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The Invention of Zero: How Ancient Mesopotamia Created 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 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

Trailblazing 18th-Century Mathematician Émilie du Châtelet, Who Popularized Newton, on Gender in Science and the Nature of Genius

“One must know what one wants to be. In the latter endeavors irresolution produces false steps, and in the life of the mind confused ideas.”

Trailblazing 18th-Century Mathematician Émilie du Châtelet, Who Popularized Newton, on Gender in Science and the Nature of Genius

A century before Ada Lovelace became the world’s first computer programmer, a century before the word “scientist” was coined for the Scottish polymath Mary Somerville, another woman of towering genius and determination subverted the limiting opportunities her era afforded her and transcended what astrophysicist and writer Janna Levin has aptly called “the enraging pointlessness of small-minded repressions of a soaring and generous human urge” — the urge to understand the nature of reality and use that understanding to expand the corpus of human knowledge.

Émilie du Châtelet (December 17, 1706–September 10, 1749), born nineteen years after the publication of Newton’s revolutionary Principia, became besotted with science at the age of twelve and devoted the remainder of her life to the passionate quest for mathematical illumination. Although she was ineligible for academic training — it would be nearly two centuries until universities finally opened their doors to women — and was even excluded from the salons and cafés that served as the era’s informal epicenters of intellectual life, open only to men, Du Châtelet made herself into a formidable mathematician, a scholar of unparalleled rigor, and a pioneer of popular science.

Émilie du Châtelet (Portrait by Maurice Quentin de La Tour)

Together with her collaborator and lover Voltaire, who considered her in possession of “a genius worthy of Horace and Newton” and referred to her jocularly as “Madame Newton du Châtelet,” she set about popularizing Newton’s then-radical ideas at a time when even gravity was a controversial notion. The resulting 1738 book, Elements of the Philosophy of Newton, listed Voltaire as the author. But without Du Châtelet’s mathematical brilliance, he — a poet, playwright, philosopher, and political essayist — would’ve been swallowed whole by Newton’s science.

Voltaire knew this and acknowledged it readily in the preface, naming Du Châtelet as an indispensable colleague. The frontispiece of the book depicted her as Minerva, the Roman goddess of truth and wisdom, beaming down upon the seated Voltaire as he wrote. Voltaire’s dedicatory poem celebrated her “vast and powerful Genius” and called her the “Minerva of France,” a “disciple of Newton and of Truth.” In a letter to his friend Crown Prince Frederick of Prussia, Voltaire was even more explicit about the division of labor in their Newtonian collaboration: “Minerva dictated and I wrote.”

Frontispiece to Elements of the Philosophy of Newton

By the end of her lamentably short life, Du Châtelet had become a dominant world authority on Newtonian physics. In her final year, she undertook her most ambitious project yet — a translation of Newton’s Principia into French, which became a centerpiece of the Scientific Revolution in Europe and remains the standard French text to this day. Du Châtelet’s accompanying commentary added a great deal of original thought and conveyed to the popular imagination the ideas that would come to shape the modern world, embodying the great Polish poet and Nobel laureate Wisława Szymborska‘s notion of “that rare miracle when a translation stops being a translation and becomes … a second original.”

But it was in another work of translation, which Du Châtelet had undertaken a decade earlier with Voltaire’s encouragement, that she first honed the art of that “rare miracle.” In the late 1730s, while living with Voltaire in her country house in Cirey and collaborating on their Newtonian primer, she read and was deeply moved by The Fable of The Bees: or, Private Vices, Public Benefits — Bernard Mandeville’s 1714 prose commentary on his 1705 satirical poem The Grumbling Hive: or, Knaves turn’d Honest, exploring ethics, economics, and the deleterious role of cultural conditioning in gender norms. It was a visionary work centuries ahead of its time in many ways, asserting that human societies prosper through collaboration rather than selfishness, outlining what psychologists now call “the power paradox,” presaging the principle that Adam Smith would term the “invisible hand” seven decades later, and making a case for equal educational opportunities for women a quarter millennium before the modern feminist movement.

Nowhere do Du Châtelet’s remarkable character and fortitude in the face of her culture’s limitations come to life more vividly than in her translator’s preface, included in her Selected Philosophical and Scientific Writings (public library) and discussed in Robyn Arianrhod’s altogether magnificent book Seduced by Logic: Emilie Du Châtelet’s, Mary Somerville and the Newtonian Revolution (public library).

Du Châtelet writes from Cirey in her early thirties:

Since I began to live with myself, and to pay attention to the price of time, to the brevity of life, to the uselessness of the things one spends one’s time with in the world, I have wondered at my former behavior: at taking extreme care of my teeth, of my hair and at neglecting my mind and my understanding. I have observed that the mind rusts more easily than iron, and that it is even more difficult to restore to its first polish.

Centuries before modern psychologists conceived of the 10,000 hours rule of genius, she argues for giving the intellect a disciplined opportunity to incline itself toward its goals through regular practice:

The fakirs of the East Indies lose the use of the muscles in their arms, because those are always in the same position and are not used at all. Thus do we lose our own ideas when we neglect to cultivate them. It is a fire that dies if one does not continually give it the wood needed to maintain it… Firmness … can never be acquired unless one has chosen a goal for one’s studies. One must conduct oneself as in everyday life; one must know what one wants to be. In the latter endeavors irresolution produces false steps, and in the life of the mind confused ideas.

In a sentiment that makes one wonder whether Schopenhauer read Du Châtelet when he conceived of his famous distinction between talent and genius a century later, she adds:

Those who have received very decided talent from nature can give themselves up to the force that impels their genius, but there are few such souls which nature leads by the hand through the field that they must clear for cultivation or improvement. Even fewer are sublime geniuses, who have in them the seeds of all talents and whose superiority can embrace and perform everything.

In a passage that calls to mind Nietzsche’s reflections on how to find yourself and the true value of education, she considers how ordinary people — that is, non-geniuses — can cultivate their talent:

It sometimes happens that work and study force genius to declare itself, like the fruits that art produces in a soil where nature did not intend it, but these efforts of art are nearly as rare as natural genius itself. The vast majority of thinking men — the others, the geniuses, are in a class of their own — need to search within themselves for their talent. They know the difficulties of each art, and the mistakes of those who engage in each one, but they lack the courage that is not disheartened by such reflections, and the superiority that would enable them to overcome such difficulties. Mediocrity is, even among the elect, the lot of the greatest number.

With an eye to the perils of self-comparison to those more fortune or more gifted than oneself, she adds:

But one must cultivate the portion one has received and not give in to despair, because one has only two arpents [French measurement] of land while others have ten lieues of land.

Émilie du Châtelet (Portrait by Nicolas de Largillière)

In a passage of courage so tremendous and so near-impossible to grasp with our modern imagination, for we have only a detached and abstract idea of what life was like for women in the early 18th century, Du Châtelet proceeds into a visionary critique of patriarchal power structures in science and in life itself:

I feel the full weight of prejudice that excludes us [women] so universally from the sciences, this being one of the contradictions of this world, which has always astonished me, as there are great countries whose laws allow us to decide their destiny, but none where we are brought up to think.

Considering the era’s standard practice of excommunicating actors from the Catholic Church, which considered them “instruments of Satan,” she adds:

Another observation that one can make about this prejudice, which is odd enough, is that acting is the only occupation requiring some study and a trained mind to which women are admitted, and it is at the same time the only one that regards its professionals as infamous.

Du Châtelet, who embodied Adrienne Rich’s notion that an education is something you claim rather than get, points to education as the fulcrum of women’s absence — for, at that point, it was an absence rather than the underrepresentation it is today — from the professional worlds of science, philosophy, and the arts, and proposes a radical vision for education reform that would bolster equality:

Why do these creatures whose understanding appears in all things equal to that of men, seem, for all that, to be stopped by an invincible force on this side of a barrier; let someone give me some explanation, if there is one. I leave it to naturalists to find a physical explanation, but until that happens, women will be entitled to protest against their education. As for me, I confess that if I were king I would wish to make this scientific experiment. I would reform an abuse that cuts out, so to speak, half of humanity. I would allow women to share in all the rights of humanity, and most of all those of the mind… This new system of education that I propose would in all respects be beneficial to the human species. Women would be more valuable beings, men would thereby gain a new object of emulation, and our social interchanges which, in refining women’s minds in the past, too often weakened and narrowed them, would now only serve to extend their knowledge.

In a bittersweet reflection on her own life, which has emboldened women in science for centuries, she adds:

I am convinced that many women are either ignorant of their talents, because of the flaws in their education, or bury them out of prejudice and for lack of a bold spirit. What I have experienced myself confirms me in this opinion. Chance led me to become acquainted with men of letters, I gained their friendship, and I saw with extreme surprise that they valued this amity. I began to believe that I was a thinking creature. But I only glimpsed this, and the world, the dissipation, for which alone I believed I had been born, carried away all my time and all my soul. I only believed in earnest in my capacity to think at an age when there was still time to become reasonable, but when it was too late to acquire talents.

Being aware of that has not discouraged me at all. I hold myself quite fortunate to have renounced in mid-course frivolous things that occupy most women all their lives, and I want to use what time remains to cultivate my soul.

Her closing words — wry, unsentimental, quietly poetic — radiate Du Châtelet’s defiant genius:

The unfairness of men in excluding us women from the sciences should at least be of use in preventing us from writing bad books. Let us try to enjoy this advantage over them, so that this tyranny will be a happy necessity for us, leaving nothing for them to condemn in our works but our names.

Seduced by Logic delves deeper into Du Châtelet’s extraordinary mind, spirit, and legacy. Complement it with pioneering physicist Lise Meitner’s only direct remarks on gender in science and this loving remembrance of astrophysicist Vera Rubin, who led the way for modern women in STEM, then revisit the story of how Voltaire fell in love with his Minerva.

BP

Truth Beyond Logic and Time Beyond Clocks: Janna Levin on the Vienna Circle and How Mathematician Kurt Gödel Shaped the Modern Mind

“The past does not exist except as a threadbare fragment in the weaker minds of the many.”

Truth Beyond Logic and Time Beyond Clocks: Janna Levin on the Vienna Circle and How Mathematician Kurt Gödel Shaped the Modern Mind

If it is true — and true it is — that creativity blooms when seemingly unrelated ideas are cross-pollinated into something novel, then its most fecund ground is an environment where minds of comparable caliber but divergent obsession come together and swirl their ideas into a common wellspring of genius. There is hardly more concrete a testament to this principle than the Vienna Circle — the collective of scientists, philosophers, and novelists, who met in Europe in the first decades of the twentieth century and shaped modern culture by bringing art and science into intimate, fertile contact. But in the 1930s, as they demolished the boundaries between these disciplines, the Vienna Circle also exposed the limits of logic as a sensemaking mechanism for the nature of reality, limitation being perhaps as necessary to creativity as freedom of thought. (“The more a person limits himself,” Kierkegaard had asserted a century earlier, “the more resourceful he becomes.”)

The paradigm-shifting ideas that emerged from that unusual petri dish are what cosmologist and novelist Janna Levin explores throughout A Mad Man Dreams of Turing Machines (public library) — her lyrical and darkly enthralling novel, partway between magical realism and poetry, yet guided by science and rigorously grounded in the real lives of two of the twentieth century’s most tragic geniuses: computing pioneer Alan Turing and trailblazing mathematician Kurt Gödel.

Inside Café Josephinum, the convening place of the Vienna Circle
Inside Café Josephinum, the convening place of the Vienna Circle

Levin casts the making of this small, enormous revolution:

A group of scientists from the university begin to meet and throw their ideas into the mix with those of artists and novelists and visionaries who rebounded with mania from the depression that follows a nation’s defeat. The few grow in number through invitation only. Slowly their members accumulate and concepts clump from the soup of ideas and take shape until the soup deserves a name, so they are called around Europe, and even as far as the United States, the Vienna Circle.

Barely a generation after Bertrand Russell shook the verdure of mysticism from the tree of knowledge to reveal the robust barren branches of logic, the Vienna Circle made it their mission to weld reality with the axe of Logical Empiricism. Levin transports us to the singular atmosphere of their gatherings:

At the center of the Circle is a circle: a clean, round, white marble tabletop. They select the Café Josephinum precisely for this table. A pen is passed counterclockwise. The first mark is made, an equation applied directly to the tabletop, a slash of black ink across the marble, a mathematical sentence amid the splatters. They all read the equation, homing in on the meaning amid the disordered drops. Mathematics is visual not auditory. They argue with their voices but more pointedly with their pens. They stain the marble with rays of symbolic logic in juicy black pigment that very nearly washes away.

They collect here every Thursday evening to distill their ideas — to distinguish science from superstition. At stake is Everything. Reality. Meaning. Their lives. They have lost any tolerance for ineffectual and embroidered attitudes, for mysticism or metaphysics.

Vienna in the 1930s. The sign, belonging to a gambling parlor, reads: "Don't let luck pass you by." A horseshoe and chimney sweep, superstitious symbols of good fortune, appear above. (Photograph:  Roman Vishniac)
Vienna in the 1930s. The sign, belonging to a gambling parlor, reads: “Don’t let luck pass you by.” A horseshoe and chimney sweep, superstitious symbols of good fortune, appear above it. (Photograph: Roman Vishniac)

The members of the Vienna Circle were endowed with minds exceeding the average not by degree but by kind — the kind of genius that risked bleeding into madness, nowhere more so than in Gödel. Levin paints his conflicting multitudes — the internal tensions that powered his, and perhaps power all, genius:

In 1931 he is a young man of twenty-five, his sharpest edges still hidden beneath the soft pulp of youth. He has just discovered his theorems. With pride and anxiety he brings with him this discovery. His almost, not-quite paradox, his twisted loop of reason, will be his assurance of immortality. An immortality of his soul or just his name? This question will be the subject of his madness.

Levin, who has written beautifully about the complex relationship between genius and madness, adds:

Here he is, a man in defense of his soul, in defense of truth, ready to alter the view of reality his friends have formulated on this marble table. He joins the Circle to tell the members that they are wrong, and he can prove it.

[…]

He is still all potential. The potential to be great, the potential to be mad. He will achieve both magnificently.

kurtgodel

In his incompleteness theorems, which he began publishing that year, Gödel set out to prove that there are limits to how much of reality mathematical logic can grasp — something many intuited but none had substantiated. (Nearly a century earlier, the pioneering astronomer Maria Mitchell articulated that intuition, if not its empirical proof, in her diary: “The world of learning is so broad, and the human soul is so limited in power! We reach forth and strain every nerve, but we seize only a bit of the curtain that hides the infinite from us.”) With poetic precision, Levin conveys Gödel’s ideas and their broader significance:

Gödel will prove that some truths live outside of logic and that we can’t get there from here. Some people — people who probably distrust mathematics — are quick to claim that they knew all along that some truths are beyond mathematics. But they just didn’t. They didn’t know it. They didn’t prove it.

Gödel didn’t believe that truth would elude us. He proved that it would. He didn’t invent a myth to conform to his prejudice of the world — at least not when it came to mathematics. He discovered his theorem as surely as if it was a rock he had dug up from the ground. He could pass it around the table and it would be as real as that rock. If anyone cared to, they could dig it up where he buried it and find it just the same. Look for it and you’ll find it where he said it is, just off center from where you’re staring. There are faint stars in the night sky that you can see, but only if you look to the side of where they shine. They burn too weakly or are too far away to be seen directly, even if you stare. But you can see them out of the corner of your eye because the cells on the periphery of your retina are more sensitive to light. Maybe truth is just like that. You can see it, but only out of the corner of your eye.

But the truth is not something everyone wants to see — it can be inconvenient, even obstructionist. In the spring of 1936, as the ideas of the Vienna Circle were becoming increasingly threatening to the Nazi party rising to power, Moritz Schlick, chair of the Vienna Circle, was shot by a former student of his on the steps of the University of Vienna, where he taught. Meanwhile, Gödel’s swirling genius was spiraling further and further into madness. Having already necessitated psychiatric care two years earlier, he was destroyed anew upon hearing of Moritz’s murder and endured an even sharper nervous breakdown that landed him in a psychiatric institution. Levin writes:

In his quiet room in the sanatorium with the narrow window over the big groomed lawn, Gödel rested alone, slumped and motionless, and wondered, where did he go? Where is Moritz?

What does it mean to say that Moritz lived in the past? Nothing. The past does not exist. The notion of a past refers to a paltry and brittle memory, incomplete and flawed. Moritz is dead. He is lost but for fragments in the minds of those who have moved around the globe since his death. The Vienna Circle died with him as the headlines condemned Moritz Schlick as a Jew sympathizer who got what he deserved at the top of the stairs in the University of Vienna at the hands of a pan-Germanic hero who rightly killed this Jew philosopher. Moritz was a Protestant. Facts of the world are sealed in minds. People wear a facade. All of reality goes on behind their eyes, and there lie secret plans and hidden agendas. A tar of false motives and intentions. Truth mauled. Because the past does not exist except as a threadbare fragment in the weaker minds of the many.

Complement the enormously invigorating A Mad Man Dreams of Turing Machines with philosopher Rebecca Goldstein on how Gödel and Einstein changed our understanding of time, then revisit Levin on free will, the vitalizing power of obsessiveness, the century-long quest to hear the sound of space-time, and her remarkable Moth story about the unlikely paths that lead us back to ourselves.

BP

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