**The History of Zero: How Ancient Mesopotamia Invented the Mathematical Concept of Nought and Ancient India Gave It Symbolic Form**

If the ancient Arab world had closed its
gates to foreign travelers, we would have no medicine, no astronomy, and no
mathematics — at least not as we know them today.

Central to humanity’s quest to grasp the
nature of the universe and make sense of our own existence is zero, which began
in Mesopotamia and spurred one of the most significant paradigm shifts in human
consciousness — a concept first invented (or perhaps discovered) in pre-Arab
Sumer, modern-day Iraq, and later given symbolic form in ancient India. This
twining of meaning and symbol not only shaped mathematics, which underlies our
best models of reality, but became woven into the very fabric of human life,
from the works of Shakespeare, who famously winked at zero in

*King Lear*by calling it “an O without a figure,” to the invention of the bit that gave us the 1s and 0s underpinning my ability to type these words and your ability to read them on this screen.
Mathematician

**Robert Kaplan**chronicles nought’s revolutionary journey in**It is, in a sense, an archetypal story of scientific discovery, wherein an abstract concept derived from the observed laws of nature is named and given symbolic form. But it is also a kind of cross-cultural fairy tale that romances reason across time and space***The Nothing That Is: A Natural History of Zero*
Kaplan writes:

If you look at zero you see nothing; but look
through it and you will see the world. For zero brings into focus the great,
organic sprawl of mathematics, and mathematics in turn the complex nature of
things. From counting to calculating, from estimating the odds to knowing
exactly when the tides in our affairs will crest, the shining tools of
mathematics let us follow the tacking course everything takes through
everything else – and all of their parts swing on the smallest of pivots, zero

With these mental devices we make visible the
hidden laws controlling the objects around us in their cycles and swerves. Even
the mind itself is mirrored in mathematics, its endless reflections now
confusing, now clarifying insight.

[…]

As
we follow the meanderings of zero’s symbols and meanings we’ll see along with
it the making and doing of mathematics — by humans, for humans. No god gave it
to us. Its muse speaks only to those who ardently pursue her.

With an eye to the eternal question of
whether mathematics is discovered or invented — a question famously debated
by Kurt
Gödel and the Vienna Circle — Kaplan observes:

The
disquieting question of whether zero is out there or a fiction will call up the
perennial puzzle of whether we invent or discover the way of things, hence the
yet deeper issue of where we are in the hierarchy. Are we creatures or
creators, less than – or only a little less than — the angels in our power to
appraise?

Like all transformative inventions, zero
began with necessity — the necessity for counting without getting bemired in
the inelegance of increasingly large numbers. Kaplan writes:

Zero began its career as two wedges pressed
into a wet lump of clay, in the days when a superb piece of mental engineering
gave us the art of counting.

[…]

The story begins some 5,000 years ago with
the Sumerians, those lively people who settled in Mesopotamia (part of what is
now Iraq). When you read, on one of their clay tablets, this exchange between
father and son: “Where did you go?” “Nowhere.” “Then why are you late?”, you
realize that 5,000 years are like an evening gone.

The Sumerians counted by 1s and 10s but also
by 60s. This may seem bizarre until you recall that we do too, using 60 for
minutes in an hour (and 6 × 60 = 360 for degrees in a circle). Worse, we also
count by 12 when it comes to months in a year, 7 for days in a week, 24 for
hours in a day and 16 for ounces in a pound or a pint. Up until 1971 the
British counted their pennies in heaps of 12 to a shilling but heaps of 20
shillings to a pound.

Tug
on each of these different systems and you’ll unravel a history of customs and
compromises, showing what you thought was quirky to be the most natural thing
in the world. In the case of the Sumerians, a 60-base (sexagesimal) system most
likely sprang from their dealings with another culture whose system of weights
— and hence of monetary value — differed from their own.

Having to reconcile the decimal and
sexagesimal counting systems was a source of growing confusion for the
Sumerians, who wrote by pressing the tip of a hollow reed to create circles and
semi-circles onto wet clay tablets solidified by baking. The reed eventually
became a three-sided stylus, which made triangular cuneiform marks at varying
angles to designate different numbers, amounts, and concepts. Kaplan
demonstrates what the Sumerian numerical system looked like by 2000 BCE:

This cumbersome system lasted for thousands
of years, until someone at some point between the sixth and third centuries BCE
came up with a way to wedge accounting columns apart, effectively symbolizing
“nothing in this column” — and so the concept of, if not the symbol for, zero
was born. Kaplan writes:

In
a tablet unearthed at Kish (dating from perhaps as far back as 700 BC), the
scribe wrote his zeroes with three hooks, rather than two slanted wedges, as if
they were thirties; and another scribe at about the same time made his with
only one, so that they are indistinguishable from his tens. Carelessness? Or
does this variety tell us that we are very near the earliest uses of the
separation sign as zero, its meaning and form having yet to settle in?

But zero almost perished with the
civilization that first imagined it. The story follows history’s arrow from
Mesopotamia to ancient Greece, where the necessity of zero awakens anew. Kaplan
turns to Archimedes and his system for naming large numbers, “myriad” being the
largest of the Greek names for numbers, connoting 10,000. With his notion
of

*orders*of large numbers, the great Greek polymath came within inches of inventing the concept of powers, but he gave us something even more important — as Kaplan puts it, he showed us “how to think as concretely as we can about the very large, giving us a way of building up to it in stages rather than letting our thoughts diffuse in the face of immensity, so that we will be able to distinguish even such magnitudes as these from the infinite.”
This concept of the infinite in a sense
contoured the need for naming its mirror-image counterpart: nothingness.
(Negative numbers were still a long way away.) And yet the Greeks had no word
for zero, though they clearly recognized its spectral presence. Kaplan writes:

Haven’t we all an ancient sense that for
something to exist it must have a name? Many a child refuses to accept the
argument that the numbers go on forever (just add one to any candidate for the
last) because names run out. For them a googol — 1 with 100 zeroes after it —
is a large and living friend, as is a googolplex (10 to the googol power, in an
Archimedean spirit).

[…]

By
not using zero, but naming instead his myriad myriads, orders and periods,
Archimedes has given a constructive vitality to this vastness — putting it just
that much nearer our reach, if not our grasp.

Ordinarily, we know that naming
is what gives meaning to existence. But
names are given to things, and zero is not a thing — it is, in fact, a
no-thing. Kaplan contemplates the paradox:

Names
belong to things, but zero belongs to nothing. It counts the totality of what
isn’t there. By this reasoning it must be everywhere with regard to this and
that: with regard, for instance, to the number of humming-birds in that bowl
with seven — or now six — apples. Then what does zero name? It looks like a
smaller version of Gertrude Stein’s Oakland, having no there there.

Zero, still an unnamed figment of the
mathematical imagination, continued its odyssey around the ancient world before
it was given a name. After Babylon and Greece, it landed in India. The first
surviving written appearance of zero as a symbol appeared there on a stone
tablet dated 876 AD, inscribed with the measurements of a garden: 270 by 50,
written as “27°” and “5°.” Kaplan notes that the same tiny zero appears on
copper plates dating back to three centuries earlier, but because forgeries ran
rampant in the eleventh century, their authenticity can’t be ascertained. He
writes:

We
can try pushing back the beginnings of zero in India before 876, if you are
willing to strain your eyes to make out dim figures in a bright haze. Why
trouble to do this? Because every story, like every dream, has a deep point,
where all that is said sounds oracular, all that is seen, an omen.
Interpretations seethe around these images like froth in a cauldron. This deep
point for us is the cleft between the ancient world around the Mediterranean
and the ancient world of India.

But if zero were to have a high priest in
ancient India, it would undoubtedly be the mathematician and astronomer
Āryabhata, whose identity is shrouded in as much mystery as Shakespeare’s.
Nonetheless, his legacy — whether he was indeed one person or many — is an
indelible part of zero’s story.

Kaplan writes:

Āryabhata wanted a concise way to store
(not calculate with) large numbers, and hit on a strange scheme. If we hadn’t
yet our positional notation, where the 8 in 9,871 means 800 because it stands
in the hundreds place, we might have come up with writing it this way:
9T8H7Te1, where T stands for ‘thousand’, H for “hundred” and Te for “ten” (in
fact, this is how we usually pronounce our numbers, and how monetary amounts
have been expressed: £3.4s.2d). Āryabhata did something of this sort, only one
degree more abstract.

He made up nonsense words whose syllables
stood for digits in places, the digits being given by consonants, the places by
the nine vowels in Sanskrit. Since the first three vowels are a, i and u, if
you wanted to write 386 in his system (he wrote this as 6, then 8, then 3) you
would want the sixth consonant, c, followed by a (showing that c was in the
units place), the eighth consonant, j, followed by i, then the third consonant,
g, followed by u: CAJIGU. The problem is that this system gives only 9 possible
places, and being an astronomer, he had need of many more. His baroque solution
was to double his system to 18 places by using the same nine vowels twice each:
a, a, i, i, u, u and so on; and breaking the consonants up into two groups,
using those from the first for the odd numbered places, those from the second
for the even. So he would actually have written 386 this way: CASAGI (c being
the sixth consonant of the first group, s in effect the eighth of the second
group, g the third of the first group)…

There
is clearly no zero in this system — but interestingly enough, in explaining it
Āryabhata says: “The nine vowels are to be used in two nines of places” — and
his word for “place” is “kha”. This

*kha*later becomes one of the commonest Indian words for zero. It is as if we had here a slow-motion picture of an idea evolving: the shift from a “named” to a purely positional notation, from an empty place where a digit can lodge to “the empty number”: a number in its own right, that nudged other numbers along into their places.
Kaplan reflects on the multicultural
intellectual heritage encircling the concept of zero:

While having a symbol for zero matters,
having the notion matters more, and whether this came from the Babylonians
directly or through the Greeks, what is hanging in the balance here in India is
the character this notion will take: will it be the idea of the absence of any
number — or the idea of a number for such absence? Is it to be the mark of the
empty, or the empty mark? The first keeps it estranged from numbers, merely
part of the landscape through which they move; the second puts it on a par with
them.

In the remainder of the fascinating and
lyrical

**, Kaplan goes on to explore how various other cultures, from the Mayans to the Romans, contributed to the trans-civilizational mosaic that is zero as it made its way to modern mathematics, and examines its profound impact on everything from philosophy to literature to his own domain of mathematics. Complement it with this Victorian love letter to mathematics and the illustrated story of how the Persian polymath Ibn Sina revolutionized modern science.***The Nothing That Is*
BRAIN PICKINGS

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