Husserl: Science, Logic, and Mathematics

Ludwig Wittgenstein’s Tractatus logico-philosophicus and Edmund Husserl’s Experience and Judgement (Erfahrung und Urteil) are based on remarkably different conceptual frameworks and methodologies. After analyzing their respective accounts on the foundations of (formal) logic, I map out their common aims and different conclusions. I hold that Husserl and Wittgenstein both use the epistemic necessity of the existence of logical relations among things as an argument against philosophical scepticism, but their different epistemological convictions lead them to decisively diverging accounts of the nature (...) of those relations. Wittgenstein assumes a syntactic correspondence theory of truth, which identifies general logical form as a necessary condition for accurate representation, apparent in the existence of local truth-functions between propositions. As logical form is the (transcendental) necessary condition of every meaningful proposition, he infers that it is itself not representable and without ontological status. Husserl, by contrast, does not draw from a correspondence theory, but from a processual theory of certainty and truth, which offers genetic instead of categorical distinctions between perception and the logical relations apparent in conceptual knowledge. His theory of formal logic ultimately offers a coherent ontology for logical objects, which avoids logical mysticism. (shrink)

Le concept de Fundierung est introduit dans la IIIe Recherche Logique de Husserl et, malgré sa fonctionnalité apparente, il n’est pratiquement plus utilisé dans la suite de son oeuvre. En revanche, ce concept manifeste une puissance latente que le travail de Gian-Carlo Rota permet d’exalter. Entre les mains du mathématicienphilosophe, la Fundierung devient l’un des piliers fondamentaux d’une logique phénoménologique encore in fieri. Une logique qui voudrait prendre ses distances tant avec la « logique philosophique » traditionnelle qu’avec la « (...) logique mathématique » qui s’est désormais imposée en tant que standard de la rigueur et fascine souvent les philosophes. Cet article se propose de décrire comment l’interprétation de Rota enrichit le concept originel de Husserl et de quelle manière sa démarche ouvre la possibilité d’une évolution.The concept of Fundierung introduced by Husserl in his III Logic Research, notwithstanding its seemingly logical functionality, very rarely appears in his subsequent works. Yet this concept shows a latent power that Gian-Carlo Rota’s work succeeds in exalting. In the hands of the mathematician-philosopher, Fundierung becomes one of the fundamental pillars of a phenomenological logic still in embryo, a logic the aim of which is to distance itself from both the traditional ‘philosophical logic’ and the ‘mathematical logic’, the latter having by now become a standard of rigour by which the philosophers are often bewitched. This paper aims to describe how Rota’s interpretation has enhanced Husserl’s original concept and how this can foster a likely great evolution in the future. Il concetto di Fundierung appare nella IIIe Ricerca Logica di Husserl e, malgrado l’apparente funzionalità logica, appare molto raramente nel seguito della sua opera. Tuttavia tale concetto manifesta una potenza latente che il lavoro di Gian-Carlo Rota riesce ad esaltare. Nelle mani del matematico-filosofo, laFundierung diventa uno dei pilastri fondamentali di una logica fenomenologica ancora in fieri. Una logica che vorrebbe distanziarsi tanto dalla “logica filosofica” tradizionale quanto dalla “logica matematica” che ormai si è imposta come standard di rigore e di cui i filosofi subiscono spesso la fascinazione. Questo articolo si propone di descrivere quanto l’interpretazione di Rota arricchisca l’originale concetto husserliano e in che modo ciò permetta una grande possibilità di evoluzione.This article is in French. (shrink)

Gian-Carlo Rota a su concilier un travail mathématique exemplaire et des recherches philosophiques largement inspirées par la phénoménologie husserlienne. Son œuvre philosophique nous semble avoir de fait deux composantes : l’une s’intéresse majoritairement à des phénomènes universels. L’autre se déploie de façon plus subtile en filigrane de ses travaux mathématiques ; sans être thématisée comme telle – comme contribution philosophique –, elle alimente très lar-gement l’aura de Rota dans la communauté mathématique et justifie le rôle qu’il y joue de père (...) fondateur d’une école combinatoire. L’analyse de ces différents moments de sa pensée permet de mieux en cerner les ressorts, l’unité et la portée.Gian-Carlo Rota was able to combine a remarkable mathematical work with philosophical research largely inspired by Husserl’s phenomenology. His philosophical work has two components. One focuses mainly on universal phenomena in mathematics. The other unfolds in a more subtle way in his mathematical work; without being characterized as a proper philosophical contribution, it feeds largely the aura of Rota in the mathematical community and contributes to justify his role of founder of a school of combinatorics.The analysis of these different moments of his thought allows to understand better its origins, unit and scope. Gian-Carlo Rota ha saputo coniugare il propio lavoro matematico esemplare ad una ricerca filosofica in gran parte ispirata dalla fenomenologia di Husserl. La sua opera filosofica ha secondo noi due componenti: una riguarda principalmente i fenomeni universali. L’altra si sviluppa in modo più sottile nel suo lavoro matematico e, senza diventare un tema in quanto tale – come vero e pro-prio contributo filosofico –, alimenta in gran parte l’aura di Rota nella co-munità matematica, contribuendo a giustificare il suo ruolo di fondatore di una scuola combinatoria. L’analisi di questi diversi momenti del suo pensi-ero permette di identificarne meglio le origini, l’unità e la portata.This article is in French. (shrink)

RésuméGian-Carlo Rota est l’un des rares grands mathématiciens de la deuxième moitié du XX e siècle dont l’intérêt pour la logique formelle soit aussi ouvertement déclaré et ne se soit jamais démenti, depuis sa formation d’étudiant à Princeton jusqu’à ses derniers écrits. Plus exceptionnel encore, il fait partie des rares lecteurs assidus de Husserl à s’être aperçu que la phé-noménologie poursuivait un projet de réforme de la logique formelle. L’article propose d’attester l’existence d’un tel projet chez Husserl ; d’en examiner (...) la réappropriation et les prolongements chez Rota.Gian-Carlo Rota is among the few great mathematicians of the second-half of the XXth century whose interest in formal logic is openly declared and has never flagged, since his training as a student in Princeton up to his last writings. Even more exceptional, he belongs to the rare diligent readers of Husserl, who noticed that phenomenology was pursuing a project of reform of formal logic. This paper propose to testify to the existence of such a project in Husserl; to examine how it is taken over and continued by Rota. Gian-Carlo Rota è uno dei pochi grandi matematici della seconda metà del ventesimo secolo, il cui interesse per la logica formale è, dalla sua formazione come studente a Princeton al suo ultimi scritti, apertamente dichiarato e mai negato. Cosa ancora più eccezionale, Rota è uno dei rari lettori assidui di Husserl ad aver percepito che la fenomenologia stava perseguendo un progetto di riforma della logica formale. L’articolo propone di attestare l’esistenza di un tale progetto in Husserl e di esaminare la sua riappropriazione e le sue estensioni proposte da Gian-Carlo Rota.This article is in French. (shrink)

The traditional view regarding the philosophy of mathematics in the twentieth century is the dogma of three schools: Logicism, Intuitionism and Formalism. The problem with this dogma is not, at least not first and foremost, that it is wrong, but that it is biased and essentially incomplete. 'Biased' because it was formulated by one of the involved parties, namely the logical empiricists - if I see it right - in order to make their own position look more agreeable with Intuitionism (...) and Formalism. 'Essentially incomplete' because there was - and still exists - beside Frege's Logicism, Brouwer's Intuitionism and Hilbert's Formalism at least one further position, namely Husserl's phenomenological approach to the foundations of arithmetic, which is also philosophically interesting. In what follows, I want to do two things: First, I will show that the standard dogma regarding the foundations of mathematics is not only incomplete, but also inaccurate, misleading and basically wrong with respect to the three schools themselves. In doing this I hope to make room for Husserl and his phenomenological approach as a viable alternative in the foundations of arithmetic. Second, I will show how Husserl's phenomenological point of view is a position that fits exactly in between Frege's "logicism", properlyunderstood, and Hilbert's mature proof theory, in which his so called "formalism" turns out to be only a means to an end and not a goal in itself. (shrink)

This paper explicates Husserl’s usage of what he calls “radical Besinnung” in Formale und transzendentale Logik. Husserl introduces radical Besinnung as his method in the introduction to FTL. Radical Besinnung aims at criticizing the practice of formal sciences by means of transcendental phenomenological clarification of its aims and presuppositions. By showing how Husserl applies this method to the history of formal sciences down to mathematicians’ work in his time, the paper explains in detail the relationship between historical critical Besinnung and (...) transcendental phenomenology. Ultimately the paper suggests that radical Besinnung should be viewed as a general methodological framework within which transcendental phenomenological descriptions are used to criticize historically given goal-directed practices. (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)

In his book, 'History as a Science and the System of the Sciences', Thomas Seebohm articulates the view that history can serve to mediate between the sciences of explanation and the sciences of interpretation, that is, between the natural sciences and the human sciences. Among other things, Seebohm analyzes history from a phenomenological perspective to reveal the material foundations of the historical human sciences in the lifeworld. As a preliminary to his analyses, Seebohm examines the formal and material presuppositions of (...) phenomenological epistemology, as well as the emergence of the human sciences and the traditional distinctions and divisions that are made between the natural and the human sciences. -/- As part of this examination, Seebohm devotes a section to discussing Husserl’s formal mereology because he understands that a reflective analysis of the foundations of the historical sciences requires a reflective analysis of the objects of the historical sciences, that is, of concrete organic wholes (i.e., social groups) and of their parts. Seebohm concludes that Husserl’s mereological ontology needs to be altered with regard to the historical sciences because the relations between organic wholes and their parts are not summative relations. Seebohm’s conclusion is relevant for the issue of the reducibility of organic wholes such as social groups to their parts and for the issue of the reducibility of the historical sciences to the lower-order sciences, that is, to the sciences concerned with lower-order ontologies. -/- In this paper, I propose to extend Seebohm’s conclusion to the ontology of chemical wholes as object of quantum chemistry and to argue that Husserl’s formal mereology is descriptively inadequate for this regional ontology as well. This may seem surprising at first, since the objects studied by quantum chemists are not organic wholes. However, my discussion of atoms and molecules as they are understood in quantum chemistry will show that Husserl’s classical summative and extensional mereology does not accurately capture the relations between chemical wholes and their parts. This conclusion is relevant for the question of the reducibility of chemical wholes to their parts and of the reducibility of chemistry to physics, issues that have been of central importance within the philosophy of chemistry for the past several decades. (shrink)

THE VIRTUAL OF PHENOMENOLOGY IN PHYSICS. THE THEORY OF RELATIVITY LIKE PROOF BENCH OF TEMPORALITY AND KINAESTHESIA. FROM EINSTEIN AND HUSSERL’S MANUSCRIPTS The search for objective knowledge purports to aim at a reality independent of our experience of it, but we find ourselves dependent upon our sense experience as the only possible access to this purportedly independent reality that is the object of science. Husserl’s phenomenological point of view reveals how this aim is understandable, and, as the major developments in (...) twentieth-century physics have shown, how science must take account of the way that virtual forms of our thinking play a key role in capturing the phenomena scientists describe and explain. What this paper claims to prove is the epistemological evidence of an unavoidable relationship between the virtual and real that are described through the formal sciences and the concrete objects that they purport to capture. In one of his manuscript entitled: “Overthrow of the Copernican theory” Husserl emphasized the role of virtuality of natural laws such as it arises from Michelson’s experiment, dwelling on natural scientific naiveté. In fact, even though we eventually come to understand the earth as a thing moving among things, the unmoving earth remains the virtual condition that makes any movement intelligible. Thus, we might consider the actual world, each of its pieces and all determinations in them, as «intentional-real» objects of «virtual-conscious» acts. (shrink)

This article attempts to broaden the phenomenologically motivated perspective of H. Weyl's Das Kontinuum in the hope of elucidating the differences between the intuitive and mathematical continuum and further providing a deeper phenomenological interpretation. It is known that Weyl sought to develop an arithmetically based theory of continuum with the reasoning that one should be based on the naturally accessible domain of natural numbers and on the classical first-order predicate calculus to found a theory of mathematical continuum free of impredicative (...) circularities only to stumble, to cite a key question, in the evident lack of intuitive support for the notion of points of the continuum. In this motivation, I set out to deal from a Husserlian viewpoint with the general notion of points as appearances reducible to individuals of pre-predicative experience in contrast with the notion of an interval of real numbers taken as an abstraction based on the intuition of time-flowing experience. I argue that the notions of points and of real intervals in the above sense are not by essence related to objective temporality and thus their incompatibility in mathematical terms is ultimately due to deeper constituting reasons independently of any causal and spatio-temporal constraints. (shrink)

This paper reconstructs and critically analyzes Husserl’s philosophical engagement with symbolic technologies—those material artifacts and cultural devices that serve to aid, structure and guide processes of thinking. Identifying and exploring a range of tensions in Husserl’s conception of symbolic technologies, I argue that this conception is limited in several ways, and particularly with regard to the task of accounting for the more constructive role these technologies play in processes of meaning-constitution. At the same time, this paper shows that a critical (...) examination of Husserl’s account of symbolic technologies, particularly as developed in his mature, genetic phenomenology, can be enduringly fruitful—if some of the specific conceptual weakness of this account are identified and properly accounted for. My discussion will proceed as follows. In the first part I briefly analyze the early Husserl’s account of the role the ‘method of sensible signs’ plays in arithmetic cognition. In the second, main part I critically examine the bearing the genetic-phenomenological concepts of sedimentation and technization have on the conceptualization of symbolic technologies in Husserl’s work. In the final part I summarize the major strengths and weaknesses of Husserl’s account of symbolic technologies, and in the process make a case for the ongoing relevance of some of the crucial elements of this account. (shrink)

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)

This essay argues against David Carr’s relativism by clarifying the in principle requirements appropriate to non-relative truths and showing that de facto differences of conceptual frameworks threaten none of them. Non-relative truths are not threatened by history. This defense of non-relative truth belongs to a larger defense of Husserlian “science” that shows how essences, even those “delivered” by history, have a universal “governance” and can be affirmed in nonrelative truths-as such science requires. If history also allows the other qualities of (...) Husserlian science to obtain, then, the essay concludes, such science can exist even as a “situated science.”. (shrink)

By tracing the general evolution of HusserI’s theory of logic and mathematics, this essay explores Husserl’s identification and strategic overcoming of the two forms of psychologism--Iogical psychologism and transcendental psychologism--that bar the way to rigorous phenomenological inquiry. In the early works “On the Concept of Number” and the Philosophie der Arithmetik Husserl himself falls victim to a particular form of logical psychologism. By the time of the Logical Investigations this problem has been dealt with: the method of eidetic intuition enables (...) an account of the “origins” of logical and mathematical concepts without reducing such concepts to mere predicates of mental acts. The task of Formal and Transcendental Logic is to disclose the more pervasive problem of transcendental psychologism, one that taints even the theory of pure logic articulated in the Logical Investigations. A radical solution is provided through the development of an “ultimate logic” of transcendental subjectivity. (shrink)

Name der Zeitschrift: Zeitschrift für Religionswissenschaft Jahrgang: 24 Heft: 2 Seiten: 213-220.

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)

This paper points to two little-discussed interrelated features—among sociologists—about the nature of the lifeworld (Lebenswelt): that the experience of transcendence is an essential component of human actions, and that lived experience (Erlebnis) is founded on the non-discursivity of the lifeworld, i.e., the pre-predicative background expectancies from which the discursive arises. I examine the intellectual route of Alfred Schutz who developed his mundane lifeworld theory from appropriating Edmund Husserl’s notions of appresentation and apperception. Harold Garfinkel later extended Schutz’s concept of lifeworld (...) to the empirical investigations of constitutive social orders. By way of conclusion, I warn against a strain of constructionism in sociology, which tends to ignore the two said features of lived experience and inaccurately conceives social realities as essentially the actor’s discursive accomplishments. (shrink)

The traditional view regarding the philosophy of mathematics in the twentieth century is the dogma of three schools: Logicism, Intuitionism and Formalism. The problem with this dogma is not, at least not first and foremost, that it is wrong, but that it is biased and essentially incomplete. 'Biased' because it was formulated by one of the involved parties, namely the logical empiricists - if I see it right - in order to make their own position look more agreeable with Intuitionism (...) and Formalism. 'Essentially incomplete' because there was - and still exists - beside Frege's Logicism, Brouwer's Intuitionism and Hilbert's Formalism at least one further position, namely Husserl's phenomenological approach to the foundations of arithmetic, which is also philosophically interesting. In what follows, I want to do two things: First, I will show that the standard dogma regarding the foundations of mathematics is not only incomplete, but also inaccurate, misleading and basically wrong with respect to the three schools themselves. In doing this I hope to make room for Husserl and his phenomenological approach as a viable alternative in the foundations of arithmetic. Second, I will show how Husserl's phenomenological point of view is a position that fits exactly in between Frege's "logicism", properlyunderstood, and Hilbert's mature proof theory, in which his so called "formalism" turns out to be only a means to an end and not a goal in itself. (shrink)

Gödel’s philosophical rationalism includes a program for “developing philosophy as an exact science.” Gödel believes that Husserl’s phenomenology is essential for the realization of this program. In this article, by analyzing Gödel’s philosophy of idealism, conceptual realism, and his concept of “abstract intuition,” based on clues from Gödel’s manuscripts, I try to investigate the reasons why Gödel is strongly interested in Husserl’s phenomenology and why his program for an exact philosophy is unfinished. One of the topics that has attracted much (...) attention recently is the development of Gödel’s philosophical thoughts and its connection with other philosophical ideas. For instance, some scholars are searching for the possible connections between Gödel’s philosophy and Husserl’s phenomenology and examining if there is any solid evidence of Husserl’s influence on Gödel from Gödel’s works :181–203, 1998; Huaser, Bull Symbolic Logic 12:529–588, 2006). Why is Gödel’ s interested in Husserl? How should this turn to Husserl be interpreted? Is it a dismissal of Leibnizian philosophy, or a different way to achieve similar goals? Way did Gödel turn specifically to Husserl’s transcendental idealism? :425–476, 2003) I believe, the reason is that Gödel has a valuable program for “developing philosophy as an exact science” and he believes that Husserl’s phenomenology is relevant to the realization of this program. So far there are no sufficient evidence to show that there is a direct inheritance relation between Gödel’s and Husserl’s thoughts. However, from the clues in Gödel’s idealistic philosophy, conceptual realism, and his concept of “abstract intuition,” we can perhaps explore some similarities between his thoughts and Husserl’s thoughts, and analyze the reason why Gödel is interested in Husserl’s phenomenology and why his program for an exact philosophy is unfinished. (shrink)

This article locates Weyl's philosophy of mathematics and its relationship to his philosophy of science within the epistemological and ontological framework of Husserl's phenomenology as expressed in the "Logical Investigations" and "Ideas". This interpretation permits a unified reading of Weyl's scattered philosophical comments in "The Continuum" and "Space-Time-Matter". But the article also indicates that Weyl employed Poincaré's predicativist concerns to modify Husserl's semantics and trim Husserl's ontology. Using Poincaré's razor to shave Husserl's beard leads to limitations on the least upper (...) bound theorem in the foundations of analysis and Dirichlet's principle in the foundations of physics. Finally, the article opens the possibility of reading Weyl as a systematic thinker, that he follows Husserl's so-called transcendental turn in the "Ideas". This permits an even more unified reading of Weyl's scattered philosophical comments. (shrink)

Gödel has argued that we can cultivate the intuition or 'perception' of abstract concepts in mathematics and logic. Gödel's ideas about the intuition of concepts are not incidental to his later philosophical thinking but are related to many other themes in his work, and especially to his reflections on the incompleteness theorems. I describe how some of Gödel's claims about the intuition of abstract concepts are related to other themes in his philosophy of mathematics. In most of this paper, however, (...) I focus on a central question that has been raised in the literature on Gödel: what kind of account could be given of the intuition of abstract concepts? I sketch an answer to this question that uses some ideas of a philosopher to whom Gödel also turned in this connection: Edmund Husserl. The answer depends on how we understand the conscious directedness toward 'objects' and the meaning of the term 'abstract' in the context of a theory of the intentionality of cognition. (shrink)

Offering a collection of fifteen essays that deal with issues at the intersection of phenomenology, logic, and the philosophy of mathematics, this 2005 book is divided into three parts. Part I contains a general essay on Husserl's conception of science and logic, an essay of mathematics and transcendental phenomenology, and an essay on phenomenology and modern pure geometry. Part II is focused on Kurt Godel's interest in phenomenology. It explores Godel's ideas and also some work of Quine, Penelope Maddy and (...) Roger Penrose. Part III deals with elementary, constructive areas of mathematics. These are areas of mathematics that are closer to their origins in simple cognitive activities and in everyday experience. This part of the book contains essays on intuitionism, Hermann Weyl, the notion of constructive proof, Poincaré and Frege. (shrink)

Originally published in 1986. This work remains of compelling interest to those concerned with the natural sciences and their social problems. It puts forward original and unorthodox ideas about the philosophy of and sociology of science, starting from the conviction that modern societies face deep problems arising from unresolved dilemmas about the meaning, content and technical applications of the theories of nature they employ. The book draws on insights developed within a variety of traditions to explore these problems, especially the (...) work of Edmund Husserl and modern critical theory. (shrink)

There are many interpretations of the birth of modern science. Most of them are, nevertheless, confined to the analysis of certain historical episodes or technical details, while leaving the very notion of mathematization unanalyzed. In my opinion this is due to a lack of a proper philosophical framework which would show the process of mathematization as something radically new. Most historians assume that the world is just like it is depicted by science. Thus they are not aware of the radical (...) novelty of the mathematization of nature and focus their attention on the details of this process. Phenomenology by its radical questioning of the traditional interpretations of reality provides an ideal means for the reconstruction of the process of mathematization of nature and the birth of mathematical natural science. In a series of papers I tried to show the power of Husserl 's concept of mathematization by filling in the historical details into his interpretation. In the present article I want to compare Husserl 's approach with the approach of Heidegger. I believe that by comparing these two phenomenological theories of mathematization the advantages of Husserl 's approach will come to the fore. _German_ Es gibt mehrere Interpretationen der Entstehung der modernen Wissenschaft. Die meisten von ihnen beschränken sich allerdings auf die Analyse bestimmter historischer Episoden oder technischer Details, während der Begriff der Mathematisierung unanalysiert bleibt. Meiner Meinung nach ist dies auf das Fehlen eines richtigen philosophischen Rahmens zurückzuführen, der den Prozess der Mathematisierung als etwas radikal Neues zeigen würde. Die meisten Historiker gehen davon aus, dass die Welt so ist, wie sie von der Wissenschaft dargestellt wird. So verkennen sie die Radikalität der Mathematisierung der Natur und konzentrieren ihre Aufmerksamkeit auf die Details dieses Prozesses. Die Phänomenologie bietet dank ihrer radikalen Kritik der traditionellen Interpretationen der Realität ein ideales Mittel für die Rekonstruktion des Prozesses der Mathematisierung der Natur und der Geburt der mathematischen Naturwissenschaft. In einer Reihe von Arbeiten habe ich versucht, die Stärke der Husserlschen Interpretation der Mathematisierung zu zeigen, indem ich die historischen Details in seine Interpretation einfügte. Im vorliegenden Artikel möchte ich Husserls Ansatz mit dem Ansatz von Heidegger vergleichen. Durch diesen Vergleich der beiden phänomenologischen Theorien der Mathematisierung der Natur werden die Vorteile von Husserls Ansatz in den Vordergrund treten. (shrink)

Erst in Husserls und Heideggers Spätzeit rückt die kritische Besinnung auf das gegenwärtige, durch die Herrschaft der naturwissenschaftlich-technischen Rationalität geprägte Zeitalter ins Zentrum ihres Denkens. Der eigentliche Ursprung dieser Besinnung liegt aber in dem phänomenologischen Denkweg, den beide Philosophen von früh an beschritten haben. Nur weil sie der Urmaxime der Phänomenologie Zu den Sachen selbst! verpflichtet waren, konnte ihnen eine Wissenschaftsanalyse gelingen, durch die sich die wissenschaftliche Erkenntnis als ein abgeleitetes Verhalten des Menschen gegenüber den Dingen herausstellt, das auf ein (...) ursprüngliche Weise der Welterfahrung zurückweist. Wenn die so angesetzte Zeitkritik bei beiden Denkern erst spät in den Vordergrund tritt, so nicht, weil sie sich vom ursprünglichen Ansatz der Phänomenologie abgekehrt hätten, sondern umgekehrt deswegen, weil sie erst allmählich zur vollen Entfaltung der in diesem Ansatz liegenden Möglichkeiten vordrangen. Allein hierdurch konnten sie die Geschichte entdecken und sich das Rüstzeug erarbeiten, um den geschichtlichen Hintergrund der modernen Wissenschaft in die Grundlagen ihrer Betrachtung einzubeziehen. Die phänomenologische Wissenschaftskritik wird so zu einer Ortsbestimmung der Gegenwart. Diese Zusammenhänge rechtfertigen einen Vergleich von Husserls und Heideggers Denken in ihrer Spätzeit. Er zeigt, daß die Positionen, von denen aus die beiden Philosophen Perspektiven für einen Ausweg aus der gegenwärtigen Situation entwerfen, entgegengesetzte Grundeinstellungen zur technisch-wissenschaftlichen Weltauffassung repräsentieren. (shrink)

We compare Gödel’s and Brouwer’s explorations of mysticism and its relation to mathematics.

Cette these divisee en trois parties, comporte deux caracteristiques essentielles: ; Elle comporte une teneure mathematique: des progres accomplis entre 1870 et 1914 par les mathematiciens au sein des savoirs que sont l'analyse infinitesimale, l'algebre, la geometrie etc. ont conferre une place sui generis a l'algebre. Sa redefinition va integrer des objets quelconques et non plus seulement les simples techniques du calcul litteral. Aussi certains mathematiciens observerent un desinteret pour ces calculs au profit d'une reflexion theorique autour du mot d'ordre (...) de Dirichlet de "substituer les idees au calcul" . L'enjeu consistait a donner un fondement sur au concept de nombre. ; Historiquement, c'est entre 1891 et 1906 que Frege et Husserl ont entretenu un echange epistolaire sur la question. Pour Husserl, eleve de Brentano, ce fondement est a trouver dans le sujet connaissant, pole constitutif de tous les ordres du savoir. Sa Philosophie de l'Arithmetique en donne les contours. Critique et oppose a cette base psychologiste, Frege va presenter son approche qui privilegie la logique. Dans les Fondements de l'arithmetique , Frege estime que l'arithmetique entiere peut etre reduite aux lois de logique generale. Cette reduction de l'arithmetique a la logique donna naissance a la tradition logiciste qui fut mortellement ebranlee par le paradoxe de Russell et inaugura la crise des fondements de type ensembliste. ;Aux alentours de 1894, Husserl abandonna le psychologisme. Faut-il y voir l'influence de Frege? Cette question est debatue dans la 2e partie de la these par Mohanty et Dummett. Ils ne s'entendent ni sur la date de l'abandon par Husserl du psychologisme ni sur l'influence que Frege aurait pu exercer sur lui. ;La troisieme partie est un recours a Wittgenstein. Elle porte sur ce qu'on peut percevoir comme esquisse d'une solution qui passe par le renoncement de la philosophie a fonder les objets mathematiques qui ne sont que des conventions, brisant ainsi 2000 ans de domination platoniste. D'aucuns taxent la position de Wittgenstein de strict finitisme. Si cela s'averait juste, n'y-a-t-il pas la un lien possible entre Wittgenstein et le constructivisme de Kronecker? (shrink)