This monograph offers a fresh perspective on the applicability of mathematics in science. It explores what mathematics must be so that its applications to the empirical world do not constitute a mystery. In the process, readers are presented with a new version of mathematical structuralism. The author details a philosophy of mathematics in which the problem of its applicability, particularly in physics, in all its forms can be explained and justified. Chapters cover: mathematics as a formal science, mathematical ontology: what (...) does it mean to exist, mathematical structures: what are they and how do we know them, how different layers of mathematical structuring relate to each other and to perceptual structures, and how to use mathematics to find out how the world is. The book simultaneously develops along two lines, both inspired and enlightened by Edmund Husserl’s phenomenological philosophy. One line leads to the establishment of a particular version of mathematical structuralism, free of “naturalist” and empiricist bias. The other leads to a logical-epistemological explanation and justification of the applicability of mathematics carried out within a unique structuralist perspective. This second line points to the “unreasonable” effectiveness of mathematics in physics as a means of representation, a tool, and a source of not always logically justified but useful and effective heuristic strategies. (shrink)

Eine vollständige Darstellung von Edmund Husserls Verhältnis zu Gottlob Frege steht noch aus, so dass es nicht verwundert, einige Missverständnisse, dieses Verhältnis betreffend, im Umlauf zu finden. Selbst scheinbar längst überwundene systematische Dogmen tauchen wieder auf, so z.B. die Auffassung, dass Husserl nicht nur entscheidend von Gottlob Frege beeinflusst wurde, sondern darüber hinaus auch seine schärfste Frege-Kritik 1891 zurückgenommen habe. Mein Beitrag enthält eine überwiegend historisch vorgehende Entgegnung auf solche fälschlich vertretenen Ansichten wie sie sich auch in dem neu erschienenen (...) und mit vielen dokumentarischen Belegen versehenen Frege-Kommentarbuch von W. Künne finden. Die wichtigsten Argumente, die in diesem Buch gegen Husserl gerichtet werden, stützen sich auf Stellen in Husserls Frühwerk, die zwar sehr aufschlussreich für die Beurteilung des philosophischen und menschlichen Habitus Husserls in den 1890er Jahre sind, nicht aber Künnes Thesen stützen oder gar beweisen können. Die als zurückgenommene Frege-Kritik verstandene Selbstkritik Husserls betrifft jedoch nach meinen Darlegungen gerade nicht die Kritik, die er an Freges Buch Grundlagen der Arithmetik geübt hatte, sondern seine eigene systematische Stellungnahme hinsichtlich der Begründung der Arithmetik. Ausgehend von dieser Einsicht werde ich auf ein wichtiges Spezifikum von Husserls Erstlingswerk (Philosophie der Arithmetik) aufmerksam machen, das voreiligen Interpretationen einen Riegel vorschiebt: Dieses Buch ist ein Mosaik aus Einsichten verschiedener Arbeits- und Denkperioden des Verfassers, die sich vor allem auch in dem zweifachen Begriff der Äquivalenz zeigen. Dies knüpft an einen langen geschichtlichen Entwicklungsprozess an, der in eine von Bolzano, Lotze, Brentano und Carl Stumpf geprägte Tradition zurückführt und das Problem der Unterscheidung zwischen Sinn und Gegenstand eines Aktes betrifft. (shrink)

Husserl is well known for his critique of the “mathematizing tendencies” of modern science, and is particularly emphatic that mathematics and phenomenology are distinct and in some sense incompatible. But Husserl himself uses mathematical methods in phenomenology. In the first half of the paper I give a detailed analysis of this tension, showing how those Husserlian doctrines which seem to speak against application of mathematics to phenomenology do not in fact do so. In the second half of the paper I (...) focus on a particular example of Husserl’s “mathematized phenomenology”: his use of concepts from what is today called dynamical systems theory. (shrink)

In the present essay we attempt to reconstruct Newtonian mechanics under the guidance of logical principles and of a constructive approach related to the genetic epistemology of Piaget and García. Instead of addressing Newton’s equations as a set of axioms, ultimately given by the revelation of a prodigious mind, we search for the fundamental knowledge, beliefs and provisional assumptions that can produce classical mechanics. We start by developing our main tool: the no arbitrariness principle, that we present in a form (...) that is apt for a mathematical theory as classical mechanics. Subsequently, we introduce the presence of the observer, analysing then the relation objective–subjective and seeking objectivity going across subjectivity. We take special care of establishing the precedence among all contributions to mechanics, something that can be better appreciated by considering the consequences of removing them: the consequence of renouncing logic and the laws of understanding is not being able to understand the world, renouncing the early elaborations of primary concepts such as time and space leads to a dissociation between everyday life and physics, the latter becoming entirely pragmatic and justified a-posteriori, changing our temporary beliefs has no real cost other than effort. Finally, we exemplify the present approach by reconsidering the constancy of the velocity of light. It is shown that it is a result of Newtonian mechanics, rather than being in contradiction with it. We also indicate the hidden assumption that leads to the contradiction. (shrink)

This essay explores an ideal notion of form (mathematical structure) that embraces logical, phenomenological, and ontological form. Husserl envisioned a correlation among forms of expression, thought, meaning, and object—positing ideal forms on all these levels. The most puzzling formal entities Husserl discussed were those he called ‘manifolds’. These manifolds, I propose, are forms of complex states of affairs or partial possible worlds representable by forms of theories (compare structuralism). Accordingly, I sketch an intentionality-based semantics correlating these four Husserlian levels of (...) form—thereby integrating logic, phenomenology, and ontology. (shrink)

The article seeks to challenge the standard accounts of how to view the difference between Husserl and Frege on the nature of ideal objects and meanings. It does so partly by using Derrida’s deconstructive reading of Husserl to open up a critical space where the two approaches can be confronted in a new way. Frege’s criticism of Husserl’s philosophy of mathematics (that it was essentially psychologistic) was partly overcome by the program of transcendental phenomenology. But the original challenge to the (...) prospect of a fulfilled intuition of idealities remained and was in fact encountered again from within the transcendental analysis by Husserl himself in his last writings on geometry and language. According to the two standard and conflicting accounts, Husserl either changed his earlier psychologistic program as a result of Frege’s criticism, or he was in fact never challenged by it in the first place. The article shows instead how Husserl continued to struggle with the problem of the constitution of ideal objects, and how his quest led him to a point where his analyses anticipate a more dialectical and deconstructive conclusion, eventually made explicit by Derrida. It also shows not only how this development constitutes a philosophical continuity from the original dispute with Frege, but also how Frege’s critique in a certain respect could be read as an anticipation of Derrida’s deconstructive elaboration of Husserl’s phenomenology. (shrink)

Quine’s criticism of the notion of analyticity applies, at best, to Carnap’s notion, not to those of Frege or Husserl. The failure of logicism is also the failure of Frege’s definition of analyticity, but it does not even touch Husserl’s views, which are based on logical form. However, some relatively concrete number-theoretic statements do not admit such a formalization salva veritate. A new definition of analyticity based not on syntactical but on semantical logical form is proposed and argued for.

Intentionality (‘directedness’, ‘aboutness’) is both a central topic in contemporary philosophy of mind, phenomenology and the cognitive sciences, and one of the themes with which both analytic and Continental philosophers have separately engaged starting from Brentano and Edmund Husserl’s ground-breaking Logical Investigations (1901) through Roderick M. Chisholm, Daniel C. Dennett’s The Intentional Stance, John Searle’s Intentionality, to the recent work of Tim Crane, Robert Brandom, Shaun Gallagher and Dan Zahavi, among many others. In this paper, I shall review recent discussions (...) of intentionality, including some recent explorations of the history of the concept (paying particular attention to Anselm), and suggest some ways the phenomenological approach of Husserl and Heidegger can still offer insights for contemporary philosophy of mind and consciousness. (shrink)

In this article I intend to show that certain aspects of the axiomatical structure of mathematical theories can be, by a phenomenologically motivated approach, reduced to two distinct types of idealization, the first-level idealization associated with the concrete intuition of the objects of mathematical theories as discrete, finite sign-configurations and the second-level idealization associated with the intuition of infinite mathematical objects as extensions over constituted temporality. This is the main standpoint from which I review Cantor’s conception of infinite cardinalities and (...) also the metatheoretical content of some later well-known theorems of mathematical foundations. These are, the Skolem-Löwenheim Theorem which, except for its importance as such, it is also chosen for an interpretation of the associated metatheoretical paradox (Skolem Paradox), and Gödel’s (first) incompleteness result which, notwithstanding its obvious influence in the mathematical foundations, is still open to philosophical inquiry. On the phenomenological level, first-level and second-level idealizations, as above, are associated respectively with intentional acts carried out in actual present and with certain modes of a temporal constitution process. (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)

In “Function and Concept” and “On Concept and Object”, Frege argued that certain differences between dependent and independent meanings were inviolable and “founded deep in the nature of things” but, in those articles, he was not explicit about the actual consequences of violating such differences. However, since by creating a law that permitted one to pass from a concept to its extension, he himself mixed dependent and independent meanings, we are in a position to study some of the actual consequences (...) of his having done so. To make certain of Frege’s ideas about the inviolability of logical form more tangible, I describe a string of very interrelated consequences that his attempt to transform dependent meanings into independent meanings actually brought in its wake. (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)

This paper is concerned with the use of logic to solve philosophical problems. Such use of logic goes counter to the prevailing empiricist tradition in analytic circles. Specifically, model-theoretic tools are applied to three fundamental issues in the philosophy of logic and mathematics, namely, to the issue of the existence of mathematical entities, to the dispute between first- and second-order logic and to the definition of analyticity.

Quine’s criticism of the notion of analyticity applies, at best, to Carnap’s notion, not to those of Frege or Husserl. The failure of logicism is also the failure of Frege’s definition of analyticity, but it does not even touch Husserl’s views, which are based on logical form. However, some relatively concrete number-theoretic statements do not admit such a formalization salva veritate. A new definition of analyticity based not on syntactical but on semantical logical form is proposed and argued for.

(No abstract is available for this citation).

El artículo plantea la actualidad y pertinencia de la Filosofía de la información de Luciano Floridi, considerada a la luz de las revoluciones científicas de Occidente y de la instauración de nuevos paradigmas, tanto en las ciencias como en la filosofía. La analogía con el “giro matemático” de la Modernidad permite establecer el alcance revolucionario de la obra de Floridi, cuya aceptación implicará superar el obstáculo epistemológico del escolasticismo, en función del dinamismo histórico inherente al progreso científico.