Charles Parsons examines the notion of object, with the aim to navigate between nominalism, denying that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a 'nature' than that confers on them. Parsons also analyzes the concept of intuition (...) and presents a conception of it distantly inspired by that of Kant, which describes a basic kind of access to abstract objects and an element of a first conception of the infinite. (shrink)

Taking Wittgenstein at His Word is an experiment in reading organized around a central question: What kind of interpretation of Wittgenstein's later philosophy emerges if we adhere strictly to his claims that he is not in the business of presenting and defending philosophical theses and that his only aim is to expose persistent conceptual misunderstandings that lead to deep philosophical perplexities? Robert Fogelin draws out the therapeutic aspects of Wittgenstein's later work by closely examining his account of rule-following and how (...) he applies the idea in the philosophy of mathematics.The first of the book's two parts focuses on rule-following, Wittgenstein's "paradox of interpretation," and his naturalistic response to this paradox, all of which are persistent and crucial features of his later philosophy. Fogelin offers a corrective to the frequent misunderstanding that the paradox of interpretation is a paradox about meaning, and he emphasizes the importance of Wittgenstein's often undervalued appeals to natural responses. The second half of the book examines how Wittgenstein applies his reflections on rule-following to the status of mathematical propositions, proofs, and objects, leading to remarkable, demystifying results.Taking Wittgenstein at His Word shows that what Wittgenstein claims to be doing and what he actually does are much closer than is often recognized. In doing so, the book underscores fundamental--but frequently underappreciated--insights about Wittgenstein's later philosophy. (shrink)

The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers.

Fur Wittgenstein haben als 'grundlegend' gedachte mathematische Bemuhungen, wie sie sich in den logizistischen Systemen Freges und Russells, der Mengentheorie und der nach Konsistenzbeweisen fur solche Systeme suchenden Beweistheorie Hilberts zeigen, keineswegs einen nennenswerten grundlegenden Charakter, sondern es handelt sich bei ihnen um normale Stucke Mathematik wie andere auch. Im Teil III der Bemerkungen uber die Grundlagen der Mathematik entwickelt er seine Kritik an jenen Grundlegungsanspruchen in vielfaltigster Weise, und dieser Kommentar zeichnet seine Gedankengange in all ihren Verastelungen nach. Dabei (...) wird nicht nur der veroffentlichte Text berucksichtigt, sondern auch samtliche relevanten Nachlasspassagen, aus denen er hervorgegangen ist und ohne die er teilweise gar nicht verstanden werden kann. Der Kommentar ist sowohl kritisch als auch konstruktiv: Mangel werden als solche benannt und Erganzungen und Verbesserungen vorgeschlagen, wenn es moglich erscheint. Das Ergebnis sind Wittgensteinsche Gedankengange, die ungleich uberzeugender und vernunftiger sind, als dies bislang in der Literatur uber Wittgensteins Philosophie der Mathematik sichtbar war. (shrink)

System Requirements System requirements Minimum 80486, 66MHz IBM PC or full compatible ; Minimum 16MB RAM 177MB hard disk space to store and run the Nachlass, an extra 12MB in addition to this should be available during installation. SVGA monitor set to 800x600 pixels, 16-bit colour, or higher setting recommended to use and display the transcription text and facsimiles; Quad-speed CD-ROM drive or higher; Windows 3.1, 3.11; Windows 95/98; Windows NT 4.0; Windows 2000. Microsoft mouse or compatible Network versions Windows (...) NT 4, and Novell networks. Customers with Wide Area Networks should ring Customer Services to discuss specific requirements. Customer Service Contact: Customer Services, Electronic Publishing Tel: +44 1865 267979 Fax: +44 1865 267990 Email: [email protected]. (shrink)

The later Wittgenstein holds that the sole function of mathematical propositions is to determine the concepts they invoke. In the paper this view is discussed by means of a single example: Wittgenstein's investigation of the concept of a regular heptagon as used in Euclidean geometry (i.e., the Euclidean constructiongame with rulerand compass) andinCartesian analytic geometry. Going on from some well-known passages in Wittgenstein's Lectures on the Foundations of Mathematics, and completing these passages, it is shown that Wittgenstein'sview makes perfectly good (...) sense and can be very well defended. (shrink)

In Part III of his Remarks on the Foundations of Mathematics Wittgenstein deals with what he calls the surveyability of proofs. By this he means that mathematical proofs can be reproduced with certainty and in the manner in which we reproduce pictures. There are remarkable similarities between Wittgenstein's view of proofs and Hilbert's, but Wittgenstein, unlike Hilbert, uses his view mainly in critical intent. He tries to undermine foundational systems in mathematics, like logicist or set theoretic ones, by stressing the (...) unsurveyability of the proof-patterns occurring in them. Wittgenstein presents two main arguments against foundational endeavours of this sort. First, he shows that there are problems with the criteria of identity for the unsurveyable proof-patterns, and second, he points out that by making these patterns surveyable, we rely on concepts and procedures which go beyond the foundational frameworks. When we take these concepts and procedures seriously, mathematics does not appear as a uniform system, but as a mixture of different techniques. (shrink)