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Mathematics and Logic Mark Kac and Stanislaw M. Ulam Mathematics and Logic Mark Kac and Stanislaw M. Ulam DOVER PUBLICATIONS, INC., NEW YORK Copyright © 1968 by Encyclopredia Britannica, Inc. All rights reserved under Pan American and International Copyright Conventions. Published in Canada by General Publishing Company, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario. Published in the United Kingdom by Constable and Company, Ltd., 3 The Lanches­ ters, 162-164 Fulham Palace Road, London W6 9ER. This Dover edition, first published in 1992, is an unabridged, unaltered republica­ tion of the work first published by Frederick A. Praeger, New York, 1968, under the title Mathematics and Logic: Retrospect and Prospects and as "a Britannica Perspec­ tive prepared to commemorate the 200th anniversary of Encyclopredia Britannica." Manufactured in the United States of America Dover Publications, Inc., 31 East 2nd Street, Mineola, N. Y. 11501 Library ofCongress Cataloging-in-Publication Data Kac, Mark. Mathematics and logic / Mark Kac and Stanislaw M. Ulam. p. em. Originally published: New York: Praeger, 1968, in series: Britannica perspective. ISBN 0-486-67085-6 (pbk.) 1. Mathematics-Popular works. I. Ulam, Stanislaw M. II. Title. QA93.K24 1992 51O-dc20 91-40429 CIP Introduction WHAT IS MATHEMATICS? How was it created and who were and are the people creating and practising it? Can one describe its development and its role in the history of scientific thinking and can one predict its future? This book is an attempt to provide a few glimpses into the nature of such questions and the scope and the depth of the subject. Mathematics is a self-contained nucrocosm, but it also has the potentiality of mirroring and modeling all the processes of thought and perhaps all of sci­ ence. It has always had, and continues to an ever increasing degree to have, great usefulness. One could even go so far as to say that mathematics was neces­ sary for man's conquest of nature and for the development of the human race through the shaping of its modes of thinking. For as far back as we can reach into the record of man's curiosity and quest of understanding, we find mathematics cultivated, cherished, and taught for transmittal to new generations. It has been considered as the most definitive expression of rational thought about the external world and also as a monu­ ment to man's desire to probe the workings of his own mind. We shall not un­ dertake to define mathematics, because to do so would be to circumscribe its domain. As the reader will see, mathematics can generalize any scheme, change it, and enlarge it. And yet, every time this is done, the result still forms only a part of mathematics. In fact, it is perhaps characteristic of the discipline that it develops through a constant self-examination with an ever increasing degree of consciousness of its own structure. The structure, however, changes con­ tinually and sometimes radically and fundamentally. In view of this, an attempt to define mathematics with any hope of completeness and finality is, in our opinion, doomed to failure. We shall try to describe some of its development historically and to survey briefly high points and trenchant influences. Here and there attention will focus on the question of how much progress in mathematics depends on "invention" and to what extent it has the nature of "discovery." Put differently, we shall dis­ cuss whether the external physical world, which we perceive with our senses and observe and measure with our instruments, dictates the choice of axioms, definitions, and problems. Or are these in essence free creations of the human mind, perhaps influenced, or even determined, by its physiological structure? v vi Introduction Like other sciences, mathematics has been subject to great changes during the past fifty years. Not only has its subject matter vastly increased, not only has the emphasis on what were considered the central problems changed but the tone and the aims of mathematics to some extent have been transmuted. There is no doubt that many great triumphs of physics, astronomy, and other "exact" sciences arose in significant measure from mathematics. Having freely borrowed the tools mathematics helped to develop, the sister disciplines recipro­ cated by providing it with new problems and giving it new sources of inspira­ tion. Technology, too, may have a profound effect on mathematics; having made possible the development of high-speed computers, it has increased immea­ surably the scope of experimentation in mathematics itself. The very foundations of mathematics and of mathematical logic have under­ gone revolutionary changes in modem times. In Chapter 2 we shall try to ex­ plain the nature of these changes. Throughout mathematical history specific themes constantly recur; their in­ terplay and variations will be illustrated in many examples. The most characteristic theme of mathematics is that of infinity. We shall de­ vote much space to attempting to show how it is introduced, defined, and dealt with in various contexts. Contrary to a widespread opinion among nonscientists, mathematics is not a closed and perfect edifice. Mathematics is a science; it is also an art. The criteria ofjudgment in mathematics are always aesthetic, at least in part. The mere truth of a proposition is not sufficient to establish it as a part of mathematics. One looks for "usefulness," for "interest," and also for "beauty." Beauty is subjective, and it may seem surprising that there is usually considerable agreement among mathematicians concerning aesthetic values. In one respect mathematics is set apart from other sciences: it knows no ob­ solescence. A theorem once proved never loses this quality though it may be­ come a simple case of a more general truth. The body of mathematical material grows without revisions, and the increase of knowledge is constant. In view of the enonnous diversity of its problems and of its modes of appli­ cation, can one discern an order in mathematics? What gives mathematics its unquestioned unity, and what makes it autonomous? To begin with, one must distinguish between its objects and its method. The most primitive mathematical objects are positive integers 1, 2, 3, ... Perhaps equally primitive are points and simple configurations (e.g., straight lines, triangles). These are so deeply rooted in our most elementary experiences going back to childhood that for centuries they were taken for granted. Not until the end of the 19th century was an intricate logical examination of arith- Introduction vii metic (Peano, Frege, Russell) and ofgeometry (Hilbert) undertaken in earnest. But even while positive integers and points were accepted uncritically, the proc­ ess (so characteristic of mathematics) of creating new objects and erecting new structures was going on. From objects one goes on to sets of these objects, to functions, and to cor­ respondences. (The idea of a correspondence or transformation comes from the still elementary tendency of people to identify similar arrangements and to abstract a common pattern from seemingly different situations.) And as the process of iteration continues, one goes on to classes of functions, to corre­ spondences between functions (operators), then to classes of such correspon­ dence, and so on at an ever accelerating pace, without end. In this way simple objects give rise to those of new and ever growing complexity. The method consists mainly ofthe formalism of proofthat hardly has changed since antiquity. The basic pattern still is to start with a small number of axioms (statements that are taken for granted) and then by strict logical rules to derive new statements. The properties of this process, its scope, and its limitations have been examined critically only in recent years. This study-metamathe­ matics-is itself a part of mathematics. The object of this study may seem a rather special set of rules-namely, those of mathematical logic. But how all­ embracing and powerful these tum out to be! To some extent then, mathe­ matics feeds on itself. Yet there is no vicious circle, and as the triumphs of mathematical methods in physics, astronomy, and other natural sciences show, it is not sterile play. Perhaps this is so because the external world suggests large classes of objects of mathematical work, and the processes of generalization and selection of new structures are not entirely arbitrary. The "unreasonable effectiveness of mathematics" remains perhaps a philosophical mystery, but this has in no way affected its spectacular successes. Mathematics has been defined as the science of drawing necessary conclu­ sions. But which conclusions? A mere chain of syllogisms is not mathematics. Somehow we select statements that concisely embrace a large class of special cases and consider some proofs to be elegant or beautiful. There is thus more to the method than the mere logic involved in deduction. There is also less to the objects than their intuitive or instinctive origins may suggest. It is in fact a distinctive feature of mathematics that it can operate effectively and efficiently without defining its objects. Points, straight lines, and planes are not defined. In fact, a mathematician of today rejects the attempts of his predecessors to define a point as something that has "neither length nor width" and to provide equally meaningless pseudo­ defihitions of straight lines or planes. The point of view as it evolved through centuries is that one need not know what things are as long as one knows what statements about them one is viii Introduction allowed to make. Hilbert's famous Grundlagen der Geometrie begins with the sentence: "Let there be three kinds of objects; the objects of the first kind shall be called 'points,' those of the second kind 'lines,' and those of the third 'planes.' "That is all, except that there follows a list of initial statements (ax­ ioms) that involve the words "point," "line," and "plane," and from which other statements involving these undefined words can now be deduced by logic alone.

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