Mathematics and Logic

Mark Kac and Stanislaw M. Ulam

Introduction

WHAT IS Mathematics? How was it created and who were and are the people creating and practicing 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 microcosm, but it also has the potentiality of mirroring and modeling all the processes of thought and perhaps all of science. 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 necessary 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 monument to man's desire to probe the workings of his own mind. We shall not undertake 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 continually 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 discuss 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?

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 reciprocated by providing it with new problems and giving it new sources of inspiration.

Technology, too, may have a profound effect on mathematics; having made possible the development of high-speed computers, it has increased immeasurably the scope of experimentation in mathematics itself.

The very foundations of mathematics and of mathematical logic have undergone revolutionary changes in modern times. In Chapter 2 we shall try to explain the nature of these changes.

Throughout mathematical history specific themes constantly recur; their interplay and variations will be illustrated in many examples.

The most characteristic theme of mathematics is that of infinity. We shall devote 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 of judgment 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 obsolescence. A theorem once proved never loses this quality though it may become 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 enormous diversity of its problems and of its modes of application, 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 arithmetic (Peano, Frege, Russell) and of geometry (Hilbert) undertaken in earnest. But even while positive integers and points were accepted uncritically, the process (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 correspondences. (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 correspondences between functions (operators), then to classes of such correspondence, 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 of the formalism of proof that 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 - metamathematics - 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 turn out to be! To some extent then, mathematics 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 conclusions. 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 pseudodefinitions 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 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 (axioms) that involve the words "point," "line," and "plane," and from which other statements involving these undefined words can now be deduced by logic alone. This permits geometry to be taught to a blind man and even to a computer!

This characteristic kind of abstraction, which leads to a nearly total disregard of the physical nature of geometric objects, is not confined to the traditional boundaries of mathematics. Ernst Mach's critical discussion (which owes much to James Clerk Maxwell) of the notion of temperature is a case in point. To define temperature one needs the notions of thermal equilibrium and thermal contact, but to define these in logically acceptable terms is, at least, awkward and perhaps not even possible. An analysis shows that all one really needs is the transitivity of thermal equilibrium; i.e., the postulate. (sometimes called the zeroth law of thermodynamics) that if (A and B) and (A and C) are in thermal equilibrium, then so are (B and C). For completeness one also needs a kind of converse of the zeroth law, namely that if A, B, and C are in thermal equilibrium, then so are (A and B) and (A and C). Again, as in geometry, one need not know the (logically) precise meaning of terms, but only how to combine them into meaningful (i.e., allowable) statements.

But while we may operate reliably with undefined (and perhaps even undefinable) objects and concepts, these objects and concepts are rooted in apparent physical (or at least sensory) reality. Physical appearances suggest and even dictate the initial axioms; the same apparent reality guides us in formulating questions and problems.

To exist (in mathematics), said Henri Poincare, is to be free from contradiction. But mere existence does not guarantee survival. To survive in mathematics requires a kind of vitality that cannot be described in purely logical terms.

In the following chapters we discuss a number of problems that not only have survived but have given birth to some of the most fruitful developments in mathematics. They range from the concrete to the abstract and from the very simple to the relatively complex. They were chosen to illustrate both the objects and the methods of mathematics, and should convince the reader that there is more to pure mathematics than is contained in Bertrand Russell's definition that "Pure mathematics is the class of all propositions of the form ‘p implies q,’ where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants."

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