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Παρασκευή 3 Φεβρουαρίου 2017

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

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

“If you look at zero you see nothing; but look through it and you will see the world.”

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

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 The Nothing That Is: A Natural History of Zero (public library).
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

Art by Paul Rand from Little 1 by Ann Rand, a vintage concept book about the numbers

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?

Art by Shel Silverstein from The Missing Piece Meets the Big O

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.”

“Archimedes Thoughtful” by Domenico Fetti, 1620

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.

Āryabhata (art by K. Ganesh Acharya)

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 The Nothing That Is,
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.

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