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)

Little is known of Edmund Husserl's direct encounter with Georg Cantor's ideas on Platonic idealism and the abstraction of number concepts during the late 19th century, when Husserl's philosophical orientation changed considerably and definitely. Closely analyzing and comparing the two men's writings during that important time in their intellectual careers, I describe the crucial shift in Husserl's views on psychologism and metaphysical idealism as it relates to Cantor's philosophy of arithmetic. I thus establish connections between their ideas which have been (...) until now been virtually unsuspected and contribute to a better understanding of the development of Husserl's thought and of the philosophical and metaphysical ideas within which Cantor chose to frame his theories. (shrink)

Edmund Husserl was one of the very first to experience the direct impact of challenging problems in set theory and his phenomenology first began to take shape while he was struggling to solve such problems. Here I study three difficulties associated with Frege's use of sets that Husserl explicitly addressed: reference to non-existent, impossible, imaginary objects; the introduction of extensions; and 'Russell's paradox'.I do so within the context of Husserl's struggle to overcome the shortcomings of set theory and to develop (...) his own theory of manifolds. I define certain issues involved and discuss how Husserl's theory of manifolds might confront them. In so doing I hope to help bring Husserl's theories about sets and manifolds out of the realm of abstract theorizing and prompt further exploration of uncharted philosophical territory rich in philosophical implications. (shrink)

Few have entertained the idea that Georg Cantor, the creator of set theory, might have influenced Edmund Husserl, the founder of the phenomenological movement. Yet an exchange of ideas took place between them when Cantor was at the height of his creative powers and Husserl in the throes of an intellectual struggle during which his ideas were particularly malleable and changed considerably and definitively. Here their writings are examined to show how Husserl's and Cantor's ideas overlapped and crisscrossed in the (...) areas of philosophy and mathematics, arithmetization, abstraction, consciousness and pure logic, psychologism, metaphysical idealism, new numbers, and sets and manifolds. (shrink)

The current resurgence of interest in cognition and in the nature of cognitive processing has brought with it also a renewed interest in the early work of Husserl, one of the most sustained attempts to come to grips with the problems of logic from a cognitive point of view. Logic, for Husserl, is a theory of science; but it is a theory which takes seriously the idea that scientific theories are constituted by the mental acts of cognitive subjects. The present (...) essay begins with an exposition of Husserl's act-based conception of what a science is, and goes on to consider his account of the role of linguistic meanings, of the ontology of scientific objects, and of evidence and truth. The essay concentrates almost exclusively on the Logical Investigations of 1900/01. This is not only because this work, which is surely Husserl's single most important masterpiece, has been overshadowed first of all by his Ideas I and then later by the Crisis. It is also because the Investigations contain, in a peculiarly clear and pregnant form, a whole panoply of ideas on logic and cognitive theory which either simply disappeared in Husserl's own later writings or became obfuscated by an admixture of that great mystery which is 'transcendental phenomenology'. (shrink)

Translator's Preface LA LOGIQUE DE HUSSERL, etude sur "Logique for- melle et logique transcendentale" the original of the present translation, was published ...

Husserl and the Logic of Experience includes both detailed work on particular aspects of logical theory (such as an inquiry into the status of the principle of excluded middle) and also detailed investigations into the nature of the logic of temporal conceptions. Demonstrating the cultural import of Husserl's work while also showing its continuing significance for logical theory, this collection is a milestone in the study of transcendental phenomenology.

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)

"Husserl's work include lengthy treatment of universals, categories, meanings, numbers, manifolds, etc. from an ontological perspective. Here, however, we shall concentrate almost exclusively on the Logical Investigations, which contain in a clear form the ontological ideas which provided the terminological and theoretical basis both for much of the detailed phenomenological description and for many of the metaphysical theses presented in Husserl's later works.

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.

According to Edmund Husserl in the Prolegomena to Pure Logic, which constitutes the preliminary rational foundation for – and also the entire first volume of – his Logical Investigations, pure logic is the a priori theoretical, nomological science of „demonstration“ (LI 1, 57; Hua XVIII, 23).1 For him, demonstration includes both consequence and provability. Consequence is the defining property of all and only formally valid arguments, i. (...) class='Hi'> e., arguments that cannot lead from true premises to false conclusions. And provability (a. k. a. „completeness“) is the property of a logical system such that, for every truth of logic in that system, there is, at least in principle, a rigorous step-by-step logically valid procedure demonstrating its validity according to strictly universal, ideal, and necessary logical laws. In this way, the laws of pure logic completely determine its internal structure. Moreover, these laws and these proofs are all knowable a priori, with selfevident insight (LI 1, 196; Hua XVIII, 185–195). So not only is pure logic independent of any other theoretical science, in that it requires no other science in order to ground its core notion of demonstration, it also provides both epistemic and semantic foundations for every other theoretical science, as well as every practical discipline or „technology.“ To the extent that pure logic is the foundation of every other.. (shrink)

(No abstract is available for this citation).

This paper discusses Jean van Heijenoort’s (1967) and Jaakko and Merrill B. Hintikka’s (1986, 1997) distinction between logic as auniversal language and logic as a calculus, and its applicability to Edmund Husserl’s phenomenology. Although it is argued that Husserl’s phenomenology shares characteristics with both sides, his view of logic is closer to the model-theoretical, logic-as-calculus view. However, Husserl’s philosophy as transcendental philosophy is closer to the universalist view. This paper suggests that Husserl’s position shows that holding a model-theoretical view of (...) logic does not necessarily imply a calculus view about the relations between language and the world. The situation calls for reflection about the distinction: It will be suggested that the applicability of the van Heijenoort and the Hintikkas distinction either has to be restricted to a particular philosopher’s views about logic, in which case no implications about his or her more general philosophical views should be inferred from it; or the distinction turns into a question of whether our human predicament is inescapable or whether it is possible, presumably by means of model theory, to obtain neutral answers to philosophical questions. Thus the distinction ultimately turns into a question about the correct method for doing philosophy. (shrink)

This book makes available to the English reader nearly all of the shorter philosophical works, published or unpublished, that Husserl produced on the way to the phenomenological breakthrough recorded in his Logical Investigations of 1900-1901. Here one sees Husserl's method emerging step by step, and such crucial substantive conclusions as that concerning the nature of Ideal entities and the status the intentional `relation' and its `objects'. Husserl's literary encounters with many of the leading thinkers of his day illuminates both the (...) context and the content of his thought. Many of the groundbreaking analyses provided in these texts were never again to be given the thorough expositions found in these early writings. Early Writings in the Philosophy of Logic and Mathematics is essential reading for students of Husserl and all those who enquire into the nature of mathematical and logical knowledge. (shrink)

The centerpiece of the Analyses is a translation from the German of notes for a series of lectures given by phenomenologist Edmund Husserl in the early twenties, which is to say some eighty years ago. Husserl designated the topic of the lectures 'transcendental logic'. In this context, the term, 'transcendental', is not to be understood in some mystical sense, but rather in a Kantian sense: pertaining to the conditions of possibility of experience. Likewise, the term, 'logic', is not to be (...) taken in the narrow sense of formal logic, but rather in the very general sense it had for Platonic dialectic: a concern with normative guidelines and critical assessment of the possibility of truth. The topic of the lectures is succinctly characterized by Husserl as 'a universal theory of science, and at the same time, a theory of science in principle,' where the latter means 'the science of the a priori of all sciences as such' (p. 1). (shrink)

Psychologism in logic holds that logic is a branch of psychology. This view has been vigorously defended by John Stuart Mill and by a number of German philosophers of logic, notably Erdmann. Its chief critics have been Husserl and Frege and, to a lesser extent, Russell. Husserl set forth a profound and detailed critique of psychologism in Logical Investigations. This paper examines this critique. First, I explain why the psychologistic theory is attractive. Then I show that Husserl's critique is not (...) convincing, partly because he does not take the theory in its most plausible form and partly because he ignores certain important distinctions (for example, between what a statement is about and what it is true in virtue of). Then I raise two new objections to the psychologistic theory. The purpose of this paper is to suggest that the psychologistic theory remains an important and serious position from which we can learn much about the status of logic. (shrink)

This article is composed of three sections that investigate the epistemological foundations of Husserl’s idea of logic from the Logical Investigations . First, it shows the general structure of this logic. Husserl conceives of logic as a comprehensive, multi-layered theory of possible theories that has its most fundamental level in a doctrine of meaning. This doctrine aims to determine the elementary categories that constitute every possible meaning (meaning-categories). The second section presents the main idea of Husserl’s search for an epistemological (...) foundation for knowledge, science and logic. Their epistemological clarification can only be reached through a detailed analysis of the structure of those intentions that give us what is meant in our intentions. To reveal the intuitive giveness of logical forms is the ultimate aim of Husserl’s epistemology of logic. Logical forms and meaning-categories can only be given in a certain higher-order intuition that Husserl calls categorical intuition. The third section of this article distinguishes different kinds of categorical intuition and shows how the most basic logical categories and concepts are given to us in a categorical abstraction. (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.

Acknowledgments I wish to express my gratitude to Dr. Jakob Amstutz for his continual feedback during my writing of this work. ...

In this book Robert Tragesser sets out to determine the conditions under which a realist ontology of mathematics and logic might be justified, taking as his starting point Husserl's treatment of these metaphysical problems. He does not aim primarily at an exposition of Husserl's phenomenology, although many of the central claims of phenomenology are clarified here. Rather he exploits its ideas and methods to show how they can contribute to answering Michael Dummet's question 'Realism or Anti-Realism?'. In doing so he (...) makes a challenging and provocative contribution to the debate. (shrink)

INTRODUCTION Phenomenology as transcendental phenomenology is centered in a description of meaning interpreted in relationship to acts of consciousness. ...

Husserl claimed that all theoretical scientific concepts originate in and are valid in reference to 'life-world' experience and that scientific traditions preserve the sense and validity of such concepts through unitary and cumulative change. Each of these claims will, in turn, be sympathetically laid out and assessed in comparison with more standard characterizations of scientific method and conceptual change as well as the history of physics, concerning particularly the challenge they may pose for scientific realism. The Husserlian phenomenological framework is (...) accepted here without defense, and hence the present project is limited to the task of asking what can and cannot be accommodated within that framework on its own terms. (shrink)

This essay is concerned with the processes of idealization as described by Husserl in his last work, "The Crisis of European Sciences and Transcendental Phenomenology". Central as the processes of idealization are to Husserl's reflections on the origin of natural scientific knowledge and his attempt to reground that knowledge in the "forgotten meaning-fundament of natural science," they have not always been well understood. One reason for this is the lack of concrete historical examples. The main purpose of this paper is (...) to correct this deficit. The paper is comprised of four sections. The first distinguishes two separate processes of idealization, one ascending from the life-world and the other descending and applying to it. The interaction of the two is then considered. The second section takes up Husserl's own discussion of Galileo's employment of idealization in his original mathematization of nature. The third section examines Galileo's analysis of freefall as a historical example of the processes of idealization. Here it is seen that the evidence clearly justifies Husserl's claims regarding the role of idealization in the origins of modern natural science. The conclusion employs the insights gained in the previous sections to exhibit the importance of understanding the processes of idealization as propaedeutic to the appreciation of the role and importance of the phenomenological methods of epoché and reduction to restoring lost layers of meaning by nullifying the idealizations which cover the life world. (shrink)

Husserl argues in the Crisis that the prevalent tradition of positive science in his time had a philosophical core, called by him "Galilean science", that mistook the quest for objective theory with the quest for truth. Husserl is here referring to Gottingen science of the Golden Years. For Husserl, theory "grows" out of the "soil" of the prescientific, that is, pretheoretical, life-world. Scientific truth finally is to be sought not in theory but rather in the pragmatic-perceptual praxes of measurement. Husserl (...) is faulted for taking measuring processes to be "infinitely perfectible". The dependence of new scientific phenomena on the existence of prior "prescientific" inductive praxis is analyzed, also Husserl's residual objectivism and failure to appreciate the hermeneutic character of measurement. Though not a scientific (theory-)realist, neither was he an instrumentalist, but he was a scientific (phenomena-)realist. (shrink)

Fundamental notions Husserl introduced in Ideen I, such as epochè, reality, and empty X as substrate, might be useful for elucidating how mathematical physics concepts are produced. However, this is obscured in the context of Husserl’s phenomenology itself. For this possibility, the author modifies Husserl’s fundamental notions introduced for pure phenomenology, which found all sciences on the absolute Ego. Subsequently, the author displaces Husserl's phenomenological notions toward the notions operating inside scientific activities themselves and shows this using a case study (...) of the construction of noncommutative geometry. The perspective in Ideen I about geometry and mathematical physics includes points that are inappropriate to modern geometry and to modern physics, especially to noncommutative geometry and to quantum physics. The first point relates to the intuitive character of geometrical objects in Husserl. The second is linked to the notion of locality related to the notion of extension, by which Husserl characterizes the essence of physical things. The points show that the notion of empty X as a substrate, developed in “Phenomenology of Reason” in Ideen I, is helpful for considering the notions of physical reality and of geometrical space, especially reality in quantum physics and space in noncommutative geometry. The salient conclusions include the proposition that aphilosophical study of the relationship between the physical object X, which imparts a unity to what is given to sensibility, and the geometrical space X, which imparts a unity of sense to various mathematical operations, opens a reinterpretation of Husserl’s interpretation, supporting an epistemology of mathematical physics. (shrink)

In this book, which remained unfinished at his death, Husserl attempts to forge a union between phenomenology and existentialism.

: This paper discusses the origins of two key notions in Foucault's work up to and including The Archaeology of Knowledge. The first of these notions is the notion of "archaeology" itself, a form of historical investigation of knowledge that is distinguished from the mere history of ideas in part by its unearthing what Foucault calls "historical a prioris". Both notions, I argue, are derived from Husserlian phenomenology. But both are modified by Foucault in the light of Jean Cavaillès's critique (...) of Husserl's theory of science. On Husserl's view, we demand that propositions holding of scientific objects be intersubjective and invariant, but this demand conflicts with our immediate experience, which is essentially bound to a subject's perspective. Thus the mathematical and physical sciences must utilise formal languages to fix these truths independently of the thoughts of a particular subject. This necessary procedure leads to the sedimentation of these formal systems: we forget their source in the concrete experiences of individuals, and use them as purely technical means. The technique of reactivating the intentional acts in which sedimented formal systems originated is thus, in Fink's terminology, an archaeological method. Foucault and Cavaillès retain the general outlines of this archaeology of the sciences, but they reject its appeal to conscious acts of meaning, to what Cavaillès calls "the philosophy of consciousness". I conclude by discussing the implicit difficulties in the "linguistic transcendentalism" proposed as an alternative by these French critics of Husserl. (shrink)

In this paper, I reconstruct Edmund Husserl's view on the relationship between formal inquiry and the life-world, using the example of formal geometry. I first outline Husserl's account of geometry and then argue that he believed that the applicability of formal geometry to intuitive space (the space of everyday-experience) guarantees the conceptual continuity between different notions of space.

This article critically reviews an outstanding collection of new essays addressing Edmund Husserl’s Crisis of European Sciences. In Science and the Life-World (Stanford, 2010), David Hyder and Hans-Jörg Rheinberger bring together an impressive range of first-rate philosophers and historians. The collection explicates key concepts in Husserl’s often obscure work, compares Husserl’s phenomenology of science to the parallel tradition of historical epistemology, and provocatively challenges Husserl’s views on science. The explications are uniformly clear and helpful, the comparative work intriguing, and the (...) criticisms interesting but uneven. The article also elaborates on Husserl’s phenomenological method as it relates to the historiography of science, and compares his views on mathematical idealisation to more recent work in the analytical tradition. (shrink)

In this essay I argue that Husserl’s development of the nineteenth century Natur/Geist distinction is grounded in the intentional correlate between the pre-theoretical natural attitude and environing world ( Umwelt ). By reconsidering the Natur/Geist distinction through its historical context in the nineteenth century debate between Wilhelm Dilthey and the Neo-Kantians from the Baden or Southwest school, it is possible to understand more clearly Husserl’s appropriations and novel contributions. One of Husserl’s contributions lies in his rigorous thematization and clarification of (...) the constitutive features proper to the natural and human sciences as they arise from the pre-theoretical experience of an environing world. This ordinary lived experience between the lived body and environing world is presupposed by and forms a unity with both Natur and Geist , thereby acting as the unified ground that is inclusive of naturalized Geist and a geistig nature. This unbuilding ( Abbau ) of the Natur/Geist distinction is necessary, according to Husserl, for the radical clarification of the respective methodologies of the natural and human sciences. (shrink)

There is a common concern in some of the writings of Husserl and Gadamer. It is the concern to defend the legitimacy and dignity of the "human sciences." They argue from the methodological standpoint that the method of the natural sciences leaves out the relationship between the object of inquiry and the inquirer. This relationship plays a key role in "understanding," which is the concem of the human sciences. In explicating it, Husserl and Gadamer stress the role of the community (...) and the lifeworld as the context of understanding. Parallels to Husserl's and Gadamer's ideas can be found in analytical philosophy. (shrink)

This paper discusses Husserl’s views on physical theories in the first volume of his Logical Investigations , and compares them with those of his contemporaries Pierre Duhem and Henri Poincaré. Poincaré’s views serve as a bridge to a discussion of Husserl’s almost unknown views on physical geometry from about 1890 on, which in comparison even with Poincaré’s—not to say Frege’s—or almost any other philosopher of his time, represented a rupture with the philosophical tradition and were much more in tune with (...) the physical geometry underlying the Einstein-Hilbert general theory of relativity developed more than two decades later. (shrink)

The idea that scientific objectivity requires a method of concept formation according to which concepts are freely created by the mind was famously propagated by Hermann Weyl. I argue that this idea, which he saw as essentially characterizing what physicists do when they do physics, led him to abandon the phenomenological view on objectivity, more particularly the strong connection between objectivity and evidence (understood in a Husserlian sense as a satisfaction of meaning intentions). The free creation of concepts, that is (...) ultimately their introduction via Hilbert-style axiomatizations, is at the heart of Weyl's account of scientific objectivity, for it allows the introduction of hypothetical elements, without which, on his view, objectivity collapses (at best) into mere intersubjectivity. (shrink)

Abstract Not much scholarly work is needed in order to stumble across many passages where Edmund Husserl seems to advocate an anti-realist attitude towards the natural sciences. This tendency, however, is not well-received within the secondary literature. While some commentators criticize Husserl for his alleged scientific anti-realism, others argue that Husserl's position is much more realist than the first impression indicates. It is against this background that I want to argue for the following theses: a) The basic outlook of Husserl's (...) epistemology as well as his more substantial comments regarding the natural sciences indeed result in a (sophisticated version of) scientific anti-realism which bears certain resemblances to Bas van Fraassen's constructive empiricism; b) This scientific anti-realism can be defended against the two most common objections raised in the secondary literature; c) It is only by means of this sophisticated version of scientific anti-realism that phenomenology can circumvent the problem of ?scientific objectivism? (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 year 1894, Husserl had not been already contaminated by Bolzano’s realism. It was then that he conceived a theory of assumptions in order to “save an existence” for mathematical objects. Here we would like to explore this theory and show in what way it represented a convincing alternative to realistic ontology and its counterpart: the correspondence theory of truth. However, as soon as he designed it, Husserl shoved away all the implications for his theory of assumptions, and merely (...) abandoned it. We would also like to consider the wrong reasons for this renouncement, which undoubtedly have to do with the dichotomy between signification and perception in his elaboration of phenomenology. (shrink)

In the present paper, we discuss Husserl's deep account of the notions of ?calculation? and of arithmetical ?operation? which is found in the final chapter of the Philosophy of Arithmetic, arguing that Husserl is ? as far as we know ? the first scholar to reflect seriously on and to investigate the problem of circumscribing the totality of computable numerical operations. We pursue two complementary goals, namely: (i) to provide a formal reconstruction of Husserl's intuitions, and (ii) to demonstrate ? (...) on the basis of our reconstruction ? that the class of operations that Husserl has in mind turns out to be extensionally equivalent to the one that, in contemporary logic, is known as the class of ?partial recursive functions? (shrink)

Résumé -/- Dans son premier livre (Philosophie de l’arithmétique 1891), Husserl élabore une très intéressante philosophie des mathématiques. Les concepts mathématiques sont interprétés comme des concepts de « deuxième ordre » auxquels on accède par une réflexion sur nos opérations mentales de numération. Il s’ensuit que la vérité de la proposition : « il y a trois pommes sur la table » ne consiste pas dans une relation mythique quelconque avec la réalité extérieure au psychique (où le nombre trois doit (...) être exemplifié de quelque manière), mais bien dans le fait que les pommes sur la table peuvent être dénombrées correctement en tant qu’elles sont au nombre de trois. Nous avons affaire ici à une position « antiréaliste » fondant la vérité mathématique non pas dans un monde platonicien, mais bien sur le concept de rectitude de nos opérations formelles. -/- Abstract -/- In his first book (Philosophy of Arithmetic 1891) Husserl develops a very interesting philosophy of mathematics. Mathematical concepts are interpreted as « second order » concepts gained by reflection on our mental operations of counting. Accordingly the truth of the proposition « There are three apples on the table » consists not in any mythical relation to the extra-mental reality (where the number three should be somehow exemplified), but rather in the fact that the apples on the table can be correctly counted as three. We have here an « anti-realist » position grounding the mathematical truth not in platonic realms, but rather in the concept of correctness of our formal operations. (shrink)

Husserl left many unpublished drafts explaining (or trying to) his views on spatial representation and geometry, such as, particularly, those collected in the second part of Studien zur Arithmetik und Geometrie (Hua XXI), but no completely articulate work on the subject. In this paper, I put forward an interpretation of what those views might have been. Husserl, I claim, distinguished among different conceptions of space, the space of perception (constituted from sensorial data by intentionally motivated psychic functions), that of physical (...) geometry (or idealized perceptual space), the space of the mathematical science of physical nature (in which science, not only raw perception has a word) and the abstract spaces of mathematics (free creations of the mathematical mind), each of them with its peculiar geometrical structure. Perceptual space is proto-Euclidean and the space of physical geometry Euclidean, but mathematical physics, Husserl allowed, may find it convenient to represent physical space with a non-Euclidean structure. Mathematical spaces, on their turn, can be endowed, he thinks, with any geometry mathematicians may find interesting. Many other related questions are addressed here, in particular those concerning the a priori or a posteriori character of the many geometric features of perceptual space (bearing in mind that there are at least two different notions of a priori in Husserl, which we may call the conceptual and the transcendental a priori). I conclude with an overview of Weyl’s ideas on the matter, since his philosophical conceptions are often traceable back to his former master, Husserl. (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.

Husserl's contributions to the nature of mathematical knowledge are opposed to the naturalist, empiricist and pragmatist tendences that are nowadays dominant. It is claimed that mainstream tendences fail to distinguish the historical problem of the origin and evolution of mathematical knowledge from the epistemological problem of how is it that we have access to mathematical knowledge.

(No abstract is available for this citation).

The paper examines the roots of Husserlian phenomenology in Weierstrass's approach to analysis. After elaborating on Weierstrass's programme of arithmetization of analysis, the paper examines Husserl's Philosophy of Arithmetic as an attempt to provide foundations to analysis. The Philosophy of Arithmetic consists of two parts; the first discusses authentic arithmetic and the second symbolic arithmetic. Husserl's novelty is to use Brentanian descriptive analysis to clarify the fundamental concepts of arithmetic in the first part. In the second part, he founds the (...) symbolic extension of the authentically given arithmetic with stepwise symbolic operations. In the process of doing so, Husserl comes close to defining the modern concept of computability. The paper concludes with a brief comparison between Husserl and Frege. While Frege chose to subject arithmetic to logical analysis, Husserl wants to clarify arithmetic as it is given to us. Both engage in a kind of analysis, but while Frege analyses within Begriffsschrift, Husserl analyses our experiences. The difference in their methods of analysis is what ultimately grows into two separate schools in philosophy in the 20th century. (shrink)

This book makes available to the English reader nearly all of the shorter philosophical works, published or unpublished, that Husserl produced on the way to the phenomenological breakthrough recorded in his Logical Investigations of 1900-1901. Here one sees Husserl's method emerging step by step, and such crucial substantive conclusions as that concerning the nature of Ideal entities and the status the intentional `relation' and its `objects'. Husserl's literary encounters with many of the leading thinkers of his day illuminates both the (...) context and the content of his thought. Many of the groundbreaking analyses provided in these texts were never again to be given the thorough expositions found in these early writings. Early Writings in the Philosophy of Logic and Mathematics is essential reading for students of Husserl and all those who enquire into the nature of mathematical and logical knowledge. (shrink)

The article examines the development of Husserl’s early philosophy from his Habilitationsschrift (1887) to the Philosophie der Arithmetik (1891). -/- An attempt will be made at reconstructing the lost Habilitationsschrift (of which only the first chapter survives, which we know as Über den Begriff der Zahl). The examined sources show that the original version of the Habilitationsschrift was by far broader than the printed version, and included most topics of the PA. -/- The article contains an extensive and detailed comparison (...) of these texts to illustrate the changes in Husserl’s position before and after February 1890. This date is taken as a turning point in his development, because of Husserl’s announcement in a letter to Carl Stumpf that he was mistaken in his basic assumption, i.e. that the (psychological) analysis of the concept of Anzahl would yield a foundation for arithmetic. Some interesting conclusions in this respect can also be drawn from an unpublished lecture that Husserl held in the WS 1889/90, in which he anticipates aspects of his position hitherto deemed to belong to the last phases of the PA. (shrink)

The article examines the development of Husserl’s early philosophy from his Habilitationsschrift (1887) to the Philosophie der Arithmetik (1891). -/- An attempt will be made at reconstructing the lost Habilitationsschrift (of which only the first chapter survives, which we know as Über den Begriff der Zahl). The examined sources show that the original version of the Habilitationsschrift was by far broader than the printed version, and included most topics of the PA. -/- The article contains an extensive and detailed comparison (...) of these texts to illustrate the changes in Husserl’s position before and after February 1890. This date is taken as a turning point in his development, because of Husserl’s announcement in a letter to Carl Stumpf that he was mistaken in his basic assumption, i.e. that the (psychological) analysis of the concept of Anzahl would yield a foundation for arithmetic. Some interesting conclusions in this respect can also be drawn from an unpublished lecture that Husserl held in the WS 1889/90, in which he anticipates aspects of his position hitherto deemed to belong to the last phases of the PA. (shrink)

In the article Husserl’s view of the infinite around 1890 is analysed. I give a survey of his mathematical background and other important influences (especially Bolzano). The article contains a short exposition on Husserl's distinction between proper and symbolic presentations in the "Philosophie der Arithmetik" and between finite and infinite symbolic collections. Subsequently Husserl’s conception of surrogate presentations in his treatise "Zur Logik der Zeichen (Semiotik)" is discussed. In this text Husserl gives a detailed account of infinity, using surrogate presentations. (...) The conclusion is that with surrogates we can only operate according to blind psychological rules. (shrink)

CHAPTER I THE EMERGENCE AND DEVELOPMENT OF HUSSERL'S 'PHILOSOPHY OF ARITHMETIC'. HISTORICAL BACKGROUND: WEIERSTRASS AND THE ARITHMETIZATION OF ANALYSIS In ...