By Maria Popova (brainpickings.org)
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 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:
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:
[…]
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:
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:
[…]
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:
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:
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:
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:
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.
