimony! As part of our argument, we identify mathematical objects as abstract individuals in the domain governed by the comprehension principle, and we show that our knowledge of mathematical truths is linked to our knowledge of that principle. (shrink)

In this work Dummett discusses, section by section, Frege's masterpiece The Foundations of Arithmetic and Frege's treatment of real numbers in the second volume ...

First published in 1971, Professor Putnam's essay concerns itself with the ontological problem in the philosophy of logic and mathematics - that is, the issue of whether the abstract entities spoken of in logic and mathematics really exist. He also deals with the question of whether or not reference to these abstract entities is really indispensible in logic and whether it is necessary in physical science in general.

John Woods' The Logic of Fiction, now thirty-five years old, is a ground-breaking event in the establishment of the semantics of fiction as a stand-alone research programme in the philosophies of language and logic. There is now a large literature about these matters, but Woods' book retains a striking freshness, and still serves as a convincing template of the treatment options for the field's key problems. The book now appears in a second edition with a new Foreword by Nicholas Griffin (...) and an extended bibliography covering the period 1969-2009. As Griffin notes in his Foreword, it is "surprising on looking back to discover how little was written on the semantics of fiction before John Woods' The Logic of Fiction was published in 1974. The surprise is the greater because Woods' book appeared after almost a quarter century of fierce philosophical debate about reference Fictional discourse, one would have thought, would be an important testing ground for philosophical theories of referential expressions and one, moreover, in which the standard theories would likely be tested to destruction. " " One of the great merits of Woods' book is that it takes seriously the wide-ranging demands that fiction imposes on logic and semantics, and does not try to force fiction into some pre-conceived logical mould. but thanks to Woods' pioneering efforts, we are much closer to one now than we were when he set out to write his book. His book was not the last word on the logic of fiction; it was much more important: it was nearly the first." NICHOLAS GRIFFIN is Canada Research Chair in Philosophy at McMaster University. Recent publications include Russell vs Meinong: The Legacy of "On Denoting," edited with Dale Jacquette. JOHN WOODS is Director of the Abductive Systems Group at the University of British Columbia and Charles S. Peirce Visiting Professor of Logic in the Group on Logic and Computational Science, King's College London. He has two forthcoming books on fiction - an edited volume, Fictions and Models: New Essays, and a research monograph, Sherlock's Member: New Perspectives on the Semantics of Fiction, both to appear in 2010. (shrink)

Numbers are abstract, certain, and eternal, but to most of us somewhat dry and bloodless. Good stories are full of life: they engage our emotions and have subtlety and nuance, but they lack rigour and the truths they tell are elusive and subject to debate. As ways of understanding the world around us, numbers and stories seem almost completely incompatible. This study aims to show that stories and numbers aren't as different as might be imagined, and in fact they have (...) surprising and fascinating connections. The concepts of logic and probability both grew out of intuitive ideas about how certain situations would play out. Now, logicians are inventing ways to deal with real world situations by mathematical means - by acknowledging, for instance, that items that are mathematically interchangeable may not be interchangeable in a story. (shrink)

_Philosophy of Mathematics_ is an excellent introductory text. This student friendly book discusses the great philosophers and the importance of mathematics to their thought. It includes the following topics: * the mathematical image * platonism * picture-proofs * applied mathematics * Hilbert and Godel * knots and nations * definitions * picture-proofs and Wittgenstein * computation, proof and conjecture. The book is ideal for courses on philosophy of mathematics and logic.

a collection of articles by philosophers of mathematics on themes associated with the work of Paul Benacceraf.

Science Without Numbers caused a stir in 1980, with its bold nominalist approach to the philosophy of mathematics and science. It has been unavailable for twenty years and is now reissued in a revised edition with a substantial new preface presenting the author's current views and responses to the issues raised in subsequent debate.

Written in an outstandingly clear and lively style, this 1969 book provokes its readers to rethink issues they may have regarded as long since settled.

Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic (...) problems. As a way out of this dilemma, Shapiro articulates a structuralist approach. On this view, the subject matter of arithmetic, for example, is not a fixed domain of numbers independent of each other, but rather is the natural number structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle. Using this framework, realism in mathematics can be preserved without troublesome epistemic consequences. Shapiro concludes by showing how a structuralist approach can be applied to wider philosophical questions such as the nature of an "object" and the Quinean nature of ontological commitment. Clear, compelling, and tautly argued, Shapiro's work, noteworthy both in its attempt to develop a full-length structuralist approach to mathematics and to trace its emergence in the history of mathematics, will be of deep interest to both philosophers and mathematicians. (shrink)

This book analyzes the different ways mathematics is applicable in the physical sciences, and presents a startling thesis--the success of mathematical physics ...

Platonism is the most pervasive philosophy of mathematics. Indeed, it can be argued that an inarticulate, half-conscious Platonism is nearly universal among mathematicians. The basic idea is that mathematical entities exist outside space and time, outside thought and matter, in an abstract realm. In the more eloquent words of Edward Everett, a distinguished nineteenth-century American scholar, "in pure mathematics we contemplate absolute truths which existed in the divine mind before the morning stars sang together, and which will continue to exist (...) there when the last of their radiant host shall have fallen from heaven." In What is Mathematics, Really?, renowned mathematician Rueben Hersh takes these eloquent words and this pervasive philosophy to task, in a subversive attack on traditional philosophies of mathematics, most notably, Platonism and formalism. Virtually all philosophers of mathematics treat it as isolated, timeless, ahistorical, inhuman. Hersh argues the contrary, that mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context. Mathematical objects are created by humans, not arbitrarily, but from activity with existing mathematical objects, and from the needs of science and daily life. Hersh pulls the screen back to reveal mathematics as seen by professionals, debunking many mathematical myths, and demonstrating how the "humanist" idea of the nature of mathematics more closely resembles how mathematicians actually work. At the heart of the book is a fascinating historical account of the mainstream of philosophy--ranging from Pythagoras, Plato, Descartes, Spinoza, and Kant, to Bertrand Russell, David Hilbert, Rudolph Carnap, and Willard V.O. Quine--followed by the mavericks who saw mathematics as a human artifact, including Aristotle, Locke, Hume, Mill, Peirce, Dewey, and Lakatos. In his epilogue, Hersh reveals that this is no mere armchair debate, of little consequence to the outside world. He contends that Platonism and elitism fit well together, that Platonism in fact is used to justify the claim that "some people just can't learn math." The humanist philosophy, on the other hand, links mathematics with geople, with society, and with history. It fits with liberal anti-elitism and its historical striving for universal literacy, universal higher education, and universal access to knowledge and culture. Thus Hersh's argument has educational and political ramifications. Written by the co-author of The Mathematical Experience, which won the American Book Award in 1983, this volume reflects an insider's view of mathematical life, based on twenty years of doing research on advanced mathematical problems, thirty-five years of teaching graduates and undergraduates, and many long hours of listening, talking to, and reading philosophers. A clearly written and highly iconoclastic book, it is sure to be hotly debated by anyone with a passionate interest in mathematics or the philosophy of science. (shrink)

This is a long-awaited new edition of one of the best known Oxford Logic Guides. The book gives an introduction to intuitionistic mathematics, leading the reader gently through the fundamental mathematical and philosophical concepts. The treatment of various topics, for example Brouwer's proof of the Bar Theorem, valuation systems, and the completeness of intuitionistic first-order logic, have been completely revised.

Presenting a novel account of singular thought, a systematic application of recent work in the theory of speech acts, and a partial revival of Russell's analysis of singular terms, this book takes an original approach to the perennial problems of reference and singular terms by separating the underlying issues into different levels of analysis.

Most philosophers of mathematics try to show either that the sort of knowledge mathematicians have is similar to the sort of knowledge specialists in the empirical sciences have or that the kind of knowledge mathematicians have, although apparently about objects such as numbers, sets, and so on, isn't really about those sorts of things as well. Jody Azzouni argues that mathematical knowledge really is a special kind of knowledge with its own special means of gathering evidence. He analyses the linguistic (...) pitfalls and misperceptions philosophers in this field are often prone to, and explores the misapplications of epistemic principles from the empirical sciences to the exact sciences. What emerges is a picture of mathematics both sensitive to mathematical practice, and to the ontological and epistemological issues that concern philosophers. (shrink)