Phenomenology And Mathematics

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Phenomenology And Mathematics

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Author : Michael Roubach
language : en
Publisher: Cambridge University Press
Release Date : 2023-12-06

Phenomenology And Mathematics written by Michael Roubach and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-12-06 with Science categories.

This Element explores the relationship between phenomenology and mathematics. Its focus is the mathematical thought of Edmund Husserl, founder of phenomenology, but other phenomenologists and phenomenologically-oriented mathematicians, including Weyl, Becker, Gödel, and Rota, are also discussed. After outlining the basic notions of Husserl's phenomenology, the author traces Husserl's journey from his early mathematical studies. Phenomenology's core concepts, such as intention and intuition, each contributed to the emergence of a phenomenological approach to mathematics. This Element examines the phenomenological conceptions of natural number, the continuum, geometry, formal systems, and the applicability of mathematics. It also situates the phenomenological approach in relation to other schools in the philosophy of mathematics-logicism, formalism, intuitionism, Platonism, the French epistemological school, and the philosophy of mathematical practice.

Mathematical Intuition

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Author : R.L. Tieszen
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Mathematical Intuition written by R.L. Tieszen and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Philosophy categories.

"Intuition" has perhaps been the least understood and the most abused term in philosophy. It is often the term used when one has no plausible explanation for the source of a given belief or opinion. According to some sceptics, it is understood only in terms of what it is not, and it is not any of the better understood means for acquiring knowledge. In mathematics the term has also unfortunately been used in this way. Thus, intuition is sometimes portrayed as if it were the Third Eye, something only mathematical "mystics", like Ramanujan, possess. In mathematics the notion has also been used in a host of other senses: by "intuitive" one might mean informal, or non-rigourous, or visual, or holistic, or incomplete, or perhaps even convincing in spite of lack of proof. My aim in this book is to sweep all of this aside, to argue that there is a perfectly coherent, philosophically respectable notion of mathematical intuition according to which intuition is a condition necessary for mathemati cal knowledge. I shall argue that mathematical intuition is not any special or mysterious kind of faculty, and that it is possible to make progress in the philosophical analysis of this notion. This kind of undertaking has a precedent in the philosophy of Kant. While I shall be mostly developing ideas about intuition due to Edmund Husser! there will be a kind of Kantian argument underlying the entire book.

Phenomenology And Mathematics

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Author : Mirja Hartimo
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-03-11

Phenomenology And Mathematics written by Mirja Hartimo and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-03-11 with Philosophy categories.

During Edmund Husserl’s lifetime, modern logic and mathematics rapidly developed toward their current outlook and Husserl’s writings can be fruitfully compared and contrasted with both 19th century figures (Boole, Schröder, Weierstrass) as well as the 20th century characters (Heyting, Zermelo, Gödel). Besides the more historical studies, the internal ones on Husserl alone and the external ones attempting to clarify his role in the more general context of the developing mathematics and logic, Husserl’s phenomenology offers also a systematically rich but little researched area of investigation. This volume aims to establish the starting point for the development, evaluation and appraisal of the phenomenology of mathematics. It gathers the contributions of the main scholars of this emerging field into one publication for the first time. Combining both historical and systematic studies from various angles, the volume charts answers to the question "What kind of philosophy of mathematics is phenomenology?"

Didactical Phenomenology Of Mathematical Structures

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Author : Hans Freudenthal
language : en
Publisher: Springer Science & Business Media
Release Date : 1986-11-30

Didactical Phenomenology Of Mathematical Structures written by Hans Freudenthal and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1986-11-30 with Education categories.

The launch ofa new book series is always a challenging eventn ot only for the Editorial Board and the Publisher, but also, and more particularly, for the first author. Both the Editorial Board and the Publisher are delightedt hat the first author in this series isw ell able to meet the challenge. Professor Freudenthal needs no introduction toanyone in the Mathematics Education field and it is particularly fitting that his book should be the first in this new series because it was in 1968 that he, and Reidel, produced the first issue oft he journal Edu cational Studies in Mathematics. Breakingfresh ground is therefore nothing new to Professor Freudenthal and this book illustrates well his pleasure at such a task. To be strictly correct the ‘ground’ which he has broken here is not new, but aswith Mathematics as an Educational Task and Weeding and Sowing, it is rather the novelty oft he manner in which he has carried out his analysis which provides us with so many fresh perspectives. It is our intention that this new book series should provide those who work int he emerging discipline of mathematicseducation with an essential resource, and at a time of considerable concern about the whole mathematics cu rriculum this book represents just such resource. ALAN J. BISHOP Managing Editor vii A LOOK BACKWARD AND A LOOK FORWARD Men die, systems last.

The Mathematical Experience Study Edition

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Author : Philip Davis
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-10-27

The Mathematical Experience Study Edition written by Philip Davis and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-10-27 with Mathematics categories.

Winner of the 1983 National Book Award! "...a perfectly marvelous book about the Queen of Sciences, from which one will get a real feeling for what mathematicians do and who they are. The exposition is clear and full of wit and humor..." - The New Yorker (1983 National Book Award edition) Mathematics has been a human activity for thousands of years. Yet only a few people from the vast population of users are professional mathematicians, who create, teach, foster, and apply it in a variety of situations. The authors of this book believe that it should be possible for these professional mathematicians to explain to non-professionals what they do, what they say they are doing, and why the world should support them at it. They also believe that mathematics should be taught to non-mathematics majors in such a way as to instill an appreciation of the power and beauty of mathematics. Many people from around the world have told the authors that they have done precisely that with the first edition and they have encouraged publication of this revised edition complete with exercises for helping students to demonstrate their understanding. This edition of the book should find a new generation of general readers and students who would like to know what mathematics is all about. It will prove invaluable as a course text for a general mathematics appreciation course, one in which the student can combine an appreciation for the esthetics with some satisfying and revealing applications. The text is ideal for 1) a GE course for Liberal Arts students 2) a Capstone course for perspective teachers 3) a writing course for mathematics teachers. A wealth of customizable online course materials for the book can be obtained from Elena Anne Marchisotto ([email protected]) upon request.

Indiscrete Thoughts

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Author : Gian-Carlo Rota
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-11-03

Indiscrete Thoughts written by Gian-Carlo Rota and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-11-03 with Mathematics categories.

Indiscrete Thoughts gives a glimpse into a world that has seldom been described that of science and technology as seen through the eyes of a mathematician. The era covered by this book, 1950 to 1990, was surely one of the golden ages of science as well as the American university. Cherished myths are debunked along the way as Gian-Carlo Rota takes pleasure in portraying, warts and all, some of the great scientific personalities of the period —Stanislav Ulam (who, together with Edward Teller, signed the patent application for the hydrogen bomb), Solomon Lefschetz (Chairman in the 50s of the Princeton mathematics department), William Feller (one of the founders of modern probability theory), Jack Schwartz (one of the founders of computer science), and many others. Rota is not afraid of controversy. Some readers may even consider these essays indiscreet. After the publication of the essay “The Pernicious Influence of Mathematics upon Philosophy” (reprinted six times in five languages) the author was blacklisted in analytical philosophy circles. Indiscrete Thoughts should become an instant classic and the subject of debate for decades to come.

Mathematics And Its Applications

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Author : Jairo José da Silva
language : en
Publisher: Springer
Release Date : 2017-08-22

Mathematics And Its Applications written by Jairo José da Silva and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-08-22 with Philosophy categories.

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.

Naturalizing Phenomenology

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Author : Jean Petitot
language : en
Publisher: Stanford University Press
Release Date : 1999

Naturalizing Phenomenology written by Jean Petitot and has been published by Stanford University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999 with Philosophy categories.

This ambitious work aims to shed new light on the relations between Husserlian phenomenology and the present-day efforts toward a scientific theory of cognition—with its complex structure of disciplines, levels of explanation, and conflicting hypotheses. The book’s primary goal is not to present a new exegesis of Husserl’s writings, although it does not dismiss the importance of such interpretive and critical work. Rather, the contributors assess the extent to which the kind of phenomenological investigation Husserl initiated favors the construction of a scientific theory of cognition, particularly in contributing to specific contemporary theories either by complementing or by questioning them. What clearly emerges is that Husserlian phenomenology cannot become instrumental in developing cognitive science without undergoing a substantial transformation. Therefore, the central concern of this book is not only the progress of contemporary theories of cognition but also the reorientation of Husserlian phenomenology. Because a single volume could never encompass the numerous facets of this dual aim, the contributors focus on the issue of naturalization. This perspective is far-reaching enough to allow for the coverage of a great variety of topics, ranging from general structures of intentionality, to the nature of the founding epistemological and ontological principles of cognitive science, to analyses of temporality and perception and the mathematical modeling of their phenomenological description. This book, then, is a collective reflection on the possibility of utilizing a naturalized Husserlian phenomenology to contribute to a scientific theory of cognition that fills the explanatory gap between the phenomenological mind and brain.

Weeding And Sowing

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Author : Hans Freudenthal
language : en
Publisher: Springer Science & Business Media
Release Date : 1977-12-31

Weeding And Sowing written by Hans Freudenthal and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1977-12-31 with Education categories.

A title that sounds like poetry, and a subtitle that seems to contradict the title! But the subtitle is right, and originally it was just the title. A strange subtitle, isn’t it? Preface to a Science of Mathematical Education. All sciences – in their prenatal stage – have known this kind of literature: only the term used was not ‘Preface’, but, for instance, ‘Prolegomena’, which * means the same though it sounds less provisional. In fact such works were thicker than the present one, by up to ten times. There is much more that can be said about a science before it comes into being than after; with the first results comes modesty. This is the preface to a book that will never be written: not by me, nor by anybody else. Once a science of mathematical education exists, it will get the preface it deserves. Nevertheless this preface – or what for honesty’s sake I have labelled so – must fulfil a function: the function of accelerating the birth of a science of mathematical education, which is seriously impeded by the unfounded view that such already exists. Against this view I have to argue: it rests on a wrong estimation – both over and under estimation at the same time – of what is to be considered as science.

Philosophy Of Arithmetic

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Author : Edmund Husserl
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Philosophy Of Arithmetic written by Edmund Husserl and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

In his first book, Philosophy of Arithmetic, Edmund Husserl provides a carefully worked out account of number as a categorial or formal feature of the objective world, and of arithmetic as a symbolic technique for mastering the infinite field of numbers for knowledge. It is a realist account of numbers and number relations that interweaves them into the basic structure of the universe and into our knowledge of reality. It provides an answer to the question of how arithmetic applies to reality, and gives an account of how, in general, formalized systems of symbols work in providing access to the world. The "appendices" to this book provide some of Husserl's subsequent discussions of how formalisms work, involving David Hilbert's program of completeness for arithmetic. "Completeness" is integrated into Husserl's own problematic of the "imaginary", and allows him to move beyond the analysis of "representations" in his understanding of the logic of mathematics. Husserl's work here provides an alternative model of what "conceptual analysis" should be - minus the "linguistic turn", but inclusive of language and linguistic meaning. In the process, he provides case after case of "Phenomenological Analysis" - fortunately unencumbered by that title - of the convincing type that made Husserl's life and thought a fountainhead of much of the most important philosophical work of the twentieth Century in Europe. Many Husserlian themes to be developed at length in later writings first emerge here: Abstraction, internal time consciousness, polythetic acts, acts of higher order ('founded' acts), Gestalt qualities and their role in knowledge, formalization (as opposed to generalization), essence analysis, and so forth. This volume is a window on a period of rich and illuminating philosophical activity that has been rendered generally inaccessible by the supposed "revolution" attributed to "Analytic Philosophy" so-called. Careful exposition and critique is given to every serious alternative account of number and number relations available at the time. Husserl's extensive and trenchant criticisms of Gottlob Frege's theory of number and arithmetic reach far beyond those most commonly referred to in the literature on their views.