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

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

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

The question of the psychologism of the theory of number developed by Husserl in his Philosophy of Arithmetic has long been debated, but it cannot be considered fully resolved. In this paper, I address the issue from a new point of view. My claim is that in the Philosophy of Arithmetic, Husserl made, albeit indirectly, a series of arguments that are worth reconstructing and clarifying since they are useful in shedding some light on the psychologism issue. More specifically, I maintain (...) that the clarification of these arguments, along with other arguments that Husserl presented against alternative theories of number as well as with some contemporary distinctions concerning the notion of ontological dependence, allows us to determine that Husserl’s theory of number is psychologistic in a minimal and precise sense: it entails a generic ontological dependence of numbers upon the mind. (shrink)

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)

La lógica es para Husserl una ciencia de la ciencia, una ciencia de lo que todas las ciencias tienen en común respecto de sus modos de validación. De este modo, la lógica trata por un lado con leyes universales relacionadas con la verdad, la deducción, la verificación y la falsación; y, por otro lado, con leyes relacionadas con la teoría como tal, y con lo que produce la unidad teorética. Ambos tipos de leyes se refieren por una parte a las (...) proposiciones de una teoría y por otra parte al dominio de objetos al que esas proposiciones se refieren. Este ensayo presenta una perspectiva sistemática del punto de vista husserliano sobre estos temas tal como él lo expone en sus Investigaciones Lógicas. Se muestra la manera en que su teoría de los significados lingüísticos en tanto especies de actos mentales, su ontología formal de la parte, el todo y la dependencia, su teoría de las categorías significativas, y su teoría de la intuición categorial combinada con su teoría de la ciencia conforman un todo singular. Finalmente, este trabajo explora la forma en que las ideas de Husserl sobre estos temas pueden ser reactivadas para resolver problemas en la filosofía del lenguaje, la lógica y la matemática de un modo que haga justicia al rol de la actividad mental en cada uno de estos dominios, evitando al mismo tiempo caer en los problemas del psicologismo. (shrink)

ABSTRACT Edmund Husserl strongly emphasized the importance of reduction to his phenomenology. For his followers, however, it has proved a formidable task to specify exactly how this intricate accomplishment that opens up the possibility for phenomenological research is to be performed. In this article, we study different approaches to gaining access to reduction and conclude by suggesting that we should read Husserliana itself as a set of accomplished reductions. In other words, our task is to pinpoint chapters where the movement (...) of reduction takes place. As an example of such a reading, I will point out events that come about in Husserl???s Idea of Phenomenology. R??SUM?? Husserl a fortement insist?? sur l???importance de la r??duction dans sa ph??nom??nologie. Pour ses disciples, cependant, s???est av??r????tre redoutable la t??che de pr??ciser comment doit??tre ex??cut??e cette r??alisation complexe qui ouvre la possibilit?? pour la recherche ph??nom??nologique. Dans cet article, nous??tudions les diff??rentes approches de la r??duction et nous concluons en sugg??rant que l???ouvrage Husserliana lui-m??me doit se lire comme un ensemble de r??ductions accomplies. Notre t??che consistera donc?? identifier les chapitres o?? a lieu ce mouvement de r??duction.?? titre d???exemple d???une telle lecture, j?????voquerai quelques-uns des??v??nements de L???id??e de la ph??nom??nologie de Husserl. (shrink)

The aim of this work is to elucidate the meaning of 'infinity' from a phenomenological perspective, especially within the framework of Husserl’s theory of knowledge and perception. In the first chapter I firstly sketch the basics of Husserl’s phenomenology of knowledge. Thereafter I delve into the questions concerning the reduction to the 'reellen Bestand', which is hold to be the ground of verification of purports in the "Logical Investigations". I then propose an interpretation of the categorial intuition as directed to (...) the laws an 'Auffassung' follows when it organizes the sensual contents related to an object. In the second chapter I briefly expound Husserl’s phenomenology of perception and I show how perception can be hold as an intentional experience. Afterwards I concentrate on the constitution of space and of thing (Ding). I show in which sense the structures of both constitutions are dependent on the sensations and kinesthesis (i.e. proprioceptions) experienced by the subject. In the third chapter I finally deal with the problem of infinity. I face some recent interpretations of Husserl’s theory of perception according to which infinity is somehow intertwined in and even intuited through thing-perception (Dingwahrnehmung). I claim that thing-perception can not be hold as an access to infinity, i.e. no thing-perception is a perception of something as infinite. Following I show how some other kinds of sensual experience can correspond to a perception of infinity. On the base of some Husserl’s manuscripts and through a brief confrontation with Kant’s Analytic of the Sublime I sustain that infinity can be sensually intuited in the senses of limitlessness and of formlessness. I then show that categorial intuition is necessary in order to constitute the meaning of mathematical infinity, i.e. an endless whole of discrete parts. In fact, the idea of such an “entity” is dependent on the capacity of grasping the law of a serial production. In the conclusion I establish that mathematical infinity is the only kind of infinity which can not find any sensual correlate. This is due to the fact that it is a meaning composed by two contradictory prescriptions: 'reach the limitless' and 'go on without an end'. I consequently argue that the idea of infinity is grounded on sensual experience and does not need any supernatural origin. (shrink)

In 1913, in a draft for a new Preface for the second edition of the Logical Investigations, Edmund Husserl reveals to his readers that "The source of all my studies and the first source of my epistemological difficul­ties lies in my first works on the philosophy of arithmetic and mathematics in general", i.e. his Habilitationsschrift and the Philosophy of Arithmetic: "I carefully studied the consciousness constituting the amount, first the collec­tive consciousness (consciousness of quantity, of multiplicity) in its simplest and (...) higher levels (consciousness of sums, sums of sums etc.). I immediately separated proper (intuitive) and symbolic consciousness, in the characteriza­tion of the former I hit the radical difference of categorial consciousness [...] and sensuous consciousness of unity." Later on, in the Third Investigation, Husserl makes some very specific claims, that are of considerable importance to assess the development of his early works and their relation to his later phenomenology: "This first work of mine (an elaboration of my Habilitationsschrift, [...], 1887) should be compared with all assertions of the present work on compounds, moments of unity, complexes, wholes and objects of higher order. I am sorry that in many recent treatments of the doctrine of "Gestalt-qualities", this work has mostly been ignored, though quite a lot of the thought-content of later treatments by Cornelius, Meinong etc., of questions of analysis, apprehension of plurality and complexion is already to be found, differently expressed, in my Philosophy of Arithmetic. I think it would still be of use today to consult this work on the phenomenological and ontological issues in question, especially since it is the first work that attached importance to acts and objects of higher order and investigated them thoroughly." Hence, at the time of the Ideas, Husserl retrospectively considers his first works4 as being still relevant for phenomenological issues. Not only does Husserl advance a very interesting priority claim with respect to Von Ehrenfels’ development of the notion of Gestalt and Meinong’s development of Gegenstandstheorie, but also a strong affirmation of continuity and coherence of his position from 1887 all the way up to 1913, encompassing the alleged “revolution” in his position from psychologism to anti-psycho­logism in the 1890s. Indeed, according to much of the recent secondary literature, in 1894, right in the middle of the ten “incubation” years between the Philosophy of Arithmetic and the Logical Investigations, Frege’s destruc­tive review would have converted Husserl to antipsychologism practically overnight. This gives us two conflicting interpretations: on the one hand, Husserl himself in 1913 still seems to approve of the Philosophy of Arithmetic and even considers it to contain valuable phenomenological material, on the other, it is routinely dismissed by much of the secondary literature as hope­lessly psychologistic. So which one is it: do we have a phenomenological arithmetic or a psychologistic arithmetic in Husserl’s first book? On balance, I think that Husserl in his Philosophy of Arithmetic developed a position that does not fall prey to the exaggerated and poorly aimed critiques of Frege, while at the same time, as a descriptive psychology of the genesis and constitution of number, it can certainly be considered as providing phenome­nologically meaningful analyses, though of course not made from within an explicitly transcendental phenomenological framework. (shrink)

This book offers an historically-informed critical assessment of Dummett's account of abstract objects, examining in detail some of the Fregean presuppositions whilst also engaging with recent work on the problem of abstract entities.

Gottlob Frege and Edmund Husserl are two seemingly different philosophers in their methodology. Both have significantly influenced Western philosophy in that their contributions established fields within philosophy that are of intensive study today. Still, their differences in methodology have, in certain instances, yielded similar or distinct results. Their results ranged from the distinction of sense and reference, objectivity, and the theory of mathematics: specifically, their definition of number. Frege and Husserl have such striking similarities in their theory of sense and (...) reference and related notions that from their apparent correspondence, Frege seems to have acknowledged the coincidences in their theories (Frege & Kluge, 1972) (Hill & Rosado Haddock Intro, 4, & 30-33). That is not to say there are exact parallels between Frege and Husserl and their results, as I will acknowledge their similarities and differences within scale. So, I intend to demonstrate their respective theories and results descriptively to show their likenesses while still recognizing their divergences in scope and methodology. Indirectly, I would also hope to illustrate the ties, in terms of subject matter, of these two philosophers who are historically considered at odds with one another; be it personally, in their schools of philosophy, or methodologically (Hartimo 47-53). (shrink)

In this talk I present the main results from Anta (2021), namely, that the theoretical division between Boltzmannian and Gibbsian statistical mechanics should be understood as a separation in the epistemic capabilities of this physical discipline. In particular, while from the Boltzmannian framework one can generate powerful explanations of thermal processes by appealing to their microdynamics, from the Gibbsian framework one can predict observable values in a computationally effective way. Finally, I argue that this statistical mechanical schism contradicts the Hempelian (...) (1958) thesis that the predictive power of a scientific theory is directly proportional to its explanatory potential, and vice versa. (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.

The recent move to naturalize phenomenology through a mathematical protocol is a significant advance in consciousness research. It enables a new and fruitful level of dialogue between the cognitive sciences and phenomenology of such a nuanced kind that it also prompts advancement in our phenomenological analyses. But precisely what is going on at this point of ‘dialogue’ between phenomenological descriptions and mathematical algorithms, the latter of which are based on dynamical systems theory? It will be shown that what is happening (...) is something more than a mere ‘passing of the baton’ from phenomenology to mathematics. For this sophisticated naturalization to prove a worthy endeavour it must produce more than just correlation, it must prove some form of interrelation to the extent that phenomenology is deterministic. But such interrelational and deterministic requirements are the start of a slippery slope, and it will be argued that this slope only loses more friction once a further demand of formal and precise descriptions is made of phenomenology. Such deterministic and formally precise demands misconstrue phenomenology’s ideal goal of a unification of genuine/originary reason and truth. Not a deductive and definitive discipline, phenomenology is rather from the outset descriptive and critical. Phenomenology’s descriptive beginnings will thus be employed as an essential barrier to the naturalization of phenomenology. (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 reconstructs and critically analyzes Husserl’s philosophical engagement with symbolic technologies—those material artifacts and cultural devices that serve to aid, structure and guide processes of thinking. Identifying and exploring a range of tensions in Husserl’s conception of symbolic technologies, I argue that this conception is limited in several ways, and particularly with regard to the task of accounting for the more constructive role these technologies play in processes of meaning-constitution. At the same time, this paper shows that a critical (...) examination of Husserl’s account of symbolic technologies, particularly as developed in his mature, genetic phenomenology, can be enduringly fruitful—if some of the specific conceptual weakness of this account are identified and properly accounted for. My discussion will proceed as follows. In the first part I briefly analyze the early Husserl’s account of the role the ‘method of sensible signs’ plays in arithmetic cognition. In the second, main part I critically examine the bearing the genetic-phenomenological concepts of sedimentation and technization have on the conceptualization of symbolic technologies in Husserl’s work. In the final part I summarize the major strengths and weaknesses of Husserl’s account of symbolic technologies, and in the process make a case for the ongoing relevance of some of the crucial elements of this account. (shrink)