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)

Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.

This massive two-volume reference presents a comprehensive selection of the most important works on the foundations of mathematics. While the volumes include important forerunners like Berkeley, MacLaurin, and D'Alembert, as well as such followers as Hilbert and Bourbaki, their emphasis is on the mathematical and philosophical developments of the nineteenth century. Besides reproducing reliable English translations of classics works by Bolzano, Riemann, Hamilton, Dedekind, and Poincare, William Ewald also includes selections from Gauss, Cantor, Kronecker, and Zermelo, all translated here for (...) the first time. (shrink)

Dieser Band enthält Husserls letzte grosse Arbeit, an der er von 1934 bis 1937 arbeitete. Husserl weist darin die Probleme auf, die seiner Ansicht nach zu der Krise geführt haben, in der die Menschheit der Gegenwart sich befindet. Er verfolgt den Ursprung dieser Krise zurück bis zur Entstehung der neuzeitlichen mathematischen Naturwissenschaften bei Galilei, um aufzuweisen, wie es zu der verhängnisvollen Spaltung des physikalistischen Objektivismus und des transzendentalen Subjektivismus gekommen ist. Die Geschichte der neuzeitlichen Philosophie wird von Descartes über Locke (...) - Berkeley - Hume bis zu Kant verfolgt. Husserl stellt dar, wie es das eigentliche Anliegen der Phänomenologie ist den verhängnisvollen Riss zu überwinden, indem sie den Rückgang zu dem Sinnesfundament vollzieht, das in den Wissenschaften selbst und für diese verborgen bleibt. In diesem Zusammenhang kommt gerade der Wissenschaft vom Menschen, der Psychologie, eine fundamentale Bedeutung zu, allerdings einer Psychologie, die auf der Phänomenologie gründet und von ihren Erkenntnissen ausgeht. Ausser dem erhaltenen Haupttext befinden sich im Husserl-Archiv noch zahlreiche "Forschungsmanuskripte", die im Zusammenhang mit der Krisis-Problematik entstanden sind. Um auch in sie Einblick zu geben wurde eine umfassende Auswahl in Form von Beilagen veröffentlicht, sowie noch einige selbständige Abhandlungen die die Entstehungsgeschichte des Werkes erhellen. (shrink)

This book, written by one of philosophy's pre-eminent logicians, argues that many of the basic assumptions common to logic, philosophy of mathematics and metaphysics are in need of change. It is therefore a book of critical importance to logical theory. Jaakko Hintikka proposes a new basic first-order logic and uses it to explore the foundations of mathematics. This new logic enables logicians to express on the first-order level such concepts as equicardinality, infinity, and truth in the same language. The famous (...) impossibility results by Gödel and Tarski that have dominated the field for the last sixty years turn out to be much less significant than has been thought. All of ordinary mathematics can in principle be done on this first-order level, thus dispensing with the existence of sets and other higher-order entities. (shrink)

Band VIII: Institutionelle Schreiben; Band IX: Familienbriefe; Band X: Einführung und Register.

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.

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)

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 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)

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)

The paper traces the development and the role of syntactic reduction in Edmund Husserl’s early writings on mathematics and logic, especially on arithmetic. The notion has its origin in Hermann Hankel’s principle of permanence that Husserl set out to clarify. In Husserl’s early texts the emphasis of the reductions was meant to guarantee the consistency of the extended algorithm. Around the turn of the century Husserl uses the same idea in his conception of definiteness of what he calls “mathematical manifolds.” (...) The paper argues that the notion anticipates the notion of reduction in term rewrite theory in computer science. The role of the reduction for Husserl is, however, primarily epistemological: its purpose is to impart clarity to formal mathematics. (shrink)