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Trees, Whales, and Our Digital Future: George Dyson on Nature, Human Nature, and the Relationship Between Our Minds and Our Machines

“Nature’s answer to those who seek to control nature through programmable machines is to allow us to build systems whose nature is beyond programmable control.”

Trees, Whales, and Our Digital Future: George Dyson on Nature, Human Nature, and the Relationship Between Our Minds and Our Machines

Long ago, in the ancient bosom of the human animal stirred a quickening of thought and tenderness at the sheer beauty of the world — a yearning to fathom the forces and phenomena behind the enchantments of birdsong and bloom, the rhythmic lapping of the waves, the cottony euphoria of clouds, the swirling patterns of the stars. When we made language to tell each other of the wonder of the world, we called that quickening science.

But our love of beauty grew edged with a lust for power that sent our science on what Bertrand Russell perceptively rued as its “passage from contemplation to manipulation.” The road forked between knowledge as a technology of control and knowledge as a technology of acceptance, of cherishing and understanding reality on its own terms and decoding those terms so that they can be met rather than manipulated.

We went on making equations and theories and bombs in an attempt to control life; we went on making poems and paintings and songs in an attempt to live with the fact that we cannot. Suspended between these poles of sensemaking, we built machines as sculptures of the possible and fed them our wishes encoded in commands, each algorithm ending in a narrowing of possibility between binary choices, having begun as a hopeful verse in the poetry of prospection.

Art by Dorothy Lathrop, 1922. (Available as a print and as stationery cards.)

Every writer, if they are lucky enough and passionate enough and dispassionate enough, reads in the course of their lifetime a handful of books they wish they had written. For me, Analogia (public library) by George Dyson is one such book — a book that traverses vast territories of fact and feeling to arrive at a promontory of meaning from which one can view with sudden and staggering clarity the past, the present, and the future all at once — not with fear, not with hope, but with something beyond binaries: with a quickening of wonderment and understanding.

Dyson is a peculiar person to tell the history and map the future of our relationship with technology. Peculiar and perfect: The son of mathematician Verena Huber-Dyson and the philosophically inclined physicist Freeman Dyson, and brother to technology investor and journalist Esther Dyson, George rebelled by branching from the family tree of science and technology at age sixteen to live, as he recounts, “in a tree house ninety-five feet up in a Douglas fir above Burrard Inlet in British Columbia, on land that had never been ceded by its rightful owners, the Tsleil-Waututh.”

Art from The Tree House by Dutch father-daughter duo Ronald Tolman and Marije Tolman, 2009

In this tree house he built with his own hands, Dyson shared the harsh winters — winters when a cup of tea poured from his perch would freeze before touching the ground — with a colony of cormorants roosting in the nextcrown fir. There, he watched a panoply of seabirds disappear underwater diving after silver swirls of fish he could see in the clear ocean all the way up from the tree. There, he learned to use, and to this day uses, his hands to build kayaks and canoes with the traditional materials and native techniques perfected over millennia. With those selfsame hands, he types these far-seeing thoughts:

There are four epochs, so far, in the entangled destinies of nature, human beings, and machines. In the first, preindustrial epoch, technology was limited to the tools and structures that humans could create with their own hands. Nature remained in control.

In the second, industrial epoch, machines were introduced, starting with simple machine tools, that could reproduce other machines. Nature began falling under mechanical control.

In the third epoch, digital codes, starting with punched cards and paper tape, began making copies of themselves. Powers of self-replication and self-reproduction that had so far been the preserve of biology were taken up by machines. Nature seemed to be relinquishing control. Late in this third epoch, the proliferation of networked devices, populated by metazoan codes, took a different turn.

In the fourth epoch, so gradually that almost no one noticed, machines began taking the side of nature, and nature began taking the side of machines. Humans were still in the loop but no longer in control. Faced with a growing sense of this loss of agency, people began to blame “the algorithm,” or those who controlled “the algorithm,” failing to realize there no longer was any identifiable algorithm at the helm. The day of the algorithm was over. The future belonged to something else.

A belief that artificial intelligence can be programmed to do our bidding may turn out to be as unfounded as a belief that certain people could speak to God, or that certain other people were born as slaves. The fourth epoch is returning us to the spirit-laden landscape of the first: a world where humans coexist with technologies they no longer control or fully understand. This is where the human mind took form. We grew up, as a species, surrounded by mind and intelligence everywhere we looked. Since the dawn of technology, we were on speaking terms with our tools. Intelligence in the cloud is nothing new. To adjust to life in the fourth epoch, it helps to look back to the first.

Born in the third epoch but identifying with the ways of the first, Dyson finds himself challenged “to reconcile the distinction, enforced by the American educational system, between those who make a living with their minds and those who make a living with their hands.” The challenge feels personal — we have each touched it in some aspect of our lives — but it is a universal challenge rooted in a long-ago bifurcation in our civilizational sensemaking: the split between digital computers, which process one thing at a time in succession, and analog computers, which process the dizzying everythingness of everything all at once. Our brains are analog computers, constantly orienting to reality by weaving a topology of connections into a three-dimensional map of patterns. Our machines hum to one-dimensional algorithms of sequential logical steps. Theirs is the time of bits, ours the time of atoms, the time of Kierkegaard, who knew that “the moment is not properly an atom of time but an atom of eternity.”

Ever/After by Maria Popova. (Available as a print.)

To be sure, there is ample digital coding at work in nature, in the building blocks of life itself — the DNA code used for information storage and information editing across time and generations. Trees, too, are digital computers, integrating myriad continuously changing inputs — available sunlight, available water, soil composition, atmospheric chemistry, wind direction, proximity of other trees — into the single-channel output of growth rings spaced in precise one-year intervals. They embody what may be the fundamental difference between the analog universe, in which time is a continuum, and the digital universe, in which there is no time — only the illusion of time woven of discrete steps, sequential but timeless. In his tree house, the teenage Dyson lived amid growth-rings dating back to the year 1426, a time when none of his European ancestors had set foot or the mind’s eye on those shores.

Imagine coming of age in such palpable contact with the continuity of time against the selective fragmentation we call history. Imagine becoming, in that crucible of awareness, an uncommonly insightful historian of science and technology.

Dyson traces the birth of the digital universe to Leibniz, who developed binary arithmetic after pondering the hexagrams of the ancient Chinese I Ching, then built on his already revolutionary work on infinitesimals to enlist the functions of binary arithmetic — functions analogous to the logical operations “and,” “or,” and “not” — in building the first universal language of binary code: a system of black and white marbles rolling along mechanical tracks, not unlike the zeroes and ones churning the Internet, that would encode into an alphabet of primes the real alphabet and all the concepts with which language is tasked. Leibniz envisioned the result as “a new kind of instrument which will increase the power of the mind much more than optical lenses strengthen the eyes.” This rudimentary digital computer would “work out, by an infallible calculus, the doctrines most useful for life, that is, those of morality and metaphysics.”

Art from Thomas Wright’s An Original Theory or New Hypothesis of the Universe, 1750. (Available as a print and as stationery cards.)

So began the modern mythos of computation as a controlled instrument for meaning-making, which we call artificial intelligence — the cult at whose altar we daily lay our faith in the ever-swifter logical processing of information, only to find ourselves empty-palmed for meaning and increasingly out of control. Dyson writes:

Leibniz’s digital universe, despite its powers, remains incomplete, just as Isaac Newton, his rival over credit for the invention of the calculus, gave us a mathematical description of nature that predicts everything correctly, but only up to a certain point. The next revolution will be the coalescence of programmable machines into systems beyond programmable control.

Every technology is a technology of thought that carries with it the ideologies of its time. Dyson builds this cautionary model of the future upon the foundation of the past, stratified with the same human tendencies that are now shaping our machines. He paints neither a techno-utopia nor a techno-dystopia but something more nuanced and complex, a kind of ominous autonomous techno-colonialism rooted in the ruthless colonial past: The first high-speed wireless communication network in North America, which manifested the contours of Leibniz’s vision and furnished the rudiments of the Internet, transmitted Morse code over sunlight across 60,000 square miles in a campaign to track down and capture the last free-roaming Apache: nineteen men, thirteen women, and six children.

Not one person alive in the spring of 1886 when the network first began firing — not Thomas Edison, who had just shut down his Menlo Park laboratory and married his second wife, not Walt Whitman who was facing his mortality while contemplating “the similitudes of the past and those of the future” — could have envisioned what would become of these rudiments, just as none of the early digital programmers high on their technicolor dreams envisioned how the algorithms they were composing might one day come to colonize the species that made them. Dyson writes:

Some inventions result from theorizing how something should work and then building it. Others result from building something that works before understanding it.

Pulsing beneath the history of our technologies of thought is the intimation that our unexamined belief in the digital universe as more efficient, powerful, and altogether superior to the analogy might be the product a colossal and catastrophic civilizational blind spot. Dyson challenges some of our basic intuitions and assumptions about analog and digital computing by contrasting our communication systems with those of whales — our evolutionary elders, predating our minds and our machines by fifty million years, whose songs were the only nonhuman sound we encoded on The Golden Record that sailed aboard the Voyager spacecraft to carry the signal of who and what we are for a thousand million years, to some other civilization in the unfathomed reaches of spacetime.

Art from Year of the Whale by Victor B. Scheffer, 1969

Dyson writes:

Whales both perceive their surroundings and communicate using sound, which behaves differently in an incompressible medium like water than in a compressible medium like air. If humans could communicate directly, brain to brain, using light, would we have developed languages based on a limited vocabulary of sounds? Human language, either evolved from or coevolved with sequences of discrete gestures, is optimized to withstand poor transmission over a noisy, low-bandwidth channel and might emerge quite differently among minds not subject to these constraints. Whales are no doubt communicating, but not necessarily by mapping their intelligence to sequences of discrete symbols the way we use language to convey our thoughts. When we play music, the whales might be thinking, “Finally, they are showing signs of trying to communicate like us!”

The difference between analog and digital computing parallels the question of whether a linear, symbolically coded language is a necessary indicator of conscious intelligence or not. In a digital computer, higher-dimensional inputs are reduced to one-dimensional strings of code that are stored, processed, and then translated back into higher-dimensional outputs, with a hierarchy of languages mediating the intervening steps. Large numbers of logical operations are transformed into waste heat along the way. Among analog computers, information can be stored, processed, and communicated directly as higher-dimensional maps.

An epoch after Einstein and Tagore contemplated the notion of a universal mind in their historic conversation, after the physicist John Ambrose Fleming — inventor of the vacuum tube and popularizer of the term “electronic” — exhorted humanity to regard the universe “not as a collection of Things or Events existing apart from any awareness of them by observers, but as manifested Thoughts in a Universal Mind,” after modern neuroscientists predicted the inevitability of a planetary übermind as the next step in the evolution of consciousness, Dyson points to what may be the most astonishing, supernatural-seeming analog sensemaking mesh-network of minds in nature:

Killer whales (Orcinus orca) are the largest of the dolphins, evolved from land mammals who returned to the sea more than fifty million years ago. They roam the entire planet as separate populations belonging to a single species, forming complex, persistent matrilineal social structures, with young males mentored by their post-reproductive grandmothers, whose life spans are known to reach a hundred years or more. Breathing, sleep, and other physiological functions are synchronized across the members of a pod. Their communication may be closer to telepathy than to language as we know it, and it could even be that orca mind and consciousness is a parallel, distributed property belonging to the pod collectively as much as to any individual whale.

These questions of consciousness and networked communication, central to our notions of artificial intelligence, grow even more rife with astonishment when we consider trees — organisms that, unlike us and unlike whales, lack minds as we understand them, minds as systems of operations conducted on nervous systems and brains, instead operating by what poet Jane Hirshfield admired as a “blind intelligence.”

Little Painting of Fir-Trees (1922) by Paul Klee, who believed that an artist is like a tree. (Available as a print and a face mask.)

Looking back on his time in the Douglas fir tree house, where he lived decades before Suzanne Simard published her epoch-making research on how trees communicate with one another, Dyson writes:

Living without telephone, computer, internet, or even electric light, I had time beyond measure to think. I found myself thinking about what, if anything, a tree might think. Not thinking the way we think, but the way a single neuron thinks, integrating information over time. It might take years to register the premonition of an idea, centuries for an entire forest, networked through synapses established by chemical signaling pathways among its roots, to form a thought. After three years I was no closer to an understanding, except to have gained a lingering suspicion that trees were, in some real and tangible way, as John Ambrose Fleming put it, “manifested Thoughts in a Universal Mind.”


Growth rings in trees are Nature’s way of digitizing time. Some of the split cedar boards paneling the walls of the tree house spanned seven hundred years. I counted the grain in one seven-inch board, and it went back to the year 1426. Halfway through that board, in 1679, Leibniz had imagined his digital computer, with marbles running along mechanical tracks. Two and a half inches ago, in 1778, James Cook had arrived on the Northwest Coast. Bering and Chirikov had arrived half an inch earlier, in 1741. My entire life, so far, spanned one-quarter of an inch.

Relief print from the cross-section of fallen tree by artist Bryan Nash Gill (1961–2013) from his project Woodcut.

With this telescopic view of time and with the hindsight of half a lifetime, having lived through the birth and euphoric adolescence of the modern digital age, Dyson suggests that the digital world will inevitably follow the trajectory of the living world as nature devised it, our algorithms commencing a kind self-referential evolutionary process that will soon altogether slip from our imperious creator-hands to take on a destiny of their own:

A digital universe is populated by two species of bits: differences that are varying in time but invariant in space, and differences that are varying in space but invariant in time. Bits can be stored over time as memory, or communicated across distance as code. Digital computers translate between these two forms of information — structure and sequence — according to definite rules. These powers of translation are more general than the arithmetical functions for which they were first invoked. Nature, too, discovered a method for translating sequences (of nucleotides) into structures (of proteins) — and back. Once this loop is established, evolution will do the rest.


Strings of bits gained the power of self-replication, just like strings of DNA. Thus began a chain reaction, with the order codes persisting largely unchanged, like the primordial alphabet of amino acids, over the seventy years since they were first released.

Nature evolved its analog computers — the nervous systems and brains that encode, store, and use information absorbed from the world, including the brain with which you are parsing this thought — so that organisms can learn to govern their own behavior and control their environment. Digital computers, being the product of our evolution-honed analog minds, cannot but follow the same course. Dyson writes:

Bits are the new electrons. Governing everything from the flow of goods to the flow of traffic to the flow of ideas, information is treated statistically, the way pulse-frequency-coded information is processed in a neuron or a brain. Analog is back, and its nature is to assume control.


Nature’s answer to those who seek to control nature through programmable machines is to allow us to build systems whose nature is beyond programmable control.

And yet something essential and essentially human is lost in our foundational assumption undergirding the digital world, in the strange certainty that binary arithmetic could ever fully represent the way we think. It is the thing all the poets and the rare poetic physicists have pondered all those epochs: the hunger for meaning beyond truth, for the beautiful beyond the binary. Dyson writes:

What if you wanted to capture what everything known to the human species means? Thanks to Moore’s law it takes little time and less and less money to capture all the information that exists. But how do you capture meaning? Even in the age of all things digital, this cannot be defined in logical terms, because meaning, among humans, isn’t logical. Leibniz’s logical utopia fails to close. The best you can do, once you have collected all possible answers, is to invite well-defined questions and compile a pulse-frequency-weighted map of how everything connects. This system, in conjunction with illogical humans, will not only be observing and mapping the meaning of things; it will start constructing meaning as well, the way a dictionary doesn’t just catalog a language, but defines the language, over time. The meaning of something is established, among humans, by the degree to which that something connects to other familiar things. A search engine, mapping those connections, isn’t just a collective model of how we think; increasingly, it is how we think. In time it will control meaning, in the same way as the traffic map controls the flow of traffic, even though no one is in control.

An exquisite mosaic of meaning, this book of subtle and unsythnesizable splendors chronicles and questions the choices we made as a civilization — not always consciously and not always conscientiously — that took us to where we are and shaped what we might become. But Analogia is also Dyson’s tender love letter to his parents, his love letter to the natural world, and his sensitive appeal, drawn both from a dispassionate scholarship of history and from the passions of his own life, for recognizing that the flow of information will neither drown out nor slake the longing for illumination in our primal search for meaning; an appeal for remembering that while the life of the mind filters our experience of the world, the mind is both function and functionary of the life of the body — not digital, not mechanical, but pulsating with analog aliveness, animated by the selfsame forces that rib the whales and ring the trees and constellate the atoms of long-dead stars into these cathedrals of consciousness that consecrate the subjective interpretation we call meaning.

Art from Trees at Night by Art Young, 1926. (Available as a print.)

Two generations ago, cybernetics pioneer Norbert Weiner made his cautionary case for “the human use of human beings,” prophesying that the world of the future — which is now our present — would be “an ever more demanding struggle against the limitations of our intelligence, not a comfortable hammock in which we can lie down to be waited upon by our robot slaves.”

Humanity was then too high on those early digital hopes and hubrises to heed his caution.

We now have another chance to listen, another chance to course-correct toward a future that cherishes whale songs above even the most efficient logical sequences of bits, another chance to branch off from the evolutionary tree of digital determinism that we ourselves have seeded.

Another Way by Maria Popova. (Available as a print, benefitting The Nature Conservancy.)

Humanity’s Most Successful Scientific Theory, Animated

How the gaps in gravity contour the next frontiers in the quest to understand the fundaments of what we are.

Humanity’s Most Successful Scientific Theory, Animated

Between the time Hypatia of Alexandria first pointed her pre-telescopic eye to the cosmos millennia before the notion of galaxies and the time Vera Rubin stood at the foot of the world’s most powerful telescope to confirm the existence of dark matter by observing how distant galaxies rotate, and in all the time before, and in all the time since, we have hungered to understand the forces that move the stars and the Moon and the mind. Ever since Galileo leaned on his artistic training in perspective to draw his astronomical observations intimating that the universe might not be what the theologians have claimed it to be, humanity has been on a passionate and disorienting quest to understand the nature of the mystery that made us.

Art from An Original Theory or New Hypothesis of the Universe, 1750. (Available as a print, as a face mask, and as stationery cards.)

In the centuries since, we have made staggering discoveries of fundamental forces swirling exotic particles into “the ricochet wonder of it all: the plain everythingness of everything, in cahoots with the everythingness of everything else.” Along the way, in our longing for a final theory of everything, we have been staggered by revelation after revelation that things are not what we previously thought them to be and beneath each layer of reality we have unpeeled lies another. The heavens are not a clockwork orrery of perfect orbs revolving around us in perfect circles. The cosmic wilderness is overgrown with a species of mystery we call dark matter and the fabric of spacetime is pocked with black holes the rims of which gape our Munchian scream at the sense that the universe remains a sweeping enigma whose native language we are only just beginning to decipher, naming our particles and composing our equations in the alphabet of a long-gone civilization that believed the Earth was flat and the stars were at its service.

Art from An Original Theory or New Hypothesis of the Universe, 1750. (Available as a print, as a face mask, and as stationery cards.)

Our yearning for a Theory of Everything has culminated in what we call the Standard Model — a conceptual map of all the known particles and the fundamental forces that govern them to make the universe cohere into everything we know and are. It is the most successful scientific theory in the history of our species. But it is rather a Theory of Everything We Know So Far, at once triumphal and tessellated with incompleteness.

The essence of that theory, its central contradictions, and how it contours the next layer of reality awaiting discovery is what theoretical physicist David Tong details in this animated primer for Quanta Magazine, drawing out discoveries and questions that punctuate the excellent anthology Alice and Bob Meet the Wall of Fire: The Biggest Ideas in Science from Quanta (public library).

Complement with an animated look at the little loophole in the Big Bang model, then revisit the remarkable story of how Johannes Kepler revolutionized our understanding of the universe while defending his mother in a witchcraft trial.


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.


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, from 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.


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