Phenomenology of Mathematics

We propose a way to explain the diversification of branches of mathematics, distinguishing the different approaches by which mathematical objects can be studied. In our philosophy of mathematics, there is a base object, which is the abstract multiplicity that comes from our empirical experience. However, due to our human condition, the analysis of such multiplicity is covered by other empirical cognitive attitudes (approaches), diversifying the ways in which it can be conceived, and consequently giving rise to different mathematical disciplines. This (...) diversity of approaches is founded on the manifold categories that we find in physical reality. We also propose, grounded on this idea, the use of Aristotelian categories as a first model for this division, generating from it a classification of mathematical branches. Finally we make a history review to show that there is consistency between our classification, and the historical appearance of the different branches of mathematics. (shrink)

From the general history of culture, with a particular attention turned towards the personal and intellectual relationships between Hermann Weyl and Edmund Husserl, it will be possible to identify certain historical-critical moments from which a philosophical reflection concerning aspects of the ontology of mathematics may be carried out. In particular, a notable epistemological relevance of group theory methods will stand out.

In this first paper we outline what possible historic-epistemological role might have played the work of Ulisse Dini on implicit function theory in formulating the structure of differentiable manifold, via the basic work of Hassler Whitney. A detailed historiographical recognition about this Dini's work has been done. Further methodological considerations are then made as regards history of mathematics.

Triadic (systemical) logic can provide an interpretive paradigm for understanding how quantum indeterminacy is a consequence of the formal nature of light in relativity theory. This interpretive paradigm is coherent and constitutionally open to ethical and theological interests. -/- In this statement: -/- (1) Triadic logic refers to a formal pattern that describes systemic (collaborative) processes involving signs that mediate between interiority (individuation) and exteriority (generalized worldview or Umwelt). It is also called systemical logic or the logic of relatives. The (...) term "triadic logic" emphasizes that this logic involves mediation of dualities through an irreducibly triadic formalism. The term "systemical logic" emphasizes that this logic applies to systems in contrast to traditional binary logic which applies to classes. The term "logic of relatives" emphasizes that this logic is background independent (in the sense discussed by Smolin ). -/- (2) An interpretive paradigm refers to a way of thinking that generates an understanding through concepts, their inter-relationships and their connections with experience. -/- (3) Coherence refers to holistic integrity or continuity in the meaning of concepts that form an interpretation or understanding. -/- (4) Constitutionally open refers to an inherent dependence in principle of an interpretation or understanding on something outside of a specific discipline's discourse or domain of inquiry (epistemic system). Interpretations that are constitutionally open are incomplete in themselves and open to responsive, interdisciplinary discourse and collaborative learning. (shrink)

Philosophers such as Nietzsche, Heidegger, Deleuze and Gendlin pronounce that difference must be understood as ontologically prior to identity. They teach that identity is a surface effect of difference, that to understand the basis of logico-mathematical idealities we must uncover their genesis in the fecundity of differentiation. In this paper, I contrast Heidegger’s analyses of the present- to-hand logico-mathematical object, which he discuses over the course of his career in terms of the ‘as’ structure, temporalization and enframing , with the (...) approaches of Gendlin and Deleuze, supplementing this discussion with Husserl’s investigations of mathematical idealities. Gendlin distinguishes between the representational power a logical pattern has in itself, apart from its virtual generative source, to exactly repeat itself, and the way this self-same pattern is generated and changed by the larger situational texture within which it is embedded. In so doing, he misconstrues the empty, meaningless temporalization of logical calculation as the explicitly preserving carrying-through of already instituted implicit sense. For Heidegger, by contrast, logical inference is less a supplement to or development of implicit experience than a narrowing of its scope , a deficient mode of handiness. Experiencing something as present to hand extension modifies the relevant usefulness of ‘as’ structured comportment by stripping away what is meaningful in our relation with beings, and in the process stripping away its intelligibility. Thus, contrary to the assertion Gendlin, extensive repetition does not carry through intelligible, relevant meaning, it dissolves understanding into the nihilism of empty calculation. While Nietzsche and Deleuze agree with Heidegger on the nihilism of countable units of extension, they retain the idea of instantaneous intrinsic presence. They put into question most aspects of traditional notions of subjectivity, objectivity and causation, except instantaneous presence. Deleuze's notion of intrinsicality as a differential “produced in a time smaller than the minimum continuous time thinkable'.” remains a notion of temporality as ‘in-timeness', as the occupying of a moment of time by a present-at-hand being. (shrink)

Negation is widely thought to be uniquely captured by the usual extensional Boolean connective in the setting of classical logic. However, there has been recent interest in a modal approach to negation. This essay examines the problem of modal negation with an Husserlian phenomenological lens. I argue that the Husserlian approach to negation contains an ambiguity which points to a pluralism about negation. On this view, negation begins its life as a modal notion with nonclassical properties, and the question of (...) classical negation is a question of its demodalization. I reconstruct a phenomenological legitimation of the demodalization, but I remain skeptical about its wider prospects. Nevertheless, the phenomenological-modal approach to negation answers the skepticism about the very possibility of debates about negation and gives valuable insight into the fundamental nature of the problem. The argument should be of interest both to those who are specifically interested in Husserl's logic and the relationship of phenomenology to intuitionistic (and other nonclassical) logics, as well as to philosophers of logic more generally interested in obtaining different angles on the problem of negation from a systematic perspective. (shrink)

In 'Intuition, iteration, induction', Mark van Atten argues that iteration is an object of intuition for Brouwer and explains the intuitive character of the act of iteration drawing from Husserl’s phenomenology. I find the arguments for this reading of Brouwer unconvincing. In this note I set out some issues with his claim that iteration is an object of intuition and his Husserlian explication of iteration. In particular, I argue that van Atten does not accomplish his goals due to tensions with (...) Brouwer’s comments on second-order mathematics and because Husserl does not understand the experience of succession as Brouwer does. (shrink)

The purpose of this study is to assess Husserl’s debt to Lotze’s philosophy during the Halle period (1886-1901). I shall first track the sources of Husserl’s knowledge of Lotze’s philosophy during his studies with Brentano in Vienna and then with Stumpf in Halle. I shall then briefly comment on Husserl’s references to Lotze in his early work and research manuscripts for the second volume of his Philosophy of Arithmetic. In the third section, I examine Lotze’s influence on Husserl’s antipsychologistic turn (...) in the mid-1890s. The fourth section is a commentary on Husserl’s manuscript entitled “Microcosmus,” to which he explicitly refers in his Prolegomena, and which he planned to publish as an annex of his Logical Investigations. This work contains a detailed analysis of the third book of Lotze’s 1874 Logic. The last section examines Husserl’s arguments against logical psychologism in his Prolegomena, which I discuss through the lens of Stumpf’s critique of psychologism in his paper “Psychology and theory of knowledge”. I argue that Stumpf’s early works on this topic make it possible to establish a connection between Lotze’s interpretation of Plato’s theory of Ideas and Husserl’s antipsychologism. My hypothesis is that Stumpf’s analyses represent the background of Husserl's critique of logical psychologism in his Logical Investigations. I shall conclude this study by showing that Husserl’s position with respect to Lotze’s philosophy remains basically unchanged after the publication of his Logical Investigations, and that Husserl’s main criticism of Lotze pertains, in the final analysis, to the absence of a theory of intentionality in Lotze’s philosophy. (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 problems of clarifying the fundamental logical and mathe­matical concepts, and hence of accomplishing a truly radical grounding of logic and mathematics, were precisely what motivated the very beginnings of Husserl’s phenomenology. This paper is divided into two main parts. The first part focuses on the meaning and structure of Husserl’s explanation of the “logical and psychological” nature of fundamental arithmetical concepts. Particular emphasis is placed on the strategy of Philosophy of Arithmetics (1891) of analysing cardinal numbers in concepts (pivotal (...) yet just as much harmful to philosophy) such as Vorstellungen, mental phenomena and repre­sentations. Together with Arend Heyting and Hermann Weyl, it was the phenomenologist-mathematician Oskar Becker who was the main actor in the second round in the complex dialogue between meta-mathematics and phenomenological philosophy. The second part of this paper aims to clarify Becker’s attempt to apply the theory of actualization of intentionality to the problem of the mode of being of the mathematical and the criterion of mathematical existence. (shrink)

In Intuition et idéalités: Phénoménologie des objets mathématiques (2020), Dominique Pradelle questioned the nature of mathematical knowledge–the status of math.

This book shows, for the first time in its full spectrum, the interconnectedness and topicality of two historically and philosophically significant developments of philosophical theories of the study of mind: that of phenomenology of Edmund Husserl and phaneroscopy of Charles S. Peirce. The chapters in this book put the two thinkers in a novel discourse while engaging in mutual scholarship on the large overlaps between the historically two largely independently developed but converging ideas of mind, cognition, consciousness, being, and experience. (...) It is the second volume in a projected series of three, the first of which is Peirce and Husserl: Mutual Insights on Logic, Mathematics, and Cognition (2019). This book consists of three parts. Part I contains studies on the basic elements and the methodological themes of both “phenomenologies” vis-à-vis each other. Part II of the book is dedicated to metaphysical and existential themes. Finally, this book contains a hitherto unpublished selection of connected texts from Charles Peirce concerning phaneroscopy, the theory of definitions, and other related historical, philosophical, and religious themes from 1910, transcribed and introduced by one of the editors of the volume. This book is of interest to scholars in phenomenology, phaneroscopy, and the history of ideas. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Husserl and Mathematics by Mirja HartimoAndrea StaitiMirja Hartimo. Husserl and Mathematics. Cambridge: Cambridge University Press, 2021. Pp. 214. Hardback, $99.99.Mirja Hartimo has written the first book-length study of Husserl's evolving views on mathematics that takes his intellectual context into full consideration. Most importantly, Hartimo's historically informed approach to the topic benefits from her extensive knowledge of Husserl's library. Throughout the book, she provides references to texts and articles (...) that Husserl read, marked, and annotated, although regrettably, very few of these sources are quoted explicitly in his published works. The overall picture that emerges from Hartimo's book shows that Husserl continued to read and engage with mathematics throughout his career, and, even though extensive discussions of mathematical issues are not frequent in his later works, some of the ideas he develops, for instance, in The Crisis of the European Sciences and Transcendental Phenomenology, can be read against the backdrop of the developments of mathematics in his time.Chapter 1 sets the method for the rest of the book. Hartimo emphasizes the importance of a methodological device that takes center stage in Husserl's work from the 1920s onward: Besinnung. Hartimo claims that Besinnung is a method to reactivate the "goals and purposes" (21) that originally motivated scientists engaging in a particular line of research. According to Husserl, as specialized research progresses and becomes standard practice, such original goals and purposes might be blurred by other concerns. When applied to mathematics, Besinnung thus amounts to a reflection "on what mathematicians should do, on what the genuine goal of their activities should be" (21–22). Hartimo argues that this kind of approach to mathematics comes close to Penelope Maddy's naturalism, which takes its departure from what mathematicians are actually doing and then proceeds to inquire into the ontological status of mathematical objects and theories. Hartimo identifies particularly in David Hilbert, a colleague of Husserl's at Göttingen, a vivid embodiment of what Husserl considered to be exemplary work oriented toward the ideal goal of mathematics.In chapters 2 and 3, Hartimo expands upon Husserl's conception of what the true goal of mathematics should be. She does so, first, by addressing the classical topic of Husserl's psychologism in his first (and only) work entirely devoted to mathematics, The Philosophy of Arithmetic, and Frege's well-known critical review of it. In Hartimo's rendition, Husserl criticizes Frege for his attempt to provide formal definitions of basic notions, such as equality. For Husserl, such basic notions are intuitively available and require no definition—rather, they should be put to work to clarify vague notions. Frege, in return, attacks what he takes to be Husserl's wishy-washy psychological account of numbers, which revolves around the concrete act of collecting items and abstracting from their specific contents. Ultimately, Hartimo argues, Husserl takes Frege's criticism to heart and in his later Prolegomena no longer pursues a psychological foundation of number, but rather emphasizes the distinction between the subjective and the objective sides of mathematics, with philosophers studying the former and mathematicians focusing on the latter. While Hartimo's verdict on the philosophical yield of the debate is in line with the standard interpretation of Frege's review as prompting Husserl's antipsychologistic turn, she makes a remark that might be worthy of further discussion: "While for Husserl the debate was about logicism, Frege shifts the debate to be about psychologism and here commentators have followed Frege's example" (50). It would be interesting to imagine what a Fregean reply to Husserl's view about the uselessness of formal definitions for basic notions like equality might look like. In any case, Hartimo's reconstruction makes it clear that, however motivated, Husserl's rejection of psychologism does not amount to an acceptance of logicism.In chapter 3, Hartimo puts forward two of her central claims: (1) Husserl was a structuralist about mathematics; and (2) for Husserl, the goal of modern mathematics is encapsulated in the concept of definiteness. According to (1), mathematics is about the formal structures of systems, which can be isolated and compared. Definiteness, in turn, is the property of systems... (shrink)

Brouwer’s view on induction has relatively recently been characterised as one on which it is not only intuitive (as expected) but functional, by van Dalen. He claims that Brouwer’s ‘Ur-intuition’ also yields the recursor. Appealing to Husserl’s phenomenology, I offer an analysis of Brouwer’s view that supports this characterisation and claim, even if assigning the primary role to the iterator instead. Contrasts are drawn to accounts of induction by Poincaré, Heyting, and Kreisel. On the phenomenological side, the analysis provides an (...) alternative to that in Tieszen’s Mathematical Intuition, and confirms a view of Gödel on his Dialectica Interpretation. (shrink)

Die Phänomenologie stellt eine der Hauptströmungen der Gegenwartsphilosophie dar und findet in zahlreichen Wissenschaften sowie in Praxis und Therapeutik starke Resonanz. Nach 120 Jahren Wirkungsgeschichte füllt die Bibliothek phänomenologischer Werke zahllose Bücherregale und selbst für Expert:innen ist die Forschungsliteratur mittlerweile unüberschaubar geworden. An allgemeinen Einführungen sowie spezialisierter Fachliteratur mangelt es dabei keineswegs, wohl aber an einem Handbuch, in dem sowohl der Vielfalt der historischen Entwicklungen als auch dem berechtigten Wunsch nach innerer systematischer Kohärenz Rechnung getragen wird. Das Handbuch Phänomenologie schließt (...) diese Lücke. Ausgewiesene Autor:innen bereiten in eigens für diesen Band verfassten Artikeln komplexe sachliche Zusammenhänge übersichtlich auf. Durch seinen Aufbau eignet sich das Handbuch sowohl für Anfänger:innen als auch für Fortgeschrittene. Anhand bündig präsentierter Grundbegriffe und Verfahren konturiert das Handbuch die Spezifik der phänomenologischen Methode, spart dabei jedoch nicht die Kontroversen und methodologischen Neuausrichtungen aus, die von ihrer Lebendigkeit und Vielstimmigkeit zeugen. Ein umfangreicher Schlussteil ist der Rezeption und Anwendung in einzelnen Wirkfeldern gewidmet. Als Hilfsmittel zur eigenständigen Erschließung der phänomenologischen Denkrichtung und zu ihrer Anwendung auf aktuelle Probleme zeichnet sich das Handbuch durch seine Leser:innenfreundlichkeit und einen stark forschungspraktischen Bezug aus. Ein Kompendium, das sich bald als ABC der Phänomenologie etablieren dürfte. (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)

I argue against the interpretation of propositions as intentions and proof-objects as fulfillments proposed by Heyting and defended by Tieszen and van Atten. The idea is already a frequent target of criticisms regarding the incompatibility of Brouwer’s and Husserl’s positions, mainly by Rosado Haddock and Hill. I raise a stronger objection in this paper. My claim is that even if we grant that the incompatibility can be properly dealt with, as van Atten believes it can, two fundamental issues indicate that (...) the interpretation is unsustainable regardless: (1) it is hard to determine, without appealing to propositional intentions on pain of circularity, what intention a proof-object should be understood as a fulfillment of; (2) due to a difficult fulfillment dilemma, it is unclear, at best, what the object of an intention corresponding to a proposition is. (shrink)

My first aim in this paper is to use time diagrams in the style of Brentano to analyze constructions in Brouwer's separable mathematics more precisely. I argue that constructions must involve not only pairing and projecting as basic operations guaranteed by the intuition of twoity, as sometimes assumed in the literature, but also a recalling operation. My second aim is to argue that Brouwer's views on the intuition of twoity and arithmetic lead to an ontological explosion. Redeveloping the constructions of (...) natural numbers and systems sketched in an appendix to Brouwer's Cambridge lectures, I observe that the only plausible way he can make some elementary arithmetic in his separable mathematics is by allowing for the same canonical number to be determined by multiple separable entities, resulting in an overabundant mathematical ontology. (shrink)

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)

The sciences are in a state of crisis. Due to factors like hyperspecialization and an all too naive and uncritical faith in their own method, the sciences have lost sight of their initial goal. The idea that sciences are in a state of crisis can of course famously be found in Edmund Husserl’s Crisis of the European Sciences. What is less well-known, however, is that Martin Heidegger also discusses and analyzes a crisis of the sciences in his 1928/29 lecture course (...) Einleitung in die Philosophie. There are interesting similarities between the nature of the crisis the two thinkers observe, but key differences when it comes to the relation between science and philosophy and the question of whether or not the crisis can be resolved. The aim of my article will be to provide a thorough comparative analysis of Husserl’s and Heidegger’s accounts of the crisis, the of Galileo’s mathematization of nature in their analyses, and what this means for their ideas concerning the relation between science and philosophy. The goal of this analysis is to provide some conceptual clarity regarding the prospect of naturalizing phenomenology. (shrink)

La propuesta fenomenológica de Husserl, aunque se inició con un estudio sobre la naturaleza del número y las funciones matemáticas, abarca mucho más que eso. Se conocen una serie de ensayos editados póstumamente en el tomo XXI de la serie Hussserliana que abarcan los años 1890-1908, y que se enfocan en comprender los avances de la geometría (euclidiana y no euclidiana) y el papel que desempeña en la constitución del espacio físico. Los estudios sobre geometría euclidiana y no euclidiana continuarían (...) así la investigación de Filosofía de la aritmética, ya que ambos comparten los Grundgedanken, es decir, el conjunto de herramientas y temas relacionados con la noción de representación simbólica. El objetivo de este artículo es presentar de manera general los principales problemas a los que Husserl se enfrenta en relación con temas geométricos y espaciales. Se intentará reconstruir, de manera histórico-crítica, aspectos esenciales sobre los temas y problemas de la filosofía de la geometría tal y como Husserl la desarrolla en sus primeras obras fenomenológicas. (shrink)

La question centrale de cet ouvrage se situe à la croisée du réalisme et de l'idéalisme : comment peut-on à la fois affirmer que les objets dont traite la mathématique possèdent un être identique en tout temps et pour tout sujet pensant et qu'ils ont été produits par un sujet mathématicien? L'idéalité des objectités formelles implique en effet leur autonomie ontologique vis-à-vis de la conscience, donc l'impossibilité de les produire. Or ces objets idéaux requièrent l'invention d'un système de notations symboliques (...) et font leur apparition à un moment de l'histoire : n'est-ce pas le signe de leur dépendance ontologique vis-à-vis de la spontanéité productrice de la conscience? Partant, la vérité mathématique doit-elle être conçue comme adéquation de la connaissance à des objets qui lui préexistent, ou comme dévoilement d'objets n'ayant ni existence en soi, ni préexistence à l'acte qui les dévoile? Ces questions fournissent l'occasion d'élaborer un discours de la méthode phénoménologique, d'analyser le mode d'être temporel des idéalisés et de conjoindre idéalisme du sens et réalisme nomologique : si le sens mathématique est engendré, propriétés et lois doivent être découvertes et démontrées."--Page 4 of cover. (shrink)

The following text is divided in four parts. The first presents the inner relation between the phenomenological concept of intentionality and space in a general mathematical sense. Following this train of though the second part brie_ly characterizes the use of the geometrical concept of manifold (Mannigfaltigkeit) in Husserl’s work. In the third part we present some examples of the use of the concept in Husserl’s analyses of space, time and intersubjectivity, pointing out some dif_iculties in his endeavor. In the fourth (...) and _inal part we offer some points of coincidence between phenomenology and category theory suggesting that the latter can work as a formal frame for ontology in the former. Our thesis is that intentionality operates in different levels as a morphism, functor and natural transformation. (shrink)

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)

RESUMEN Un atento repaso por la concepción de la violencia chulhaniana en el contexto de las formas tradicionales que han estudiado este fenómeno social permite exponer con mayor detalle y claridad esta propuesta en cuanto a la relación con el otro se refiere. Cuando lo sucedido durante el colonialismo que azotó al mundo en el transcurso de los siglos XVIII y XIX se asumía como las más peligrosas acciones cometidas en procura de la supresión de las características propias de cada (...) uno, en la medida en que los fenómenos de invasión suprimían materialmente la diferencia que poblaba el mundo, apareció la violencia de la positividad como la disposición sistémica que parte de la libertad y el consentimiento individual para, tácitamente, eliminar cualquier tipo de sentido de otredad presente en su constitución. ABSTRACT A careful review of the conception of Chulhanian violence in the context of the traditional forms that have studied this social phenomenon makes it possible to explain in greater detail and clarity this proposal as to the relationship with the other refers. When what happened during colonialism that hit the world during the 18th and 19th centuries was assumed to be the most dangerous actions committed in seeking the suppression of each other's own characteristics to the extent that invasion phenomena materially suppressed the difference that populated the world, the violence of positivity appeared as the systemic disposition that begins with freedom and individual consent to, tacitly, to eliminate any kind of sense of otherness present in its constitution. (shrink)

This paper argues that the shared intersubjective accessibility of mathematical objects has its roots in a stratum of experience prior to language or any other form of concrete social interaction. On the basis of Husserl’s phenomenology, I demonstrate that intersubjectivity is an essential stratum of the objects of mathematical experience, i.e., an integral part of the peculiar sense of a mathematical object is its common accessibility to any consciousness whatsoever. For Husserl, any experience of an objective nature has as its (...) correlate a “we,” which he terms the “community of monads”. Thus, even before mathematical objects gain expression, formalization, and axiomatization through natural and scientific language, from a phenomenological viewpoint their objectivity has its roots in raw pre-linguistic though intersubjective experience. Accordingly, I demonstrate the different senses in which the experience of mathematical objects is permeated by intersubjectivity, suggesting a picture of mathematical intersubjectivity as pre-linguistic common experience based on Husserl’s idea of a “community of monads”. (shrink)

This paper aims to clarify Merleau-Ponty’s contribution to an embodied-enactive account of mathematical cognition. I first identify the main points of interest in the current discussions of embodied higher cognition and explain how they relate to Merleau-Ponty and his sources, in particular Husserl’s late works. Subsequently, I explain these convergences in greater detail by more specifically discussing the domains of geometry and algebra and by clarifying the role of gestalt psychology in Merleau-Ponty’s account. Beyond that, I explain how, for Merleau-Ponty, (...) mathematical cognition requires not only the presence and actual manipulation of some concrete perceptible symbols but, more strongly, how it is fundamentally linked to the structural transformation of the concrete configurations of symbolic systems to which these symbols appertain. Furthemore, I fill a gap in the literature by explaining Merleau-Ponty’s claim that these structural transformations are operated through motor intentionality. This makes it possible, in turn, to contrast Merleau-Ponty’s approach to ontologically idealistic and realistic views on mathematical objects. On Merleau-Ponty’s account, mathematical objects are relational entities, that is, gestalts that necessarily imply situated cognizers to whom they afford a specific type of engagement in the world and on whom they depend in their eventual structural transformations. I argue that, by attributing a strongly constitutive role to phenomenal configurations and their motor transformation in mathematical thinking, Merleau-Ponty contributes to clarifying the worldly, historical, and socio-cultural aspects of mathematical truths without compromising what we perceive as their universality, certainty, and necessity. (shrink)

1. INTRODUCTIONIt has been some time now since the philosophical community has learned to appreciate Husserl’s contribution to the philosophies of logic, mathematics, and science in general, despite still some prejudices and misinterpretations in certain academic circles incapable of reading Husserl beyond the incompetent and malicious review which Frege wrote in 1894 of his Philosophie der Arithmetik (PA) [1891/2003], hereafter Hua XII.Husserl’s philosophy of mathematics, in particular, has been the subject of many articles and books and has attracted the attention (...) of many interpreters and philosophers who have found in Husserl fruitful ideas with which to tackle the many philosophical problems elicited by mathematics — its ontology, its epistemology, and its somehow surprising utility, even when not indispensable, in modern science. This original and interesting work by Professor Hartimo is a welcome addition to this already respectable bibliography. The book is essentially exegetical and presents a reading of Husserl’s philosophy, particularly his philosophy of mathematics, as found in some of his most important published works, which were reactions to the development of mathematics in his time, of which he was, as the mathematician he used to be, an attentive and well-informed observer. (shrink)

The phenomenological origins of the historian and epistemologist Alexandre Koyré are not entirely evident or sufficiently recognized by scholars interested in his thinking. The main aim of this paper is therefore to highlight the belonging of Koyré to the phenomenological movement and to establish lines of continuity between the investigations of Husserl and Koyré, namely mathematics and philosophy. The proposal is divided into two parts. First, on a biographical level, we reconstruct the formation of the young Koyré at the phenomenological (...) school of Göttingen. Secondly, at the thematic level, we present some notions of continuity between Phenomenology and Koyré’s historical studies. (shrink)

Hermann Weyl was one of the most important figures involved in the early elaboration of the general theory of relativity and its fundamentally geometrical spacetime picture of the world. Weyl’s development of “pure infinitesimal geometry” out of relativity theory was the basis of his remarkable attempt at unifying gravitation and electromagnetism. Many interpreters have focused primarily on Weyl’s philosophical influences, especially the influence of Husserl’s transcendental phenomenology, as the motivation for these efforts. In this article, I argue both that these (...) efforts are most naturally understood as an outgrowth of the distinctive mathematical-physical tradition in Göttingen and also that phenomenology has little to no constructive role to play in them. (shrink)

This study purports a unifying view of the ontology of mathematics and fiction presented in Husserl’s 1894 manuscript “Intentional Objects” [Intentionale Gegenstände] in relation to his theory of manifolds. In particular, I clarify that Husserl’s argument supposes deductive systems of mathematical theories and fictional work as well as their “correlates,” which are mathematical manifolds in the former cases. This unifying view concretizes the concept of manifolds as an ontological concept that is not bound to mathematics. Although mathematical and fictional objects (...) differ in whether they are purely formal or materially filled, the concept of manifolds can be extended to admit materially filled objects, thus encompassing the two into the unitary theory of manifolds. By contrast, reviewing the subsequent developments of Husserl’s theory of manifolds naturally leads to restricting its concept to be strictly formal. This notion is consistent with the view that describes manifolds as forms of worlds and states of affairs; however, I suggest that the concept of the manifold is more fundamental than the concept of the world and state of affairs. (shrink)

The following text is divided in four parts. The first presents the inner relation between the phenomenological concept of intentionality and space in a general mathematical sense. Following this train of though the second part brie_ly characterizes the use of the geometrical concept of manifold (Mannigfaltigkeit) in Husserl’s work. In the third part we present some examples of the use of the concept in Husserl’s analyses of space, time and intersubjectivity, pointing out some dif_iculties in his endeavor. In the fourth (...) and _inal part we offer some points of coincidence between phenomenology and category theory suggesting that the latter can work as a formal frame for ontology in the former. Our thesis is that intentionality operates in different levels as a morphism, functor and natural transformation. (shrink)

After a period in which the relation between mathematics and phenomenology was little researched, and sometimes assumed to be antagonistic, in recent decades the place of mathematics in Husserl’s t...

Since the end of the last century, there have been several ambitious attempts to naturalize Husserlian phenomenology by way of mathematization. To justify themselves in view of Husserl’s adamant antinaturalism, many of these attempts appeal to the new physico-mathematical tools that were unknown in Husserl’s time and thus allegedly make his position outdated. This paper critically addresses these mathematization proposals and aims to show that Husserl had, in fact, sufficiently good arguments that make his antinaturalistic position sound even today. The (...) starting point of the discussion presented in this paper is the mathematization project introduced by Jean-Michel Roy, Jean Petitot, Bernard Pachoud, and Francisco Varela in their introduction to the book Naturalizing Phenomenology. This proposal was followed by a number of critiques but also by several alternative naturalization attempts clearly inspired by Roy et al.’s ambitious project. The review of some of Husserl’s important arguments often overlooked or misinterpreted by both the naturalization advocates and their critics leads the author of the paper to the twofold conclusion which, on the one hand, explores the deeper reasons for the impossibility of a physical and mathematical treatment of phenomenology, on the other hand, clarifies the sense in which such treatments are possible, namely by way of restriction of the variety of experiential aspects that undergo naturalization and substitution of the aspects amenable to the direct mathematization for the directly unmathematizable ones. In the fourth section of this paper, the author attempts to demonstrate that, contrary to widespread belief, Husserl’s arguments are not obsolete by the standards of the contemporary physico-mathematical approaches employed in the mathematization of phenomenology and indeed stand the test of time. (shrink)

In a 2022 paper, Hamami claimed that the orthodox view in mathematics is that a proof is rigorous if it can be translated into a derivation. Hamami then developed a descriptive account that explains how mathematicians check proofs for rigor in this sense and how they develop the capacity to do so. By exploring introductory texts in computability theory, we demonstrate that Hamami’s descriptive account does not accord with actual mathematical practice with respect to computability theory. We argue instead for (...) an alternative account in which imagination, anticipation, and interpretations of natural language play roles in establishing mathematical rigor. (shrink)

본 연구는 기하학에 대한 구조-구성주의 인식론을 정당화하는 것이다. 즉 구조주의와 구성주의를 융합하는 필자의 고유한 인식론으로 기하학적 인식의 본성을 해명하는 것이다. 그러나 현재의 연구는 이러한 큰 주제에 접근하기 위한 예비적 연구로서 칸트의 기하학적 직관 개념에 대한 역사-비판적 분석을 제공하는 데 한정된다. 잘 알려져 있는 것처럼, 칸트는 수학적 인식, 특히 기하학적 인식을 위해서 직관이 핵심적으로 중요하다고 주장했다. 그런데 칸트가 말하는 기하학적 인식을 위한 직관의 역할이 무엇인지는 여전히 논쟁적이다. 이점과 관련해서 역사적으로 대립적인 두 가지 해석이 있다. 하나는 베스(E. Beth), 힌티카(J. Hintikka), 프리드만(M. Fridman) (...) 등으로 이어지는 논리적 해석이며, 다른 하나는 파슨스(Parsons)와 칼슨(E. Carson)에 의해 제공된 현상학적 해석이다. 이에 필자는 특히 프리드만과 칼슨의 논쟁을 중심으로 두 해석의 내용을 재구성하고, 다음에 두 해석의 종합 가능성을 해명하고자 한다. 이를 위해 헬름홀츠-피아제의 인식론, 즉 “군 이론”에 근거한 새로운 구성주의 인식론을 제안할 것이다. 그러나 본고에서는 1차적 근사의 차원에서 그 개략적 구조만을 소개하는 데 만족하고, 종합의 구체적 내용인, 헬름홀츠-피아제의 공간 인식론에 근거한 구조-구성주의 공간 이론은 다른 지면에서 전개될 것이다. (shrink)

The paper is divided into two parts. In the first one, I set forth a hypothesis to explain the failure of Husserl’s project presented in the Philosophie der Arithmetik based on the principle that the entire mathematical science is grounded in the concept of cardinal number. It is argued that Husserl’s analysis of the nature of the symbols used in the decadal system forces the rejection of this principle. In the second part, I take into account Husserl’s explanation of why, (...) albeit independent of natural numbers, the system is nonetheless correct. It is shown that its justification involves, on the one hand, a new conception of symbols and symbolic thinking, and on the other, the recognition of the question of “the formal” and formalization as pivotal to understand “the mathematical” overall. (shrink)

The purpose of this study is to assess Husserl’s debt to Lotze’s philosophy during the Halle period. I first track the sources of Husserl’s knowledge of Lotze’s philosophy during his studies with Brentano in Vienna and then with Stumpf in Halle. I then briefly comment on Husserl’s references to Lotze in his early work and research manuscripts for the second volume of his Philosophy of Arithmetic. In the third section, I examine Lotze’s influence on Husserl’s anti-psychologistic turn in the mid-1890s. (...) The fourth section is a commentary on Husserl’s manuscript titled “Mikrokosmos,” to which he explicitly refers in his Prolegomena, and which he planned to publish as an appendix of his Logical Investigations. This work contains a detailed analysis of the third book of Lotze’s 1874 Logic. The last section examines Husserl’s arguments against logical psychologism in his Prolegomena, which I discuss through the lens of Stumpf’s critique of psychologism in his paper “Psychology and theory of knowledge”. I argue that Stumpf’s early works on this topic make it possible to establish a connection between Lotze’s interpretation of Plato’s theory of Ideas and Husserl’s anti-psychologism. My hypothesis is that Stumpf’s analyses represent the background of Husserl’s critique of logical psychologism in his Logical Investigations. I conclude by showing that Husserl’s position with respect to Lotze’s philosophy remains basically unchanged after the publication of his Logical Investigations, and that Husserl’s main criticism of Lotze pertains, in the final analysis, to the absence of a theory of intentionality in Lotze’s philosophy. (shrink)

RESUMEN La fenomenología de Husserl está inmersa en un entramado de regresiones y revisiones conceptuales que dificultan la identificación de una estructura sistémica. Los conceptos característicos de la fenomenología carecen de univocidad y no son propios de algunas obras de Husserl. Philosophie der Arithmetik ilustra el problema referido. ¿Puede inscribirse esta obra dentro de la categoría de texto fenomenológico? ¿Es posible hablar de una fenomenología de la matemática en Husserl? y ¿qué sentido tendría esto? Los objetivos del presente ensayo son: (...) primero, analizar la dificultad que tiene Philosophie der Arithmetik para responder a las inquietudes de la filosofía de las matemáticas, y segundo, proponer la posibilidad de una fenomenología de la matemática condicionada a la clarificación conceptual y al estudio crítico de la influencia de la tradición analítica sobre la filosofía de las matemáticas. ABSTRACT Husserl's phenomenology is immersed in a network of regressions and conceptual revisions that hinder the identification of a systemic structure. The characteristic concepts of phenomenology fall short of uniqueness and are not appropriate for some of Husserl's works. Philosophie der Arithmetik illustrates the aforementioned problem. Can this work be classified under the category of phenomenological texts? Is it possible to talk about a phenomenology of mathematic in Husserl? And what would be the point of this? The objectives of the present essay are: first, to analyze the difficulty that Philosophie der Arithmetik has responding to the concerns of the philosophy of mathematics, and second, to propose the possibility of a phenomenology of mathematics, conditioned to conceptual clarification and critical study of the influence of the analytical tradition on the philosophy of mathematics. (shrink)

Husserl and Mathematics explains the development of Husserl's phenomenological method in the context of his engagement in modern mathematics and its foundations. Drawing on his correspondence and other written sources, Mirja Hartimo details Husserl's knowledge of a wide range of perspectives on the foundations of mathematics, including those of Hilbert, Brouwer and Weyl, as well as his awareness of the new developments in the subject during the 1930s. Hartimo examines how Husserl's philosophical views responded to these changes, and offers a (...) pluralistic and open-ended picture of Husserl's phenomenology of mathematics. Her study shows Husserl's phenomenology to be a method capable of both shedding light on and internally criticizing scientific practices and concepts. (shrink)

This essay presents Husserl’s Formal and Transcendental Logic (1929) in three main sections following the layout of the work itself. The first section focuses on Husserl’s introduction where he explains the method and the aim of the essay. The method used in FTL is radical Besinnung and with it an intentional explication of proper sense of formal logic is sought for. The second section is on formal logic. The third section focuses on Husserl’s “transcendental logic,” which is needed to make (...) Husserl’s conception complete, i.e., to understand what makes Besinnung radical, how the proper sense of formal logic has been reached and how all this relates to transcendental constitution of an object in the world. Husserl’s work is connected to his context (in exact sciences) and the relevance of his views for the contemporary approaches is discussed. (shrink)

The paper discusses Husserl’s criticism of Frege in Philosophy of Arithmetic (1891) and then his later attitude towards logicism as expressed in Logical Investigations (1900-01). In Philosophy of Arithmetic Husserl holds that logicists offer needless and artificial definitions of notions such as equivalence and number. Frege criticized Husserl’s approach in Philosophy of Arithmetic as psychological, thus shifting the focus of the debate away from logicism. However, Frege’s criticism could be seen to lead Husserl to his later transcendental phenomenological concept of (...) correlation and, with it, to a more developed criticism of logicism. In this regard the division of labor between mathematics and philosophy established in Prolegomena to the Logical Investigations (1900) is central. It implies that in Husserl’s view formal methods fail to shed philosophical insight into the essence of mathematics. In his picture the central notions in need of philosophical clarification are, instead of number and quantity, the notions that are central to abstract structural mathematics. Contrary to Frege who can only elucidate the primitive concepts, Husserl engages in detailed description of their constitution. (shrink)

La publicación de Filosofía de la aritmética (1891) trajo consigo un notable avance en las investigaciones fenomenológicas de E. Husserl, a la vez que se abrió paso entre las indagaciones ya existentes sobre la fundamentación de las matemáticas. Fue precisamente dentro de esta constelación de publicaciones donde se originó una de las polémicas filosóficas más interesantes de las postrimerías del siglo XIX: la recensión de Filosofía de la aritmética hecha por Frege en 1894. Lo que ofreceré en este texto son (...) algunas claves de interpretación que permitirán hacer una revisión juiciosa de aquellos aspectos filosóficos no vistos por Frege en su crítica. Todo ello a través del intercambio epistolar entre ambos, de la exposición de la reseña de 1894 y de las recensiones de Filosofía de la aritmética que han permanecido desconocidas hasta ahora. PALABRAS CLAVE: Filosofía de la aritmética, Fenomenología, Filosofía analítica, Husserl, Frege. (shrink)

Reviewed by F. PATRAS, Laboratoire J.-A. Dieudonné, Université Côte d'Azur and CNRS, Nice, France. [email protected] those who are interested in Husserl’s philosophy of mathematics and its links w...