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The Enchantment of Mathematics

A 19th-century love letter to the most limitless medium of thought.

The Enchantment of Mathematics

Mathematics is at once the most precise and the most abstract instrument of thought — a convergence of symbol and sentience utterly poetic in its ability to convey the most complex underlying laws of the universe in stunning simplicity of expression. It mirrors the world back to itself both condensed and expanded, granting us an enlarged understanding through the art of distillation. Ada Lovelace considered mathematics the “poetical science” and, in contemplating the nature of the imagination, called it “the language of the unseen relations between things.” Perhaps E = mc2 is the greatest line of poetry ever written, then. At the very least, it inhabits the same world as “To be, or not to be”; it is the mathematical counterpart to “the still point of the turning world.” So is it any wonder that mathematics renders some of humanity’s most potent minds nothing short of besotted?

Hardly anyone has captured the mesmerism of mathematics more beautifully than the pioneering 19th-century English mathematician James Joseph Sylvester (September 3, 1814–March 15, 1897) in a magnificent speech he delivered on February 22, 1877 in Baltimore.

I often say that literature is the original Internet: A footnote — that ancient analog hyperlink — in Oliver Sacks’s masterwork of science and spirit, Awakenings, led me to Sylvester’s speech, included in The Collected Mathematical Papers of James Joseph Sylvester: Volume III (public library) under the title “Address on Commemoration Day at Johns Hopkins University.”

jamesjosephsylvester

Sylvester considers this difficult art of conceptual condensation:

It is the constant aim of the mathematician to reduce all his expressions to the lowest terms, to retrench every superfluous word and phrase, and to condense the Maximum of meaning into the Minimum of language.

And yet the joy of mathematics, he argues, isn’t an esoteric pursuit reserved for academically trained mathematicians — rather, it is a supreme and universal delight of the human mind at play with itself:

I have reason to think that the taste for mathematical science, even in its most abstract form, is much more widely diffused than is generally supposed…

This wide appeal of the mathematical spirit, Sylvester observes, stems from its immensity of scope and its infinite range of intimacies with the nature of the world:

Mathematics is not a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its contour defined: it is limitless as that space which it finds too narrow for its aspirations; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the astronomer’s gaze: it is as incapable of being restricted within assigned boundaries or being reduced to definitions of permanent validity, as the consciousness, the life, which seems to slumber in each monad, in every atom of matter, in each leaf and bud and cell, and is forever ready to burst forth into new forms of vegetable and animal existence.

[…]

Every science becomes more perfect, approaches more closely to its own ideal, in proportion as it imitates or imbibes the mathematical form and spirit.

Complement with the illustrated story of legendary mathematician Paul Erdős and mathematical genius John Horton Conway on the art of being a professional nonunderstander.

BP

Genius at Play: A Brilliant Mathematician on Tinkering, Thinkering, and the Art of Being a Professional Nonunderstander

Anatomy of thought at the fault line of invention and discovery.

Genius at Play: A Brilliant Mathematician on Tinkering, Thinkering, and the Art of Being a Professional Nonunderstander

“Mathematical Science,” wrote Ada Lovelace in contemplating the nature of the imagination, “is the language of the unseen relations between things.” Few have mastered that language and transmuted it into Lovelace’s “poetical science” more deftly than the trailblazing English mathematician John Horton Conway, best known for the invention of the 1960s cellular automaton Game of Life.

A fine addition to the best science books of the year, Genius at Play: The Curious Mind of John Horton Conway (public library) by Siobhan Roberts is noteworthy for many reasons, among them the non-negligible fact of being a biography of a living subject — a task generally self-defeating (Susan Sontag famously proclaimed that “no biography makes sense that isn’t written after its subject is dead”) and rarely approached with as much tenacious graciousness as Roberts’s. That the subject is a man of enormous complexity and contradiction, with Richard Feynman’s charisma and Slavoj Žižek’s contrarian edge, only adds to the feat.

John Horton Conway (Photograph: Thane Plambeck)
John Horton Conway (Photograph: Thane Plambeck)

Alongside her intense intellectual admiration for Conway’s genius, Roberts becomes a keen observer of the elemental human psychology that bedevils even a mind as superhuman as his. She describes her subject as an “insecure egotist” — a redundant phrase absolutely perfect in its redundancy, for it’s hard to think of an egotist who isn’t at bottom insecure, brimming with what psychoanalysts call “narcissistic vulnerability.”

This paradoxical orientation of self to world comes into play in Conway’s conflicted attitude toward the biography itself. Roberts writes:

He very much cares what other people think, and he worries that a self-portrait might come off as too egotistical. And partly because he’d have a hard time with “the fiction of humility that the conventional autobiographer must at every moment struggle to maintain,” as the occasional biographer Janet Malcolm describes the dilemma. So he’ll stick to doing what he does best. Gnawing on his left index finger with his chipped old British teeth, temporal veins bulging and brow pensively squinched beneath the day before yesterday’s hair, Conway unapologetically whiles away his hours tinkering and thinkering — which is to say he’s ruminating, or maybe he is doing some work, but he’ll insist he’s doing nothing, being lazy, playing games.

For his part, Conway is rather precise about the particular allure of this tinkering and thinkering. Reflecting on the governing rule of Subprime Fibs, one of the innumerable games he invented, he tells Roberts:

I’ll tell you what interests me about this — it’s really what interests me about mathematics. Nobody else in the whole history of the world has been stupid enough to invent this rule. That’s the first thing. But then, if they had, they would find exactly this behavior that I’m finding.

[…]

That’s a curious thing about the nature of mathematical existence. This rule hasn’t physically existed in any sense in the world before a month ago, before I invented it, but it sort of intellectually existed forever. There is this abstract world which in some strange sense has existed throughout eternity.

Imagine an uninhabited planet, full of interesting things. You land on it, and it existed for a million years, but no people have ever been there, no sentient beings. There are such places, I’m sure. Go to some remote star and there will be something. But you don’t have to go there. You can sit in this very chair and find something that has existed throughout all of eternity and be the first person to explore it.

Art by Anatolii Fomenko from Mathematical Impressions
Art by Anatolii Fomenko from Mathematical Impressions

Conway arrives at this bewitching intersection of discovery and invention by being at once a naturalist of numbers, an algebraic adventurer, and an unflinching empiricist. Roberts captures the singular spirit of his endeavor:

He turns numbers over, upside down, and inside out, observing how they behave. Why is it that when you pick a number, any number, then double it, add 6, halve it, and take away the number you started with, your answer is always 3? Above all he loves knowledge, and he seeks to know everything about the universe. Conway’s charisma lies in his desire to share his incurable lust for learning, to spread the contagion and the romance. He is dogged and undaunted in explaining the inexplicable, and even when the inexplicable remains so, he leaves his audience elevated, fortified by the failed attempt and feeling somehow in cahoots, privy to the inside dope, satisfied at having flirted with a glimmer of understanding. For his own part, he calls himself a professional nonunderstander. The pursuit is what counts…

This notion, of course, is as central to genius in the sciences as it is in the arts — something Grace Paley articulated beautifully in her advice to aspiring writers. Conway himself examines the cognitive machinery of this essential disposition of nonunderstanding:

In a fundamental way my job is thinking. You can’t see it from the outside. What does the thinking consist of? I think about how to explain whatever I am thinking about to someone. Then I explain it to someone and it doesn’t work. So I think about it some more. I tinker with it, with thinking, until I’ve simplified it. I personally can only understand things after I’ve thought about them for ages and made them very, very simple.

In a sentiment that calls to mind I, Pencil — that brilliant 1958 allegory of the division of knowledge, illustrating how everything is connected — Conway adds:

Most people just understand enough to work. For example, a mechanic doesn’t necessarily understand the physics or engineering of how a car works. I’m not putting down a car mechanic. We need practical people. I’m not sure we need theoretical people. Though I’m not going to campaign for my own abolishment.

Genius at Play is a tremendous read in its totality. Complement it with this wonderful picture-book biography of the eccentric mathematician Paul Erdős, a close collaborator of Conway’s, then revisit John Dewey on how we think.

BP

Alexander von Humboldt and the Invention of Nature: How One of the Last True Polymaths Pioneered the Cosmos of Connections

“In this great chain of causes and effects, no single fact can be considered in isolation.”

Alexander von Humboldt and the Invention of Nature: How One of the Last True Polymaths Pioneered the Cosmos of Connections

No thinker has shaped our understanding of the astounding interconnectedness of the universe more profoundly than the great Prussian naturalist, explorer, and geographer Alexander von Humboldt (September 14, 1769–May 6, 1859), who pioneered the notion that the natural world is a web of intricately entwined elements, each in constant dynamic dialogue with every other — a concept a century ahead of its time. His legacy isn’t so much any single discovery — although he did discover the magnetic equator, invented isotherms, and came up with climate zones — as it is a mindset, a worldview, a singular sensemaking sublimity.

Alexander von Humboldt by Friedrich Georg Weitsch, 1806
Alexander von Humboldt by Friedrich Georg Weitsch, 1806

Goethe, in his conversations with Eckermann, remarked that a single day with Humboldt enriched him more than years spent alone, enthusing:

What a man he is! … He has not his equal in knowledge and living wisdom. Then he has a many-sidedness such as I have found nowhere else. On whatever point you approach him, he is at home, and lavishes upon us his intellectual treasures. He is like a fountain with many pipes, under which you need only hold a vessel, and from which refreshing and inexhaustible streams are ever flowing.

Darwin asserted that Humboldt’s writings kindled in him a zeal without which he wouldn’t have boarded the Beagle or written On the Origin of Species. Thoreau was an ardent admirer of Humboldt’s “habit of close observation,” without the influence of which there might have been no Walden. Trailblazing astronomer Maria Mitchell, who met Humboldt weeks before his death, marveled in her diary that “no young aspirant in science ever left Humboldt’s presence uncheered,” and his ideas reverberate through her famous assertion that science is “not all mathematics, nor all logic, but it is somewhat beauty and poetry.” Emerson, in his essays and lectures, called Humboldt “a man whose eyes, ears, and mind are armed by all the science, arts, and implements which mankind have anywhere accumulated” and saw him as living proof that “a certain vastness of learning, or quasi omnipresence of the human soul in nature, is possible.”

Goethe's diagram of the comparative table elevations of the Old and New World, inspired by Humboldt
Goethe’s diagram of the comparative table elevations of the Old and New World, inspired by Humboldt

In informing and impressing the greatest minds of his time, Humboldt invariably influenced the course of science and its intercourse with the rest of culture in ways innumerable, enduring, and profound. His visionary understanding of nature’s interconnected sparked the basic ecological awareness that gave rise to the environmental movement. His integrated approach to science, incorporating elements of art, philosophy, poetry, politics, and history, provided the last bold counterpoint to the disconnected and dysfunctional “villages” of specialization into which science would fragment a mere generation later. And yet Humboldt, despite his enormous contribution to our most fundamental understanding of life, is largely forgotten today.

In The Invention of Nature: Alexander von Humboldt’s New World (public library), London-based design historian and writer Andrea Wulf sets out to liberate this extraordinary man’s legacy from the grip of obscurity and short-termism, illuminating the myriad threads of influence through which he continues to shape our present thinking about science, society, and life itself.

Alexander von Humboldt in his home library at at 67 Oranienburger Strasse, Berlin. Chromolithograph copy of watercolor drawing by Eduard Hildebrant, 1856.
Alexander von Humboldt in his home library at at 67 Oranienburger Strasse, Berlin. Chromolithograph copy of watercolor drawing by Eduard Hildebrant, 1856.

Wulf paints the backdrop for Humboldt’s enduring genius:

Described by his contemporaries as the most famous man in the world after Napoleon, Humboldt was one of the most captivating and inspiring men of his time. Born in 1769 into a wealthy Prussian aristocratic family, he discarded a life of privilege to discover for himself how the world worked. As a young man he set out on a five-year exploration to Latin America, risking his life many times and returning with a new sense of the world. It was a journey that shaped his life and thinking, and that made him legendary across the globe. He lived in cities such as Paris and Berlin, but was equally at home on the most remote branches of the Orinoco River or in the Kazakh Steppe at Russia’s Mongolian border. During much of his long life, he was the nexus of the scientific world, writing some 50,000 letters and receiving at least double that number. Knowledge, Humboldt believed, had to be shared, exchanged and made available to everybody.

But knowledge, for Humboldt, wasn’t merely an intellectual faculty — it was an embodied, holistic presence with life in all of its dimensions. A rock-climber, volcano-diver, and tireless hiker well into his eighties, Humboldt saw observation as an active endeavor and continually tested the limits of his body in his scientific pursuits. For him, mind, body, and spirit were all instruments of inquiry into the nature of the world. Two centuries before Carl Sagan sold us on the idea that “science invariably elicits a sense of reverence and awe,” Humboldt advocated for this then-radical notion amid a culture that drew a thick line between reason and emotion.

Wulf writes:

Fascinated by scientific instruments, measurements and observations, he was driven by a sense of wonder as well. Of course nature had to be measured and analysed, but he also believed that a great part of our response to the natural world should be based on the senses and emotions. He wanted to excite a “love of nature.” At a time when other scientists were searching for universal laws, Humboldt wrote that nature had to be experienced through feelings.

Alexander von Humboldt by Joseph Karl Stieler, 1843
Alexander von Humboldt by Joseph Karl Stieler, 1843

Out of this integrated approach to knowledge sprang Humboldt’s revolutionary view of life — the scientifically informed counterpart to Ada Lovelace’s famous assertion that “everything is naturally related and interconnected.” Wulf captures his greatest legacy:

Humboldt revolutionized the way we see the natural world. He found connections everywhere. Nothing, not even the tiniest organism, was looked at on its own. “In this great chain of causes and effects,” Humboldt said, “no single fact can be considered in isolation.” With this insight, he invented the web of life, the concept of nature as we know it today.

When nature is perceived as a web, its vulnerability also becomes obvious. Everything hangs together. If one thread is pulled, the whole tapestry may unravel.

Humboldt's 1806 drawing of the geographic distribution of plants based on mountain height and air temperature
Humboldt’s 1806 drawing of the geographic distribution of plants based on mountain height and air temperature

Given his attentiveness to this interconnectedness across all scales and dimensions of life, it is hardly surprising that Humboldt became the first scientist to admonish against the grave consequences of human-induced climate change after witnessing the environmental devastation of deforestation brought on by the spread of colonial plantations across South America in the 1800s.

Wulf writes:

Deforestation there had made the land barren, water levels of the lake were falling and with the disappearance of brushwood torrential rains had washed away the soils on the surrounding mountain slopes. Humboldt was the first to explain the forest’s ability to enrich the atmosphere with moisture and its cooling effect, as well as its importance for water retention and protection against soil erosion. He warned that humans were meddling with the climate and that this could have an unforeseeable impact on “future generations.”

[…]

We are shaped by the past. Nicolaus Copernicus showed us our place in the universe, Isaac Newton explained the laws of nature, Thomas Jefferson gave us some of our concepts of liberty and democracy, and Charles Darwin proved that all species descend from common ancestors. These ideas define our relationship to the world.

Humboldt gave us our concept of nature itself. The irony is that Humboldt’s views have become so self-evident that we have largely forgotten the man behind them. But there exists a direct line of connection through his ideas, and through the many people whom he inspired. Like a rope, Humboldt’s concept of nature connects us to him.

Wulf pulls on that rope with both hands:

There are many reasons why Humboldt remains fascinating and important: not only was his life colourful and packed with adventure, but his story gives meaning to why we see nature the way we see it today. In a world where we tend to draw a sharp line between the sciences and the arts, between the subjective and the objective, Humboldt’s insight that we can only truly understand nature by using our imagination makes him a visionary.

Humboldt’s disciples, and their disciples in turn, carried his legacy forward — quietly, subtly and sometimes unintentionally. Environmentalists, ecologists and nature writers today remain firmly rooted in Humboldt’s vision — although many have never heard of him. Nonetheless, Humboldt is their founding father.

As scientists are trying to understand and predict the global consequences of climate change, Humboldt’s interdisciplinary approach to science and nature is more relevant than ever. His beliefs in the free exchange of information, in uniting scientists and in fostering communication across disciplines, are the pillars of science today. His concept of nature as one of global patterns underpins our thinking.

[…]

It feels as if we’ve come full circle. Maybe now is the moment for us and for the environmental movement to reclaim Alexander von Humboldt as our hero.

The Invention of Nature is a riveting read in its entirety, bringing back to life the remarkable man who gave shape to life as we know it. Complement it with the equally enchanting story of Luke Howard — the young amateur meteorologist who classified the clouds and who, like his contemporary Humboldt, bewitched Goethe with his genius — then trace Humboldt’s legacy to our present-day understanding of how everything connects.

BP

Order, Disorder, and Oneself: French Polymath Paul Valéry on How to Never Misplace Anything

“Disorder comes of putting things in places you have laboriously thought up or finally discovered after a series of experiments, calculations, deviations, and successive swerves from your natural bent.”

Anyone who has experienced the profound satisfaction of alphabetizing a bookshelf or organizing a kitchen cabinet knows the psychological rewards of transmuting physical chaos into physical order — something philosopher Gaston Bachelard captured beautifully in his 1957 meditation on the poetics of space. But the relationship between material and mental order isn’t always linear — the most brilliant person I know also happens to be magnificently messy.

Count on the great French poet, essayist, and philosopher Paul Valéry (October 30, 1871–July 21, 1945) — an intellectual titan who influenced such luminaries as Susan Sontag and André Gide, and one of humanity’s greatest crusaders for nuance — to offer a counterintuitive solution to the problem of order and chaos.

Art by Ursus Wehrli from The Art of Cleanup

In one of his “analects” — aphorisms and short moral reflections from his notebooks, assembled in Collected Works of Paul Valery, Volume 14: Analects (public library) — Valéry writes under the heading Order, Disorder, and Oneself:

I have unearthed this notebook which I thought I’d lost. It had not been mislaid; quite the contrary, put in so “right” a place that I could hardly believe my eyes. To have put it there wasn’t like me. I’d lost touch with my Ariadne’s clew, my “disorder.” I mean a private, personal, familiar disorder.

From this personal anecdote Valéry proceeds to extrapolate a universal tenet of order, arguing that intentional organization — like all of our control strategies — is more likely to backfire and create disorder rather than order due to its forced nature:

If you don’t want things to get lost, always put them where your instinct is to put them. You don’t forget what you would always do.

Real disorder is a breach of this rule — a waiving of the principle of frequency. Disorder comes of putting things in places you have laboriously thought up or finally discovered after a series of experiments, calculations, deviations, and successive swerves from your natural bent. And you hail each new cache as a discovery, a New World, a marvelous solution!

So when I want to find the object again, I am obliged to retrieve one particular train of thought, without anything to guide me back to it.

But if it was placed instinctively, all I need is to rediscover myself, lock, stock, and barrel — that’s to say I have only to be myself.

If disorder is the rule with you, you will be penalized for installing order.

So — keep to your rule!

Collected Works of Paul Valery is a trove of timeless wisdom in its totality. Complement it with Valéry on the three-body problem — one of the most insightful things ever written about how we relate to our physical selves — then revisit Swiss artist Ursus Wehrli’s playful project The Art of Cleanup.

BP

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