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Jairo José Da Silva Mathematics and Its Applications a Transcendental-Idealist Perspective Synthese Library Synthese Library 385 Studies in Epistemology, Logic, Methodology, and Philosophy of Science Jairo José da Silva Mathematics and Its Applications A Transcendental-Idealist Perspective Synthese Library Studies in Epistemology, Logic, Methodology, and Philosophy of Science Volume 385 Editor-in-Chief Otávio Bueno, University of Miami, Department of Philosophy, USA Editors Berit Brogaard, University of Miami, USA Anjan Chakravartty, University of Notre Dame, USA Steven French, University of Leeds, UK Catarina Dutilh Novaes, University of Groningen, The Netherlands More information about this series at http://www.springer.com/series/6607 Jairo José da Silva Mathematics and Its Applications A Transcendental-Idealist Perspective Jairo José da Silva Researcher - CNPq, Ministry of Science, Technology, Innovation and Communication - Brazil Brasilia, DF, Brazil Synthese Library ISBN 978-3-319-63072-4 ISBN 978-3-319-63073-1 (eBook) DOI 10.1007/978-3-319-63073-1 Library of Congress Control Number: 2017945752 © Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland To the memory of my sweet Shoshana April 1, 2006–August 7, 2016 A little fat star that shone across my sky Contents 1 Introduction.............................................................................................. 1 2 Phenomenology........................................................................................ 11 3 Logic.......................................................................................................... 67 4 Numbers.................................................................................................... 93 5 Sets............................................................................................................. 131 6 Space.......................................................................................................... 153 7 Structures.................................................................................................. 185 8 Science....................................................................................................... 217 9 Final Considerations................................................................................ 251 References......................................................................................................... 273 vii Chapter 1 Introduction The philosophy of mathematics is not, or should not be a therapeutics; its task is not, or should not be to sanitize mathematics. Nor should it be a sub-product of founda- tional projects, or part of mathematics itself. Mathematics can take care of itself; it has been doing it marvelously well for the last 4000 years or so. Philosophy of mathematics is a chapter of epistemology; its task, that of making sense of mathe- matics as actually practiced and traditionally conceived as a legitimate aspect of human knowledge. Mathematics is extremely useful as an instrument of scientific investigation, not only in the empirical sciences, but in the human and social sci- ences as well. Mathematics permeates science to the point of sometimes being essential to it. Not only as a “language” for expressing ideas that could be expressed otherwise but itself a source of ideas that cannot be expressed in any other way. It is the philosopher’s task to explain what kind of knowledge mathematical knowledge is, what its objects are and how they can be accessed, but also to clarify and justify, in some sense of the term, the applicability of mathematics in science. What kind of connection mathematical objects have with objects of other sciences, empirical real- ity in particular, that it is sometimes impossible, as in modern physics, to have a theory of the latter independently of the former? A philosophy of mathematics that does not answer this question conveniently, or does not even raise it, fails its pur- pose. This will be my guiding question here. What is mathematics, I ask, which can be so widely and successfully applied in virtually all fields of science, besides so many aspects of our human lives? How can that which seems to be the epitome of pure reasoning play such an important role in the organization of our experience of reality, and do it so well to the point of predicting the outcome of future experi- ences? This cannot be a mystery; the question must have a rational and ordinary answer. Philosophy of mathematics, moreover, just like any intellectual endeavor, must be practiced without prejudices and preconceptions. Philosophical theses cannot precede or condition philosophical inquiry. One cannot embrace a readymade ontol- ogy without considering the specificity of mathematical existence, for example. Unfortunately, it is not so in the traditional philosophies of mathematics. Despite © Springer International Publishing AG 2017 1 J.J. da Silva, Mathematics and Its Applications, Synthese Library 385, DOI 10.1007/978-3-319-63073-1_1 2 1 Introduction their differences, which are very real, the three or four “schools” that come imme- diately to mind when we think of philosophy of mathematics, namely, logicism, formalism, intuitionism, and maybe also predicativism, besides being all born as parts of foundational projects, share important philosophical prejudices that obscure the comprehension of the real nature of mathematical objects, mathematical knowl- edge and the applicability of mathematics, i.e. the ontology, the epistemology and the pragmatics of mathematics. These schools of thought, which have been fighting each other for over a century without any clear sign of advantage to any of the litigants, are too well known to require yet another exposition. However, digging into the soil where they are rooted and from where they draw their nutrients can be enlightening. One conclusion stands out, no matter their differences, all traditional philosophies of mathematics share a naturalist, or more specifically, an empiricist conception of existence which is particularly misleading when applied to mathematics. Mathematical existence cannot be thought on the model of existence of the natural object. By so doing, the traditional foundational approaches to the philosophy of mathematics condemn themselves to failure. This book will fulfill its goal to a substantial amount if I con- vince my reader of the truth of this claim. The fact that the philosophy of mathemat- ics is recognized and practiced as a legitimate and independent topic of philosophical investigation mainly in analytic philosophy has the consequence that no matter the particular orientation they take, philosophies of mathematics inevitably share the philosophical partis-pris that define this philosophical tradition, empiricism in particular. Consider the following quote1: To account for the indubitability, objectivity and timelessness of mathematical results, we are tempted to regard them as true descriptions of a Platonic world outside of space-time. This leaves us with the problem of explaining how human beings can make contact with this reality. Alternatively, we could abandon the idea of a Platonic realm and view mathematics as simply a game played with formal symbols. This would explain how human beings can do mathematics, since we are game players par excellence, but it leaves us with the task of specifying the rules of the game and explaining why the mathematical games is so useful – we don’t ask chess players for help in designing bridges. This quote illustrates perfectly the alternatives that analytic philosophy with its empiricist parti-pris gives to the philosopher of mathematics. From an empiricist perspective, the model of existence is the existence of the empirical object, physical or mental. Physical objects exist and subsist independently in space and time; men- tal objects, on the other hand, are temporal entities whose existence depends on the mental lives of subjects. In empiricist philosophies of mathematics, there are, con- sequently, three modes of existence available for mathematical objects, all modeled on the empirical mode of existence. A mathematical object can exist independently, albeit not in space-time, subject-dependently as
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