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

158 Pages·2004·1.76 MB·English

by  Alfred North Whitehead

Table of Contents

Title Page
Copyright Page
Introduction

CHAPTER I - THE ABSTRACT NATURE OF MATHEMATICS
CHAPTER II - VARIABLES
CHAPTER III - METHODS OF APPLICATION
CHAPTER IV - DYNAMICS
CHAPTER V - THE SYMBOLISM OF MATHEMATICS
CHAPTER VI - GENERALIZATIONS OF NUMBERS
CHAPTER VII - IMAGINARY NUMBERS
CHAPTER VIII - IMAGINARY NUMBERS(Continued)
CHAPTER IX - COORDINATE GEOMETRY
CHAPTER X - CONIC SECTIONS
CHAPTER XI - FUNCTIONS
CHAPTER XII - PERIODICITY IN NATURE
CHAPTER XIII - TRIGONOMETRY
CHAPTER XIV - SERIES
CHAPTER XV - THE DIFFERENTIAL CALCULUS
CHAPTER XVI - GEOMETRY
CHAPTER XVII - QUANTITY

BIBLIOGRAPHY
ENDNOTES
INDEX
SUGGESTED READING

Introduction and Suggested Reading Copyright © 2005 by Barnes & Noble Books

Originally published in 1911

This edition published by Barnes & Noble Publishing, Inc.

All rights reserved. No part of this book may be used or reproduced in any manner whatsoever without
written permission of the Publisher.
Cover Design by Stacey May

2005 Barnes & Noble Publishing, Inc.

ISBN 0-7607-6588-X
eISBN : 978-1-41143142-3

Printed and bound in the United States of America

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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