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)

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". I believe that this interpretation of Weyl can clarify the views on mathematical existence and mathematical intuition which (...) are implicit in "Das Kontinuum". (shrink)

Husserl left many unpublished drafts explaining (or trying to) his views on spatial representation and geometry, such as, particularly, those collected in the second part of Studien zur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. (shrink)

I carry out in this paper a philosophical analysis of the principle of excluded middle (or, as it is often called in the version I favor here, principle of bivalence: any meaningful assertion is either true or false). 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. DOI:10.5007/1808-1711.2011v15n2p333. (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.

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. (shrink)

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. (shrink)

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. (shrink)

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. (shrink)

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. (shrink)

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. (shrink)

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. (shrink)

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. (shrink)

Neste artigo, relato os aspectos mais salientes do affair Sokal-Bricmont - uma paródia que evoluiu para uma crítica articulada dos excessos de um certo pensamento pós-modernista - e analiso algumas das reações que suscitou em artigos publicados na Folha de S. Paulo. Termino com algumas reflexões sobre a nefasta negligência para com as ciências exatas na educação em geral e, em particular, na formação dos profissionais das áreas de filosofia e ciências humanas.In this paper I summarize some of the most (...) relevant aspects of the so-called Sokal and Bricmont affair - a parody that evolved to a full-fledged criticism of the excesses of post-modernism - and analyze the reactions it elicited in some articles published in Folha de São Paulo. I close with a reflection on the unfortunate neglect of the exact sciences in education in general and, in particular, the education of philosophers and social scientists. (shrink)

In this paper I show the possibility of an ontology of mathematics that keeps some points in common with platonism and constructivism while diverging from them in other essencial ones. I understand that mathematical objects are simply the referential focus of mathematical discourse, I also understand that their existence is merely intentional but none the less objective, in the sense of being shared by all those who are engaged in the mathematical activity. However, the objective existence of mathematical entities is (...) not secured once and for all but only in so far as the mathematical discourse is consistent. This is the core of the criterium of objective existence put forward and that I believe should sustain a mathematical ontology without the presupposition of the independent existence of a domain of mathematical objects, and without the restrictions imposed on it by constructivism and formalism in their various versions.Neste artigo quero apontar para a possibilidade de uma ontologia da matemática que, mesmo mantendo alguns pontos em comum com o platonismo e com o construtivismo, desliga-se destes em outros pontos essenciais. Por objeto matemático entendo o foco referencial do discurso matemático, ou seja, aquilo sobre o qual a matemática fala. Entendo que a existência destes objetos é meramente intencional, presuntiva, mas, simultaneamente, objetiva, no sentido de ser uma existência comunalizada, compartilhada por todos aqueles engajados no fazer matemático. A existência objetiva das entidades matemáticas não está, entretanto, garantida de uma vez por todas, mas apenas enquanto o discurso matemático for consistente. Este é o espírito do critério de existência objetiva enunciado que, acredito, deve sustentar uma ontologia matemática sem o pressuposto da existência independente de um domínio de objetos matemáticos, sem o empobrecimento que lhe impõem as diferentes versões construtivistas e sem a aniquilação que lhe infringe o formalismo sem objetos. (shrink)

Este artigo procura mostrar que as idéias filosóficas de Husserl não apenas influenciaram o trabalho de alguns dos maiores matemáticos do século XX, mas foram decisivas para aproximarem uma epistemologia das ciências formais de uma fenomenologia do significado.

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.

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. (shrink)

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. (shrink)

For different reasons, Husserl's original, thought-provoking ideas on the philosophy of logic and mathematics have been ignored, misunderstood, even despised, by analytic philosophers and phenomenologists alike, who have been content to barricade themselves behind walls of ideological prejudices. Yet, for several decades, Husserl was almost continuously in close professional and personal contact with those who created, reshaped and revolutionized 20th century philosophy of mathematics, logic, science and language in both the analytic and phenomenological schools, people whom those other makers of (...) 20th century philosophy, Russell, Frege, Wittgenstein and their followers, rarely, if ever, met. Independently of them, Husserl offered alternatives to the well-trodden paths of logicism, nominalism, formalism and intuitionism. He presented a well-articulated, thoroughly argued case for logic as an objective science, but was not philosophically naive to the point of not seeing the role of subjectivity in shaping the sense of the reality facing objective science. Given the preeminent role that philosophy of logic and philosophy of mathematics have played in transforming the way philosophy has been done since Husserl's time, and given the depth of his insights and his obvious expertise in those fields, his ideas need to be integrated into present-day, mainstream philosophy. Here, philosopher Claire Ortiz Hill and mathematician-philosopher Jairo da Silva offer a wealth of interesting insights intended to subvert the many mistaken idees recues about the development of Husserl's thought and reestablish broken ties between it and philosophy now. (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)

Resenha do livro de: SILVA, Jairo José da. Filosofias da matemática . Sáo Paulo: Editora UNESP, 2007.

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. (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.

O Brasil foi um dos maiores traficantes de africanos destinados à escravidão nas Américas; e a Bahia um dos seus principais destinos. Porém, juntamente com a escravidão, os africanos trouxeram também a rebelião. Uma das principais formas de negação da escravidão pela revolta trazida pelos negros africanos às Américas foi o quilombo. O artigo discute as mudanças ocorridas na literatura histórica sobre a escravidão e suas formas de negação após o centenário da abolição, em 1988. O quilombo de Palmares surgido (...) no interior de Alagoas em finais do século XIV, veio a se constituir, durante o século XVII, no maior e mais bem organizado quilombo das Américas. Visto de formas diferentes pela historiografia ao longo do tempo, Palmares acabou por se transformar, no imaginário dos explorados de todos os matizes e dos negros de forma especial, num dos maiores símbolos de liberdade e de resistência a qualquer forma de exploração, opressão e discriminação. Palavras-chave: Escravidão; negros; quilombo de Palmares; resistência. (shrink)

Background: Small-sided games are an adjusted form of official games that are often used in training scenarios to introduce a specific tactical issue to team sports players. Besides the acute effects of SSGs on players' performance, it is expectable that the consistent use of these drill-based games induces adaptations in the technical execution and tactical behaviors of youth team sports players.Objective: This systematic review with meta-analysis was conducted to assess the effects of SSG programs on the technical execution and tactical (...) behaviors of young and youth team sports players.Data Sources: The data sources utilized were PubMed, Scopus, SPORTDiscus, and Web of Science.Study Eligibility Criteria: Young and youth team sports players (i.e., 17 and <17 SSG training sessions, respectively.Conclusions: This systematic review and meta-analysis revealed a significant beneficial effect of using SSG training programs for enhancing technical execution in young and youth players. The benefits were similar despite the various numbers of training sessions applied. Further studies should add tactical behaviors as one of the outcomes for controlling the effects of SSG training programs. (shrink)

This article intends to explore the philosophical potency in the work of the black writer Carolina Maria de Jesus (1914-1977). Carolina de Jesus is best known for her work Quarto de Despejo, diary of a favelada (1960), our approach, however, focuses specifically on her short story “Socrates Africano”, in which she deals with her experience with her grandfather Benedito and the relationship between her wisdom and that of the Greek philosopher Sócrates (5th century BC). Her reflection starts from the attempt (...) to understand the reason why her grandfather was compared to Socrates and called “African Socrates”. In this sense, we intend to expand the idea of an African Socrates and the origins of one of the African philosophical matrices, the Yoruba tradition, dialoguing with the thought of the Nigerian philosopher Sophie Oluwole (1935-2018) and her approximation between Socrates and Orunmilá, as a motto to think about the philosophical potentialities of Carolina's work and the traits of an Afro-Brazilian philosophy. (shrink)

Por ocasião do centenário de nascimento de Paul Ricoeur, a Universidade de Coimbra juntou num mesmo volume um conjunto de estudos académicos em tributo a este pensador. Reuniram-se deste modo comunicações de professores, de novos investigadores e de estudantes de doutoramento em redor de temas que marcaram a receção da obra e do filósofo francês em Coimbra. Cada capítulo do presente livro demonstra a fecundidade de um pensamento que, enraizado na “tradição reflexiva”, assumindo a mensagem das “filosofias da existência”, articulando (...) Fenomenologia e Hermenêutica, adquire, hoje, crescente impacto e relevância. Esta é uma filosofia que promove o diálogo entre disciplinas, autores e culturas e supera divisões teóricas em ordem a enfrentar as exigências da praxis. O desafio da praxis fez assim emergir um pensamento exigente e, simultaneamente, capaz de dialogar com diversas áreas do saber como: a psicoterapia, a história, as neurociências, a bioética, as ciências sociais e os estudos literários. At the occasion of the Paul Ricoeur’s centenary this volume joined in tribute several scholars, young researchers, and Ph.D students around the main themes that marked the reception of the philosopher’s work in Portugal and, especially in the University of Coimbra. Each chapter demonstrates the vigour Ricoeur’s thought as it is rooted in the "reflexive tradition", as it assumed the "existentialist" horizon, as it articulated Phenomenology and Hermeneutics. This philosophical project as, today, an enormous range of influence. This is a project that promotes dialogue between disciplines, authors, and cultures, overcoming theoretical divisions in face the demands of praxis. From this rich and plural inspiration comes the ability to meet with other fields of thought, namely psychotherapy, history, neuroscience, bioethics, social sciences, and literary studies. (shrink)

Robert Kilwardby is a central figure in late medieval philosophy and theology, but key areas of his thought still remain unexamined in a systematic way. This book offers a comprehensive overview of his works, ranging from topics in logic to theology, done in a way that is accessible to non-specialists and to anyone interested in medieval thought.

How we come to know the external world has intrigued thinkers throughout the history of philosophy. Medieval philosophers understood that a theory of perception requires an account of the categorization of sensory information: to perceive things as being dangerous or beneficial and even as being individuals that belong to certain kinds (e.g., 'this is a dog'). A key question is whether this requires the intervention of rational cognitive capacities, cooperating with sensory ones in normal instances of perception. The contributions to (...) this volume investigate how thinkers from the thirteenth and fourteenth centuries answer this and other related questions about human perception. Contributors are Fabrizio Amerini, Joel Biard, Veronique Decaix, Christian Kny, Lydia Schumacher, Josee Filipe Silva, and Jorg Alejandro Tellkamp. (shrink)