Ghadar Jari Hai, Vol 2, Issue 1 2007
Book Review
Cultural Foundations of Mathematics
It has been common understanding that mathematical proof based on deduction is
universal and is the ultimate proof and also that mathematical truths are eternal universal
truths. C K Raju argues that this is a narrow European view of mathematics and the Indian
view was very different and empirical. Thus he has raised the important issue of cultural
foundation of mathematics. We present here a summary of his startling new book.
ritish scholars have known
since 1832 that traditional In-
dian mathematicians had de-
veloped a way to handle infinite se-
ries, a key component of the calculus.
However, Western historians have
denied that this amounted to the
calculus proper, and many aspects of
this fascinating Indian contribution
to science have remained unclear for
the last 175 years.
In this book, Raju asks four
questions that have not been asked
before. (1) How were infinite series
useful to Indian society? (2) Did the
Indian infinite series amount to the
calculus? (3) Was this Indian math-
ematics transmitted to Europe be-
fore Newton and Leibniz? (4) Does
the traditional Indian approach to
mathematics have any practical ap-
plications today? Raju’s answers are
as follows:
1) The main source of wealth in
India is agriculture which depends
on the monsoons. The monsoons are
“erratic”, so a good calendar is indis-
pensable to Indian agriculture. The
traditional Indian calendar identi-
fies the months of Sawan and Bha-
don as the rainy season, unlike the
common Gregorian calendar which
has no rainy season. Traditional In-
dian festivals like Rakhi and Holi do
not occur on “fixed” days of the Gre-
gorian calendar (such as 25 Decem-
ber or 15 August), and are related
to agriculture. Constructing this
specialised Indian calendar required
complex planetary models. Calibrat-
ing this calendar across the length
and breadth of India required precise
knowledge of the size of the earth,
and of ways of determining latitude
and longitude of any place and all
this required precise trigonometric
values. (This knowledge was use-
ful also for navigation and overseas
trade with Alexandria, Arabs, Africa,
and China was also a key source of
wealth in India.) The required trigo-
nometric values were developed in
India since the Surya Siddhanta (3rd
c.) and Aryabhata (5th c.). Over the
next thousand years these trigono-
metric values were gradually made
more precise, and that led to the
development of the Indian infinite
series. Thus, Raju concludes that
the social utility for agriculture and
navigation drove the development of
the Indian infinite series.
2) Western scholars have dubbed
the Indian infinite series as “pre-cal-
culus”, claiming that the calculus
proper emerged with the “funda-
mental theorem of calculus” which
was absent in India. Raju responds
to this criticism in various ways. (a)
First, he questions the premise that
mathematics means theorem-prov-
ing rather than calculation. This re-
quires a re-examination of all West-
ern history and philosophy. Raju
argues that “Euclid” is a historical
concoction, and that the Elements,
attributed to “Euclid”, is actually a
Neo-Platonic religious book. It was
radically reinterpreted by Christian
rational theologians after the 12th c.
CE, to support their agenda of con-
verting Arabs, during the Crusades.
To this end they declared reason
(and mathematics) to be universal.
However, since Buddhists and Jains
have used a different logic from that
used in mathematical proof today,
this notion of proof can never be uni-
versal. Different principles of proof
or pramana were used in Indian
tradition, where mathematics is not
singled out as requiring a special
sort of proof. Raju concludes that