Ghadar Jari Hai, Vol 2, Issue 1 2007

26

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.

B

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