Introduction to Mathematics (Barnes & Noble Library of Essential Reading)

By Alfred North Whitehead and Amit Hagar

Alfred North Whitehead’s Introduction to Mathematics is one of the most charming books ever written about the queen of the sciences. It gives a delightful and intellectually stimulating exposition of mathematical concepts, the history of their development, and the application thereof. Written with razor-sharp clarity, this little gem of a book is filled with precious insights and lively prose.  Everyone from high-school students and undergraduates to laymen wishing to re-familiarize themselves with the fundamental ideas of mathematics will find this tutorial rewarding.

• Language: English
• Publisher: Barnes & Noble
• Release date: Sep 1, 2009
• ISBN: 9781411431423

Book preview

INTRODUCTION

ALFRED North Whitehead’s Introduction to Mathematics is one of the most charming books ever written about the queen of the sciences. In it the great British mathematician and philosopher gives a delightful and intellectually stimulating exposition of mathematical concepts, the history of their development, and the application thereof. Expertly written with razor-sharp clarity by a brilliant man, Introduction to Mathematics is filled with precious insights and lively prose. A required reading for all those who never understood how anyone could find the subject even remotely interesting, this little gem of a book demonstrates the immense pedagogical talent of its author. While it can be easily adopted as a basis for a short class or tutorial for high-school students or science and humanities undergraduates, laymen wishing to re-familiarize themselves with the fundamental ideas of mathematics will find it more enjoyable than any other contemporary introductory book in the field.

The youngest of four children of an Anglican vicar, Whitehead (1861-1947)—one of the most interesting and imaginative scholars of our era—showed no sign of the genius that he was later in life. As a scholar, his academic interests spanned mathematics, science, and metaphysics, all of which were thoroughly and carefully treated by him with a unique and original style. Guided by the intellectual honesty and the personal touch that is so characteristic of his writings, only few follow Whitehead in the rarely taken path that combines mathematics, physics, and philosophy. His ingenuity and creativity shall remain an inspiration to generations of scholars.

After winning two scholarships for studying mathematics in Trinity College, Cambridge, Whitehead was offered a fellowship as an assistant lecturer there. Despite a poor publication record, Whitehead displayed remarkable teaching skills, as readers of Introduction to Mathematics will surely appreciate, and was soon promoted to lecturer. The shift of emphasis in his career, from teaching to publishing, came with his marriage in late 1890. It was also marked by his decision to renounce Christianity. He himself stated that the biggest factor in his becoming an agnostic was the rapid developments in science; particularly his view that Newton’s physics was false. It may seem surprising to many that the correctness of Newton’s physics could be a major factor in deciding anyone’s religious views. However, one has to understand the complex person that Whitehead was and, in particular, the interest which he was developing in philosophy and metaphysics.

Whitehead left Cambridge in 1910 and went to London, and then to Harvard, where he was the chair of the philosophy department until his retirement. Apart from his metaphysics, he is perhaps best known for his collaboration with Bertrand Russell, who came to Cambridge in 1890 as an undergraduate and was immediately spotted by the talented lecturer. A decade later, the student and the master began collaborating on one of the most ambitious projects in the philosophy of mathematics, Principia Mathematica (1910), which was an attempt to supply mathematics with rigorous logical foundations. When the first volume of this monumental work was finished, Whitehead and Russell began to go their separate ways. Perhaps inevitably, Russell’s anti-war activities during World War I, in which Whitehead lost his youngest son, also led to something of a split between the two men. Nevertheless, they remained on relatively good terms for the rest of their lives. It was then that Whitehead turned his attention to the philosophy of science. This interest arose out of the attempt to explain the relation of formal mathematical theories in physics to their basis in experience, and was sparked by the revolution brought on by Einstein’s GTR, to which he had developed an alternative in another famous book, The Principle of Relativity (1922).

Most of Whitehead’s work prior to 1911 was intended exclusively for a professional audience, namely mathematicians. This was especially evident in his Treatise on Universal Algebra (1898) and in Principia Mathematica, but also in minor publications where Whitehead had mathematical physicists in mind. Introduction to Mathematics, however, was written for a broader readership. Although he no longer could assume such intense mathematical familiarity as with his previous readers, and although the subjects of higher mathematics (such as mathematical logic, group theory, analytic, non-Euclidian, and projective geometries, and integral calculus) could not be dealt with here, mathematics cognoscenti will surely agree that Whitehead’s exposition of mathematics is anything but superficial. Rather, we find him profoundly dealing afresh with philosophical, historical, and applied mathematics, insofar as these topics had already appeared in his earlier works on pure mathematics.

Whitehead declares in the beginning of Introduction to Mathematics that [t]he object of the following chapters is not so much to teach mathematics, but to enable the students from the very beginning of their course to know what the science is about, and why it is necessarily the foundation of exact thought as applied to natural phenomena. The book fulfills this promise by intertwining three basic problems that had occupied Whitehead in his work up to that point: the question about the applicability of mathematics to the physical world; its essence, or nature ; and its unity, generality, and internal structure.

The unique position of mathematics is that it is apparently unconstrained by physical reality: Unlike physical laws, mathematical theorems need no empirical ratification. Their discovery, however, is equally emotionally stirring, as, says Whitehead, the discovery of the Western shore by Columbus, or the Pacific Ocean by Pizzaro. By and large, the development of mathematical concepts is driven by the innate human curiosity and by the pleasure derived from the exercise of the human faculties. But these concepts are widely applied in modern physics, and it is this successful applicability that was always at the center of attention of theoretical physicists and philosophers. The Physics Nobel Laureate Eugene Wigner, for one, called this applicability a miracle in his celebrated paper The Unreasonable Effectiveness of Mathematics in the Natural Sciences (1960), and Whitehead, a trained mathematician, has chosen this miracle to be one of the threads that unite the chapters of Introduction to Mathematics.

The question about the relation of mathematics to physics is an old one. One may even say that the mere possibility of mathematical physics is a major continental divide between rationalism and empiricism—the two famous philosophical schools that constitute early modern philosophy from Descartes to Kant. Interestingly, none of the traditional philosophies of mathematics (e.g., formalism, intuitionism, logicism) ever made an especially serious attempt to explain why mathematics works when applied to the world or even gave much of a sign that they could offer such an explanation. In contrast, this question must have occupied Whitehead since the beginning of his scientific career. His dissertation on Maxwell’s electromagnetic field theory as well as his first two scientific publications on special problems of the hydrodynamics of incompressible fluids testify to his strong awareness to the longstanding debate on the possibility of mathematical physics. As Christoph Wassermann—a contemporary German Whitehead scholar—remarks, it may also explain why Whitehead devotes so much attention to questions concerning the methods and principles of applying mathematical ideas to the phenomena of nature, and why he sees himself obliged to write that all science as it grows to perfection becomes mathematical in its ideas.

Whitehead’s analysis of the applicability of mathematics to the physical world involves a threefold distinction between symbolism and interpretation, between variables and form, and between mathematical ideas and mathematical correlations. Formal as this analysis is, Whitehead, in his unique style, portrays matters in their most humane: It was an act eminently characteristic of the age that Galileo, a philosopher, should have dropped the weights from the leaning tower of Pisa. There are always men of thought and men of action; Mathematical physics is the product of an age which combined in the same men impulses to thought with impulses to action.

The miraculous applicability of mathematics to the physical world is conceived as such because of the unique nature of mathematics itself. This nature, according to Whitehead, is abstract in character: It is the fact that mathematics .... deals with properties and ideas which are applicable to things just because they are things, and apart from any feelings, or emotions, or sensations in any way connected with them. Not surprisingly, says Wassermann, whose research aims to establish a continuity in Whitehead’s work, Whitehead had been engaged with this abstract character more than two decades prior to Introduction to Mathematics. This preoccupation had found an exact establishment in Whitehead’s monumental Principia Mathematica, where he and Russell aimed to show that the concepts thus far regarded as basic to mathematics, e.g., numbers or geometrical points, were not that basic at all, nor constituted by our intuition of nature, but could be deduced from the axioms of mathematical logic. It is noteworthy that even in this new foundational level, the axioms and definitions of formal logic could be formulated without any direct reference to the content of assertions, thereby assuring complete independence from any particular context in which they were applied.

In connection with the abstract nature of mathematics Whitehead specifies three concepts that unite all mathematical disciplines, while using a somewhat personal, religious analogy:

These three notions of the variable, of form, and of generality, compose a sort of mathematical trinity which preside over the whole subject. They all really spring from the same root, namely from the abstract nature of the science.

Of these three, the notion of the variable is the most fundamental. In Whitehead’s earlier works it is defined in the context of the calculus of propositional functions. In Introduction to Mathematics Whitehead brilliantly condenses this notion in the two concepts any and some: "Mathematics as a science commenced when first someone, probably a Greek, proved propositions about any things or about some things, without specification of definite particular things." These concepts represent a non-technical rendering of existential and universal quantifiers, the cornerstones of the rigorous formalism of the theory he dealt with in Principia Mathematica.

Another condensation of Whitehead’s earlier work is found in the notion of generality. Using this notion Whitehead points out that mathematics always seeks expressions which, taking up the notions of the variable and of form, are able to unite as great a subdivision of mathematics as possible, using only one uniform formalism. The best example for such unification and generality is the correspondence between algebra and geometry regarding their main abstractive processes. In Introduction to Mathematics Whitehead shows how just as in algebra the variables are an abstraction from specific numbers, so in geometry variable points are generalized from points. The same can be said of the algebraic transition from special equations to general algebraic forms, and the geometrical extension of figures to general geometrical loci. Only on this more abstract or general level of the two branches of mathematics is unification possible. Variable points and variable numbers are thus united in the idea of coordinates, making it possible to identify algebraic correlations with geometrical loci.

The study of mathematics is apt to commence in disappointment is the startling opening sentence of this wonderful book. Those who shall read through it would probably applaud Whitehead for proving himself wrong. Guided by the author’s firm but gentle hand they would finally arrive at the special state of mind so cherished by the eminent mathematician—that of extending the number of important operations which they can perform without even thinking about them.

Amit Hagar is a philosopher of physics with a Ph.D. from the University of British Columbia, Vancouver. His area of specialization is the conceptual foundations of modern physics, especially in the domains of statistical and quantum mechanics.

CHAPTER I

THE ABSTRACT NATURE OF MATHEMATICS

THE study of mathematics is apt to commence in disappointment. The important applications of the science, the theoretical interest of its ideas, and the logical rigour of its methods, all generate the expectation of a speedy introduction to processes of interest. We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet’s father, this great science eludes the efforts of our mental weapons to grasp it—’Tis here, ’tis there, ’tis gone—and what we do see does not suggest the same excuse for illusiveness as sufficed for the ghost, that it is too noble for our gross methods. A show of violence, if ever excusable, may surely be offered to the trivial results which occupy the pages of some elementary mathematical treatises.

The reason for this failure of the science to live up to its reputation is that its fundamental ideas are not explained to the student disentangled from the technical procedure which has been invented to facilitate their exact presentation in particular instances. Accordingly, the unfortunate learner finds himself struggling to acquire a knowledge of a mass of details which are not illuminated by any general conception. Without a doubt, technical facility is a first requisite for valuable mental activity: we shall fail to appreciate the rhythm of Milton, or the passion of Shelley, so long as we find it necessary to spell the words and are not quite certain of the forms of the individual letters. In this sense there is no royal road to learning. But it is equally an error to confine attention to technical processes, excluding consideration of general ideas. Here lies the road to pedantry.

The object of the following chapters is not to teach mathematics, but to enable students from the very beginning of their course to know what the science is about, and why it is necessarily the foundation of exact thought as applied to natural phenomena. All allusion in what follows to detailed deductions in any part of the science will be inserted merely for the purpose of example, and care will be taken to make the general argument comprehensible, even if here and there some technical process or symbol which the reader does not understand is cited for the purpose of illustration.

The first acquaintance which most people have with mathematics is through arithmetic. That two and two make four is usually taken as the type of a simple mathematical proposition which everyone will have heard of. Arithmetic, therefore, will be a good subject to consider in order to discover, if possible, the most obvious characteristic of the science. Now, the first noticeable fact about arithmetic is that it applies to everything, to tastes and to sounds, to apples and to angels, to the ideas of the mind and to the bones of the body. The nature of the things is perfectly indifferent, of all things it is true that two and two make four. Thus we write down as the leading characteristic of mathematics that it deals with properties and ideas which are applicable to things just because they are things, and apart from any particular feelings, or emotions, or sensations, in any way connected with them. This is what is meant by calling mathematics an abstract science.

The result which we have reached deserves attention. It is natural to think that an abstract science cannot be of much importance in the affairs of human life, because it has omitted from its consideration everything of real interest. It will be remembered that Swift, in his description of Gulliver’s voyage to Laputa, is