This book contains seminal discussions of central issues in the philosophy of language, mathematics, mind, religion and time. Is common language conceptually prior to idiolectics? What is a theory of meaning? Does constructivism provide a satisfactory account of mathematics? What are indefinitely extensible concepts? Can we change the past? These are only some of the very important questions addressed here. Both the papers written by the contributors and Dummett's replies provide a great wealth of stimulating ideas for those who currently (...) do research in the respective areas touched upon without making the reading exceedingly tedious. This feature, common to most of the papers in this book, makes it possible to use the material presented in undergraduate courses at university level. (shrink)

general, abstract situation. On the other side, I know that graduate students and all mathematicians sometimes falter because their intuitive, ..

This paper attempts to show that mathematical knowledge does not grow by a simple process of accumulation and that it is possible to provide a quasi-empirical (in Lakatos's sense) account of mathematical theories. Arguments supporting the first thesis are based on the study of the changes occurred within Eudidean geometry from the time of Euclid to that of Hilbert; whereas those in favour of the second arise from reflections on the criteria for refutation of mathematical theories.

The present article aims at showing that it is possible to construct a realist philosophy of mathematics which commits one neither to dream the dreams of Platonism nor to reduce the word ''realism'' to mere noise. It is argued that mathematics is a science of patterns, where patterns are not objects (or properties of objects), but aspects, or aspects of aspects, etc. of objects. (The notion of aspect originates from ideas sketched by Wittgenstein in the Philosophical Investigations.).

The present paper aims at showing that there are times when set theoretical knowledge increases in a non-cumulative way. In other words, what we call ‘set theory’ is not one theory which grows by simple addition of a theorem after the other, but a finite sequence of theories T1, ..., Tn in which Ti+1, for 1 ≤ i < n, supersedes Ti. This thesis has a great philosophical significance because it implies that there is a sense in which mathematical theories, (...) like the theories belonging to the empirical sciences, are fallible and that, consequently, mathematical knowledge has a quasi-empirical nature. The way I have chosen to provide evidence in favour of the correctness of the main thesis of this article consists in arguing that Cantor–Zermelo set theory is a Lakatosian Mathematical Research Programme (MRP). (shrink)

Peter Abelard (1079–1142 ce) was the most wide‐ranging philosopher of the twelfth century. He quickly established himself as a leading teacher of logic in and near Paris shortly after 1100. After his affair with Heloise, and his subsequent castration, Abelard became a monk, but he returned to teaching in the Paris schools until 1140, when his work was condemned by a Church Council at Sens. His logical writings were based around discussion of the “Old Logic”: Porphyry's Isagoge, aristotle'S Categories and (...) On Interpretation and boethius'S textbook on topical inference. They comprise a freestanding Dialectica (“Logic”; probably c.1116), a set of commentaries (known as the Logica [Ingredientibus], c. 1119) and a later (c. 1125) commentary on the Isagoge (Logica Nostrorum Petititoni Sociorum or Glossulae). In a work Abelard called his Theologia, issued in three main versions (between 1120 and c.1134), he attempted a logical analysis of trinitarian relations and explored the philosophical problems surrounding God's claims to omnipotence and omniscience. The Collationes (“Debates,” also known as “Dialogue between a Christian, a Philosopher and a Jew”; probably c.1130) present a rational investigation into the nature of the highest good, in which the Christian and the Philosopher (who seems to be modeled on a philosopher of pagan antiquity) are remarkably in agreement. The unfinished Scito teipsum (“Know thyself,” also known as the “Ethics”; c.1138) analyses moral action. (shrink)

The present collection of essays aims at casting light on important elements of Wittgenstein's thought and intellectual development. Some of Wittgenstein's ideas are here discussed using the «Tractatus» as a frame of reference, others by relating them to the views of G. Frege, G.E.M. Anscombe and D. Davidson.

Bertrand Russell's contributions to last century's philosophy and, in particular, to the philosophy of mathematics cannot be overestimated.Russell, besides being, with Frege and G.E. Moore, one of the founding fathers of analytical philosophy, played a major rôle in the development of logicism, one of the oldest and most resilient1 programmes in the foundations of mathematics.Among his many achievements, we need to mention the discovery of the paradox that bears his name and the identification of its logical nature; the generalization to (...) the whole of mathematics of Frege's idea that it is not possible to draw a demarcation line between logic and arithmetic; the programme, carried out with Whitehead, of derivation of mathematics from the logical system of Principia Mathematica ; and the ramified theory of types, devised by Russell to protect the system of PM from the known paradoxes.Although there is an ample literature on these topics, it is quite important to reconsider Russell's contributions to the foundations of mathematics at a time when, as a consequence of the crisis of the classical programmes in the foundations of mathematics, new trends are beginning to develop within the philosophy of mathematics. These are trends which move in a very different direction from that of logicism, intuitionism, and Hilbert's programme.To see this we need to consider that, in spite of profound disagreements on the nature of mathematical activity, on the relationship existing between logic and mathematics, on the causes of and therapies for the paradoxes, etc., logicism, intuitionism, and Hilbert's programme share an important metaphor: the idea that mathematics is an edifice built on unshakable foundations,2 an edifice which makes possible only a cumulative growth of mathematical knowledge.Such a metaphor—which, together with more specific theses belonging to these schools of thought, remained unsupported …. (shrink)

Available from UMI in association with The British Library. Requires signed TDF. ;The central position defended in this work is that Analytical philosophy is the best realization so far of Descartes' concern for the construction of a method of scientific investigation in philosophy. ;In Frege, the father of Analytical philosophy, such concern was embodied in his Conceptual Notation, which should have provided philosophy with a rigorous and powerful deductive system, and in the 'linguistic turn', i.e. the study of the structure (...) of thought conducted by means of an analysis of the structure of language. This 'linguistic turn' should have produced, in Frege's view, definitive conclusions about the limits of what is thinkable not by setting up arbitrary systems of Pure Reason, but through the controllable investigation of the logical structure of an entity--language--of which we have conceptual experience. ;Wittgenstein's later philosophy is seen here as an attack against the aims and tenets of Analytical philosophy. In fact, what Wittgenstein shows in Philosophical Investigations is, among other things, that by means of an investigation of language it is impossible to justify the assumption that language is an expression of thought. But, if this were true, it would follow that we were not justified in assuming that there exists an isomorphism between language and thought. That there is such an isomorphism is the presupposition which lies at the basis of Frege's 'linguistic turn'. ;In conclusion I attempt to prove that there is no solution to Wittgenstein's objections unless the original exclusively linguistic Analytical philosophy model is modified by a 'pragmatic turn'. The key element of the 'pragmatic turn' is that the truth of certain assumptions--and therefore the justification for their introduction--can be confirmed by the impact that the system based on them has on reality in controlling and transforming it successfully through techniques which are logically dependent on the system. (shrink)

In this paper we intend to argue that: the question ‘True V or not True V’ is central to both the philosophical and mathematical investigations of the foundations of mathematics; when posed within a framework in which set theory is seen as a science of objects, the question ‘True V or not True V’ generates a dilemma each horn of which turns out to be unacceptable; a plausible way out of the dilemma mentioned at is provided by an approach to (...) set theory according to which this is considered to be a science of structures. (shrink)