- Derbyshire, John.
Unknown Quantity.
Washington: Joseph Henry Press, 2006.
ISBN 0-309-09657-X.
-
After exploring a renowned mathematical conundrum (the
Riemann
Hypothesis) in all its profundity in
Prime Obsession
(June 2003), in this book the author recounts
the history of algebra—an intellectual quest sprawling
over most of recorded human history and occupying some
of the greatest minds our species has produced.
Babylonian cuneiform tablets
dating from the time of
Hammurabi, about 3800 years ago,
demonstrate solving quadratic equations, extracting square
roots, and finding
Pythagorean
triples. (The methods in the Babylonian texts are recognisably
algebraic but are expressed as “word problems” instead of
algebraic notation.)
Diophantus,
about 2000 years later, was the first to write equations in
a symbolic form, but this was promptly forgotten. In fact,
twenty-six centuries after the Babylonians were solving quadratic
equations expressed in word problems,
al-Khwārizmī
(the word “algebra” is derived from the title of his
book,
الكتاب
المختصر
في حساب
الجبر
والمقابلة
al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala,
and “algorithm” from his name) was
solving quadratic equations in word problems.
It wasn't until around 1600 that anything resembling
the literal symbolism of modern algebra came into use, and
it took an intellect of the calibre of
René Descartes
to perfect it. Finally, equipped with an expressive notation,
rules for symbolic manipulation, and the slowly dawning realisation
that this, not numbers or geometric figures, is ultimately
what mathematics is about, mathematicians embarked on a
spiral of abstraction, discovery, and generalisation which has never ceased to
accelerate in the centuries since. As more and more mathematics
was discovered (or, if you're an anti-Platonist, invented), deep
and unexpected connections were found among topics once considered
unrelated, and this is a large part of the story told here, as
algebra has “infiltrated” geometry, topology,
number theory, and a host of other mathematical fields while,
in the form of algebraic geometry and group theory, providing the
foundation upon which the most fundamental theories of modern
physics are built.
With all of these connections, there's a strong temptation for an
author to wander off into fields not generally considered part of
algebra (for example, analysis or set theory); Derbyshire is admirable
in his ability to stay on topic, while not shortchanging the reader
where important cross-overs occur. In a book of this kind, especially
one covering such a long span of history and a topic so broad, it is
difficult to strike the right balance between explaining the
mathematics and sketching the lives of the people who did it, and
between a historical narrative and one which follows the evolution of
specific ideas over time. In the opinion of this reader, Derbyshire's
judgement on these matters is impeccable. As implausible as it may
seem to some that a book about algebra could aspire to such a
distinction, I found this one of the more compelling page-turners I've
read in recent months.
Six “math primers” interspersed in the text provide the
fundamentals the reader needs to understand the chapters which
follow. While excellent refreshers, readers who have never
encountered these concepts before may find the primers difficult to
comprehend (but then, they probably won't be reading a history of
algebra in the first place). Thirty pages of end notes not only cite
sources but expand, sometimes at substantial length, upon the main
text; readers should not deprive themselves this valuable lagniappe.
January 2007 