It is advisable to treat some sorts of discourse about fiction with the aid of an intensional operator "in such-And-Such fiction...." the operator may appear either explicitly or tacitly. It may be analyzed in terms of similarity of worlds, As follows: "in the fiction f, A" means that a is true in those of the worlds where f is told as known fact rather than fiction that differ least from our world, Or from the belief worlds of the community in (...) which the fiction originated. (shrink)

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.

The twentieth century has witnessed an unprecedented 'crisis in the foundations of mathematics', featuring a world-famous paradox, a challenge to 'classical' mathematics from a world-famous mathematician, a new foundational school, and the profound incompleteness results of Kurt Gödel. In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably with Bertrand Russell, W. V. Quine, and Gödel himself, and which remains at the focus of Anglo-Saxon philosophical discussion. The present collection (...) brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers. It is a substantially revised version of the edition first published in 1964 and includes a revised bibliography. The volume will be welcomed as a major work of reference at this level in the field. (shrink)

Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre (...) Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations. (shrink)

An examination of the emergence of the phenomenon of deductive argument in classical Greek mathematics.

Charles Crittenden here offers an original solution to one of the traditional dilemmas of philosophy—whether there can be any thing not existing, since to say that some thing does not exist seems to presuppose its existence. Drawing on the tools of Wittgensteinian philosophy and speech act theory, Crittenden argues that we can and often do make reference to unreal objects such as fictional characters, though they do not exist in any sense at all.

Originally published in 1966. This volume analyzes the general structure of scientific theories, their relation to experience and to non-scientific thought. Part One is concerned with the logic underlying empirical discourse before its subjection to the various constraints, imposed by the logico-mathematical framework of scientific theories upon their content. Part Two is devoted to an examination of this framework and, in particular, to showing that the deductive organization of a field of experience is by that very act a modification of (...) empirical discourse and an idealization of its subject matter. Part Three analyzes the concordance between theories and experience and the relevance of science to moral and religious beliefs. (shrink)

In this book, Zalta attempts to lay the axiomatic foundations of metaphysics by developing and applying a (formal) theory of abstract objects. The cornerstones include a principle which presents precise conditions under which there are abstract objects and a principle which says when apparently distinct such objects are in fact identical. The principles are constructed out of a basic set of primitive notions, which are identified at the end of the Introduction, just before the theorizing begins. The main reason for (...) producing a theory which defines a logical space of abstract objects is that it may have a great deal of explanatory power. It is hoped that the data explained by means of the theory will be of interest to pure and applied metaphysicians, logicians and linguists, and pure and applied epistemologists. (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)

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)

It is argued that there can only be a small-finite number of mathematical objects; that these objects range from the very concrete to the very abstract; and that mathematics is essentially not concerned with objects but with concepts. This viewpoint is described as mentalist and is upheld over Platonism, intuitionism, and formalism.

This paper contains a defense against anti-realism in mathematics in the light both of incompleteness and of the fact that mathematics is a ‘cultural artifact.’. Anti-realism (here) is the view that theorems, say, of aritltmetic cannot be taken at face value to express true propositions about the system of numbers but must be reconstrued to be about somctliiiig else or about nothing at all. A ‘bite-the-bullet’ aspect of the defease is that, adopting new axioms, liitherto independent, is not. a matter (...) of recognizing trutlis wliich had previoasly been unrecognized, but of extending the domain of what is true. (shrink)

By a `denoting phrase' I mean a phrase such as any one of the following: a man, some man, any man, every man, all men, the present King of England, the present King of France, the center of mass of the solar system at the first instant of the twentieth century, the revolution of the earth round the sun, the revolution of the sun round the earth. Thus a phrase is denoting solely in virtue of its form. We may distinguish (...) three cases: (1) A phrase may be denoting, and yet not denote anything; e.g., `the present King of France'. (2) A phrase may denote one definite object; e.g., `the present King of England' denotes a certain man. (3) A phrase may denote ambiguously; e.g. `a man' denotes not many men, but an ambiguous man. The interpretation of such phrases is a matter of considerably difficulty; indeed, it is very hard to frame any theory not susceptible of formal refutation. All the difficulties with which I am acquainted are met, so far as I can discover, by the theory which I am about to explain. (shrink)

The configurational dimension, in turn, displays temporal features that may be opposed to these "features" of episodic time. The configurational arrangement makes the succession of events into significant wholes that are the correlate of the act of grouping together. Thanks to this reflective act—in the sense of Kant's Critique of Judgment—the whole plot may be translated into one "thought." "Thought," in this narrative context, may assume various meanings. It may characterize, for instance, following Aristotle's Poetics, the "theme" that accompanies the (...) "fable" or "plot" of a tragedy.1 "Thought" may also designate the "point" of the Hebraic maschal or of the biblical parable, concerning which Jeremias observes that the point of the parable is what allows us to translate it into a proverb or an aphorism. The term "thought" may also apply to the "colligatory terms" used in history writing, such terms as "the Renaissance," "the Industrial Revolution," and so on, which, according to Walsh and Dray, allow us to apprehend a set of historical events under a common denominator. In a word, the correlation between thought and plot supersedes the "then" and "and then" of mere succession. But it would be a complete mistake to consider "thought" as a-chronological. "Fable" and "theme" are as closely tied together as episode and configuration. The time of fable-and theme, if we may make of this a hyphenated expression, is more deeply temporal than the time of merely episodic narratives. · 1. It may be noted in passing that this correlation between "theme" and "plot" is also the basis of Northup Frye's "archetypal" criticism. Paul Ricoeur is professor of philosophy at the Université de Paris and John Nuveen Professor at the University of Chicago. Some of his works to appear in English are Husserl: An Analysis of His Phenomenology, Main Trends in Philosophy, and The Conflict of Interpretations: Essays on Hermeneutics. His previous contribution to Critical Inquiry, "The Metaphorical Process as Cognition, Imagination, and Feeling," appeared in the Autumn 1978 issue. (shrink)

This book expounds a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics. He links this to a defense of realism about the metaphysics of mathematics--the view that mathematics (...) is about things that really exist. (shrink)

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.