This volume provides an articulate restatement of many of the themes of Husserlian phenomenology.

In the following, I argue that L. E. J. Brouwer's notion of the construction of purely mathematical objects and Edmund Husserl's notion of their constitution coincide.

The notion of cognitive act is of importance for an epistemology that is apt for constructive type theory, and for epistemology in general. Instead of taking knowledge attributions as the primary use of the verb 'to know' that needs to be given an account of, and understanding a first-person knowledge claim as a special case of knowledge attribution, the account of knowledge that is given here understands first-person knowledge claims as the primary use of the verb 'to know'. This means (...) that a cognitive act is an act that counts as cognitive from a first-person point of view. The method of linguistic phenomenology is used to explain or elucidate our epistemic notions. One of the advantages of the theory is that an answer can be given to some of the problems in modern epistemology, such as the Gettier problem. (shrink)

Edmund Husserl is the founder of phenomenology. The Logical Investigations is Edmund Husserl's most famous work and has had a decisive impact on the direction of twentieth century philosophy. This is the first time both volumes of this classic work, translated by J.N. Findlay, have been available in paperback. They include a new introduction by Dermot Moran, placing the Logical Investigations in historical context and bringing out its importance for contemporary philosophy.

Science in a new sense arises in the first instance from Plato's establishing of logic, as a place for exploring the essential requirements of "genuine" ...

Luitzen Egburtus Jan Brouwer founded a school of thought whose aim was to include mathematics within the framework of intuitionistic philosophy; mathematics was to be regarded as an essentially free development of the human mind. What emerged diverged considerably at some points from tradition, but intuitionism has survived well the struggle between contending schools in the foundations of mathematics and exact philosophy. Originally published in 1981, this monograph contains a series of lectures dealing with most of the fundamental topics such (...) as choice sequences, the continuum, the fan theorem, order and well-order. Brouwer's own powerful style is evident throughout the work. (shrink)

The aim of this paper is to establish a phenomenological mathematical intuitionism that is based on fundamental phenomenological-epistemological principles. According to this intuitionism, mathematical intuitions are sui generis mental states, namely experiences that exhibit a distinctive phenomenal character. The focus is on two questions: what does it mean to undergo a mathematical intuition and what role do mathematical intuitions play in mathematical reasoning? While I crucially draw on Husserlian principles and adopt ideas we find in phenomenologically minded mathematicians such as (...) Hermann Weyl and Kurt Gödel, the overall objective is systematic in nature: to offer a plausible approach towards mathematics. (shrink)

I argue that Brouwer’s notion of the construction of purely mathematical objects and Husserl’s notion of their constitution by the transcendental subject coincide. Various objections to Brouwer’s intuitionism that have been raised in recent phenomenological literature are addressed. Then I present objections to Gödel’s project of founding classical mathematics on transcendental phenomenology. The problem for that project lies not so much in Husserl’s insistence on the spontaneous character of the constitution of mathematical objects, or in his refusal to allow an (...) appeal to higher minds, as in the combination of these two attitudes. (shrink)

Provability, Computability and Reflection.

Brouwer's demonstration of his Bar Theorem gives rise to provocative questions regarding the proper explanation of the logical connectives within intuitionistic and constructivist frameworks, respectively, and, more generally, regarding the role of logic within intuitionism. It is the purpose of the present note to discuss a number of these issues, both from an historical, as well as a systematic point of view.

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)

Can the straight line be analysed mathematically such that it does not fall apart into a set of discrete points, as is usually done but through which its fundamental continuity is lost? And are there objects of pure mathematics that can change through time? The mathematician and philosopher L.E.J. Brouwer argued that the two questions are closely related and that the answer to both is "yes''. To this end he introduced a new kind of object into mathematics, the choice sequence. (...) But other mathematicians and philosophers have been voicing objections to choice sequences from the start. This book aims to provide a sound philosophical basis for Brouwer's choice sequences by subjecting them to a phenomenological critique in the style of the later Husserl. "It is almost as if one could hear the two rebels arguing their case in a European café or on a terrace, and coming to a common understanding, with both men taking their hat off to the other, in admiration and gratitude. Dr. van Atten has convincingly applied Husserl's method to Brouwer's program, and has equally convincingly applied Brouwer's intuition to Husserl's program. Both programs have come out the better." Piet Hut, professor of Interdisciplinary Studies, Institute for Advanced Study, Princeton, U.S.A. (shrink)

First published in 2002. Routledge is an imprint of Taylor & Francis, an informa company.

This volume is a window on a period of rich and illuminating philosophical activity that has been rendered generally inaccessible by the supposed "revolution" attributed to "Analytic Philosophy" so-called. Careful exposition and critique is given to every serious alternative account of number and number relations available at the time.

"Intuition" has perhaps been the least understood and the most abused term in philosophy. It is often the term used when one has no plausible explanation for the source of a given belief or opinion. According to some sceptics, it is understood only in terms of what it is not, and it is not any of the better understood means for acquiring knowledge. In mathematics the term has also unfortunately been used in this way. Thus, intuition is sometimes portrayed as (...) if it were the Third Eye, something only mathematical "mystics", like Ramanujan, possess. In mathematics the notion has also been used in a host of other senses: by "intuitive" one might mean informal, or non-rigourous, or visual, or holistic, or incomplete, or perhaps even convincing in spite of lack of proof. My aim in this book is to sweep all of this aside, to argue that there is a perfectly coherent, philosophically respectable notion of mathematical intuition according to which intuition is a condition necessary for mathemati cal knowledge. I shall argue that mathematical intuition is not any special or mysterious kind of faculty, and that it is possible to make progress in the philosophical analysis of this notion. This kind of undertaking has a precedent in the philosophy of Kant. While I shall be mostly developing ideas about intuition due to Edmund Husser! there will be a kind of Kantian argument underlying the entire book. (shrink)

the Logische Untersuchungen,l phenomenology has been conceived as a substratum of empirical psychology, as a sphere comprising "imma nental" descriptions of psychical mental processes, a sphere compris ing descriptions that - so the immanence in question is understood - are strictly confined within the bounds of internal experience. It 2 would seem that my protest against this conception has been oflittle avail; and the added explanations, which sharply pinpointed at least some chief points of difference, either have not been understood (...) or have been heedlessly pushed aside. Thus the replies directed against my criticism of psychological method are also quite negative because they miss the straightforward sense of my presentation. My criticism of psychological method did not at all deny the value of modern psychology, did not at all disparage the experimental work done by eminent men. Rather it laid bare certain, in the literal sense, radical defects of method upon the removal of which, in my opinion, must depend an elevation of psychology to a higher scientific level and an extraordinary amplification ofits field of work. Later an occasion will be found to say a few words about the unnecessary defences of psychology against my supposed "attacks. (shrink)

ON BROUWER, like other titles in the Wadsworth Philosopher's Series, offers a concise, yet comprehensive, introduction to this philosopher's most important ideas. Presenting the most important insights of well over a hundred seminal philosophers in both the Eastern and Western traditions, the Wadsworth Philosophers Series contains volumes written by scholars noted for their excellence in teaching and for their well-versed comprehension of each featured philosopher's major works and contributions. These titles have proven valuable in a number of ways. Serving as (...) standalone texts when tackling a philosophers' original sources or as helpful resources for focusing philosophy students' engagements with these philosopher's often conceptually daunting works, these titles have also gained extraordinary popularity with a lay readership and quite often serve as "refreshers" for philosophy instructors. (shrink)

The thesis is a study of the notion of intuition in the foundations of mathematics which focuses on the case of natural numbers and hereditarily finite sets. Phenomenological considerations are brought to bear on some of the main objections that have been raised to this notion. ;Suppose that a person P knows that S only if S is true, P believes that S, and P's belief that S is produced by a process that gives evidence for it. On a phenomenological (...) view the relevant evidence is provided by intuition , and this should be the case in either ordinary perceptual knowledge or in mathematical knowledge. Intuition is to be understood in terms of fulfillments of intentions. Knowledge is a product of intuition and intention. In the case of mathematical knowledge it is said that there is a construction for a mathematical statement S if and only if the intention expressed by S is fulfilled . Constructions are thus viewed as intuition processes that could actually or possibly be carried out. In elementary parts of mathematics they might be characterized in terms of certain classes of recursive functions. This view is discussed in the case where S is taken to be a singular statement about natural numbers or finite sets, and also where S is taken to be a general statement about such objects. The distinction between intuition of and intuition that is also investigated in this context. ;It is pointed out how on a phenomenological view a number of central problems about mathematical intuition can be avoided: problems about the analogousness of perceptual and mathematical intuition, about causal accounts of knowledge in mathematics, and about structuralism in mathematics. The bearing of the account on issues concerning constructivism and platonism is also discussed. (shrink)

La littérature témoigne d’une tendance croissante à soutenir l’intuitionisme par la phénoménologie. Le disciple de Brouwer Arend Heyting est considéré comme un précurseur de cette tendance, parce qu’il usait d’une terminologie phénoménologique en vue de définir la négation intuitioniste, en élaborant la première logique intuitioniste. Dans cet article, l’auteur tente d’explorer — en référence aux matériaux inédits conservés aux Archives Heyting — ce qui, dans la pensée de Heyting, est compatible avec la phénoménologie. Dans la conclusion, l’auteur suggère que Heyting (...) et Husserl, pour autant qu’ils pensent tous deux que la conscience doit être le véritable commencement de la connaissance, partagent une même attitude antipsychologiste coexistant avec une tentative de dépassement du solipsisme. Toutefois, le concept phénoménologique de degré d’évidence ne peut être appliqué à l’échelle de l’évidence de Heyting (incluant les petits nombres naturels, les grands nombres naturels, les suites infinies, le quantificateur universel), d’un côté parce qu’on ne voit pas clairement si la seconde est partagée par tous les intuitionistes, et de l’autre parce que le premier présuppose une évidence révisable qui ne cadre pas avec le point de vue de Heyting. En outre, les conceptions de la nature de la mathématique et de la logique et de leurs relations mutuelles défendues par Husserl et Heyting sont essentiellement différentes. D’un point de vue intuitioniste, la mathématique est le domaine de l’évidence, alors que la logique transcrit ses régularités. D’un point de vue phénoménologique, la mathématique demeure en dehors du domaine de l’évidence. La logique apophantique coïncide avec la mathématique (sans que l’une absorbe l’autre), mais la logique transcendantale se situe à un niveau plus élevé. (shrink)