In seiner letzten Schrift unternimmt Husserl den Versuch, auf dem Wege einer teleologisch-historischen Besinnung auf die Ursprünge unserer kritischen wissenschaftlichen und philosophischen Situation die Notwendigkeit einer transzendentalphänomenologischen Umwendung der Philosophie zu begründen. Er geht von seinem Begriff der "Lebenswelt" aus und entwickelt eine auf diesen Zentralbegriff seiner Spätphilosophie gegründete eigenständige Einleitung in die transzendentale Phänomenologie.

First published in 2002. Routledge is an imprint of Taylor & Francis, an informa company.

First published in 2002. Routledge is an imprint of Taylor & Francis, an informa company.

This is a book that no one but Weyl could have written--and, indeed, no one has written anything quite like it since.

This course on logic and theory of knowledge fell exactly midway between the publication of the Logical Investigations in 1900-01 and Ideas I in 1913. It constitutes a summation and consolidation of Husserl’s logico-scientific, epistemological, and epistemo-phenomenological investigations of the preceding years and an important step in the journey from the descriptivo-psychological elucidation of pure logic in the Logical Investigations to the transcendental phenomenology of the absolute consciousness of the objective correlates constituting themselves in its acts in Ideas I. In (...) this course Husserl began developing his transcendental phenomenology as the genuine realization of what had only been realized in fragmentary form in the Logical Investigations. Husserl considered that in the courses that he gave at the University of Göttingen he had progressed well beyond the insights of the Logical Investigations. Once he exposed the objective theoretical scaffolding needed to keep philosophers from falling into the quagmires of psychologism and skepticism, he set out on his voyage of discovery of the world of the intentional consciousness and to introduce the phenomenological analyses of knowledge that were to yield the general concepts of knowledge needed to solve the most recalcitrant problems of theory of knowledge understood as the investigation of the thorny problems involving the relationship of the subjectivity of the knower to the objectivity of what is known. This translation appears at a time when philosophers in English-speaking countries have heartily embraced the thoughts of Husserl’s German contemporary Gottlob Frege and his concerns. It is replete with insights into matters that many philosophers have been primed to appreciate out of enthusiasm for Frege’s ideas. Among these are: anti-psychologism, meaning, the foundations of mathematics, logic, science, and knowledge, his questions about sets and classes, intensions, identity, calculating with concepts, perspicuity, and even his idealism. (shrink)

Most areas of philosopher Edmund Husserl’s thought have been explored, but his views on logic, mathematics, and semantics have been largely ignored. These essays offer an alternative to discussions of the philosophy of contemporary mathematics. The book covers areas of disagreement between Husserl and Gottlob Frege, the father of analytical philosophy, and explores new perspectives seen in their work.

8.3 The consistency proof -- 8.4 Applications of the consistency proof -- 8.5 Second-order arithmetic -- Problems -- Chapter 9: Set Theory -- 9.1 Axioms for sets -- 9.2 Development of set theory -- 9.3 Ordinals -- 9.4 Cardinals -- 9.5 Interpretations of set theory -- 9.6 Constructible sets -- 9.7 The axiom of constructibility -- 9.8 Forcing -- 9.9 The independence proofs -- 9.10 Large cardinals -- Problems -- Appendix The Word Problem -- Index.

This volume provides an articulate restatement of many of the themes of Husserlian phenomenology.

The first part of the essay describes how mathematics, in particular mathematical concepts, are applicable to nature. mathematical constructs have turned out to correspond to physical reality. this correlation between the world and mathematical concepts, it is argued, is a true phenomenon. the second part of this essay argues that the applicability of mathematics to nature is mysterious, in that not only is there no known explanation for the correlation between mathematics and physical reality, but there is a good reason (...) to except no such correlation. it is argued that there is a subjective element in the decision as to what constitutes a mathematical concept. a number of purported solutions to the mystery of the applicability of mathematics to nature are discarded, until we are left with eugene wigner's thesis that we are here confronted with a "miracle that we neither understand nor deserve.". (shrink)

Discussions of the applicability of mathematics in the natural sciences have been flawed by failure to realize that there are multiple senses in which mathematics can be ‘applied’ and, correspondingly, multiple problems that stem from the applicability of mathematics. I discuss semantic, metaphysical, descriptive, and and epistemological problems of mathematical applicability, dwelling on Frege's contribution to the solution of the first two types. As for the remaining problems, I discuss the contributions of Hartry Field and Eugene Wigner. Finally, I argue (...) that there are epistemological problems concerning the applicability of mathematics that nobody in the philosophical community has yet confronted, though the problems are well known to physicists. (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)

We investigate the form of mathematical structuralism that acknowledges the existence of structures and their distinctive structural elements. This form of structuralism has been subject to criticisms recently, and our view is that the problems raised are resolved by proper, mathematics-free theoretical foundations. Starting with an axiomatic theory of abstract objects, we identify a mathematical structure as an abstract object encoding the truths of a mathematical theory. From such foundations, we derive consequences that address the main questions and issues that (...) have arisen. Namely, elements of different structures are different. A structure and its elements ontologically depend on each other. There are no haecceities and each element of a structure must be discernible within the theory. These consequences are not developed piecemeal but rather follow from our definitions of basic structuralist concepts. (shrink)

CHAPTER I THE EMERGENCE AND DEVELOPMENT OF HUSSERL'S 'PHILOSOPHY OF ARITHMETIC'. HISTORICAL BACKGROUND: WEIERSTRASS AND THE ARITHMETIZATION OF ANALYSIS In ...

Translator's Preface LA LOGIQUE DE HUSSERL, etude sur "Logique for- melle et logique transcendentale" the original of the present translation, was published ...

According to realist structuralism, mathematical objects are places in abstract structures. We argue that in spite of its many attractions, realist structuralism must be rejected. For, first, mathematical structures typically contain intra-structurally indiscernible places. Second, any account of place-identity available to the realist structuralist entails that intra-structurally indiscernible places are identical. Since for her mathematical singular terms denote places in structures, she would have to say, for example, that 1 = − 1 in the group (Z, +). We call this (...) the identity problem and conclude that nominalism is presently the safest route for the structuralist. (shrink)

Edmund Husserl is the founder of phenomenology. The Logical Investigations is Edmund Husserl's most famous work and has had a decisive impact on the direction of twentieth century philosophy. This is the first time both volumes of this classic work, translated by J.N. Findlay, have been available in paperback. They include a new introduction by Dermot Moran, placing the Logical Investigations in historical context and bringing out its importance for contemporary philosophy.

Science in a new sense arises in the first instance from Plato's establishing of logic, as a place for exploring the essential requirements of "genuine" ...

Develops a structuralist understanding of mathematics, as an alternative to set- or type-theoretic foundations, that respects classical mathematical truth while ...

00 Drawing on the phenomenological tradition in the philosophy of science and philosophy of nature, Patrick Heelan concludes that perception is a cognitive, ...

Important study focuses on the revival and assimilation of ancient Greek mathematics in the 13th–16th centuries, via Arabic science, and the 16th-century development of symbolic algebra. This brought about the crucial change in the concept of number that made possible modern science — in which the symbolic "form" of a mathematical statement is completely inseparable from its "content" of physical meaning. Includes a translation of Vieta's Introduction to the Analytical Art. 1968 edition. Bibliography.

Derrida's introduction to his French translation of Husserl's essay "The Origin of Geometry," arguing that although Husserl privileges speech over writing in an account of meaning and the development of scientific knowledge, this privilege is in fact unstable.

In the first year of the twentieth century, in Gottingen, Husserl delivered two talks dealing with a problem that proved central in his philosophical development, that of imaginary elements in mathematics. In order to solve this problem Husserl introduced a logical notion, called “definiteness”, and variants of it, that are somehow related, he claimed, to Hilbert’s notions of completeness. Many different interpretations of what precisely Husserl meant by this notion, and its relations with Hilbert’s ones, have been proposed, but no (...) consensus has been reached. In this paper I approach this question afresh and thoroughly, taking into consideration not only the relevant texts and context, as others have also done before, but, more importantly, Husserl’s philosophy, his intuition-based epistemology in particular. Based on a system of clearly defined concepts that I here present, I reinforce an interpretation—definiteness as a form of syntactic completeness—that has, I believe, some advantages vis-à-vis alternative interpretations. It is in conformity with the available texts; it makes clear that Husserl’s notion of definiteness is indeed close to Hilbert’s notions of completeness; it solves the important problem of imaginaries for which it was created; and last, but not least, it fits naturally into Husserl’s system of concepts and ideas. (shrink)