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How Pythagoras and Sappho Radicalized Music and Revolutionized the World

The story of the invention of the love song, the world’s first algorithm, and the mathematics of transcendence.

How Pythagoras and Sappho Radicalized Music and Revolutionized the World

“To create today is to create dangerously,” Albert Camus told a gathering of young people at the peak of the Cold War, shortly after becoming the second-youngest laureate of the Nobel Prize. “The question, for all those who cannot live without art and what it signifies, is merely to find out how, among the police forces of so many ideologies… the strange liberty of creation is possible.” A generation before him, while policed by the forces about to unworld humanity in its first global war, the revolutionary artist Egon Schiele observed that true visionaries tend to come from the minority.

Millennia and civilizations earlier, two such visionaries who lived a generation apart, one born the day the other threw herself into the sea — Sappho (c. 630–c. 570 BC) and Pythagoras (c. 570–c. 495 BC) — revolutionized the deepest undertone of modern thought with their repugnant ideas about the most delicate, most beloved, and most elemental of the arts: music.

Pythagoras (Art by J. Augustus Knapp, circa 1926)

In an era when the most widespread musical instrument was the tetrachord — the Hellenic four-string lyre — and musicians had no standardized system of tuning their instruments, no understanding of the underlying tonal patterns, and nothing more than a vague intuitive sense about how to strum melodies rather than discord, Pythagoras discovered the relationship between musical harmony and the mathematical harmony of numbers. According to his foremost biographer, the fourth-century Syrian scholar Iamblichus, Pythagoras took it upon himself to devise a mechanical aid for musical tuning.

One day, Iamblichus’s account goes, Pythagoras was strolling past a blacksmith’s forge and was captivated by the sound of the many hammers pounding in a pattern that suddenly sounded harmonious. He rushed into the forge and immediately began investigating the cause of the harmony, testing the various hammers in various stroke combinations — some producing harmony, others discord. After analyzing the patterns and weighing the hammers, he discovered a simple mathematical relationship between those that produced harmony — their masses were exact ratios of one another’s.

Although the anecdote may belong to that murky shoreline between the apocryphal and the factual that marks many biographies of genius, Pythagoras did eventually test these ratios on the lyre. They proved to be perfectly predictive of harmony — the first discovery of a mathematical rule undergirding a physical phenomenon, and the basis of what became known as the Music of the Spheres.

Celestial harmonics of the planets, from The Harmony of the World (1619) by Johannes Kepler, based on the Pythagorean concept of the Music of the Spheres.

In our time, Pythagoras, known to every schoolchild for his famous triangle theorem, is celebrated as the pioneer who set the golden age of mathematics into motion with the development of numeric logic. Having coined the word philosophy and defined the very meaning of wisdom, he seeded scientific ideas that fomented the later revolutions ushered in by visionaries as far-ranging and far-reaching as Plato, Copernicus, Descartes, Kepler, Newton, and Einstein.

But in his time, Pythagoras was very much a radical, a dissident, an intellectual deviant. His progressive views on social reform led him to flee the tyrannical rule of his native Samos. After arriving in the Greek colony of Croton as a refugee, he founded a philosophical school whose disciples, known as the Pythagoreans, devised an unexampled model of the universe, placing at its center a ball of fire more than a thousand years before Copernicus upended the geocentric Ptolemaic system with his heliocentricity.

In another stroke of radicalism, the Pythagoreans admitted into their school a class of sub-citizens denied education and excluded from the newborn civic system of democracy: women. One of them became the world’s first known female astronomer — Hypatia, who lived her trailblazing life and died her savage death in the city where nearly every trace of Sappho vanished.

Death of Sappho by Miguel Carbonell Selva, 1881. (Available as a print.)

When the Library of Alexandria was burned, the flames consumed the nine-volume set of Sappho’s collected works, leaving only fragments copied by fans and scholars throughout the ancient world. From this handful of surviving ashes, Sappho rose with her lyre and her verse to be remembered as the Tenth Muse, the inventor of the love song and the personal lyric, the first great beacon of women’s right to creative expression, and the first great champion of the right to love whom we love. Unlike Emily Dickinson, who deliberately changed the gender pronouns in her poems to conceal the same-sex passion that fomented her poetry, Sappho kept the female pronouns in the beautiful and heartbreaking odes she wrote to the women she loved. In doing so, she pioneered a radical shift in musical culture — the permission to sing not about the gods, the seasons, and the wars, but about oneself: about the stunning interior universe of subjective human experience. Without Sappho, there would be no Nina Simone to pose in song the central question of consciousness: “I wish you could know what it means to be me.”

The epoch-making contributions of Pythagoras and Sappho come alive in Ted Gioia’s altogether wonderful book Music: A Subversive History (public library) — the story of our species told through its most consummate and intimate art-form, traced through the lives of the visionaries and radicals who shaped it, from Pythagoras and Sappho to Bob Dylan and N.W.A.

In a sentiment that calls to mind Iris Murdoch’s astute observation that “tyrants always fear art because tyrants want to mystify while art tends to clarify,” Gioia frames his approach:

At every stage in human history, music has been a catalyst for change, challenging conventions and conveying coded messages — or, not infrequently, delivering blunt, unambiguous ones. It has given voice to individuals and groups denied access to other platforms for expression, so much so that, in many times and places, freedom of song has been as important as freedom of speech, and far more controversial.

Art from an 1878 book about the history of science and literature, depicting some of Pythagoras’s influences and inspirations. (Available as a print and as a face mask.)

Holding up Pythagoras as the most significant figure in the history of music, whose controversial contribution was both a liberation and a limitation, Gioia writes:

Greek culture before his arrival revered what we call nowadays Orphic thought (named after Orpheus, the mythical musician, but almost certainly considered a historical personage in those distant days), and believed songs possessed powerful magic. The rise of Pythagorean music theory, circa 500 BC, changed all that by conceptualizing music as a rational science of sounds that could be described in mathematical terms.

Looking back to antiquity from our own era, in which algorithms are mediating the relationship between music, feeling, and transcendence, Gioia adds:

The very first algorithm entered Western music with this philosophical rupture that happened more than 2,500 years ago.

And yet while the Pythagorean model of mathematically distinguishing melody from noise liberated music by giving it a new language of codified expression, it also limited music by excluding from the musical canon styles that didn’t conform to these proportionate structures of scales and rhythms — styles like those that emerged from the African diaspora or from my own native Balkans. Gioia writes:

The ratios and proportions that initially helped us grasp songs turned into the rules and constraints that defined them. The strategies and schemas were often seen as the ‘authentic’ music, and the actual sounds only got validation through their allegiance to what was written on the printed page… The eventual result was a conceptualization of music that excluded far more than it allowed.

[…]

The very practice of legitimization is an act of distortion.

Still, the Pythagorean conceptualization of music had profound and beneficent consequences, stretching far beyond the realm of music and into the entire landscape of culture: By bringing mathematics to an art-form previously regarded as mystical, it catalyzed the slow shift from a world of superstition and magic to a world of science and reason — a cultural evolution that would unfold on the scale of epochs. Two millennia after Pythagoras, Kepler would spend years defending his mother in a witchcraft trial while revolutionizing our understanding of the universe with his epoch-making laws of planetary motion, drawing on the Music of the Spheres to discover the proportional relationships of planetary orbits.

Solids from Kepler’s Harmony of the World, exploring the relationship between harmony and geometry. (Available as a print and as a face mask.)

From this foundation laid in the Pythagorean past, Gioia leaps across the millennia to the present:

Today, these three spheres — science, music, and magic — appear as self-contained and unrelated disciplines, but in the context of 500 BC, the connections between them were obvious to the leading minds. Anyone who hoped to dislodge magical thinking in a traditional society and replace it with a scientific worldview was forced to address music theory, because it, too, could be conceptualized as either magic or science. Any choice between these two models would have profound implications. And not just for theory: society would be altered by how this matter was decided. Before Pythagoras, songs possessed magical potency. If Pythagoras and his followers hoped to eradicate superstition and elevate a more rational and logical worldview, they were almost forced to redefine all the parameters of musical practice.

But while this reconfiguration of musical practice as a mathematical language advanced the world toward science, it also repressed a central animating force of music — its elemental humanity, ablaze with feeling, sensuality, and a sense of the sacred. Even Schopenhauer, so very German and so unfaltering in his central tenet of the will as the supreme instrument of the human spirit, considered music capable of reaching beyond the reach of will, into “the inner being, the in-itself, of the world”; even Kierkegaard, for all his ceaseless cerebration, his Nordic reserve, and his lifelong virginity, exulted in the unparalleled sensuality of music.

It was Sappho who feathered the other great wing by which music took flight toward modernity, carrying the whole of human culture on its back. Gioia writes:

Sappho has two obvious concerns, and they dominate her worldview even as they expose a hidden rift in Western thought: the emotional bonds of love, and communal obligations to the gods. In the later evolution of Western music, these two approaches will veer off into their separate traditions and have little to do with each other. You could hardly imagine two music genres with less in common than love songs and religious hymns, but for Sappho these are intimately connected.

While Pythagoras took the mysticism of music and turned it into a mathematical language, Sappho took the ancient tradition of sacred singing and turned it into a new literary genre of personal poetics. By pioneering the love song and the self-permission for telling our own stories, drawn from our most intimate experiences, she gave the world an immense and abiding gift — the ability to preserve our stories in song as a fundament of identity and survival, from the African spirituals that sustained the souls of the enslaved to the folk ballads by which refugees hold on to culture and community. In fact, Sappho’s home island has always been a nexus of cultures and remains a major portal into Europe for refugees from the Middle East. Gioia writes:

At the height of the Syrian crisis in 2016, new arrivals would show up on Lesbos almost every day, making their treacherous journey on small boats, rafts, and inflatable crafts… Songs are the possessions most likely to survive long journeys, remaining the property of the newcomer even when everything else has been taken away.

Complement this fragment of the altogether fascinating Music: A Subversive History with a haunting choral invocation of Sappho’s timeless elegy for heartbreak, then revisit Maurice Sendak on the shape of music as the key to storytelling and some of humanity’s most beloved storytellers on the singular power of music.

BP

How Reading Is Like Love: Italo Calvino on the Ecstasy of Surrendering to Other Dimensions of Experience

“Lovers’ reading of each other’s bodies… differs from the reading of written pages in that it is not linear… What makes lovemaking and reading resemble each other most is that within both of them times and spaces open, different from measurable time and space.”

How Reading Is Like Love: Italo Calvino on the Ecstasy of Surrendering to Other Dimensions of Experience

“I function only by falling in love: with French and France; with the 15th Century; with microbiology, cosmology, sleep research,” Ursula K. Le Guin wrote in her daybook, capturing the necessary passion that makes writing akin to falling in love. But reading is where the parallel begins. Some of us read in order to write — one must first read about the fifteenth century and microbiology and sleep research before writing about it — and some read purely for the private joy of a world enlarged. Reading is the real fulcrum that lifts us up into new realms of thought and feeling, new atmospheres of reality, from which we free-fall into a deeper love of life itself. And whenever we read, we read the way we love — with our whole being, bringing to the book every experience we’ve ever had, every vestige of half-remembered impressions and half-survived heartbreaks, the imprint every other book we’ve ever read has left on our conscience.

From Italo Calvino (October 15, 1923–September 19, 1985) comes an uncommonly insightful, tender, and sensual celebration of this parallel between reading and love — the making of it, the falling into it — in a wonderful passage from 1979 novel If on a winter’s night a traveler (public library). From the frame narrative about a reader trying to read a book to the novel’s very title, deliberately styled like a sentence and not like a caption of capitalized words, this book is the ultimate meta-homage to reading — a book by and for the unabashed, obsessive lover of books; a book that exemplifies all of Calvino’s fourteen criteria for a classic, but especially the fourth: “a book which with each rereading offers as much of a sense of discovery as the first reading.”

Art by Violeta Lópiz from A Velocity of Being: Letters to a Young Reader. Available as a print, benefiting The New York Public Library.

Drawing a central parallel between a story in literature and a love story in life, Calvino writes:

How to establish the exact moment in which a story begins? Everything has already begun before, the first line of the first page of every novel refers to something that has already happened outside the book. Or else the real story is the one that begins ten or a hundred pages further on, and everything that precedes it is only a prologue. The lives of individuals of the human race form a constant plot, in which every attempt to isolate one piece of living that has a meaning separate from the rest — for example, the meeting of two people, which will become decisive for both — must bear in mind that each of the two brings with himself a texture of events, environments, other people, and that from the meeting, in turn, other stories will be derived which will break off from their common story.)

He considers how reading, like physical intimacy, is an act of total immersion that at its best requires a delicate osmotic balance of total surrender and unassailable sovereignty — one of the mind, the other of the body:

Now, since your bodies are trying to find, skin to skin, the adhesion most generous in sensations, to transmit and receive vibrations and waves, to compenetrate the fullnesses and the voids, since in mental activity you have also agreed on the maximum agreement, you can be addressed with an articulated speech that includes you both in a sole, two-headed person. First of all the field of action, or of existence, must be established for this double entity you form. Where is the reciprocal identification leading? What is the central theme that recurs in your variations and modulations? A tension concentrated on not losing anything of its own potential, on prolonging a state of reactivity, on exploiting the accumulation of the other’s desire in order to multiply one’s own charge? Or is it the most submissive abandonment, the exploration of the immensity of strokable and reciprocally stroking spaces, the dissolving of one’s being in a lake whose surface is infinitely tactile?

In a sentiment evocative of Rilke’s poignant observation that “even between the closest human beings infinite distances continue to exist” and that a healthy love is one of spacious union between two neighboring solitudes, Calvino concludes of this necessary negotiation between separateness and unity:

In both situations you certainly do not exist except in relation to each other, but, to make those situations possible, your respective egos have not so much to erase themselves as to occupy, without reserve, all the void of the mental space, invest in itself at the maximum interest or spend itself to the last penny. In short, what you are doing is very beautiful but grammatically it doesn’t change a thing. At the moment when you most appear to be a united voi, a second person plural, you are two tu’s, more separate and circumscribed than before.

Art by Ping Zhu for A Velocity of Being: Letters to a Young Reader. Available as a print, benefiting The New York Public Library.

In what may be the most sensuous passage ever composed on the subject, he likens the act of reading to the act of making love, addressing the reader-lover:

Now you are being read. Your body is being subjected to a systematic reading, through channels of tactile information, visual, olfactory, and not without some intervention of the taste buds. Hearing also has its role, alert to your gasps and your trills. It is not only the body that is, in you, the object of reading: the body matters insofar as it is part of a complex of elaborate elements, not all visible and not all present, but manifested in visible and present events: the clouding of your eyes, your laughing, the words you speak, your way of gathering and spreading your hair, your initiatives and your reticences, and all the signs that are on the frontier between you and usage and habits and memory and prehistory and fashion, all codes, all the poor alphabets by which one human being believes at certain moments that he is reading another human being… The Other Reader now is reviewing your body as if skimming the index, and at some moments she consults it as if gripped by sudden and specific curiosities, then she lingers, questioning it and waiting till a silent answer reaches her, as if every partial inspection interested her only in the light of a wider spatial reconnaissance. Now she dwells on negligible details, perhaps tiny stylistic faults… and she exploits them to establish a margin of detachment, critical reserve, or joking intimacy; now instead the accidentally discovered detail is excessively cherished — for example, the shape of your chin or a special nip you take at her shoulder — and from this start she gains impetus, covers (you cover together) pages and pages from top to bottom without skipping a comma.

Art by Margaret C. Cook from a rare 1913 edition of Walt Whitman’s Leaves of Grass. Available as a print.

But then Calvino anchors the analogy in a crucial difference within the similarity of the two experiences:

Lovers’ reading of each other’s bodies (of that concentrate of mind and body which lovers use to go to bed together) differs from the reading of written pages in that it is not linear. It starts at any point, skips, repeats itself, goes backward, insists, ramifies in simultaneous and divergent messages, converges again, has moments of irritation, turns the page, finds its place, gets lost. A direction can be recognized in it, a route to an end, since it tends toward a climax, and with this end in view it arranges rhythmic phases, metrical scansions, recurrence of motives. But is the climax really the end? Or is the race toward that end opposed by another drive which works in the opposite direction, swimming against the moments, recovering time?

If one wanted to depict the whole thing graphically, every episode, with its climax, would require a three-dimensional model, perhaps four-dimensional, or, rather, no model: every experience is unrepeatable. What makes lovemaking and reading resemble each other most is that within both of them times and spaces open, different from measurable time and space.

Art by Lia Halloran from A Velocity of Being: Letters to a Young Reader. Available as a print, benefiting The New York Public Library.

Complement this fragment of the thoroughly delicious If on a winter’s night a traveler with Jeanette Winterson on reading as self-liberation, Anne Lamott on reading as healing, Alain de Botton on reading as a portal to empathy, and Rebecca Solnit on reading as an existential toolkit for transformation, then revisit Calvino on the unbearable lightness of language, literature, and life and philosopher Martha Nussbaum on what reading Proust reveals about the litmus test for true love.

BP

Art and the Human Spirit: Olivia Laing on What the Lives of Great Artists Reveal About Vulnerability, Love, Loneliness, Resistance, and Our Search for Meaning

“We’re so often told that art can’t really change anything. But… it shapes our ethical landscapes; it opens us to the interior lives of others. It is a training ground for possibility. It makes plain inequalities, and it offers other ways of living.”

Art and the Human Spirit: Olivia Laing on What the Lives of Great Artists Reveal About Vulnerability, Love, Loneliness, Resistance, and Our Search for Meaning

The composite creation of a doctor, a philosopher, a poet, and a sculptor, the word empathy in the modern sense only came into use at the dawn of the twentieth century as a term for the imaginative act of projecting yourself into a work of art, into a world of feeling and experience other than your own. It vesselled in language that peculiar, ineffable way art has of bringing you closer to yourself by taking you out of yourself — its singular power to furnish, Iris Murdoch’s exquisite phrasing, “an occasion for unselfing.” And yet this notion cinches the central paradox of art: Every artist makes what they make with the whole of who they are — with the totality of experiences, beliefs, impressions, obsessions, childhood confusions, heartbreaks, inner conflicts, and contradictions that constellate a self. To be an artist is to put this combinatorial self in the service of furnishing occasions for unselfing in others.

That may be why the lives of artists have such singular allure as case studies and models of turning the confusion, complexity, and uncertainty of life into something beautiful and lasting — something that harmonizes the disquietude and dissonance of living.

Red Hill and White Shell by Georgia O’Keeffe, 1938

In Funny Weather: Art in an Emergency (public library), Olivia Laing — one of the handful of living writers whose mind and prose I enjoy commensurately with the Whitmans and the Woolfs of yore — occasions a rare gift of unselfing through the lives and worlds of painters, poets, filmmakers, novelists, and musicians who have imprinted culture in a profound way while living largely outside the standards and stabilities of society, embodying of James Baldwin’s piercing insight that “a society must assume that it is stable, but the artist must know, and he must let us know, that there is nothing stable under heaven.”

Punctuating these biographical sketches laced with larger questions about art and the human spirit are Laing’s personal essays reflecting, through the lens of her own lived experience, on existential questions of freedom, desire, loneliness, queerness, democracy, rebellion, abandonment, and the myriad vulnerable tendrils of aliveness that make life worth living.

What emerges is a case for art as a truly human endeavor, made by human beings with bodies and identities and beliefs often at odds with the collective imperative; art as “a zone of both enchantment and resistance,” art as sentinel and witness of “how truth is made, diagramming the stages of its construction, or as it may be dissolution,” art as “a direct response to the paucity and hostility of the culture at large,” art as a buoy for loneliness and a fulcrum for empathy.

Summer 1964 by Agnes Martin

Laing writes:

Empathy is not something that happens to us when we read Dickens. It’s work. What art does is provide material with which to think: new registers, new spaces. After that, friend, it’s up to you.

I don’t think art has a duty to be beautiful or uplifting, and some of the work I’m most drawn to refuses to traffic in either of those qualities. What I care about more… are the ways in which it’s concerned with resistance and repair.

A writer — a good writer — cannot write about art without writing about those who make it, about the lives of artists as the crucible of their creative contribution, about the delicate, triumphant art of living as a body in the world and a soul outside standard society. Olivia Laing is an excellent writer. Out of lives as varied as those of Basquiat and Agnes Martin, Derek Jarman and Georgia O’Keeffe, David Bowie and Joseph Cornell, she constructs an orrery of art as a cosmos of human connection and a sensemaking mechanism for living.

In a sentiment to which I relate in my own approach to historical lives, Laing frames her method of inquiry:

I’m going as a scout, hunting for resources and ideas that might be liberating or sustaining now, and in the future. What drives all these essays is a long-standing interest in how a person can be free, and especially in how to find a freedom that is shareable, and not dependent upon the oppression or exclusion of other people.

[…]

We’re so often told that art can’t really change anything. But I think it can. It shapes our ethical landscapes; it opens us to the interior lives of others. It is a training ground for possibility. It makes plain inequalities, and it offers other ways of living.

Throughout these short, scrumptiously insightful and sensitive essays, Laing draws on the lives of artists — the wildly uneven topographies of wildly diverse interior worlds — to contour new landscapes of possibility for life itself, as we each live it, around and through and with art. In the essay about Georgia O’Keeffe — who revolutionized modern art while living alone and impoverished in the middle of the desert, in the middle of the world’s first global war — Laing observes:

How do you make the most of what’s inside you, your talents and desires, when they slam you up against a wall of prejudice, of limiting beliefs about what a woman must be and an artist can do?

[…]

From the beginning, New Mexico represented salvation, though not in the wooden sense of the hill-dominating crosses she so often painted. O’Keeffe’s salvation was earthy, even pagan, comprised of the cold-water pleasure of working unceasingly at what you love, burning anxiety away beneath the desert sun.

Georgia O’Keeffe, Grey Lines with Black, Blue and Yellow, 1923 (Georgia O’Keeffe Museum)

In an essay about another artist — the painter Chantal Joffe, for whom Laing sat — she echoes Jackson Pollock’s’s observation that “every good artist paints what he is,” and writes:

You can’t paint reality: you can only paint your own place in it, the view from your eyes, as manifested by your own hands.

A painting betrays fantasies and feelings, it bestows beauty or takes it away; eventually, it supplants the body in history. A painting is full of desire and love, or greed, or hate. It radiates moods, just like people.

[…]

Paint as fur, as velvet, as brocade, as hair. Paint as a way of entering and becoming someone else. Paint as a device for stopping time.

Art by Basquiat from Life Doesn’t Frighten Me by Maya Angelou

In another essay, Laing offers an exquisite counterpoint to the barbed-wire fencing off of identities that has increasingly made the free reach of human connection — that raw material and final product of all art — dangerous and damnable in a culture bristling with ready indignations and antagonisms:

A writer I was on a panel with said, and I’m paraphrasing here, that it is no longer desirable to write about the lives of other people or experiences one hasn’t had. I didn’t agree. I think writing about other people, making art about other people, is both dangerous and necessary. There are moral lines. There are limits to the known. But there’s a difference between respecting people’s right to tell or not tell their own stories and refusing to look at all.

[…]

It depends whether you believe that we exist primarily as categories of people, who cannot communicate across our differences, or whether you think we have a common life, an obligation to regard and learn about each other.

In a sense, the entire book is a quiet manifesto for unselfing through the art we make and the art we cherish — a subtle and steadfast act of resistance to the attrition of human connection under the cultural forces of self-righteousness and sanctimonious othering, a stance against those fashionable and corrosive forces that so often indict as appropriation the mere act of learning beautiful things from each other.

In another essay — about Ali Smith, the subject to whom Laing feels, or at least reads, the closest — she quotes a kindred sentiment of Smith’s:

Art is one of the prime ways we have of opening ourselves and going beyond ourselves. That’s what art is, it’s the product of the human being in the world and imagination, all coming together. The irrepressibility of the life in the works, regardless of the times, the histories, the life stories, it’s like being given the world, its darks and lights. At which point we can go about the darks and lights with our imagination energised.

Joseph Cornell, Untitled (Celestial Navigation), 1958. (U.S. Department of State / Joseph and Robert Cornell Memorial Foundation)

Among the subjects of a subset of essays Laing aptly categorizes as “love letters” is John Berger, whose lovely notion of “hospitality” radiates from Laing’s own work — a notion she defines as “a capacity to enlarge and open, a corrective to the overwhelming political imperative, in ascendance once again this decade, to wall off, separate and reject.” She reflects on being stopped up short by Berger’s embodiment of such hospitality when she saw him speak at the British Library at the end of his long, intellectually generous life:

It struck me then how rare it is to see a writer on stage actually thinking, and how glib and polished most speakers are. For Berger, thought was work, as taxing and rewarding as physical labour, a bringing of something real into the world. You have to strive and sweat; the act is urgent but might also fail.

He talked that evening about hospitality. It was such a Bergerish notion. Hospitality: the friendly and generous reception and entertainment of guests, visitors or strangers, a word that shares its origin with hospital, a place to treat sick or injured people. This impetus towards kindness and care for the sick and strange, the vulnerable and dispossessed is everywhere in Berger’s work, the sprawling orchard of words he planted and tended over the decades.

[…]

Art he saw as a communal and vital possession, to be written about with sensual exactness… Capitalism, he wrote in Ways of Seeing, survives by forcing the majority to define their own interests as narrowly as possible. It was narrowness he set himself against, the toxic impulse to wall in or wall off. Be generous to the strange, be open to difference, cross-pollinate freely. He put his faith in the people, the whole host of us.

Agnes Martin, With My Back to the World, 1997
With My Back to the World by Agnes Martin, 1997

In a superb 2015 essay titled “The Future of Loneliness” — an essay that bloomed into a book a year later, the splendid and unclassifiable book that first enchanted me with Laing’s writing and the mind from which it springs — she considers how technology is mediating our already uneasy relationship to loneliness, and how art redeems the simulacra of belonging with which social media entrap us in this Stockholm syndrome of self-regard. In a passage of chillingly intimate resonance to all of us alive in the age of screens and selfies and the vacant, addictive affirmation of people we have never dined with tapping heart- and thumb-shaped icons on cold LED screens, she writes:

Loneliness centres around the act of being seen. When a person is lonely, they long to be witnessed, accepted, desired, at the same time as becoming intensely wary of exposure. According to research carried out over the past decade at the University of Chicago, the feeling of loneliness triggers what psychologists term hypervigilance for social threat. In this state, which is entered into unknowingly, the individual becomes hyperalert to rejection, becoming inclined to perceive their social interactions as tinged with hostility or scorn. The result of this shift in perception is a vicious circle of withdrawal, in which the lonely person becomes increasingly suspicious, intensifying their sense of isolation.

This is where online engagement seems to exercise its special charm. Hidden behind a computer screen, the lonely person has control. They can search for company without the danger of being revealed or found wanting. They can reach out or they can hide; they can lurk and they can show themselves, safe from the humiliation of face-to-face rejection. The screen acts as a kind of protective membrane, a scrim that permits invisibility and also transformation. You can filter your image, concealing unattractive elements, and you can emerge enhanced: an online avatar designed to attract likes. But now a problem arises, for the contact this produces is not quite the same thing as intimacy. Curating a perfected self might win followers or Facebook friends, but it will not necessarily cure loneliness, since the cure for loneliness is not being looked at, but being seen and accepted as a whole person: ugly, unhappy and awkward as well as radiant and selfie-ready.

Illustration by Mimmo Paladino for a rare edition of James Joyce’s Ulysses

Having met with Ryan Trecartin — “a baby-faced thirty-four-year-old” of whom I had never heard (saying more about my odd nineteenth-century life than about his art) but whose early-twenty-first-century films about the lurid and discomposing thrill of digital culture prompted The New Yorker to describe him as “the most consequential artist to have emerged since the nineteen-eighties” — Laing reflects:

My own understanding of loneliness relied on a belief in solid, separate selves that he saw as hopelessly outmoded. In his world view, everyone was perpetually slipping into each other, passing through perpetual cycles of transformation; no longer separate, but interspersed. Perhaps he was right. We aren’t as solid as we once thought. We’re embodied but we’re also networks, expanding out into empty space, living on inside machines and in other people’s heads, memories and data streams as well as flesh. We’re being watched and we do not have control. We long for contact and it makes us afraid. But as long as we’re still capable of feeling and expressing vulnerability, intimacy stands a chance.

Vulnerability — which Laing unfussily terms “the necessary condition of love” — is indeed the bellowing undertone of these essays: vulnerability as frisson and function of art, of life itself, of the atavistic impulse for transmuting living into meaning that we call art.

Complement the thoroughly symphonic Funny Weather with Paul Klee on creativity and why an artist is like a tree, Kafka on why we make art, Egon Schiele on why visionary artists tend to come from the minority, and Virginia Woolf’s garden epiphany about what it means to be an artist — which remains, for me, the single most beautiful and penetrating thing ever written on the subject — then revisit Laing on life, loss, and the wisdom of rivers.

BP

The Pattern Inside the Pattern: Fractals, the Hidden Order Beneath Chaos, and the Story of the Refugee Who Revolutionized the Mathematics of Reality

“In the mind’s eye, a fractal is a way of seeing infinity.”

The Pattern Inside the Pattern: Fractals, the Hidden Order Beneath Chaos, and the Story of the Refugee Who Revolutionized the Mathematics of Reality

I have learned that the lines we draw to contain the infinite end up excluding more than they enfold.

I have learned that most things in life are better and more beautiful not linear but fractal. Love especially.

In a testament to Aldous Huxley’s astute insight that “all great truths are obvious truths but not all obvious truths are great truths,” the polymathic mathematician Benoit Mandelbrot (November 20, 1924–October 14, 2010) observed in his most famous and most quietly radical sentence that “clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”

An obvious truth a child could tell you.

A great truth that would throw millennia of science into a fitful frenzy, sprung from a mind that dismantled the mansion of mathematics with an outsider’s tools.

The Mandelbrot set. (Illustration by Wolfgang Beyer.)

A self-described “nomad-by-choice” and “pioneer-by-necessity,” Mandelbrot believed that “the rare scholars who are nomads-by–choice are essential to the intellectual welfare of the settled disciplines.” He lived the proof with his discovery of a patterned order underlying a great many apparent irregularities in nature — a sweeping symmetry of nested self-similarities repeated recursively in what may at first read as chaos.

The revolutionary insight he arrived at while studying cotton prices in 1962 became the unremitting vector of revelation a lifetime long and aimed at infinity, beamed with equal power of illumination at everything from the geometry of broccoli florets and tree branches to the behavior of earthquakes and economic markets.

Fractal Flight by Maria Popova. Available as a print.

Mandelbrot needed a word for his discovery — for this staggering new geometry with its dazzling shapes and its dazzling perturbations of the basic intuitions of the human mind, this elegy for order composed in the new mathematical language of chaos. One winter afternoon in his early fifties, leafing through his son’s Latin dictionary, he paused at fractus — the adjective from the verb frangere, “to break.” Having survived his own early life as a Jewish refugee in Europe by metabolizing languages — his native Lithuanian, then French when his family fled to France, then English as he began his life in science — he recognized immediately the word’s echoes in the English fracture and fraction, concepts that resonated with the nature of his jagged self-replicating geometries. Out of the dead language of classical science he sculpted the vocabulary of a new sensemaking model for the living world. The word fractal was born — binominal and bilingual, both adjective and noun, the same in English and in French — and all the universe was new.

In his essay for artist Katie Holten’s lovely anthology of art and science, About Trees (public library) — trees being perhaps the most tangible and most enchanting manifestation of fractals in nature — the poetic science historian James Gleick reflects on Mandelbrot’s titanic legacy:

Mandelbrot created nothing less than a new geometry, to stand side by side with Euclid’s — a geometry to mirror not the ideal forms of thought but the real complexity of nature. He was a mathematician who was never welcomed into the fraternity… and he pretended that was fine with him… In various incarnations he taught physiology and economics. He was a nonphysicist who won the Wolf Prize in physics. The labels didn’t matter. He turns out to have belonged to the select handful of twentieth century scientists who upended, as if by flipping a switch, the way we see the world we live in.

He was the one who let us appreciate chaos in all its glory, the noisy, the wayward and the freakish, form the very small to the very large. He gave the new field of study he invented a fittingly recondite name: “fractal geometry.”

It was Gleick who, in his epoch-making 1980 book Chaos: The Making of a New Science (public library), did for the notion of fractals what Rachel Carson did for the notion of ecology, embedding it in the popular imagination both as a scientific concept and as a sensemaking mechanism for reality, lush with material for metaphors that now live in every copse of culture.

Illustration from Chaos by James Gleick.

He writes of Mandelbrot’s breakthrough:

Over and over again, the world displays a regular irregularity.

[…]

In the mind’s eye, a fractal is a way of seeing infinity.

Imagine a triangle, each of its sides one foot long. Now imagine a certain transformation — a particular, well-defined, easily repeated set of rules. Take the middle one-third of each side and attach a new triangle, identical in shape but one-third the size. The result is a star of David. Instead of three one-foot segments, the outline of this shape is now twelve four-inch segments. Instead of three points, there are six.

As you incline toward infinity and repeat this transformation over and over, adhering smaller and smaller triangles onto smaller and smaller sides, the shape becomes more and more detailed, looking more and more like the contour of an intricate perfect snowflake — but one with astonishing and mesmerizing features: a continuous contour that never intersects itself as its length increases with each recursive addition while the area bounded by it remains almost unchanged.

Plate from Wilson Bentley’s pioneering 19th-century photomicroscopy of snowflakes

If the curve were ironed out into a straight Euclidean line, its vector would reach toward the edge of the universe.

It thrills and troubles the mind to bend itself around this concept. Fractals disquieted even mathematicians. But they described a dizzying array of objects and phenomena in the real world, from clouds to capital to cauliflower.

Against Euclid by Maria Popova. Available as a print.

It took an unusual mind shaped by unusual experience — a common experience navigated by uncommon pathways — to arrive at this strange revolution. Gleick writes:

Benoit Mandelbrot is best understood as a refugee. He was born in Warsaw in 1924 to a Lithuanian Jewish family, his father a clothing wholesaler, his mother a dentist. Alert to geopolitical reality, the family moved to Paris in 1936, drawn in part by the presence of Mandelbrot’s uncle, Szolem Mandelbrojt, a mathematician. When the war came, the family stayed just ahead of the Nazis once again, abandoning everything but a few suitcases and joining the stream of refugees who clogged the roads south from Paris. They finally reached the town of Tulle.

For a while Benoit went around as an apprentice toolmaker, dangerously conspicuous by his height and his educated background. It was a time of unforgettable sights and fears, yet later he recalled little personal hardship, remembering instead the times he was befriended in Tulle and elsewhere by schoolteachers, some of them distinguished scholars, themselves stranded by the war. In all, his schooling was irregular and discontinuous. He claimed never to have learned the alphabet or, more significantly, multiplication tables past the fives. Still, he had a gift.

When Paris was liberated, he took and passed the month-long oral and written admissions examination for École Normale and École Polytechnique, despite his lack of preparation. Among other elements, the test had a vestigial examination in drawing, and Mandelbrot discovered a latent facility for copying the Venus de Milo. On the mathematical sections of the test — exercises in formal algebra and integrated analysis — he managed to hide his lack of training with the help of his geometrical intuition. He had realized that, given an analytic problem, he could almost always think of it in terms of some shape in his mind. Given a shape, he could find ways of transforming it, altering its symmetries, making it more harmonious. Often his transformations led directly to a solution of the analogous problem. In physics and chemistry, where he could not apply geometry, he got poor grades. But in mathematics, questions he could never have answered using proper techniques melted away in the face of his manipulations of shapes.

Benoit Mandelbrot as a teenager. (Photograph courtesy of Aliette Mandelbrot.)

At the heart of Mandelbrot’s mathematical revolution, this exquisite plaything of the mind, is the idea of self-similarity — a fractal curve looks exactly the same as you zoom all the way out and all the way in, across all available scales of magnification. Gleick describes the nested recursion of self-similarity as “symmetry across scale,” “pattern inside of a pattern.” In his altogether splendid Chaos, he goes on to elucidate how the Mandelbrot set, considered by many the most complex object in mathematics, became “a kind of public emblem for chaos,” confounding our most elemental ideas about simplicity and complexity, and sculpting from that pliant confusion a whole new model of the world.

Couple with the story of the Hungarian teenager who bent Euclid and equipped Einstein with the building blocks of relativity, then revisit Gleick on time travel and his beautiful reading of and reflection on Elizabeth Bishop’s ode to the nature of knowledge.

BP

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