**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 to 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.