The paper examines Husserl’s interactions with logicians in the 1930s in order to assess Husserl’s awareness of Gödel’s incompleteness theorems. While there is no mention about the results in Husserl’s known exchanges with Hilbert, Weyl, or Zermelo, the most likely source about them for Husserl is Felix Kaufmann (1895–1949). Husserl’s interactions with Kaufmann show that Husserl may have learned about the results from him, but not necessarily so. Ultimately Husserl’s reading marks on Friedrich Waismann’s Einführung in das mathematische Denken: die (...) Begriffsbildung der modernen Mathematik, 1936, show that he knew about them before his death in 1938. (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)

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 ...

The article introduces and discusses an unpublished manuscript by Edmund Husserl, conserved at the Husserl-Archives Leuven with signature K I 26, pp. 73a–73b.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

Although mathematical philosophy is flourishing today, it remains subject to criticism, especially from non-analytical philosophers. The main concern is that even if formal tools serve to clarify reasoning, they themselves contribute nothing new or relevant to philosophy. We defend mathematical philosophy against such concerns here by appealing to its metaphysical foundations. Our thesis is that mathematical philosophy can be founded on the phenomenological theory of ideas as developed by Roman Ingarden. From this platonist perspective, the “unreasonable effectiveness of mathematics in (...) philosophy”—to adapt Wigner’s phrase—is analogous to that of mathematical explanations in science. As success-criteria for mathematical philosophy, we propose that it should be correct, responsive, illuminating, promising, relevant, and adequate. (shrink)

As a discipline of its own, the philosophy of science can be traced back to the founding of its academic journals, some of which go back to the first half of the twentieth century. While the discipline has been the object of many historical studies, notably focusing on specific schools or major figures of the field, little work has focused on the journals themselves. Here, we investigate contemporary philosophy of science by means of computational text-mining approaches: we apply topic-modeling algorithms (...) to eight major philosophy of science journals, from the 1930s up until 2017. Based on the full-text content of some 15,897 articles, we identified 25 research themes and 8 thematic clusters that show how the research agenda of the philosophy of science has changed in its content over the course of the last eight decades, up to the philosophy of science we now know. We also show how each one of the journals contributed in its own way to this thematic evolution. (shrink)

It is beginning to be rather well known that Edmund Husserl, the founder of phenomenological philosophy, was originally a mathematician; he studied with Weierstrass and Kronecker in Berlin, wrote his doctoral dissertation on the calculus of variations, and was then a colleague of Cantor in Halle until he moved to the Göttingen of Hilbert and Klein in 1901. Much of Husserl’s writing prior to 1901 was about mathematics, and arguably the origin of phenomenology was in Husserl’s attempts to give philosophical (...) foundations first for analysis and later for the formal sciences in general. However, what exactly Husserl’s thoughts about mathematics were is relatively little known. Stefania Centrone’s book Logic and Philosophy of Mathematics in the Early Husserl fills this lacuna. Centrone deciphers Husserl’s early texts about mathematics and logic somewhat selectively, but also extremely accurately and carefully into lucid English and in terms we know, without however reading contemporary views into Husserl’s views. Her analysis reveals for example that Husserl’s view of logic is close to what is today taught in the standard textbooks and that he viewed abstract mathematics as a theory of structures. Her work also uncovers Husserl’s relationship to the algebraists of logic as well as to Bolzano, Frege, and Hilbert.The book is composed of three rather independent chapters. In contrast to most of the other commentaries on early Husserl, the organization of the book is thematic rather than an attempt to document the various stages in the development of Husserl’s views. The first chapter discusses Husserl’s major work on arithmetic, the second the idea of pure logic, and the last the imaginary in mathematics. (shrink)

Kant’s Prize Essay of 1764 emphasizes the importance for mathematical cognition of manipulating signs according to rules, which has led some recent commentators to ask whether Kant’s position there is a species of mathematical formalism. While most have hesitated to find formalism in the Prize Essay, this hesitation derives from misconceptions about what formalists actually believe. I therefore examine some nineteenth century formalists who were in dialogue with Kant, using their views as a model against which to compare the Prize (...) Essay. I argue that Kant’s view in the Prize Essay is continuous with their formalism, since the Prize Essay shares the essential aspects of their views about the sign-signified relation in mathematics and the role of signs in mathematical reasoning. (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)

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)

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.

The following two review papers have a common origin. Pietarinen’s book Signs of Logic and Stjernfelt’s book Diagrammatology were both published in the same Synthese Library Series being published by Springer. The two books also share the common topic of diagrammatic reasoning in Charles Peirce’s work. Beginning in a conference Applying Peirce held in Helsinki in conjunction with the World Congress of Semiotics in June 2007, two authors have commented upon these books under the headline of Synthese Library Book Session (...) on several occasions, including the Aarhus meeting on Signs and Meaning held in February 2008, the Diagrammatology and Diagram Praxis workshop in Lisboa in March 2009, and the Peirce and Early Analytic Philosophy symposium in Helsinki in June 2009. Therefore, these two review papers form a continuing discussion on the contributions Peirce’s diagrammatic epistemology and logic will have to a broad range of issues at the intersections of philosophy, logic, cognitive sciences and beyond. (shrink)

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)