The Marginalian
The Marginalian

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

The Invention of Zero: How Ancient Mesopotamia Created 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 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.

BP

The Thing Itself: C.S. Lewis on What We Long for in Our Existential Longing

The Thing Itself: C.S. Lewis on What We Long for in Our Existential Longing

Nothing kidnaps our capacity for presence more cruelly than longing. And yet longing is also the most powerful creative force we know: Out of our longing for meaning came all of art; out of our longing for truth all of science; out of our longing for love the very fact of life. We may give this undertone of being different names — Susan Cain calls it “the bittersweet” and Portuguese has the lovely word saudade: the vague, constant longing for something or someone beyond the horizon of reality — but we recognize it in our marrow, in the strata of the soul beyond the reach of words.

No one has explored the paradoxical nature of longing more sensitively than the philosopher, storyteller, beloved Narnia creator, and modern mystic C.S. Lewis (November 29, 1898–November 22, 1963) in a sermon he delivered on June 8, 1941, which later lent its title to his 1949 collection of addresses The Weight of Glory (public library).

Illustration by Margaret C. Cook for a rare 1913 edition of Walt Whitman’s Leaves of Grass. (Available as a print.)

Lewis — who thought deeply about the significance of suffering and the secret of happiness — writes:

This desire for our own far off country [is] the secret which hurts so much that you take your revenge on it by calling it names like Nostalgia and Romanticism and Adolescence; the secret also which pierces with such sweetness that when, in very intimate conversation, the mention of it becomes imminent, we grow awkward and affect to laugh at ourselves; the secret we cannot hide and cannot tell, though we desire to do both. We cannot tell it because it is a desire for something that has never actually appeared in our experience. We cannot hide it because our experience is constantly suggesting it, and we betray ourselves like lovers at the mention of a name. Our commonest expedient is to call it beauty and behave as if that had settled the matter. Wordsworth’s expedient was to identify it with certain moments in his own past. But all this is a cheat. If Wordsworth had gone back to those moments in the past, he would not have found the thing itself, but only the reminder of it; what he remembered would turn out to be itself a remembering.

As Lewis considers the illusory nature of these shorthands for our longing, we are left with the radiant intimation that “the thing itself” is not something we reach for, something beyond us, but something we are:

The books or the music in which we thought the beauty was located will betray us if we trust to them; it was not in them, it only came through them, and what came through them was longing. These things — the beauty, the memory of our own past — are good images of what we really desire; but if they are mistaken for the thing itself they turn into dumb idols, breaking the hearts of their worshipers. For they are not the thing itself; they are only the scent of a flower we have not found, the echo of a tune we have not heard, news from a country we have never visited.

For Lewis, who was religious, this notion of “the thing itself” — the ultimate object of longing — was anchored in his understanding of God. For me, it calls to mind Virginia Woolf’s exquisite epiphany about the meaning of art and life, found while strolling through her flower-garden:

Behind the cotton wool is hidden a pattern… the whole world is a work of art… there is no Shakespeare… no Beethoven… no God; we are the words; we are the music; we are the thing itself.

BP

A Lei for Your Losses: Joan Didion on the Art of Letting Go

A Lei for Your Losses: Joan Didion on the Art of Letting Go

“We cannot deeply love anything without becoming vulnerable to loss,” Judith Viorst wrote in her defense of life’s necessary losses, “and we cannot become separate people, responsible people, connected people, reflective people without some losing and leaving and letting go.”

We know that loving anything or anyone is a contract with loss, and yet we love anyway, our lives a ceaseless osmosis of holding on and letting go, an endless apprenticeship in the art of losing.

What we keep of our losses scaffolds who we become. What we let die within us — the tortured patterns, the ill-serving habits, the limiting beliefs — makes it possible to be reborn. And yet it is not easy, letting go of something we have lived with — a love, a dream, an idea of who we are. It can never be willed, only surrendered to, and not without the thrashes of denial, dissociation, and the thousand little lies we tell ourselves in order to hold on. To finally let go is to discover that life holds, that we can be held in something larger than fear, larger than grief, larger than ourselves.

Joan Didion (December 5, 1934–December 23, 2021) offers a spare, stirring field guide to doing that in the final pages of The Year of Magical Thinking (public library) — her classic reckoning with grief following the death of her husband of four decades, John Gregory Dunne.

Joan Didion, 1977 (Photograph: Mary Lloyd Estrin)

Loosening the grip begins with the recognition that there is nothing solid enough in this world to hold on to, that transience is not a defect of things but their fundamental nature. Drawing on how plate tectonics revolutionized our understanding of life, Didion writes:

As the grandchild of a geologist I learned early to anticipate the absolute mutability of hills and waterfalls and even islands. When a hill slumps into the ocean I see the order in it. When a 5.2 on the Richter Scale wrenches the writing table in my own room in my own house… I keep on typing. A hill is a transitional accommodation to stress… A waterfall is a self-correcting maladjustment of stream to structure… [The island] Oahu, an emergent post-erosional land mass along the Hawaiian Ridge, is a temporary feature, and every rainfall or tremor along the Pacific plates alters its shape and shortens its tenure as Crossroads of the Pacific.

When she took her husband’s ashes to St. John the Divine, Didion brought along a lei — the traditional flower garland of the South Pacific, made to be worn around the neck, often given to loved ones arriving or departing. Looking back on her life with John in the idyll of their young love in the Brentwood Park neighborhood of Los Angeles, looking back on her time living and writing in Hawaii, she leans on a poetic native ritual for letting go:

During the years when people still left Honolulu on the Matson Lines the custom at the moment of departure was to throw leis on the water, a promise that the traveler would return. The leis would get caught in the wake and go bruised and brown, the way the gardenias in the pool filter at the house in Brentwood Park had gone bruised and brown.

The other morning when I woke I tried to remember the arrangement of the rooms in the house in Brentwood Park. I imagined myself walking through the rooms, first on the ground floor and then on the second. Later in the day I realized that I had forgotten one.

The lei I left at St. John the Divine would have gone brown by now.

Leis go brown, tectonic plates shift, deep currents move, islands vanish, rooms get forgotten.

The final card from An Almanac of Birds: 100 Divinations for Uncertain Days, also available as stand-alone print.

The daily deaths of hopes and plans, the failed projects and failed relationships, are our training ground for letting go, a scale model of life’s most difficult losses — the death of a parent, the death of a child, the death of a soul mate. Because the way we meet any death is the way we meet every death, Didion’s reckoning with losing the love of her life maps the disorienting terrain of navigating any loss, letting go of anything that has died, or must die, within us so we may go on living a deeper, truer life. In a poignant counterpoint to Hemingway’s insistence that “no one you love is ever dead,” she writes:

I know why we try to keep the dead alive: we try to keep them alive in order to keep them with us. I also know that if we are to live ourselves there comes a point at which we must relinquish the dead, let them go, keep them dead… Let go of them in the water.

Couple with poet Jane Kenyon on the art of letting go, then revisit Carl Jung on death and how to be more alive.

BP

The 3 Buddhist Steps to Repairing a Relationship

The 3 Buddhist Steps to Repairing a Relationship

One fact that never fails to astound me: Despite the immense cultural changes and leaps in knowledge over the epochs, the human brain — that crucible of consciousness, roiling with the psychologies that govern the behaviors we call human nature — has remained virtually unchanged for the past hundred thousand years. How humbling to consider that what is cognitively true of our ancestors — who, lacking a knowledge of astronomy as the correct frame of reference for planetary motion, explained eclipses as acts of god and comets as omens of ill fortune — is as true of us.

The explanatory contexts in which this tendency manifests today may be different, but it manifests just the same — especially in our interpersonal relationships, where so much of the correct frame of reference that is the other person’s inner reality is invisible to us. It helps to remember that between our feelings and anything in the external world that causes the ripples of consciousness we call feelings — any difficult situation, any painful event, any hurtful action of another — there lie myriad possible causal explanations.

One fact I have learned about life through the empiricism of living: When we are hurt in a relationship, when we are spinning in the blooming buzzing confusion of sensemaking, the explanation we elect as correct usually has more to do with our own fears and vulnerabilities than it does with the reality of the situation; almost always, that explanation is wrong; almost always, the true explanation has more to do with the fears and vulnerabilities roiling in the other person invisibly to us.

The Dreaming Horses by Franz Marc, 1913. (Available as a print and as stationery cards, benefitting The Nature Conservancy.)

And so, sensemaking and storytelling creatures that we are, we move through the real world in a self-generated dream, responding not to reality but to the stories we tell ourselves about what is true — stories at best incomplete and at worst injuriously incorrect, stories about what we do and don’t deserve, stories the cost of which is connection, trust, love. This is why without charity of interpretation and without candor — the vulnerability of it, the courage of it, the kindness of it — all relationships become a ricochet of unspoken resentments based mostly on misapprehended motives, and crumble.

The great Buddhist teacher and peace activist Thich Nhat Hanh (October 11, 1926–January 22, 2022) offers a three-step remedy for this elemental human tendency in a portion of his slender, potent book Fear: Essential Wisdom for Getting Through the Storm (public library), which also gave us his warm wisdom on the four Buddhist mantras for turning fear into love.

Thich Nhat Hanh

He writes:

Much of our suffering comes from wrong perceptions. To remove that hurt, we have to remove our wrong perception.

Whenever we see another person take an action, he notes, we must remain aware that there could be a number of invisible motive forces behind it and we must be willing to listen in order to better understand them — not only out of the vain self-referential transactionalism masquerading as the Golden Rule, in the hope that others would be just as willing not to misunderstand our own motives by their perception and interpretation of our actions, but because correcting our wrong perceptions is a basic and vital form of caring for ourselves:

When you make the effort to listen and hear the other side of the story, your understanding increases and your hurt diminishes.

Half a century after the great humanistic philosopher and psychologist Erich Fromm detailed the six rules of listening and unselfish understanding, Hanh offers a three-step process for correcting wrong perception in relationship conflict and emerging victorious with deeper love:

The first thing we can do in these situations is to acknowledge internally that the pictures we have in our head, what we think happened, may not be accurate. Our practice is to breathe and walk until we are more calm and relaxed.

The second thing we can do, when we are ready, is to tell the people who we think have hurt us that we are suffering and that we know our suffering may have come from our own wrong perception. Instead of coming to the other person or people with an accusation, we can come to them for help and ask them to explain, to help us understand why they have said or done those things.

There is a third thing we need to do, if we can. The third thing is very hard, perhaps the hardest. We need to listen very carefully to the other person’s response to truly understand and try to correct our perception. With this, we may find that we have been the victim of our wrong perceptions. Most likely the other person has also been a victim of wrong perceptions.

Total eclipse of the sun, observed July 29, 1878, at Creston, Wyoming Territory
One of Étienne Léopold Trouvelot’s groundbreaking astronomical drawings. (Available as a print, a face mask, and as stationery cards.)

Part of why this is so challenging to the Western mind, with its individualistic ideal of self-reliance that too readily metastasizes into self-righteousness, is that we grow incredibly insecure at the prospect of being wrong and feel incredibly unmoored by the fact of having been wrong. In a culture conflating who we are with what we know and what we stand for, the Eastern contemplative traditions can be so salutary with their gentle, steady practice of releasing the clutch of selfing and unclenching the fist of righteousness into an open palm of receptivity.

Drawing on two powerful Buddhist practices that effect this release — deep listening and loving speech — Hanh writes:

If we are sincere in wanting to learn the truth, and if we know how to use gentle speech and deep listening, we are much more likely to be able to hear others’ honest perceptions and feelings. In that process, we may discover that they too have wrong perceptions. After listening to them fully, we have an opportunity to help them correct their wrong perceptions. If we approach our hurts that way, we have the chance to turn our fear and anger into opportunities for deeper, more honest relationships.

Art from the 1750 book An Original Theory or New Hypothesis of the Universe by Thomas Wright, who originated the “island universes” concept. (Available as a print, as a face mask, and as stationery cards.)

This, he observes, applies to romantic relationships, to politics, to family and workplace dynamics — in other words, to all possible configurations of one consciousness embarking on the touching, terrifying endeavor of being known and understood by another.

With an eye to the ultimate aim of this process, he adds:

The intention of deep listening and loving speech is to restore communication, because once communication is restored, everything is possible, including peace and reconciliation.

[…]

We are all capable of recognizing that we’re not the only ones who suffer when there is a hard situation. The other person in that situation suffers as well, and we are partly responsible for his or her suffering. When we realize this, we can look at the other person with the eyes of compassion and let understanding bloom. With the arrival of understanding, the situation changes and communication is possible.

Any real peace process has to begin with ourselves… We have to practice peace to help the other side make peace.

Shortly after he wrote Fear: Essential Wisdom for Getting Through the Storm, Hanh placed this insight at the center of his now-classic teachings about how to love — an insight that also animates Alain de Botton’s soulful wisdom on what makes a good communicator. Perhaps Walt Whitman, writing with ecstatic immediacy, best captured this in his intimation that the secret of Being is “to do nothing but listen,” so that the song of life — which is the song of love — may be heard.

BP

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