BOOK REVIEW
A Backward Look at Modern Algebra
Judith Grabiner
Unknown Quantity: A Real and Imaginary History of Algebra.
John Derbyshire. viii + 374 pp. Joseph Henry Press, 2006. $27.95.
It is hard to know what to make of John Derbyshire's new book,
Unknown Quantity. The subtitle suggests that it is a
history of algebra, and developments are indeed presented in
chronological order. However, it is really three books: a history of
algebraic ideas, a set of "mathematical primers"
explaining the ideas, and a lot of interesting stories about
algebraists—many true, some embellished and some misleading.
The book is at its best in explaining the mathematics of the period
from 1590 to 1860—from the invention of general algebraic
symbolism by François Viète, or Vieta, through groups
of permutations and their application to the solution of polynomial
equations, to the invention of vector spaces and fields. Reading
those sections can teach somebody who doesn't know the algebra what
the basic ideas are and something about the historical relationship
among them.
Derbyshire is also the author of the well-received 2003 book
Prime Obsession, a popularized account of the Riemann
Hypothesis and some of the people who worked on it. But a caution is
in order about his current book: Although Unknown Quantity
claims to be a history, Derbyshire, when he hears a good story,
sometimes prefers telling it to checking it.
Here are just two examples of the way he handles evidence:
Derbyshire says that when Descartes chose the letter x to
represent the principal unknown, he did so for the convenience of
the printer, because x is less used in French than
y or z. In fact, though, x is used
more often than y in French, at least according to
cryptography texts by Fletcher Pratt and Helen Fouché Gaines
that I consulted. Derbyshire's source for his assertion is
Classic Math, a book for high-school students, whose
author, Art Johnson, gives no footnote for the claim but who may
have misunderstood a conjecture made in 1905—almost 300 years
after Descartes—by Gustav Eneström that appears in a book
in Johnson's bibliography, Florian Cajori's History of
Mathematical Notations. Eneström supposed that
x was chosen because it occurs more often than y
and z, and printers therefore would have had more
x's on hand.
Also, Derbyshire repeats the canard that J. J. Sylvester might have
been a homosexual, in part because—as if this constituted
appropriate supporting evidence—Sylvester wrote poetry and
"enjoyed singing in a high register" (Sylvester was indeed
a tenor, and serious enough to have taken singing lessons from the
French composer Charles Gounod).
In addition, on several occasions when Derbyshire doesn't seem to
know much about some relevant historical development—say, the
degree to which the ninth-century algebra of al-Khwarizmi goes
beyond that of ancient Babylonia, or the meaning of the
"geometrical algebra" of Book II of Euclid's
Elements—he dismisses it with a putdown.
There is also considerable carelessness and indifference to detail.
For instance, in Figure 15-1, the caption labels the title page on
the left as belonging to the 1941 (first) edition of Garrett
Birkhoff and Saunders MacLane's A Survey of Modern
Algebra—despite the fact that the words "Revised
Edition" appear clearly below the title. Contrary to a claim
Derbyshire makes, Euclid was not the first to give the standard
proof that the square root of 2 is irrational (apparently Derbyshire
is unaware of Aristotle's earlier discussion in the Prior
Analytics). Nor is Immanuel Kant the ultimate source of the
distinction between analytic and synthetic methods in geometry (a
claim that neglects Newton, for instance).
The amount of space Derbyshire devotes to a historical topic or a
mathematician seems to vary with how interesting he thinks it is,
rather than how important it is; for instance, six full pages are
devoted to the biography of Alexander Grothendieck (admittedly, he
has led an interesting and colorful life), but only one sentence to
the biography of Richard Dedekind. The endnotes are even more chatty
and tongue-in-cheek than the text. There is no bibliography,
although the assiduous reader can compile one from the endnotes, and
it will be a mixture of the first-rate (Michael Artin's
Algebra, Harold M. Edwards's Galois Theory,
histories of algebra by Isabella Bashmakova and by B. L. van der
Waerden, to name some examples) and the idiosyncratic (such as Rudy
Rucker'sMathenauts and Shakespeare's Henry VI, Part
I).
The mathematical primers Derbyshire provides in Unknown
Quantity are generally valuable, but some are too elementary
for a reader trained in science, whereas others are too difficult
and too brief for nonalgebraists. Unfortunately, some important
results whose proofs are accessible to a general reader, and which
would clarify the exposition of algebraic ideas, are not proved
here. For instance, Derbyshire states that the rational numbers are
countable (not explaining how we know this) but that the real
numbers are not; he does not even refer the reader to a proof
elsewhere. Again, he asserts that the general cubic equation can be
transformed into one with no x ^{2} term, but he
does not justify the assertion by reviewing the binomial expansion
for (a + b)^{3}. When he covers more
advanced material of the late 19th century and the 20th century,
things come by too fast and are sometimes only mentioned and not
explained; for example, although he gamely tries, he does not fully
specify for a general reader how Galois theory addresses the
solvability of polynomial equations. And someone ignorant of
category theory will not be helped by the explanation here, although
Derbyshire does have fun with the term "forgetful functor."
Nonetheless, a reader who wants to learn some theory of equations
and modern algebra in a relatively painless way will find this book
attractive. The explanations of many algebraic topics are accessible
and clear, especially those of the following: how Vieta's new
general symbolic notation demonstrates the relations between roots
and coefficients of polynomials; symmetric polynomials and
solvability; the roots of unity; how Paolo Ruffini's proof of the
unsolvability of the quintic worked; how studying permutations of
roots of polynomial equations gave rise to group theory; how linear
transformations and matrices are related; the nature of quaternions
and octonions; what an invariant is; an introduction to algebraic
geometry; what a vector space is; and what the differences are
between groups, division rings and fields. Those explanations also
make clear why mathematicians cared about these problems and how
these concepts were used. The anecdotes, although not reliable,
certainly make the book fun to read.
Still, a reader wanting to learn the history of algebra accurately
while also learning the algebra in question would be better served
by reading the sections on algebraic topics in Victor Katz's
magisterial History of Mathematics: An Introduction
(Addison-Wesley, second edition, 1998). Anyone seeking
fascinating—and reliable—biographical material on past
algebraists should start with the Dictionary of Scientific
Biography (a source Derbyshire himself has used) and, for
contemporary mathematicians, the highly readable interviews in
Gerald L. Alexanderson and Donald J. Albers's Mathematical
People and More Mathematical People.
But perhaps I'm being a little harsh about the contents of the book
not being entirely dependable. After all, Derbyshire's subtitle,
"a real and imaginary history of algebra," does give the
reader fair warning.