Wittgenstein's work remains, undeniably, now, that off one of those few philosophers who will be read by all future generations.

Editorial preface to the fourth edition and modified translation -- The text of the Philosophische Untersuchungen -- Philosophische untersuchungen = Philosophical investigations -- Philosophie der psychologie, ein fragment = Philosophy of psychology, a fragment.

This handbook with exercises reveals the mathematical beauty of formalisms hitherto mostly used for software and hardware design and verification.

Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.

First published in 1974. Despite the tendency of contemporary analytic philosophy to put logic and mathematics at a central position, the author argues it failed to appreciate or account for their rich content. Through discussions of such mathematical concepts as number, the continuum, set, proof and mechanical procedure, the author provides an introduction to the philosophy of mathematics and an internal criticism of the then current academic philosophy. The material presented is also an illustration of a new, more general method (...) of approach called substantial factualism which the author asserts allows for the development of a more comprehensive philosophical position by not trivialising or distorting substantial facts of human knowledge. (shrink)

First published in 1974. Despite the tendency of contemporary analytic philosophy to put logic and mathematics at a central position, the author argues it failed to appreciate or account for their rich content. Through discussions of such mathematical concepts as number, the continuum, set, proof and mechanical procedure, the author provides an introduction to the philosophy of mathematics and an internal criticism of the then current academic philosophy. The material presented is also an illustration of a new, more general method (...) of approach called substantial factualism which the author asserts allows for the development of a more comprehensive philosophical position by not trivialising or distorting substantial facts of human knowledge. (shrink)

Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set (...) theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science. (shrink)

This engaging and accessible introduction to the philosophy of language provides an important guide to one of the liveliest and most challenging areas of study in philosophy. Interweaving the historical development of the subject with a thematic overview of the different approaches to meaning, the book provides students with the tools necessary to understand contemporary analytical philosophy. The second edition includes new material on: Chomsky, Wittgenstein and Davidson as well as new chapters on the causal theory of reference, possible worlds (...) semantics and semantic externalism. (shrink)

Starting with Gottlob Frege's foundational theories of sense and reference, Miller provides a useful introduction to the formal logic used in all subsequent philosophy of language. He communicates a sense of active philosophical debate by confronting the views of the early theorists concerned with building systematic theories - such as Frege, Bertrand Russell, and the logical positivists - with the attacks mounted by sceptics - such as W.O. Quine, Saul Kripke, and Ludwig Wittgenstein. This leads to important excursions into related (...) areas of metaphysics, philosophy of mind, and cognitive science that present the more recent attempts to save the notions of sense and meaning by philosophers such as Paul Grice, John Searle, Jerry Fodor, Colin McGinn, and Crispin Wright. Miller then returns to the systematic program by examining the formal theories of Donald Davidson, concluding with a chapter surveying the relevance of philosophy of language to the broader metaphysical debates between realists and anti-realists. Miller's clear, engaged, and coherently structured approach makes Philosophy of Language an ideal text for undergraduate courses. The guides to further reading provided in each chapter help the reader pursue interesting topics further and facilitate using the book in conjunction with primary sources. (shrink)

This engaging and accessible introduction to the philosophy of language provides an important guide to one of the liveliest and most challenging areas of study in philosophy. Interweaving the historical development of the subject with a thematic overview of the different approaches to meaning, the book provides students with the tools necessary to understand contemporary analytical philosophy.

Mathematicians tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism about mathematics raises serious questions: What are mathematical things? Where are they? How do we know about them? Offering a scrupulously fair treatment of both mathematical and philosophical concerns, Penelope Maddy here delineates and defends a novel version of mathematical realism. (...) She answers the traditional questions and poses a challenging new one, refocusing philosophical attention on the pressing foundational issues of contemporary mathematics. (shrink)

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)

Combinatory logic and lambda-conversion were originally devised in the 1920s for investigating the foundations of mathematics using the basic concept of 'operation' instead of 'set'. They have now developed into linguistic tools, useful in several branches of logic and computer science, especially in the study of programming languages. These notes form a simple introduction to the two topics, suitable for a reader who has no previous knowledge of combinatory logic, but has taken an undergraduate course in predicate calculus and recursive (...) functions. The key ideas and basic results are presented, as well as a number of more specialised topics, and man), exercises are included to provide manipulative practice. (shrink)

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)