Title: Taming the Infinite – The
Story of Mathematics
Author: Ian Stewart
Publisher: Quercus,
2009 (First published 2008)
ISBN: 978-1-84724-768-1
Pages: 373
Professor Ian Stewart is a noted
scholar with over 170 published papers and a world-renowned popularizer of
mathematics. He has authored many books on the subject. This book describes the
journey of mathematics from ancient prehistorical times to the present in a
neatly classified version which lists important branches of mathematics
individually and traces their origins and growth over the ages. Mathematics
divides the society into two camps, those who love it and those who detest it.
Unfortunately, most of those who dislike it had had exposure only to
arithmetic, which forms only a small part of the gigantic whole. The subject
has grown far and wide encompassing all walks of life including science,
engineering, economy and even biological systems. Stewart succinctly narrates
the development of each idea in turn and in detail.
Numbers form the foundations on
which civilizations are built. All ancient ones, like Mesopotamia and Egypt had
one, but the number system of the former was more advanced – some of which we
still employ, like 365 days to a year and 60 minutes to an hour. They
introduced a notation for numbers for the first time and used it to record
planetary data and to calculate eclipses. Geometry, which was another important
segment, originated from the springs of Greek intellect. Pythagoras, Eudoxus,
Euclid and Archimedes were the founding fathers of the new science, who
emphasized on logical proof to demonstrate an assertion which forms the basis
of modern mathematical theory even now. Eratosthenes found the circumference of
earth by measuring the deviation of the shadow cast by a pole from the
vertical. His estimate was 250,000 stadia, but unfortunately we don’t know how
much a stadium was long. However, his logic was impeccable and contained
elements of trigonometry. Number notation developed independently in many
countries, but the most widely used system at present came from India and
diffused by Arabs around the known world. Leonardo of Pisa, also known as
Fibonacci learned it while trading with Arabs in North Africa in the 13th
century and introduced it to Europe where it spread rapidly after Renaissance.
This development developed Simon Stevin to formulate the now familiar decimal
point notation in 16th century, as an alternative to fractional
notation.
Moving on to algebra, we find that
traces of algebraic formulations are seen in ancient Babylonian clay tablets,
including a general recipe for solving quadratic equations. But the term algebra,
comes from Arabic al-jabr which means ‘adding equal amounts to both
sides’, proposed by scholar Muhammad ibn Musa al-Khwarizmi around 820 CE. He is
the eponym for the term algorithm too. Renaissance Europe proved to be a
fertile ground for new concepts like analytical geometry and number theory.
From here onwards, what we now study in schools and colleges are the sole
products of great mathematicians and scientists from 16th century. A
fundamental change in the treatment of geometry came along after Rene Descartes
proposed the coordinate system. Till that time, figures and shapes were the
results of rotations, or slicing of some shapes by others, but then the forms
themselves were recognized to be graphs of functions with profound
significance.
Number theory, a crucial component
of modern mathematics also developed around this time, particularly under the
able Fermat who got the impetus from Greek mathematicians Euclid and
Diophantus. This was further developed and systematized by Gauss, the Prince of
Mathematicians. The theory was without any practical applications until digital
communication came along in late 20th century where it is
extensively used for encrypting and decrypting messages. However, of all
mathematical concepts which directly apply to nature, none is more profound
than calculus, the principle of variation. Newton and Leibniz developed it from
first principles quite independently, but Leibniz published first. In the
modern setting, this fact would have clinched the deal in favour of Leibniz.
But Newton being Newton and the British being British, it kindled a controversy
which raged for a century between the British and Continental mathematicians
with the result that England turned into a backwater as far as mathematics was
concerned. The symbols we use today in calculus was proposed by Leibniz, which
comes in two forms – the differential and the integral. The former deals with
rates of change of a quantity, tangent to a curve and finding maxima and
minima, while the latter is concerned with calculation of area or volume under
a curve or surface. All branches of exact sciences use calculus in one form or
the other.
Stewart then gives an account of
modern mathematics, which developed after 1800, like imaginary numbers, group
theory, topology and abstract mathematics, which is not very absorbing for the
general reader. The curious fact we get to know is that though these ideas seem
so pedantic or not relevant in a practical sense, they quite unexpectedly turn
up to provide a solution to a vexing problem or supply proof to a long standing
unresolved conjecture. A case in point is Fermat’s last theorem, which was
proved by Andrew Wiles in 1995, after 350 years since it was first proposed,
using concepts developed in the 20th century. Details of the
interesting quest for proof may be obtained from Simon Singh’s impressive book,
Fermat’s Last Theorem, which was reviewed earlier in this blog.
The final part of the book deals
with new vistas opened up in mathematics during 20th century. The
quantity of theories and new areas developed during the last two centuries in
the field outnumber all that has gone before in the previous 4000 years. Chaos
theory, complexity theory and algorithmic theory are only a few arrows in the
mathematician’s quiver. Many of them don’t find much use at present, but as was
the case with several other theories which proved to be immensely practical,
this phenomenon is not something new.
The book is neatly pigeonholed
into component categories. The author convincingly answers the question often
posed by some against the teaching of arithmetic to students in the present era
when electronic calculators and computers obviate the need for manual
calculations. He argues that though most people don’t need arithmetic to
perform calculations as such, it is essential for those future scientists and
engineers who will be building newer computers and calculators. Modern
civilization would quickly break down if arithmetic is not taught and
technology allowed to stagnate. The time span covered by the book is immense –
4000 years, right from the beginning in the uncertain light of a prehistoric
dawn to modern concepts like chaos theory.
Nevertheless, the book is burdened
with several drawbacks to be pointed out against, the least of which is the
carelessness in faithfully reproducing a critical number. The base of natural
logarithm is given as 2.7128… (on page 101) where it should be 2.71828…. The
error is obviously a printing mistake, but when you introduce the number as the
one of the most important numbers in mathematics, you have to be more
careful. The book supplies a lot of informatory asides – in fact a
little more than what was necessary. While providing details of the topic under
discussion, the multitude of such boxes detracts readers from pursuing the main
thread. These include biographical sketches of mathematicians on which Stewart
does not forget to include every female mathematician who most often had only a
fleeting relevance to the theme under survey. The volume is arranged into
several chapters, perhaps mutually exclusive. It provides for subject-wise
continuity but not the chronological coherence when taken as a whole. It
appears to be a collection of chapters, not the development of an integrated
idea. The most disheartening feature is that the book turns complex and devoid
of interest after the midpoint. Only serious readers or students of higher
mathematics might find the part useful.
The book is recommended only for
serious readers who are mathematically inclined in an earnest way.
Rating: 2 Star
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