This volume will be of particular interest to researchers working in the history, and in the philosophy, of logic and mathematics, and more generally, to ...

Mit den "Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie" von 1913, von ihm selbst nur als eine "Allgemeine Einführung in die reine Phänomenologie" angezeigt, zog Edmund Husserl die Konsequenz aus seinen Logischen Untersuchungen (PhB 601), die ihn 1900/01 berühmt gemacht hatten: Ausgehend von der dort entwickelten Phänomenologie der intentionalen Erlebnisse sieht er jetzt in der Aufdeckung der Leistungen des "reinen Bewußtseins", dem die uns bekannte natürliche Welt nur als "Bewußtseinskorrelat" gegeben ist, den eigentlichen Gegenstand philosophischer Erkenntnis und in den (...) von ihm eingeführten methodologischen Begriffen der "Reduk tion" und der "Epoché" den Weg, sich über die Beschaffenheit dieses "reinen Bewußtseins", aus dem alle Erkenntnis entspringt, Klarheit zu verschaffen. Die Ausgabe bietet Husserls Schrift text- und seitengleich nach Band III/1 der Husserliana, herausgegeben von Karl Schuhmann; beigegeben sind Husserls Nachwort zu den Ideen von 1930 aus Band V der Husserliana, herausgegeben von Marly Biemel, sowie ein umfangreiches Namen- und Sachregister von Elisabeth Ströker. (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.

The article introduces and discusses an unpublished manuscript by Edmund Husserl, conserved at the Husserl-Archives Leuven with signature K I 26, pp. 73a–73b. The article is followed by the text of the manuscript in German and in an English translation. The manuscript, titled “The Transition through the Impossible” ( Der Durchgang durch das Unmögliche ), was part of the material Husserl used for his 1901 Doppelvortrag in Göttingen. In the manuscript, the impossible is characterized as the “sphere of objectlessness” ( (...) Sphäre der Gegenstandslosigkeit ) and Husserl addresses the question whether and when it is warranted to perform a transition through the impossible to obtain valid results for the sphere of objectivity. (shrink)

The concept of a Mannigfaltigkeit in Husserl has been given various interpretations, due to its shifting role in his works. Many authors have been misled by this term, placing it in the context of Husserl’s early period in Halle, while writing the Philosophy of Arithmetic, as a friend and colleague of Georg Cantor.Yet at the time, Husserl distanced himself explicitly from Cantor’s definition and rather took Bernhard Riemann as example, having studied and lectured extensively on Riemann’s theories of space. Husserl’s (...) Mannigfaltigkeitslehre would then not be a Cantorian set-theory, but come rather closer to topology. Then, in the Prolegomena, Husserl introduces the idea of a pure Mannigfaltigkeitslehre as a meta-theoretical enterprise which studies the relations among theories, e.g. how to derive or found one upon another. When Husserl announces that in fact the best example of such a pure theory of manifolds is what is actually practiced in mathematics, this sounds slightly misleading. The pure theory of theories cannot simply be the mathematics underlying topology, but should rather be considered as a mathesis universalis. Indeed, while this might not have been fully clear yet in 1900/1901, Husserl will explicitly tie together the notions of pure theory of manifolds and mathesis universalis. The mathesis universalis in this sense is formal, a priori and analytic, as theory of theory in general. It is an analysis of the highest categories of meaning and their correlative categories of objects. In my paper I try to understand the development of the notion of Mannigfaltigkeit in Husserl’s thought from its mathematical beginnings to its later central philosophical role, taking into account the mathematical background and context of Husserl’s own development. (shrink)

Husserls »Logische Untersuchungen« sind eines der folgenreichsten Werke der neueren Philosophiegeschichte. Mit dem ersten Erscheinen in den Jahren 1900 und 1901 (Max Niemeyer Verlag, Halle/Saale) nimmt jene Schule ihren Anfang, deren Name im Untertitel des zweiten Bandes zum ersten Mal sinnfällig wird: die Phänomenologie. Husserl sah damals in diesem Werk »Versuche zur Neubegründung der reinen Logik und Erkenntnistheorie«, die den Grund zu einem größeren Gedankengebäude zu legen imstande waren. Sie wollten freilich kein bloßes Programm sein, sondern »Fundamentalarbeit an den unmittelbar (...) erschauten und ergriffenen Sachen« darstellen. Die weitere Arbeit an der Ausprägung der phänomenologischen Philosophie führte schon nach wenigen Jahren zu neuen Problemen, zu vertieften Erkenntnissen und zu schärferen Formulierungen. Ihnen trägt die zweite, zum Teil tiefgreifend umgearbeitete Auflage Rechnung, deren Text seither in unveränderter Form immer wieder nachgedruckt wurde. (shrink)

In seinen ALogischen UntersuchungenA (1900/01) hatte Husserl die EigenstAndigkeit und IdealitAt logischer Gebilde gegenA1/4ber einem falschen Subjektivismus und Psychologismus geltend gemacht. Es ging ihm dabei nicht um die formale Logik selbst als solche, sondern um die BegrA1/4ndungsfunktion der Logik fA1/4r eine apriorische Wissenschaftslehre. Diese in den AIdeenA (1913) weitergefA1/4hrte Konstitutionsproblematik wird nun in dem Werk A1/4ber AFormale und transzendentale LogikA im Sinne einer AKritik der logischen VernunftA vertieft und durch den Aoebergang von der formalen zur transzendentalen Logik begrifflich entschiedener gefaAt.

Edmund Husserl was one of the very first to experience the direct impact of challenging problems in set theory and his phenomenology first began to take shape while he was struggling to solve such problems. Here I study three difficulties associated with Frege's use of sets that Husserl explicitly addressed: reference to non-existent, impossible, imaginary objects; the introduction of extensions; and 'Russell's paradox'.I do so within the context of Husserl's struggle to overcome the shortcomings of set theory and to develop (...) his own theory of manifolds. I define certain issues involved and discuss how Husserl's theory of manifolds might confront them. In so doing I hope to help bring Husserl's theories about sets and manifolds out of the realm of abstract theorizing and prompt further exploration of uncharted philosophical territory rich in philosophical implications. (shrink)

Pour projeter de la lumière dans de nombreux coins et recoins obscurs de la logique pure de Husserl et dans les rapports entre sa logique formelle et sa logique transcendantale, et combler des lacunes empêchant qu’on arrive à une appréciation juste de sa Mannigfaltigkeitslehre, ou théorie de multiplicités, on examine comment, en prônant une théorie des systèmes déductifs, ou systèmes d’axiomes, comme tâche suprême de la logique pure, Husserl cherchait à résoudre certains problèmes épineux auxquels il s’était heurté en écrivant (...) Philosophie de l’arithmétique. Ces problèmes sont décrits. Ensuite, on rassemble les éléments nécessaires pour caractériser ce que Husserl, à travers les textes présentement disponibles, voulait dire des Mannigfaltigkeiten. Pour conclure, il est indiqué comment Husserl pouvait considérer que sa théorie représentait une solution aux problèmes qui avaient conduit à son élaboration.To shed light on the numerous dark areas surrounding Husserl’s ideas about pure logic and the relationship between his formal logic and his transcendental logic, and so provide an accurate characterisation of his Mannigfaltigkeitslehre, a close look is taken at how, by advocating a theory of deductive systems, or axiomatic systems as the highest task of pure logic, Husserl sought to resolve certain thorny problems he encountered when writing the Philosophy of Arithmetic. Those problems are described. Then, his ideas about axiomatization and manifolds are drawn together from the various sources now available to provide a more complete picture of his Mannigfaltigkeitslehre. In conclusion, it is shown how Husserl considered his theory to be a solution to the problems leading to its development. (shrink)

Few have entertained the idea that Georg Cantor, the creator of set theory, might have influenced Edmund Husserl, the founder of the phenomenological movement. Yet an exchange of ideas took place between them when Cantor was at the height of his creative powers and Husserl in the throes of an intellectual struggle during which his ideas were particularly malleable and changed considerably and definitively. Here their writings are examined to show how Husserl's and Cantor's ideas overlapped and crisscrossed in the (...) areas of philosophy and mathematics, arithmetization, abstraction, consciousness and pure logic, psychologism, metaphysical idealism, new numbers, and sets and manifolds. (shrink)

Husserl’s notion of definiteness, i.e., completeness is crucial to understanding Husserl’s view of logic, and consequently several related philosophical views, such as his argument against psychologism, his notion of ideality, and his view of formal ontology. Initially Husserl developed the notion of definiteness to clarify Hermann Hankel’s ‘principle of permanence’. One of the first attempts at formulating definiteness can be found in the Philosophy of Arithmetic, where definiteness serves the purpose of the modern notion of ‘soundness’ and leads Husserl to (...) a ‘computational’ view of logic. Inspired by Gauss and Grassmann Husserl then undertakes a further investigation of theories of manifolds. When Husserl subsequently renounces psychologism and changes his view of logic, his idea of definiteness also develops. The notion of definiteness is discussed most extensively in the pair of lectures Husserl gave in front of the mathematical society in Göttingen (1901). A detailed analysis of the lectures, together with an elaboration of Husserl’s lectures on logic beginning in 1895, shows that Husserl meant by definiteness what is today called ‘categoricity’. In so doing Husserl was not doing anything particularly original; since Dedekind’s ‘Was sind und sollen die Zahlen’ (1888) the notion was ‘in the air’. It also characterizes Hilbert’s (1900) notion of completeness. In the end, Husserl’s view of definiteness is discussed in light of Gödel’s (1931) incompleteness results. (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)

" Nous avons tenté dans cet ouvrage de tracer le chemin qui va de la logique traditionnelle à la logique transcendantale [...] à la logique transcendantale qui n'est pas une seconde logique mais qui est seulement la logique elle-même, radicale et concrète, qui doit son développement à la méthode phénoménologique. En vérité, pour s'exprimer plus précisément, nous n'avons justement eu en vue, comme logique transcendantale, que la logique telle qu'elle est délimitée traditionnellement, la logique analytique qui sans contredit grâce à (...) sa généralité vide et formelle embrasse toutes les sphères d'êtres et d'objets, corrélativement toutes les sphères de connaissances. Toutefois, étant astreints à reconsidérer la caractérisation du sens et de l'ampleur de la recherche transcendantale, nous avons par avance acquis aussi du même coup le moyen de comprendre les " logiques " (qu'il reste à fonder) ayant un autre sens, conçues comme doctrines de la science, mais comme doctrines matérielles ; doctrines de la science parmi lesquelles la plus élevée et la plus étendue serait la logique de la science absolue, la logique de la philosophie phénoménologico-transcendantale elle-même ". E. Husserl. (shrink)

Situees a peu pres a mi-distance entre la publication des Recherches logiques (1900-1901) et celle des Idees directrices (1913), ces lecons appartiennent a une periode decisive durant laquelle Husserl publie tres peu, mais ou se determine le sens et la finalite de la phenomenologie: de psychologie descriptive metaphysiquement neutre, celle-ci devient progressivement une philosophie transcendantale. Le cours de 1906/07 thematise cette evolution et la justifie en montrant l'impossibilite, pour la theorie de la connaissance, d'en rester au niveau de la connaissance (...) naturelle sous peine de ne pouvoir lever l'hypothese du scepticisme et du relativisme. (shrink)

In search of the origins of some of the most fundamental problems that have beset philosophers in English-speaking countries in the past century, Claire Ortiz Hill maintains that philosophers are treating symptoms of ills whose causes lie buried in history. Substantial linguistic hurdles have blocked access to Gottlob Frege's thought and even to Bertrand Russell's work to remedy the problems he found in it. Misleading translations of key concepts like intention, content, presentation, idea, meaning, concept, etc., severed analytic philosophy from (...) its roots. Hill argues that once linguistic and historical barriers are removed, Edmund Husserl's critical study of Frege's logic in his 1891 _Philosophy of Arithmetic_ provides important insights into issues in philosophy now. She supports her conclusions with analyses of Frege's, Husserl's, and Russell's works, including _Principia Mathematica_, and with linguistic analyses of the principal concepts of analytic philosophy. She re-establishes links that existed between English and Continental thought to show Husserl's expertise as a philosopher of mathematics and logic who had been Weierstrass's assistant and had long maintained ties with Cantor, Hilbert, and Zermelo. (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)