Jairo Da Silva

I here present and discuss Husserl’s clarification of the genesis of modern empirical science, particularly its mathematical methods, as presented in his last work, The Crisis of European Sciences and Transcendental Phenomenology. Although Husserl’s analyses have as their goal to redirect science to the lifeworld and to reposition man and his immediate experiences at the foundation of the scientific project so as to overcome the “crisis” of science, I approach them from a different perspective. The problem that interests me here is the applicability of mathematics in empirical science, to assess Husserl’s treatment of this issue in order to see if it can be sustained from a strictly scientific point of view regardless of philosophical adequacy. My conclusion is that it cannot. What Husserl takes as the “crisis” of science is inherent to the best scientific methodology.

It is a philosophical naiveté to believe that the object of science is some ready-made world out there that the scientist, free of any preconceptions, simply stumbles upon. Of course, there is a world out there, given to us through the senses, but that must be intentionally elaborated to become a world for us and a possible object of scientific inquiry. The intentional constitution of the world of science supports and “justifies” a priori conceptions about the empirical world, even those of a logical nature, that are, then, properly transcendental rather than metaphysical. My goal here is to investigate what these presuppositions are and on what they are based.

I present and discuss in this paper Husserl’s investigation of the genesis of the modem conception of empirical reality as carried out in his last work The Crisis of European Sciences and Transcendental Phenomenology. The goal of Husserl’s genetic investigation was to uncover the rnany layers of constitution that frorn the life-world (the Lebenswelt) the modem scientific conception of Nature was originated and to point out the need to ground the scientific project of rnodemity in the life-world so as to overcome the “alienation” that, for him, characterized the “crisis” of European science. I, however, approach his analyses from a different perspective. The problem that interests me here is the applicability of mathematics in the empirical science. My airn is to assess Husserl’s treatment of this question in order to see whether it can be sustained from a strictly scientific perspective. My conclusion is that it cannot. What Husserl takes for the “crisis” of science is inherent to the best scientific methodology. Nonetheless, Husserl’s analyses offer irnportant insights that I incorporate in what I believe to be a more satisfactory treatment of the problern concerning the role of mathematics in the ernpirical science.

O aparecimento do coronavírus fez ressurgir um antigo debate no âmbito da filosofia política: o debate entre liberdade e segurança. A maioria dos países atingidos precisou adotar medidas que restringiram a liberdade dos cidadãos para conter o avanço da doença. Esse artigo tem o objetivo de apresentar a posição do filósofo inglês, Thomas Hobbes exposta no _Leviatã_, para enfrentar esse problema. O texto está dividido em três partes. Em um primeiro momento, apresento a tese de Hobbes sobre a segurança e contra a liberdade irrestrita para evitar o estado de natureza. Em um segundo momento, apresento a tese de Hobbes sobre a liberdade limitada no estado político. Por fim, apresento uma possível solução para conjugar liberdade e segurança a partir do conceito hobbesiano de razão pública.

O objetivo do presente artigo é apresentar e confrontar os pressupostos de duas tradições interpretativas _Leviatã _de Thomas Hobbes. Pretendo demonstrar como, a partir dos pressupostos e dos critérios de interpretação de cada uma delas, teremos não apenas duas abordagens diversas, mas resultados e soluções distintas para problemas políticos e morais que o próprio Hobbes buscou solucionar com essa obra. Na primeira parte do artigo, apresentarei os aspectos inovadores da nova interpretação do _Leviatã_ em contraposição às interpretações tradicionais ou ortodoxas. Na segunda parte, confrontarei essa abordagem com a abordagem tradicional sobre um problema específico, mas central na filosofia de Hobbes: o problema da desordem social. Apresentarei a tese tradicional segundo a qual a solução para o problema da desordem seria criar um poder capaz de impor sanções que não permitissem que o interesse racional egoísta atuasse de forma irrestrita. Em seguida, apresentarei a solução revisionista segundo a qual a solução seria produzir uma estável e apropriada reconfiguração dos diversos interesses morais e religiosos dos cidadãos através de um processo de educação contínua responsável pelo consenso social.

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

The notion of mathesis universalis appears in many of Edmund Husserl’s works, where it corresponds essentially to “a universal a priori ontology”. This paper has two purposes; one, largely exegetical, of clarifying how Husserl elaborates on Leibniz’ concept of mathesis universalis and associated notions like symbolic thinking and symbolic knowledge filtering them through the lesson of the so called “bohemian Leibniz”, Bernard Bolzano; another, more properly philosophical, of examining the role that the universal mathesis is allowed to play, and the space it occupies in Husserl’s intuition-based epistemology.

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.

I discuss here the pragmatic problem in the philosophy of mathematics, that is, the applicability of mathematics, particularly in empirical science, in its many variants. My point of depart is that all sciences are formal, descriptions of formal-structural properties instantiated in their domain of interest regardless of their material specificity. It is, then, possible and methodologically justified as far as science is concerned to substitute scientific domains proper by whatever domains —mathematical domains in particular— whose formal structures bear relevant formal similarities with them. I also discuss the consequences to the ontology of mathematics and empirical science of this structuralist approach.

In this paper I discuss the version of predicative analysis put forward by Hermann Weyl in Das Kontinuum. I try to establish how much of the underlying motivation for Weyl's position may be due to his acceptance of a phenomenological philosophical perspective. More specifically, I analyze Weyl's philosophical ideas in connexion with the work of Husserl, in particular Logische Untersuchungen} and Ideen.I believe that this interpretation of Weyl can clarify the views on mathematical existence and mathematical intuition which are implicit in Das Kontinuum.

In this paper I discuss Husserl's solution of the problem of imaginary elements in mathematics as presented in the drafts for two lectures hegave in Göttingen in 1901 and other related texts of the same period,a problem that had occupied Husserl since the beginning of 1890, whenhe was planning a never published sequel to Philosophie der Arithmetik(1891). In order to solve the problem of imaginary entities Husserl introduced,independently of Hilbert, two notions of completeness (definiteness in Husserl'sterminology) for a formal axiomatic system. I present and discuss these notionshere, establishing also parallels between Husserl's and Hilbert's notions ofcompleteness.

In this paper I argue for the view that the axioms of ZF are analytic truths of a particular concept of set. By this I mean that these axioms are true by virtue only of the meaning attached to this concept, and, moreover, can be derived from it. Although I assume that the object of ZF is a concept of set, I refrain from asserting either its independent existence, or its dependence on subjectivity. All I presuppose is that this concept is given to us with a certain sense as the objective focus of a phenomenologically reduced intentional experience. The concept of set that ZF describes, I claim, is that of a multiplicity of coexisting elements that can, as a consequence, be a member of another multiplicity. A set is conceived as a quantitatively determined collection of objects that is, by necessity, ontologically dependent on its elements, which, on the other hand, must exist independently of it. A close scrutiny of the essential characters of this conception seems to be sufficient to ground the set-theoretic hierarchy and the axioms of ZF

I discuss here the pragmatic problem in the philosophy of mathematics, that is, the applicability of mathematics, particularly in empirical science, in its many variants. My point of depart is that all sciences are formal, descriptions of formal-structural properties instantiated in their domain of interest regardless of their material specificity. It is, then, possible and methodologically justified as far as science is concerned to substitute scientific domains proper by whatever domains —mathematical domains in particular— whose formal structures bear relevant formal similarities with them. I also discuss the consequences to the ontology of mathematics and empirical science of this structuralist approach.

In this paper, I carry out a comparative study of the philosophical views of Edmund Husserl and Hermann Weyl on issues such as mathematical existence and mathematical intuition, the validity of classical logic, the concept of logical definiteness, the nature of symbolic mathematics, the role of mathematics in empirical science, the relation of scientific theories with perception, space representation and the philosophy of geometry, and intentional constitution in general. My main goal is not simply to assess the extent of Husserl’s influence on Weyl, although this is an ever present concern, but to clarify the views of one by contrasting them with those of the other.

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.

In the first year of the twentieth century, in Gottingen, Husserl delivered two talks dealing with a problem that proved central in his philosophical development, that of imaginary elements in mathematics. In order to solve this problem Husserl introduced a logical notion, called “definiteness”, and variants of it, that are somehow related, he claimed, to Hilbert’s notions of completeness. Many different interpretations of what precisely Husserl meant by this notion, and its relations with Hilbert’s ones, have been proposed, but no consensus has been reached. In this paper I approach this question afresh and thoroughly, taking into consideration not only the relevant texts and context, as others have also done before, but, more importantly, Husserl’s philosophy, his intuition-based epistemology in particular. Based on a system of clearly defined concepts that I here present, I reinforce an interpretation—definiteness as a form of syntactic completeness—that has, I believe, some advantages vis-à-vis alternative interpretations. It is in conformity with the available texts; it makes clear that Husserl’s notion of definiteness is indeed close to Hilbert’s notions of completeness; it solves the important problem of imaginaries for which it was created; and last, but not least, it fits naturally into Husserl’s system of concepts and ideas.

In the first year of the twentieth century, in Gottingen, Husserl delivered two talks dealing with a problem that proved central in his philosophical development, that of imaginary elements in mathematics. In order to solve this problem Husserl introduced a logical notion, called “definiteness”, and variants of it, that are somehow related, he claimed, to Hilbert’s notions of completeness. Many different interpretations of what precisely Husserl meant by this notion, and its relations with Hilbert’s ones, have been proposed, but no consensus has been reached. In this paper I approach this question afresh and thoroughly, taking into consideration not only the relevant texts and context, as others have also done before, but, more importantly, Husserl’s philosophy, his intuition-based epistemology in particular. Based on a system of clearly defined concepts that I here present, I reinforce an interpretation—definiteness as a form of syntactic completeness—that has, I believe, some advantages vis-à-vis alternative interpretations. It is in conformity with the available texts; it makes clear that Husserl’s notion of definiteness is indeed close to Hilbert’s notions of completeness; it solves the important problem of imaginaries for which it was created; and last, but not least, it fits naturally into Husserl’s system of concepts and ideas.

I carry out in this paper a philosophical analysis of the principle of excluded middle. This principle has been criticized, and sometimes rejected, on the charge that its validity depends on presuppositions that are not, some believe, universally obtainable; in particular, that any well-posed problem is solvable. My goal here is to show that, although excluded middle does indeed rest on certain presuppositions, they do not have the character of hypotheses that may or may not be true, or matters of fact that may or may not be the case. These presuppositions have, I claim, a transcendental character. Hence, the acceptance of excluded middle does not necessarily require, as some have claimed, an allegiance to ontological realism or some sort of cognitive optimism, construed as factual theses concerning the ontological status of domains of objects and our capability of accessing them cognitively.