It has been observed that whereas painters and musicians are likely to be embarrassed by references to the beauty in their work, mathematicians instead like to engage in discussions of the beauty of mathematics. Professional artists are more likely to stress the technical rather than the aesthetic aspects of their work. Mathematicians, instead, are fond of passing judgment on the beauty of their favored pieces of mathematics. Even a cursory observation shows that the characteristics of mathematical beauty are (...) at variance with those of artistic beauty. For example, courses in art appreciation are fairly common; it is however unthinkable to find any mathematical beauty appreciation courses taught anywhere. The purpose of the present paper is to try to uncover the sense of the term beauty as it is currently used by mathematicians. (shrink)

The paper discusses Husserl's phenomenology of mathematics in his Formal and Transcendental Logic (1929). In it Husserl seeks to provide descriptive foundations for mathematics. As sciences and mathematics are normative activities Husserl's attempt is also to describe the norms at work in these disciplines. The description shows that mathematics can be given in several different ways. The phenomenologist's task is to examine whether a given part of mathematics is genuine according to the norms that (...) pertain to the approach in question. The paper will then examine the intuitionistic, formalistic, and structural features of Husserl's philosophy of mathematics. (shrink)

This volume aims to establish the starting point for the development, evaluation and appraisal of the phenomenology of mathematics.

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)

This article attempts to link the notion of absolute ego as the ultimate subjectivity of consciousness in continental tradition with a phenomenology of Mathematical Continuum as this term is generally established following Cantor’s pioneering ideas on the properties and cardinalities of sets. My motivation stems mainly from the inherent ambiguities underlying the nature and properties of this fundamental mathematical notion which, in my view, cannot be resolved in principle by the analytical means of any formal language not even by (...) the addition of any new axioms to a consistent first-order axiomatical system such as the Zermelo-Fraenkel Set Theory. In this phenomenologically motivated approach I deal, to some extent, with the undecidability of a fundamental statement about the cardinality of Continuum inside ZFC theory, namely the Continuum Hypothesis, and also with the underlying root of Gödel’s first incompleteness result.Este artigo procura conectar a noção de ego absoluto enquanto subjetividade última da consciência na tradição continental com a fenomenologia da matemática do contínuo tal como este termo foi estabelecido seguindo as idéias pioneiras de Cantor sobre as propriedades e cardinalidade de conjuntos. Minha motivação deriva basicamente das ambigüidades subjacentes à natureza e propriedades desta noção matemática fundamental as quais, no meu entender, não podem, em princípio, ser resolvidas pelos meios analíticos de qualquer linguagem formal e nem mesmo pela adição de novos axiomas a um sistema axiomático consistente de primeira ordem como a teoria de conjuntos de Zermelo Fraenkel . Neste tratamento fenomenológica-mente motivado eu lido em certa medida com a indecidibilidade de uma tese fundamental sobre a cardinalidade do contínuo em ZFC, a saber, a Hipótese do Contínuo, e também com aquilo que está na base do primeiro resultado de incompletude de Gödel. (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)

Kurt Gdel was the most outstanding logician of the 20th century and a giant in the field. This book is part of a five volume set that makes available all of Gdel's writings. The first three volumes, already published, consist of the papers and essays of Gdel. The final two volumes of the set deal with Gdel's correspondence with his contemporary mathematicians, this fourth volume consists of material from correspondents from A-G.

This book offers a phenomenological conception of experiential justification that seeks to clarify why certain experiences are a source of immediate justification and what role experiences play in gaining (scientific) knowledge. Based on the author's account of experiential justification, this book exemplifies how a phenomenological experience-first epistemology can epistemically ground the individual sciences. More precisely, it delivers a comprehensive picture of how we get from epistemology to the foundations of mathematics and physics. The book is unique as it utilizes (...) methods and insights from the phenomenological tradition in order to make progress in current analytic epistemology. It serves as a starting point for re-evaluating the relevance of Husserlian phenomenology to current analytic epistemology and making an important step towards paving the way for future mutually beneficial discussions. This is achieved by exemplifying how current debates can benefit from ideas, insights, and methods we find in the phenomenological tradition. (shrink)

According to Becker, the dispute between the intuitionistic (construction as the guarantor of mathematical existence) and the formalistic (non-contradiction as the guarantor of mathematical existence) definition should be resolved in a phenomenological perspective on the problem. The question of the legitimacy of the transfinite should also be resolved in the perspective of a phenomenological constitutive analysis. This analysis provides the key to the problematic of mathematical existence: the result of Becker’s investigations on the logic and ontology of the mathematical is (...) a solution in favour of the “genuine phenomena” of intuitionistic mathematics and the infinite. “Genuine phenomena” are those which turn out to be accessible for constitutive analysis precisely as phenomena of “pure consciousness” and even of the “concrete historical Dasein.” The final solution in favour of the “genuine phenomena” of intuitionistic mathematics and the infinite is motivated by the fact that Hilbert’s formalism does not satisfy the “chief phenomenological principle of demonstrability” (Ausweisbarkeit). (shrink)

_Dominique Pradelle. ** Intuition et idéalités. Phénoménologie des objets mathématiques _ [Intuition and idealities: Phenomenology of mathematical objects.] Collection Épiméthée. Paris: PUF [Presses universitaires de France], 2020. Pp. 550. ISBN: 978-2-13-082237-0.

The paper shows how to use the Husserlian phenomenological method in contemporary philosophical approaches to mathematical practice and mathematical ontology. First, the paper develops the phenomenological approach based on Husserl's writings to obtain a method for understanding mathematical practice. Then, to put forward a full-fledged ontology of mathematics, the phenomenological approach is complemented with social ontological considerations. The proposed ontological account sees mathematical objects as social constructions in the sense that they are products of culturally shared and historically developed (...) practices. At the same time the view endorses the sense that mathematical reality is given to mathematicians with a sense of independence. As mathematical social constructions are products of highly constrained, intersubjective practices and accord with the phenomenologically clarified experience of mathematicians, positing them is phenomenologically justified. The social ontological approach offers a way to build mathematical ontology out of the practice with no metaphysical magic. (shrink)

A phenomenological approach to mathematical practice is sketched out, and some problems with this sort of approach are considered. The approach outlined takes mathematical practices as its data, and seeks to provide an empirically adequate philosophy of mathematics based on observation of these practices. Some observations are presented, based on two case studies of some research into the classification of C*-algebras. It is suggested that an anti-realist account of mathematics could be developed on the basis of these and (...) other studies, locating the substance of mathematics in the various informal argument methods used by mathematicians. (shrink)

This paper deals with Husserl’s idea of pure logic as it is coined in the Logical Investigations. First, it exposes the formation of pure logic around a conception of completeness ; then, it presents intentionality as the keystone of such a structuring ; and finally, it provides a systematic reconstruction of pure logic from the semiotic standpoint of intentionality. In this way, it establishes Husserlian pure logic as a phenomenological epistemology of mathematical logic.

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)

Brouwer and Weyl recognized that the intuitive continuum requires a mathematical analysis of a kind that set theory is not able to provide. As an alternative, Brouwer introduced choice sequences. We first describe the features of the intuitive continuum that prompted this development, focusing in particular on the flow of internal time as described in Husserl's phenomenology. Then we look at choice sequences and their logic. Finally, we investigate the differences between Brouwer and Weyl, and argue that Weyl's conception (...) of choice sequences is defective on several counts. (shrink)

Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable (...) epistemic problems. As a way out of this dilemma, Shapiro articulates a structuralist approach. On this view, the subject matter of arithmetic, for example, is not a fixed domain of numbers independent of each other, but rather is the natural number structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle. Using this framework, realism in mathematics can be preserved without troublesome epistemic consequences. Shapiro concludes by showing how a structuralist approach can be applied to wider philosophical questions such as the nature of an "object" and the Quinean nature of ontological commitment. Clear, compelling, and tautly argued, Shapiro's work, noteworthy both in its attempt to develop a full-length structuralist approach to mathematics and to trace its emergence in the history of mathematics, will be of deep interest to both philosophers and mathematicians. (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)

"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)

Jean Dieudonné, the spokesman of the group of French mathematicians named Bourbaki, called mathematics the music of reason. This metaphor invites a phenomenological account of the affective, in contrast to the epistemic and discursive, nature of mathematics: What constitutes its charm? Mathematical reasoning is described as a perceptual experience, which in Husserl’s late philosophy would be a case of passive synthesis. Like a melody, a mathematical proof is manifest in an affective identity of a temporal object. Rather than (...) an exercise for its own sake, this account sheds a different light on both the epistemic limitation of mathematical science, and the discursive problem of social responsibility in mathematics – two issues at the heart of Husserl’s critique of science as well as of mid-20th century mathematics, for which Nicolas Bourbaki stands as a monument of rigor. (shrink)

Brouwer and Weyl recognized that the intuitive continuum requires a mathematical analysis of a kind that set theory is not able to provide. As an alternative, Brouwer introduced choice sequences. We first describe the features of the intuitive continuum that prompted this development, focusing in particular on the flow of internal time as described in Husserl's phenomenology. Then we look at choice sequences and their logic. Finally, we investigate the differences between Brouwer and Weyl, and argue that Weyl's conception (...) of choice sequences is defective on several counts. (shrink)

Brouwer and Weyl recognized that the intuitive continuum requires a mathematical analysis of a kind that set theory is not able to provide. As an alternative, Brouwer introduced choice sequences. We first describe the features of the intuitive continuum that prompted this development, focusing in particular on the flow of internal time as described in Husserl's phenomenology. Then we look at choice sequences and their logic. Finally, we investigate the differences between Brouwer and Weyl, and argue that Weyl's conception (...) of choice sequences is defective on several counts. (shrink)

The paper is devoted to phenomenological ideas in conceptions of modern philosophy of mathematics. Views of Husserl, Weyl, Becker andGödel will be discussed and analysed. The aim of the paper is to show the influence of phenomenological ideas on the philosophical conceptions concerning mathematics. We shall start by indicating the attachment of Edmund Husserl to mathematics and by presenting the main points of his philosophy of mathematics. Next, works of two philosophers who attempted to apply Husserl’s (...) phenomenological ideas to the philosophy of mathematics, namely Hermann Weyl and Oskar Becker, will be briefly discussed. Lastly, the connections between Husserl’s ideas and the philosophy of mathematics of Kurt Gödel will be studied. (shrink)

This article highlights the mathematical structure of Henri Bergson’s method. While Bergson has been historically interpreted as an anti-scientific and irrationalist philosopher, he modeled his philosophical methodology on the infinitesimal calculus developed by Leibniz and Newton in the seventeenth century. His philosophy, then, rests on the science of number, at least from a methodological standpoint. By looking at how he conscripted key mathematical concepts into his philosophy, this article invites us to re-imagine Bergson’s place in the history of Western philosophy.

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)

In a recent article I detailed at length the methodology employed to explore the reflective and pre-reflective contents of singular intuitive experiences in contemporary mathematics in order to propose an essential structure of intuition arousal in mathematics. In this paper I present the phenomenological assessment of the essential structure according to the three formal structures as proposed by Sokolowski's scheme and show their relevance in the description of the intuitive experience in mathematics. I also show that this (...) essential structure acknowledges the perceptualist view of intuition, as proposed by Chudnoff, and discuss the analogy that is often proposed and argued between mathematical intuition and ordinary perceptive intuition. (shrink)

This article derives from a project attempting to show that Western formal logic, from Aristotle onward, has both been partially constituted by, and partially constitutive of, what has become known as racism. In the present article, I will first discuss, in light of Frege’s honorary role as founder of the philosophy of mathematics, Reuben Hersh’s What is Mathematics, Really? Second, I will explore how the infamous section of Frege’s 1924 diary (specifically the entries from March 10 to April (...) 9) supports Hersh's claim regarding the link between political conservatism and the (historically and currently) dominant school of the philosophy of mathematics (to which Frege undeniably belongs). Third, I will examine Frege’s attempt at a more reader-friendly introduction to his philosophy of mathematics, The Foundations of Arithmetic. And finally, I will briefly analyze Frege’s Begriffsschrift to see how questions of race arise even at the heights of his logical abstraction. (shrink)

In this paper I argue for the view that structuralism offers the best perspective for an acceptable account of the applicability of mathematics in the empirical sciences. Structuralism, as I understand it, is the view that mathematics is not the science of a particular type of objects, but of structural properties of arbitrary domains of entities, regardless of whether they are actually existing, merely presupposed or only intentionally intended.

Mathematical objects are divided into (1) those which are autonomous, i.e., not dependent for their existence upon mathematicians’ conscious acts, and (2) intentional objects, which are so dependent. Platonist philosophy of mathematics argues that all objects belong to group (1), Brouwer’s intuitionism argues that all belong to group (2). Here we attempt to develop a dualist ontology of mathematics (implicit in the work of, e.g., Hilbert), exploiting the theories of Meinong, Husserl and Ingarden on the relations between autonomous (...) and intentional objects. In particular we develop a phenomenology of mathematical works, which has the stratified intentional structure discovered by Ingarden in his study of the literary work. (shrink)

When reassessing the role of Debreu’s axiomatic method ineconomics, one has to explain both its success and unpopularity; onehas to explain the “bright shadow” Debreu cast on the discipline:sheltering, threatening, and difficult to pin down. Debreu himself didnot expect to have such an influence. Before he received the Bank ofSweden Prize in 1983 he had never openly engaged with themethodology or politics of mathematical economics. When in severalspeeches he later rigorously distinguished mathematical form fromeconomic content and claimed this as the (...) virtue of mathematicaleconomics, he did both: he defended mathematical reasoning againstthe theoretical innovations since the 1970s and expressed remorse forhaving promised too much because it cannot support claims abouteconomic content. The analysis of this twofold role of Debreu’saxiomatic method raises issues of the social and political responsibilityof economists over and above standard epistemic issues. (shrink)

History and Philosophy of Logic, Volume 32, Issue 4, Page 399-400, November 2011.

Both phenomenologists and analytical philosophers of mathematics should profit from this excellent exposition and defense of Husserl's account of mathematical knowledge. Tieszen places Philosophie der Arithmetik in the context of Husserl's later phenomenological thinking and demonstrates thereby that Husserl's contributions to the philosophy of mathematics are much more important than most of us, influenced by Frege's denigrating critique of Husserl's early psychologism, had thought. He also makes a convincing case for the phenomenological approach to constructive mathematics.

A critical appraisal of husserl's lectures on internal time-Consciousness and passive synthesis (touching the theme of memory) is followed by an appreciation of merleau-Ponty's "problem of passivity". I argue that husserl's descriptions of memory processes embody prejudices stemming from the 'objective time' he claims to have bracketed out and that his phenomenological method is itself a phenomenon of the mathematical imagination. The latter pursues inherited ideals of clarity, Evidence, Immanence and presence which distort all mnemonic phenomena. Merleau-Ponty eschews the representational (...) thought, Objective-Linear time, Evidence and immanence of husserlian epistemology. Merleau-Ponty's philosophy of ambiguity responds with greater suppleness and subtlety to the most ambiguous of phenomena--Memory. (shrink)

This article deals with a question of a most general, comprehensive and profound content as it is the nature of mathematical–logical objects insofar as these are considered objects of knowledge and more specifically objects of formal mathematical theories. As objects of formal theories they are dealt with in the sense they have acquired primarily from the beginnings of the systematic study of mathematical foundations in connection with logic dating from the works of G. Cantor and G. Frege in the last (...) decades of the nineteenth century. Largely motivated by a phenomenologically founded view of mathematical objects/states-of-affairs, I try to consistently argue for their character as objects shaped to a certain extent by intentional forms exhibited by consciousness and the modes of constitution of inner temporality and at the same time as constrained in the form of immanent ‘appearances’ by what stands as their non-eliminable reference, that is, the world as primitive soil of experience and the mathematical intuitions developed in relations of reciprocity within-the-world. In this perspective and relative to my intentions I enter, in the last section, into a brief review of certain positions of G. Sher’s foundational holism and R. Tieszen’s constituted platonism, among others, respectively presented in Sher and Tieszen. (shrink)

The collection of essays in this special issue point toward the rich and diverse themes under which the phenomenologist might analyze quantum mechanics. The authors in the collection demonstrate that the tradition inaugurated by Husserl promises to dispel the many experiential quandaries of quantum mechanics. They interrogate the meaning of the theoretical entities described by the mathematical equations and analyze their manner of appearing to the physicist. To this end, the efforts of the authors show that increased clarity at forefront (...) of physics requires more than progressively refined models and instruments, but understanding how the subject grasps nature. (shrink)

The paper gives an impression of the multi-dimensionality of mathematics as a human activity. This 'phenomenological' exercise is performed within an analytic framework that is both an expansion and a refinement of the one proposed by Kitcher. Such a particular tool enables one to retain an integrated picture while nevertheless welcoming an ample diversity of perspectives on mathematical practices, that is, from different disciplines, with different scopes, and at different levels. Its functioning is clarified by fitting in illustrations based (...) on actual research. (shrink)

Richard Tieszen's new book1 is a collection of fifteen articles and reviews, spanning fifteen years, presenting the author's approach to philosophical questions about logic and mathematics from the point of view of phenomenology, as developed by Edmund Husserl in the later phase2 of his philosophical thinking known as transcendental phenomenology, starting in 1907 with the Logical Investigations and characterized by the introduction of the notions of ‘reduction’. Husserlian transcendental phenomenology as philosophy of mathematics is described (...) as one that ‘cuts across’ different philosophical positions, such as platonism, nominalism, fictionalism, Hilbertian formalism, etc. but, at the same time, as having built in the conceptual tools which allow one not to incur the kinds of problems which are usually related to one's preferred approach. Phenomenology centers around the notion of intentionality3 or aboutness, i.e. the characteristic of acts of cognition of being about something. The ‘something’ a cognitive act is about is called its ‘intentional object’, meaning the object of an intentional act. Any kind of object can be seen as an intentional object irrespective of whether it is concrete, illusory, abstract, etc. Indeed the emphasis is on the intentional act, as it is in the intentional act that the object—which need not be claimed to exist—is present. In order to achieve knowledge of an object the phenomenologist investigates the consciousness of the knowing subject when performing the act directed to that particular object.Characterized in this way, it is not difficult to see that phenomenology has a straightforward bearing on almost everything, philosophy of mathematics included. As Tieszen puts it , ‘[…] our mathematical beliefs are always about something. They are about certain objects, such as numbers, sets, functions or groups […]’. …. (shrink)