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.

L’ambition de cet article est de reformuler l’ontologie des propositions proposée par Husserl dans ses Prolegomena zur reinen Logik (1900). Dans cet ouvrage, Husserl affirme que les propositions, auxquelles a trait ce qu’il appelle la “logique pure”, sont des propriétés (des “species”) d’actes, mettons d’actes de jugement. En outre, il considère les propriétés comme circonscrivant toutes leurs instantiations possibles. Sur cette base, on comprend mieux en quel sens la réflexion de Husserl sur la nature de la logique dépend de son (...) ontologie des propositions. Dans cet article, les deux points suivants sont discutés dans cette perspective. D’abord, la structure de son argument négatif contre le psychologisme n’est dans son ensemble correctement comprise que si l’on prend en considération l’ontologie husserlienne des propositions. Cela explique pourquoi la conception de la logique comme une science normative a priori ne satisfait pas Husserl lorsqu’il vise à réfuter le psychologisme. Ensuite, sa conception positive des lois logiques serait aussi rendue plus intelligible par cette théorie. Celle-ci fournit tout ce qui est nécessaire pour réfuter le psychologisme sans renoncer pour autant à une conception globalement aristotélicienne de la logique, que Husserl tient pour indispensable mais qu’il serait particulèrement difficile de maintenir dans le cadre de la conception normative a priori. (shrink)

Burt C. Hopkins, The Origin of the Logic of Symbolic Mathematics. Edmund Husserl and Jacob Klein. Bloomington and Indianapolis: Indiana University Press. 2011. 592 pp. $49.95. ISBN 978-0-253-35671-...

Husserl’s early picture of explanation in the sciences has never been completely provided. This lack represents an oversight, which we here redress. In contrast to currently accepted interpretations, we demonstrate that Husserl does not adhere to the much maligned deductive-nomological (DN) model of scientific explanation. Instead, via a close reading of early Husserlian texts, we reveal that he presents a unificationist account of scientific explanation. By doing so, we disclose that Husserl’s philosophy of scientific explanation is no mere anachronism. It (...) is, instead, tenable and relevant. We discuss how Husserl and other contemporary thinkers draw theoretical inspiration from the same source—namely, Bernard Bolzano. Husserl’s theory of scientific explanation shares a common language and discusses the same themes as, for example, Phillip Kitcher and Kit Fine. To advance our novel reading, we discuss Husserl’s investigations of grounding, inter-lawful explanation, intra-mathematical explanation, and scientific unification. (shrink)

In this work we argue that there is no strong demarcation between pure and applied mathematics. We show this first by stressing non-deductive components within pure mathematics, like axiomatization and theory-building in general. We also stress the “purer” components of applied mathematics, like the theory of the models that are concerned with practical purposes. We further show that some mathematical theories can be viewed through either a pure or applied lens. These different lenses are tied to different communities, which endorse (...) different evaluative standards for theories. We evaluate the distinction between pure and applied mathematics from a late Wittgensteinian perspective. We note that the classical exegesis of the later Wittgenstein’s philosophy of mathematics, due to Maddy, leads to a clear-cut but misguided demarcation. We then turn our attention to a more niche interpretation of Wittgenstein by Dawson, which captures aspects of the aforementioned distinction more accurately. Building on this newer, maverick interpretation of the later Wittgenstein’s philosophy of mathematics, and endorsing an extended notion of meaning as use which includes social, mundane uses, we elaborate a fuzzy, but more realistic, demarcation. This demarcation, relying on family resemblance, is based on how direct and intended technical applications are, the kind of evaluative standards featured, and the range of rhetorical purposes at stake. (shrink)

Husserl’s official account of essence is modal. It is also, I submit, incompatible with the role that essence is supposed to play, especially relative to necessity, in his overall philosophy. In the Husserlian framework, essence should rather be treated as a non-modal notion. The point, while not generally acknowledged, has been made before (by Kevin Mulligan for one); yet the arguments given for it, though perhaps sound, are not Husserlian. In this paper I present a thoroughly Husserlian argument for that (...) claim, as well as a Husserlian essentialist account of necessity. I also discuss the role of grounding within the account. (shrink)

The question of the psychologism of the theory of number developed by Husserl in his Philosophy of Arithmetic has long been debated, but it cannot be considered fully resolved. In this paper, I address the issue from a new point of view. My claim is that in the Philosophy of Arithmetic, Husserl made, albeit indirectly, a series of arguments that are worth reconstructing and clarifying since they are useful in shedding some light on the psychologism issue. More specifically, I maintain (...) that the clarification of these arguments, along with other arguments that Husserl presented against alternative theories of number as well as with some contemporary distinctions concerning the notion of ontological dependence, allows us to determine that Husserl’s theory of number is psychologistic in a minimal and precise sense: it entails a generic ontological dependence of numbers upon the mind. (shrink)

There are quite a few studies on late Bolzano’s notion of a collection (Inbegriff). We try to broaden the perspective by introducing the forerunner of collections in Bolzano’s early writings, namely the entities referred to by expressions with the technical term ‘et’. Special emphasis is laid on the question whether these entities are set-theoretical or mereological plenties. Moreover, similarities and differences to Bolzano’s mature conception are pointed out.

Husserl’s Logical Grammar is intended to explain how complex expressions can be constructed out of simple ones so that their meaning turns out to be determined by the meanings of their constituent parts and the way they are put together. Meanings are thus understood as structured contents and classified into formal categories to the effect that the logical properties of expressions reflect their grammatical properties. As long as linguistic meaning reduces to the intentional content of pre-linguistic representations, however, it is (...) not trivial to account for how semantics relates to syntax in this context. In this paper, I analyze Husserl’s Logical Grammar as a system of recursive rules operating on representations and suggest that the syntactic form of representations contributes to their semantics because it carries information about semantic role. I further discuss Husserl’s syntactic account of the unity of propositions and argue that, on this account, logical form supervenes on syntactic form. In the last section I draw some implications for the phenomenology of thought and conjecture that the structural features it displays are likely to convey the syntactic structures of an underlying language-like representational system. (shrink)

The paper is the final, fifth part of a series of studies introducing the new conceptions of “Hilbert mathematics” and “ontomathematics”. The specific subject of the present investigation is the proper philosophical sense of both, including philosophy of mathematics and philosophy of physics not less than the traditional “first philosophy” (as far as ontomathematics is a conservative generalization of ontology as well as of Heidegger’s “fundamental ontology” though in a sense) and history of philosophy (deepening Heidegger’s destruction of it from (...) the pre-Socratics to the Pythagoreans). Husserl’s phenomenology and Heidegger’s derivative “fundamental ontology” as well as his later doctrine after the “turn” are the starting point of the research as established and well known approaches relative to the newly introduced conception of ontomathematics, even more so that Husserl himself started criticizing his “Philosophy of arithmetic” as too naturalistic and psychological turning to “Logical investigations” and the foundations of phenomenology. Heidegger’s “Aletheia” is also interpreted ontomathematically: as a relation of locality and nonlocality, respectively as a motion from nonlocality to locality if both are physically considered. Aristotle’s ontological revision of Plato’s doctrine is “destructed” further from the pre-Socratics' “Logos” or Heideger’s “Language” (after the “turn”) to the Pythagoreans “Numbers” or “Arithmetics” as an inherent and fundamental philosophical doctrine. Then, a leap to contemporary physics elucidates the essence of ontomathematics overcoming the Cartesian abyss inherited from Plato’s opposition of “ideas” versus “things”, and now unifying physics and mathematics, particularly allowing for the “creation from nothing” instead of the quasi-scientific myth of the “Big Bang”. Furthermore, ontomathematics needs another interpretation of arithmetic, propositional logic and set theory in the foundations of mathematics, where the latter two ones are both identified with Boolean algebra, and the former is considered to be a “half of Boolean algebra” in the exact meaning to be equated to it after doubling by a dual anti-isometric counterpart of Peano arithmetic. That unified algebraic realization of the foundations of mathematics is related to Hilbert mathematics in both “narrow and wide senses” where the latter is isomorphic to the qubit Hilbert space, thus underlying all the physical world by the newly introduced substance of quantum information being physically dimensionless and generalizing classical information measured in bits. The substance of information, whether classical or quantum, visualizes the way of the unification of physics and mathematics by merging their foundations in Hilbert arithmetic and Hilbert mathematics: thus how ontomathetics is a “first philosophy”. The relation of ontomathematics to the Socratic “human problematics”, furthermore being fundamental for Western philosophy in Modernity, is discussed. Ontomathematics implies its “substitution” by abstract information (or by “subjectless choice” relevant to it), thus “obliterating the human outline on the ocean beach sand” (by Michel Foucault’s metaphor). A reflection back from the viewpoint of mathematic to Western philosophy as the philosophy of locality ends the study. (shrink)

The paper’s central theme is the link between phenomenology and the notion of the mathesis universalis, a link articulated by Husserl in the third volume of the Ideas: “My way to phenomenology was essentially determined by the mathesis universalis.” The paper suggests three interpretations of the phenomenology—mathesis universalis nexus: the first is related to the development of Husserl’s conception of the foundations of arithmetic; the second is based on the role of the theory of manifolds in Husserl’s Logical Investigations; and (...) the third reflects the importance of the distinction between “generalization” and “formalization” for phenomenology. After examining these interpretations, the paper explores which is most helpful for understanding why Husserl distanced himself from Bolzano, arguing that the third interpretation provides the most edifying answer. (shrink)

The recent investigations into knowledge and its elements viz facts, skills and objects have become prominent in various subfields of philosophy and other areas like linguistics, cognitive science, neuroscience and artificial intelligence. These investigations have been mainly on understanding the relation between the intentionality and its referential entities to know how they enrich knowledge with their existence. This article starts with an exploration of the fundamental aspects of judgemental sense from the knowledge origins perspective. To explain the consequences of this, (...) I start with reconstructing the notion of categorial intuitions described in Husserl’s sixth Logical Investigations by relating it to ‘knowledge origination’ as an intuitive fulfilment of signifying acts. This offers a relation of perceptual synthesis, operating implicitly among the signifying intentions upon grasping intended objects, in view of language structures such as copulas. This leads to the synthesis of intuitive aspects mainly from the categorial elements for a judgemental sense or thinking and the attitude from the very sphere of perception. The tension between categorial intuitions is understood as foundational acts and founded acts. Husserl’s sixth Logical Investigation describes the notion of passive synthesis, the genesis of categoriality in every realm of sensibility that allows one to formulate the relational role among the ‘act of knowing’ and phenomenological approaches to the judgemental sense. Thus, the investigation focuses on perceptual judgements and categories, which fulfils the judgemental sense of categorial elements and attitude. (shrink)

Husserl’s philosophy of mathematics, his metatheory, and his transcendental phenomenology have a sophisticated and systematic interrelation that remains relevant for questions of ontology today. It is well established that Husserl anticipated many aspects of model theory. I focus on this aspect of Husserl’s philosophy in order to argue that Thomasson’s recent pleonastic reconstruction of Husserl’s approach to essences is incompatible with Husserl’s philosophy as a whole. According to the pleonastic approach, Husserl can appeal to essences in the absence of a (...) positive metaphysical account of their nature. I show, using central results from recent model theory, that the pleonastic approach undermines Husserl’s approach to formalization and categoricity, an effect that will ripple out from Husserl’s philosophy of mathematics into Husserl’s metatheory and transcendental phenomenology. The result is that Husserl cannot appeal to formal essences without metaphysical commitments. However, the very observations Thomasson makes about the nature of eidetic intuition in Husserl lead to a general strategy for responding to the problem. The article thus illustrates that the pleonastic and the model-theoretic routes for making Husserl relevant to present-day ontology are competing approaches, but I conclude that Husserl scholars seeking to set Husserl’s present-day relevance into sharp relief could do worse than to emphasize the model-theoretic nature of Husserl’s enterprise. (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.

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)

In his 1896 lecture course on logic–reportedly a blueprint for the Prolegomena to Pure Logic –Husserl develops an explicit account of logic as an independent and purely theoretical discipline. According to Husserl, such a theory is needed for the foundations of logic (in a more general sense) to avoid psychologism in logic. The present paper shows that Husserl’s conception of logic (in a strict sense) belongs to the algebra of logic tradition. Husserl’s conception is modeled after arithmetic, and respectively logical (...) inferences are viewed as analogical to arithmetical calculation. The paper ends with an examination of Husserl’s involvement with the key characters of the algebra of logic tradition. It is concluded that Ernst Schröder, but presumably also Hermann and Robert Grassmann influenced Husserl most in his turn away from psychologism. (shrink)

RESUMEN La fenomenología de Husserl está inmersa en un entramado de regresiones y revisiones conceptuales que dificultan la identificación de una estructura sistémica. Los conceptos característicos de la fenomenología carecen de univocidad y no son propios de algunas obras de Husserl. Philosophie der Arithmetik ilustra el problema referido. ¿Puede inscribirse esta obra dentro de la categoría de texto fenomenológico? ¿Es posible hablar de una fenomenología de la matemática en Husserl? y ¿qué sentido tendría esto? Los objetivos del presente ensayo son: (...) primero, analizar la dificultad que tiene Philosophie der Arithmetik para responder a las inquietudes de la filosofía de las matemáticas, y segundo, proponer la posibilidad de una fenomenología de la matemática condicionada a la clarificación conceptual y al estudio crítico de la influencia de la tradición analítica sobre la filosofía de las matemáticas. ABSTRACT Husserl's phenomenology is immersed in a network of regressions and conceptual revisions that hinder the identification of a systemic structure. The characteristic concepts of phenomenology fall short of uniqueness and are not appropriate for some of Husserl's works. Philosophie der Arithmetik illustrates the aforementioned problem. Can this work be classified under the category of phenomenological texts? Is it possible to talk about a phenomenology of mathematic in Husserl? And what would be the point of this? The objectives of the present essay are: first, to analyze the difficulty that Philosophie der Arithmetik has responding to the concerns of the philosophy of mathematics, and second, to propose the possibility of a phenomenology of mathematics, conditioned to conceptual clarification and critical study of the influence of the analytical tradition on the philosophy of mathematics. (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 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)

En el apartado Freges Versuch, incluido en Filosofía de la aritmética, Husserl abiertamente señala que en los Fundamentos de la aritmética de G. Frege no existe un análisis lógico adecuado del concepto de número en términos de equinumerosidad. Según Husserl, la caracterización de Frege del concepto de número cardinal, en estrecha conexión con la noción de correspondencia uno- a-uno, es errónea. El objetivo principal de este artículo es mostrar que esta interpretación de Husserl sobre la obra de Frege, específicamente en (...) este apartado, contiene una serie de errores y confusiones graves. Palabras clave: Husserl, Frege, psicologismo, Filosofía de la aritmética, equinumerosidad, fenomenología, correspondencia uno-a-uno. (shrink)

In his booklet "Contributions to a better founded presentation of mathematics" of 1810 Bernard Bolzano made his first serious attempt to explain the notion of a rigorous proof. Although the system of logic he employed at that stage is in various respects far below the level of the achievements in his later Wissenschaftslehre, there is a striking continuity between his earlier and later work as regards the methodological constraints on rigorous proofs. This paper tries to give a perspicuous and critical (...) account of the fragmentary logic of Beyträge, and it shows that there is a tension between that logic and Bolzano's methodological ban on ?kind crossing? (shrink)

This explorative article is organized around a set of questions concerning the concept of a function. First, a summary of certain general facts about functions that are a common coin in contemporary logic is given. Then Frege's attempt at clarifying the nature of functions in his famous paper Function and Concept and in his Grundgesetze is discussed along with some questions which Freges' approach gave rise to in the literature. Finally, some characteristic uses of functional notions to be found in (...) the work of Bernard Bolzano and in Edmund Husserl's early work are presented and elucidated. (shrink)