What is “Vedic” Mathematics?
I am referring here
what has popularly come to be understood by that name, and not what it should
strictly mean. Going by the propaganda reports “Vedic Maths”’ is an “amazingly
compact and powerful system of calculation”, and one also hears things like
“once you have learnt the 16 sutras by heart, you can solve any long problem
orally”, and so on. Incredible hyperbole indeed !! Actually it is nothing of
the sort. It is essentially a compilation of some tricks in simple arithmetic
and algebra. The term “trick” is not used here in any pejorative sense, meaning
to convey deceit of some sort. It just means a short cut, with a bit of
psychological appeal to it. “Vedic” mathematics is indeed a collection of short
cut procedures, applicable to only an assortment of special situations, and
quite an incoherent one at that. It is thus really not a “system”, though there
is a popular misconception to that effect. The so called Vedic mathematics (VM)
first made its appearance in print in the book ‘Vedic Mathematics’, first
published in 1965, authored by Sri Bharati Krishna Tirtha, who was the
Shankaryacharya of the Puri Mutt from 1925 until he passed away in 1960. The
book was published posthumously by some disciples of the Shankaracharya. The
book presents some “sutras”, claimed to be from the ancient Vedic lore, and
describes some procedures, supposedly coming from the sutras, to deal with some
specific arithmetical/algebraic problems. Some background about the genesis of
the book may be found in various write-ups included in the beginning of the
book. Apparently Tirthaji had earlier been giving demonstrations on the theme
for some decades, and there is a specific mention of some regular classes held
at the Nagpur University in 1952. The write-ups provide also a variety of other
details. Though on the whole one finds many specific details missing, and many
statements there would not stand logical scrutiny, the accounts are readable to
get an overall picture. In what follows we shall examine various issues about
VM, both internal to it and concerning its significance in the present overall
context.
Are
the “sutras” from the Vedas?
The Vedas have been
well-documented and studied thoroughly by scholars, both Indian and western.
There is nothing akin to the sutras of VM in the genuine Vedic literature.
Indeed, there isn’t any text in Sanskrit containing the sutras of VM. The
sutras, adding up to barely 50 words in Sanskrit, first made their appearance
in Tirthaji’s book, which is in English. Even if there were to be a text that
was somehow escaped attention of the interested scholars, it would have
similarities in style of language, presentation, overall contents etc. It is
far from the case here. It has already been pointed out by the General Editor
Dr. V.S. Agrawala in his Foreword to Tirthaji’s book, concluding that “the
style of language [of the sutras] points to their discovery by Sri Swamiji
himself ”. When confronted by Prof. K.S. Shukla, a scholar of ancient Indian
mathematics, to show the sutras in a copy of the Atharvaveda (as claimed by
Tirthaji), the Shankaracharya said that they were only in his own Parishishta
to the Atharvaveda!. Even a Shankaracharya can not add to the Vedas; or else
they would cease to be ancient texts. Unfortunately however Tirthaji seems, as
seen from the preface of the book he authored, to be given to using the term
“Vedic” in a vague all-encompassing way, while at the same time treating it as
ancient. Such a mixup is illogical and inadmissible even from a basic
intellectual point of view.
Do
the sutras convey mathematical ideas?
There is a naive
belief, fostered by the author and later by others, that the “sutras”
incorporate in them some mathematical ideas. This is blatantly absurd. For
instance “Ekadhikena purvena”, which just means “by one more than the previous
one” does not convey any full sense in any context, mathematical or otherwise.
That if it can have the mathematical meaning as claimed, it can also be about
family planning; while I meant to highlight the illogic of the contention, one
of the well-known English proponents of VM thought that was a serious idea
giving another application of the “sutra”! A string of words which can be
interpreted in widely varied contexts in essence can not be said to have any of
the meanings. Additional contextual factors can sometimes enable one to focus
on a particular meaning, but no such context exists for the sutras of VM, which
as we have seen are stand-alone strings of words.
What
do the tricks of VM signify?
The so called sutras
are nothing but tabs stuck to certain procedures, as short phrases or titles,
which help recall the procedure or trick. The trick itself had to be known
separately. The tabs could just as well have been in any other language. Their
being in Sanskrit would (and was perhaps designed to) impress people, but has
no contextual significance. It is absurd to imagine that the 16 sutras hold the
key for everything. They stand for specific procedures in each case, with an
occasional variation here and there. Leave alone anything outside, the sutras
do not even cover many things from the book itself; the author keeps adding
what he calls “sub-sutra” and even with that many things are left out. Some of
the “sub-sutras” (“veshtanam”, “vilokanam”) are too general even to tab a
procedure. (“vilokanam” could as well be an instruction to watch out at the traffic
lights!). They serve no purpose other than to show off with a bit of Sanskrit
to the laity. On the whole, the exercise may be compared to calling, say “kanda
batata thalipithu” a “sutra” from say Jnaneswari, and claiming that Jnaneswari
teaches you how to make masala dosas, magically just with those words, without
having to describe anything of the recipe or procedure involved!
How
ancient are the tricks?
The tricks by and large
could not be very old. In the first place for a trick to be known the relevant
concepts need to exist. The decimal point representations, involved in the
opening trick, came to be introduced only in the sixteenth century. There are
no decimal fractions even in the works of Bhaskaracharya from the 12 th century
(or any of the earlier Indian mathematicians over the centuries like Aryabhata,
Brahmagupta, Sridhara and so on) leave alone the ancient Vedic times. Many
other operations involved in VM also have no relation to ancient Indian
mathematics, all the way from the Shulbasutras from the Vedic times, to the
Kerala mathematics of the fourteenth to sixteenth centuries; Tirthaji’s book
also shows acute lack of knowledge of the genuine ancient Indian mathematics,
given especially the context that the author is trying to be superlative about
Indian achievements.
The kind of
calculations dealt with could not have been of frequent occurrence until the
twentieth century. Tirthaji’s exposition of the tricks (perhaps only some of
them to begin with) came in around the middle of the twentieth century. They
may in fact have come into general use just around then, and Tirthaji may have
been the first expositor. It is interesting to note that similar compilation
also came about in Germany at the hands of Trachtenberg, at around the same
time.
What
is role of VM in mathematics?
Mathematics in all its
majesty consists of deep ideas concerning numbers, computation, shapes,
symmetry, movement, interrelation between various structures etc., and has
several branches such as, number theory, group theory, topology, algebraic
geometry, harmonic analysis, combinatorial mathematics - to name just a few. VM
on the other hand is a handful of tricks involving elementary arithmetic and
algebra. Moreover these lack generality in applicability, and theoretical
coherence. Tirthaji in his book talks of applications to calculus, and there
are attempts by proponents of VM to show that it is applicable to other areas
as well, but the results are shabby and superficial, and do not reach anywhere
near the heart of the matter. They have hardly found any acceptance in the
professional mathematical community, as can be seen from even a cursory look at
such professional sources as the Mathematical Reviews. In mathematics it is
important to focus on concepts and their interrelations. VM simply tends to
spoon-feed a few concoctions which are far from being needful to the
mathematical diet of the students.
How
worthwhile is it to pursue VM?
The only use of tricks
of VM is to help speed calculations in certain special situations. In
particular the main aspects of clarification of concepts, logical thinking etc.
are not served by VM. Excessive focus on VM can therefore be positively harmful
to the overall growth of a child’s mathematical talents. Being able to
calculate fast has lost whatever importance it had, with the advent of
calculators. It may be compared with developing the ability to run fast on a
special race track. While it could indeed be useful on occasion, only a few
would find it worth investing effort and money on such an endeavour, when for
all your practical needs you can use a vehicle to go much faster, rather than
slog over a bumpy and rough road to which you would not have got used to. Both
of these are worthwhile if you either have a flare for it, or your particular
context demands it in some way. However you have to consider how much time,
effort and money are worth spending on it, and not be carried away by the
propaganda around. It is often claimed that VM is well-appreciated in other
countries, and even taught in some schools in UK etc.. St. James Independent
School, London which is often quoted in this context, is a school run by the
’School of Economic Science’ which is, as I learnt from a letter I received
from Mr. James Glover, the Head of Mathematics at the School (and writer of the
recent books on VM), “engaged in the practical study of Advaita philosophy”. Of
course it is a matter of individual choice as to how to view such a connection,
but the point I wish to make is that general claims of appreciation abroad can
thus be very misleading. If attaining skill in speed arithmetic is the aim, VM
is really but one stream. There are also other systems, going back to the “Trachtenberg
speed arithmetic” from the 1950’s and its later day variants. One ought to take
a comprehensive and objective view, and not be carried away by the load of
“ancient wisdom” ballyhoo, which is not even remotely true in this instance.
While of course the
utility of VM can not be totally refuted (as with many other things), its
cost-effectiveness is a serious issue. It is in the nature of things in VM that
the principle of diminishing returns sets in very fast. The examples that you
learn at the outset, while being introduced to it in introductory exposures,
could easily be the last worthwhile things you may learn, making the rest of
the pursuit quite futile.
Credit:
S.G. Dani Tata Institute of Fundamental Research Homi Bhabha Road, Mumbai 400 005
