Strict Finitism and the Logic of Mathematical Applications
This book intends to show that radical naturalism (or physicalism), nominalism and strict finitism account for the applications of classical mathematics in current scientific theories. The applied mathematical theories developed in the book include the ba
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SYNTHESE LIBRARY
STUDIES IN EPISTEMOLOGY, LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE
Editors-in-Chief: VINCENT F. HENDRICKS, University of Copenhagen, Denmark JOHN SYMONS, University of Texas at El Paso, U.S.A.
Honorary Editor: JAAKKO HINTIKKA, Boston University, U.S.A.
Editors: DIRK VAN DALEN, University of Utrecht, The Netherlands THEO A.F. KUIPERS, University of Groningen, The Netherlands TEDDY SEIDENFELD, Carnegie Mellon University, U.S.A. PATRICK SUPPES, Stanford University, California, U.S.A. ´ JAN WOLENSKI, Jagiellonian University, Kraków, Poland
VOLUME 355 For further volumes: http://www.springer.com/series/6607
Strict Finitism and the Logic of Mathematical Applications
by
Feng Ye Peking University, Beijing, P. R. China
123
Prof. Feng Ye Department of Philosophy Peking University 100871 Beijing P. R. China [email protected]
ISBN 978-94-007-1346-8 e-ISBN 978-94-007-1347-5 DOI 10.1007/978-94-007-1347-5 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2011930748 c Springer Science+Business Media B.V. 2011
No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
In almost all mathematical applications, the physical entities we deal with are finite and discrete. Macroscopically, the universe is believed to be finite; microscopically, current well-established physics theories describe only things above the Planck scale (about 10−35 m, 10−45 s etc.). Except for the theories about the microscopic structure of spacetime below the Planck scale, all scientific theories in a broad sense, from physics to cognitive psychology and population studies, describe only finite things within the finite range from the Planck scale to the cosmological scale. In these theories, infinity and continuity in mathematics are idealizations to gloss over microscopic details or generalize beyond an unknown finite limit, in order to get simplified mathematical models of finite and discrete natural phenomena. Scientists are guided by their intuitions and experiences in searching for appropriate infinite and continuous mathematical models to simulate finite and discrete phenomena, and they rely on observations and experiments to confirm that their models can represent those phenomena sufficiently accurately. However, as logicians and philosophers, we have a few questions: 1. What are the logically minimum premises that imply a scientific conclusion about a finite and discrete phenomenon in the universe, and in particular, are the mathematical axioms that apparently refer to infinite mathematical entities logically strictly indispensable f
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