This paper deals with Husserl’s idea of pure logic as it is coined in the Logical Investigations. First, it exposes the formation of pure logic around a conception of completeness ; then, it presents intentionality as the keystone of such a structuring ; and finally, it provides a systematic reconstruction of pure logic from the semiotic standpoint of intentionality. In this way, it establishes Husserlian pure logic as a phenomenological epistemology of mathematical logic.

The purpose of this paper is to provide a unitary typology for the incompatibilities of meanings at stake on different levels of Husserlian pure logic—namely, between systems of axioms and pure morphology of meanings; I show that they perfectly match by converging on the notion of Widersinn.

Au terme des Prolégomènes, Husserl formule son idée de la logique pure en la structurant sur deux niveaux: l'un, supérieur, de la logique formelle fondé transcendantalement et d'un point de vue épistémologique par l'autre, inférieur, d'une morphologie des catégories. Seul le second de ces deux niveaux est traité dans les Recherches logiques, tandis que les travaux théoriques en logique formelle menés par Husserl à la même époque en paraissent plutôt indépendants. Cet article est consacré à ces travaux tels que recueillis (...) dans les appendices VI-X du volume 12 des Husserliana. Mettant en évidence la théorie de la signification qui les sous-tend par le biais d'une analyse de la question dite de l'extension des systèmes d'axiomes et de sa résolution au moyen d'une notion de complétude, son objectif est d'expliciter les modalités de l'intégration de la logique formelle dans l'idée husserlienne de la logique pure au tournant du xx siècle.When formulating his idea of pure logic in the Prole.. (shrink)

The article introduces and discusses an unpublished manuscript by Edmund Husserl, conserved at the Husserl-Archives Leuven with signature K I 26, pp. 73a–73b.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)

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)

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)

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)

The paper discusses Husserl’s notion of definiteness as presented in his Göttingen Mathematical Society Double Lecture of 1901 as a defense of two, in many cases incompatible, ideals, namely full characterizability of the domain, i.e., categoricity, and its syntactic completeness. These two ideals are manifest already in Husserl’s discussion of pure logic in the Prolegomena: The full characterizability is related to Husserl’s attempt to capture the interconnection of things, whereas syntactic completeness relates to the interconnection of truths. In the Prolegomena (...) Husserl argues that an ideally complete theory gives an independent norm for objectivity for logic and experiential sciences, hence the notion is central to his argument against psychologism. In the Double Lecture the former is captured by non-extendibility, that is, categoricity of the domain, from which, so Husserl assumes, syntactic completeness is thought to follow. In the so-called ‘mathematical manifolds’ the expressions of the theory are equations that are reducible to equations between elements of the theory. With such an equational reduction structure individual elements of the domain are given criteria of identity and hence they are fully determined. (shrink)

Richard Tieszen [Tieszen, R. (2005). Philosophy and Phenomenological Research, LXX(1), 153–173.] has argued that the group-theoretical approach to modern geometry can be seen as a realization of Edmund Husserl’s view of eidetic intuition. In support of Tieszen’s claim, the present article discusses Husserl’s approach to geometry in 1886–1902. Husserl’s first detailed discussion of the concept of group and invariants under transformations takes place in his notes on Hilbert’s Memoir Ueber die Grundlagen der Geometrie that Hilbert wrote during the winter 1901–1902. (...) Husserl’s interest in the Memoir is a continuation of his long-standing concern about analytic geometry and in particular Riemann and Helmholtz’s approach to geometry. Husserl favored a non-metrical approach to geometry; thus the topological nature of Hilbert’s Memoir must have been intriguing to him. The task of phenomenology is to describe the givenness of this logos, hence Husserl needed to develop the notion of eidetic intuition. (shrink)

This paper offers an exposition of Husserl's mature philosophy of mathematics, expounded for the first time in Logische Untersuchungen and maintained without any essential change throughout the rest of his life. It is shown that Husserl's views on mathematics were strongly influenced by Riemann, and had clear affinities with the much later Bourbaki school.

This paper offers an exposition of Husserl's mature philosophy of mathematics, expounded for the first time in Logische Untersuchungen and maintained without any essential change throughout the rest of his life. It is shown that Husserl's views on mathematics were strongly influenced by Riemann, and had clear affinities with the much later Bourbaki school.

Dans l’essai Objets intentionnels de 1894, Husserl développe en réaction à Twardowski une théorie originale de l’assomption comme solution au problème des représentations sans objet. Après avoir examiné le détail de cette théorie et en avoir soulevé les difficultés, je montre dans cet article que la solution proposée par cette théorie doit être abordée de manière indépendante de celle qui sera développée plus tard dans les Recherches logiques et j’expose dans quelle mesure elle est ancrée dans la psychologie descriptive brentanienne (...) tout en mettant à profit certains outils de la logique de Bolzano. Enfin, j’indique que Husserl continuera à développer cette théorie après les Recherches logiques, confirmant ainsi qu’elle occupe une place de choix dans la théorie phénoménologique du jugement.In his essay Intentional Objects of 1894, Husserl develops in response to Twardowski an original theory of assumptions as a solution to the problem of objectless presentations. First, I analyze in this paper the main points of his theory and point out some of the difficulties it raises. I then suggest that the solution presented in this theory must be addressed independently of the one developed by Husserl later in the Logical Investigations and try to show in which extent his theory of assumptions is rooted in the brentanian descriptive psychology, although it makes good use of some logical tools from Bolzano’s Wissenschaftslehre. Since Husserl continues to work on this theory even after the Logical Investigations, it confirms the important place one should give to the theory of assumptions it the phenomenological theory of judgment. (shrink)

This presentation aims to clarify the historical and theoretical background of the studies included in this issue of Philosophiques, which focus on the work of Husserl during the period of Halle . After a brief description of Husserl’s early years of apprenticeship in philosophy between 1876 and his studies with Brentano in Vienna, I identify several steps that marked the development of his philosophy from his arrival in Halle to the publication of the Logical Investigations : his studies under the (...) direction of C. Stumpf, the publication of the first volume of Philosophy of Arithmetic, the researches belonging to the project of the second volume of this work, the abandonment of this project and the development of the phenomenology of the Logical Investigations. The last part of this paper is an examination of the philosophical issues underlying the disputatio on D. W. Smith’s recent book on Husserl’s philosophy. (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)