"Intuition" has perhaps been the least understood and the most abused term in philosophy. It is often the term used when one has no plausible explanation for the source of a given belief or opinion. According to some sceptics, it is understood only in terms of what it is not, and it is not any of the better understood means for acquiring knowledge. In mathematics the term has also unfortunately been used in this way. Thus, intuition is sometimes portrayed as (...) if it were the Third Eye, something only mathematical "mystics", like Ramanujan, possess. In mathematics the notion has also been used in a host of other senses: by "intuitive" one might mean informal, or non-rigourous, or visual, or holistic, or incomplete, or perhaps even convincing in spite of lack of proof. My aim in this book is to sweep all of this aside, to argue that there is a perfectly coherent, philosophically respectable notion of mathematical intuition according to which intuition is a condition necessary for mathemati cal knowledge. I shall argue that mathematical intuition is not any special or mysterious kind of faculty, and that it is possible to make progress in the philosophical analysis of this notion. This kind of undertaking has a precedent in the philosophy of Kant. While I shall be mostly developing ideas about intuition due to Edmund Husser! there will be a kind of Kantian argument underlying the entire book. (shrink)

The twelve original studies collected in this volume examine different aspects of Edmund Husserl's Logical Investigations. They are authored by scholars and specialists internationally recognized for their expertise in the fields of phenomenology, logic, history of philosophy and philosophy of mind. They approach Husserl's groundwork from different angles and perspectives and shed new light on a number of issues such as meaning, intentionality, ontology, logic, etc. They also explore questions such as the place of the Logical Investigations within the whole (...) of Husserl's work, its sources in 19th century philosophy and in particular in the philosophical work of Franz Brentano and Bernard Bolzano, its reception amongst the so-called members of the Munich Circle and its influence on contemporary philosophy. In short, this volume constitutes a "companion" to Husserl's Logical Investigations. (shrink)

Category Theory has developed rapidly. This book aims to present those ideas and methods which can now be effectively used by Mathe­ maticians working in a variety of other fields of Mathematical research. This occurs at several levels. On the first level, categories provide a convenient conceptual language, based on the notions of category, functor, natural transformation, contravariance, and functor category. These notions are presented, with appropriate examples, in Chapters I and II. Next comes the fundamental idea of an adjoint (...) pair of functors. This appears in many substantially equivalent forms: That of universal construction, that of direct and inverse limit, and that of pairs offunctors with a natural isomorphism between corresponding sets of arrows. All these forms, with their interrelations, are examined in Chapters III to V. The slogan is "Adjoint functors arise everywhere". Alternatively, the fundamental notion of category theory is that of a monoid -a set with a binary operation of multiplication which is associative and which has a unit; a category itself can be regarded as a sort of general­ ized monoid. Chapters VI and VII explore this notion and its generaliza­ tions. Its close connection to pairs of adjoint functors illuminates the ideas of universal algebra and culminates in Beck's theorem characterizing categories of algebras; on the other hand, categories with a monoidal structure lead inter alia to the study of more convenient categories of topological spaces. (shrink)

Recently discovered correspondence from Oskar Becker to Hermann Weyl sheds new light on Weyl's engagement with Husserlian transcendental phenomenology in 1918-1927. Here the last two of these letters, dated July and August, 1926, dealing with issues in the philosophy of mathematics are presented, together with background and a detailed commentary. The letters provide an instructive context for re-assessing the connection between intuitionism and phenomenology in Weyl's foundational thought, and for understanding Weyl's term ‘symbolic construction’ as marking his own considered position (...) in the foundational controversy of the 1920s. In addition, they reveal Weyl's hitherto unknown objections to Becker's detailed attempt (Mathematische Existenz, 1927) to ground the transfinite phenomenologically. (shrink)

Der vorliegende Band enthält den Text der vierstündigen Vorlesung, die Husserl im Wintersemester 1908/09 unter dem Titel "Alte und neue Logik" in Göttingen gehalten hat. Es handelt sich bei dieser Vorlesung zum einen um eine Umarbeitung und Neugestaltung seiner Logikvorlesung von 1902/03, die im Band 2 der Materialien veröffentlicht wurde, zum anderen um eine Vorstufe der in Band XXX der Gesammelten Werke unter dem späteren Titel "Logik und Allgemeine Wissenschaftslehre" veröffentlichten Vorlesung "Einleitung in die Logik und Erkenntnistheorie" von 1910/11. Husserl (...) konfrontiert in der Vorlesung nicht, wie der Titel anzukündigen scheint, die traditionelle Logik mit den zeitgenössischen Reformversuchen dieser Logik, sondern er entwickelt in kritischer Abwehr des Psychologismus, zu dem er ausdrücklich auch Brentanos Reform der Logik rechnet, seine eigene Konzeption der reinen Logik. Konstitutiv für diese Konzeption ist die Zweischichtung in Bedeutungslehre als reine Formenlehre der logischen Bedeutungen ("reine Grammatik") und Geltungslehre. Die konkreten Ausführungen sind vor allem einer umfassenden Darstellung der Formenlehre sowie der kritischen Erörterung von bedeutungstheoretischen Streitfragen gewidmet. (shrink)

TO THE LOGICAL INVESTIGATIONS A DRAFT OF A PREFACE TO THE LOGICAL INVESTIGATIONS ( 1913) Edited by EUGEN FINK Translated with Introductions by PHILIP J. BOSSERT and CURTIS H. PETERS • MARTINUS NIJHOFF THE HAGUE 1975 © I975 by Martinus Nijhoff. The Hague. Netherlands All rights reserved. including the right to translate or to reproduce this book or parts thereof in any form ISBN-I3: 978-90-247-1711-8 e-ISBN-I3: 978-94-010-1655-1 DOl: 10. 1007/978-94-010-1655-1 TO HERBERT SPIEGELBERG ESTEEMED SCHOLAR, MENTOR, FRIEND ACKNOWLEDGEMENTS We would like (...) to express our thanks to H. L. Van Breda, director of the Husserl-Archiv (Louvain), for his approval and encouragement of this project, and to Professor Dr. Gerhart Husserl, Professor Dr. Eugen Fink and the editors of Tijdschrift voor Philosophie for their permission to undertake it. We also owe a debt of appreciation to Dr. Karl Schuhmann of the Catholic University of Louvain and to Dr. Elmar Holenstein, Dr. Edi Marbach and Mr. Rudolf Bernet of the Husserl-Archiv (Louvain) for their help in reading the original manuscripts and for putting their excellent knowledge of the Husserl "Nachlass" preserved at the Archives at our disposal. We especially wish to thank Professor Herbert Spiegelberg whose careful and critical reading of our manuscript at an earlier stage resulted in numerous suggestions for its improvement; and, last but not least, our wives, Jane and Pam, for their help in preparing the typescripts. TABLE OF CONTENTS Translator's Introductions XI I. HISTORICAL INTRODUCTION XI II. THEMATIC INTRODUCTION XX III. (shrink)

In this paper I discuss Husserl's solution of the problem of imaginary elements in mathematics as presented in the drafts for two lectures hegave in Göttingen in 1901 and other related texts of the same period,a problem that had occupied Husserl since the beginning of 1890, whenhe was planning a never published sequel to Philosophie der Arithmetik(1891). In order to solve the problem of imaginary entities Husserl introduced,independently of Hilbert, two notions of completeness (definiteness in Husserl'sterminology) for a formal axiomatic (...) system. I present and discuss these notionshere, establishing also parallels between Husserl's and Hilbert's notions ofcompleteness. (shrink)

In this paper I study the variants of the notion of completeness Husserl pre-sented in “Ideen I” and two lectures he gave in Göttingen in 1901. Introduced primarily in connection with the problem of imaginary numbers, this notion found eventually a place in the answer Husserl provided for the philosophically more im-portant problem of the logico-epistemological foundation of formal knowledge in sci-ence. I also try to explain why Husserl said that there was an evident correlation between his and Hilbert’s notion (...) of completeness introduced in connection with the axiomatisation of geometry and the theory of real numbers when, as many commen-tators have already observed, these two notions are independent. I show in this paper that if a system of axioms is complete in Husserl’s sense, then its formal domain, the manifold of formal objects it determines, does not admit any extension. This is precisely the idea behind Hilbert’s notion of completeness in question. Therefore, the correlation Husserl noted indeed exists. But, in order to see it, we must consider the formal domain determined by a formal theory, not its models. (shrink)