All for Nought
By accident, it
records the oldest "0" in
India for which one can assign a definite
date...
Bill Casselman
University of British Columbia, Vancouver, Canada
cass at math.ubc.ca
The history of zero is a bit complicated.
So is the history of "0".
The scholarly literature on the subject
is only tentative because much of the
historical record is very sparse, whereas
the popular literature is unfortunately
and frequently both confident and in
error.
The first problem to come along is
deciding exactly what one means by "zero" or,
for that matter, "0". Is
it a number in the mathematical sense
- that is to say, the cardinality of
the empty set? The length of a point?
The result of subtracting 1 from 1?
I am not going to engage in such deep
matters, but rather in a much more
pedestrian business. The digit "0" was
a basic part of decimal place value
notation. There is no doubt that it
was invented in India, but exactly
how and for what purpose is unclear,
and probably always will be.
This is not at first sight very complicated
mathematics, but in truth it took far
longer for humans to develop a convenient
notation for calculating than it did
for them to develop rigourous mathematical
reasoning. The apparent simplicity
of our current system is indeed a sign
of its elegance. It often happens that
the best mathematics, once seen, is
seen to be obvious.
I'm going to cover a very small part
of the story.
Mathematical
tourism
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The city of Gwalior in India is located on the main
rail line south from Delhi, just a bit below Agra, the
site of the Taj Mahal. It is in the far north of Madyha
Pradesh, and lies very near where the three northern
states of Rajasthan, Uttar Pradesh, and Madhya Pradesh
meet. It is in a region that has few hills and -
for most of the year, like much of India - little
water.
Gwalior does not seem to be well known outside India,
although it is certainly mentioned favourably in
guide books. The reason for this favourable mention
is that it happens to be the site of one of the most
impressive of all medieval forts in a country full of
impressive forts. The fort is famous inside the
country not only for its size and beauty, but also
for being the site of the last and futile stand of
the Princess (Rani) of Jhansi during the rising of
1857-1858 against the British.
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From an
appendix in Cunningham's Reports
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The city is now both sprawling
and crowded, with a population
of perhaps 2,000,000, but until
recently it was relatively
small, and designed to be small.
The fort occupies a plateau
in what is now the center of
the city, but was once on its
western boundary. The plateau
is about 300 feet high built
of basalt, rising steeply from
the plain below. It is a bit
less than two miles in length
from north to south, maybe
an average of a half mile from
west to east.
The site is of mathematical
interest because of what is
written on a tablet recording
the establishment of a small
9th century Hindu temple on
the eastern side of the plateau
(marked by the  on
the nineteenth century map
at the left). By accident,
it records the oldest "0" in
India for which one can assign
a definite date.
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The temple
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The temple is dated to 876
A. D. and is much older than
the current fort, whose construction
was begun in the late 15th
century, although it was built
quite a while after the original
one constructed on the plateau.
It is, like many temples in
India, monolithic -
that is to say, originally
carved out of one single chunk
of stone. It was dedicated
to Vishnu, but is no longer
an active site of worship.
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The temple is named Chatur-bhuja,
that of the four-armed god.
Who was reponsible for the (literal) defacement of the statue
is not known to me.
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The tablet
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Just inside the inner
chamber, on Vishnu's right hand
side, is the dedication tablet.
The tablet records the date (in
the local era, which started in
57 B. C.), the dimension of a land
grant to a neighbouring temple,
and the size of a daily gift of
flowers to be paid for from an
endowment made to this temple.
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... the whole town gave
to the temple ... which Alla,
the son of Vaillabhatta, had
caused to be built ... a piece
of land ... 270 hastas
in length ...
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... and 187 hastas
in breadth, for a flower
garden ...
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... the town gave
in perpetual endowment
... for a daily gift
of 50 garlands of flowers ...
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What is surprising about these numbers
is that they are so similar to what
modern civilization uses currently.
The more you learn about how our current
number symbols developed - transmitted
from the Hindus to the Persians, then
to Mediterranean Islam, and differently
in East and West - the more remarkable
this appears. Here, for comparison,
are some numbers from the bus system
of Mumbai:
Photographs taken
by Leslie Saper
Go figure.
The background
What the Gwalior tablet shows is that
by 876 A. D. our current place-value
system with a base of 10 had become
part of popular culture in at least
one region of India.
We know almost nothing of how this
decimal place-value notation came about,
although there are many suggestive
facts. One feature of Hindu culture
in the middle centuries of the first
millennium was that its texts were
largely in verse, and preserved through
oral tradition. It is hard to fit a
useful numerical notation into such
a scheme, and in fact what we see is
a large literature, written down only
much later than it originated, with
numbers - often very, very large numbers
- written in a kind of decimal place-value
notation, but in words instead of symbols.
Furthermore, the demands of the metric
of the verses required that the exact
words chosen to represent a given digit
might vary from one point to another,
so as to scan correctly. Whether this
usage overlay more convenient calculation
with symbols is not known to us, although
it is almost inconceivable that it
did not.
Another problem is that the climate
of India is harsh. Paper was introduced
to India late, and until then the materials
on which things were written were birch
bark in the north and palm leaves in
the south. These are both extremely
fragile. There are many extant manuscripts
written on these, but nearly all of
relatively recent date.
One of the more intriguing questions
about the origin of decimal place-value
notation is what connection it had
to a much older tradition from a nearby
region. The Babylonians began writing
in about 3000 B.C., and had the good
fortune to write on clay tablets, which
can last for a very long time. We have
extensive records from several thousand
years of their development. They used
an extremely sophisticated place-value
system, remarkably much the one we
use today, from very roughly 2000 B.C.
on, but with a base of 60 instead of
10, and without "0". All
the evidence that I am aware of suggests
that this was technology acquired only
by an elite group through rigourous
training. This somewhat ambiguous notation
persisted to about 300 B.C. when Babylonian
astronomical tables started to incorporate
a symbol that to some extent performed
as zero, that is to say as a sign to
indicate a space between two "digits".
This was adopted in modified form by
Greek astronomers after the conquests
of Alexander, and this science in turn
was transmitted (along with astrology!)
to India sometime in the first few
centuries of the current era. Exactly
how these transmissions occurred is
lost to us.
References
- Alexander Cunningham, Four reports
made during the years 1862-63-64-65,
Archaeological Survey of India,
1865.
Section XVI of volume II contains
the only substantial history of the
city and principality of Gwalior
that I have been able to locate.
Cunningham mentions the temple and
the tablet as well as its date 933,
but does not mention the other numbers.
- E. Hultsch, The two inscriptions
of the Vaillabhattasvamin temple
at Gwalior, Epigraphia
Indica I, pages 154--161.
There are two inscriptions in the
temple at Gwalior, one just above
the entrance in a small domed porch,
and the other on the left inside
wall. The first is, as Hultsch says,
written in a more attractive style
(and, he also says, a more stylish
Sanskrit), but has no mathematical
interest, contrary to what is sometimes
said.
Hultsch's article contains a transcription
of the tablet into modern Sanskrit
script, an English translation, and
a reproduction of a rubbing of the
tablet. Aside from the numerals,
the tablet does not seem to be of much historical
interest.
- George Ifrah, The universal
history of numbers, Penguin,
2000.
This book is useful, perhaps even
indispensable, for someone interested
in the history of numbers. It is
a huge compendium of material, some
fascinating and much - alas - of
very little interest. One problem
is that the author fails to warn
you when he is relying on secondary
material and when on first hand.
This problem actually arises in his
account of the temple of Gwalior
- he has apparently misread Hultsch's
transliteration and thought that
the numbering of the Sanskrit verses
found there was part of the inscription.
One very valuable feature of Ifrah's
books is the extensive bibliography.
- Robert Kaplan, The nothing that is, Oxford University Press, 1999.
This book mentions Gwalior, but it is an uninteresting account,
and seems to be passing on only
third-hand information
(as I have said, a frequent phenomnon
in popular accounts of the history
of science). I doubt
that he has bothered to read Hultsch's article.
but instead seems to rely principally on Ifrah.
Even taking this into account,
the sketches of
the numerals at Gwalior have strangely little
resemblance to the originals.
- Shunya's
collection of photographs of Gwalior will
give you a good idea of the beauty
of the fort as well as a look at
the rest of the town. The eastern
approach to the fort is shown in
the image Pedestrian
entrance, and the temple
is just at the bend in the road
at middle left.
Acknowledgements
Dipendra Prasad of the Tata Institute
in Mumbai arranged my visit to Gwalior
at very short notice, and in particular
arranged for me to meet Renu Jain,
head of the mathematics department
at Jiwaji University in Gwalior. She,
at even shorter notice, gathered a
small group to guide me, among them
V. P. Saxena (mathematics) and A. K.
Singh (archaeology). I wish that I
had had more time to talk with them
about the history of Gwalior.
Images
Images by the author (mostly) and
Leslie Saper. Personal use of the above
images is allowed. Inquiries about
publication of any of the above images
should be sent to the AMS Public Awareness
Office paoffice@ams.org.
Bill
Casselman
University of British Columbia, Vancouver,
Canada
cass
at math.ubc.ca
NOTE: Those who can access JSTOR can find some of the
papers mentioned above there. For those with access, the American Mathematical
Society's MathSciNet can be used to get
additional bibliographic information and reviews of some these materials. Some of the
items above can be accessed via the ACM
Portal, which also provides bibliographic services.
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