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Probability Theory Pioneer Mark Kac on the Duality of the Creative Life, the Singular Enchantment of Mathematics, and the Two Types of Geniuses

“Creative people live in two worlds. One is the ordinary world which they share with others and in which they are not in any special way set apart from their fellow men. The other is private and it is in this world that the creative acts take place.”

Probability Theory Pioneer Mark Kac on the Duality of the Creative Life, the Singular Enchantment of Mathematics, and the Two Types of Geniuses

The great Polish-American mathematician Mark Kac (August 3, 1914–October 26, 1984) possessed one of the most dazzling minds of the twentieth century. In pioneering probability theory, he paved the way for a radical new conception of truth and ushered in the first generation of scientists trained to think probabilistically — a more accurate assessment of knowledge, making room for uncertainty, be it scientific or otherwise. This probabilistic mode of judgment is all the more necessary today as the growing complexity of the world is swirling us into exponentially increasing uncertainty, which we attempt to tame through artificial absolutism.

Mathematics literally saved Kac’s life. His student work earned him a post-doctoral fellowship to study abroad, so he left Poland for Johns Hopkins University in December of 1938. World War II broke out months later. His entire family, along with millions of other Jews, was killed by the Nazis.

Kac went on to lead a long and creatively fertile life — one he considered, despite this unfathomable share of misfortune, a tremendously fortunate one. “I must pay tribute to that powerful but capricious lady, Chance, who chose to bestow her beneficence on my personal life even though I spent much of my mathematical life trying to prove that she does not really exist,” he wrote with his characteristic mix of wit and wisdom in Enigmas of Chance: An Autobiography (public library) — a small, wonderful 1976 book I discovered via a passing mention in an interview with the trailblazing astronomer Vera Rubin. (Here is further proof of my longstanding conviction that literature is the original Internet — such citations, allusions, and cross-references between books are the wondrous “hyperlinks” connecting human knowledge throughout our “common record.”)

Mark Kac
Mark Kac

Kac writes:

Creative people live in two worlds. One is the ordinary world which they share with others and in which they are not in any special way set apart from their fellow men. The other is private and it is in this world that the creative acts take place. It is a world with its own passions, elations and despairs, and it is here that, if one is as great as Einstein, one may even hear the voice of God. The two worlds are intimately and intricately connected. Jealousy, the desire for recognition and competitiveness, for example, are part of the ordinary world but they are among the forces which propel into the second. Similarly, dreams and triumphs in the second have a way of merging with less than lofty thoughts of rewards in the first.

With an eye to the particular challenge that autobiography presents to the creative person, he adds:

To create a coherent and truthful picture of life in the two disparate and yet interrelated worlds is a nearly impossible task.

In discussing his great heroes and influences, Kac delineates another dichotomy in creative culture — the bifurcation of brilliance by degree and by kind:

In science, as well as in other fields of human endeavor, there are two kinds of geniuses: the “ordinary” and the “magicians.” An ordinary genius is a [person] that you and I would be just as good as, if we were only many times better. There is no mystery as to how his mind works. Once we understand what he has done, we feel certain that we, too, could have done it. It is different with the magicians. They are, to use mathematical jargon, in the orthogonal complement of where we are and the working of their minds is for all intents and purposes incomprehensible. Even after we understand what they have done, the process by which they have done it is completely dark. They seldom, if ever, have students because they cannot be emulated and it must be terribly frustrating for a brilliant young mind to cope with the mysterious ways in which the magician’s mind works.

He points to the nuclear physicist Hans Bethe as an example of an “ordinary genius” and to Richard Feynman as a “magician.” (Kac’s distinction appears in James Gleick’s superb biography of Feynman and there is a high probability that it inspired the title of BBC’s documentary about the legendary physicist, No Ordinary Genius.) I would add Alan Turing to the “magicians” category, and of course Albert Einstein.

Illustration by Vladimir Radunsky for On a Beam of Light: A Story of Albert Einstein by Jennifer Berne

In the postscript, Kac considers what lends mathematics its enduring enchantment — what renders people besotted with it:

Mathematics is an ancient discipline. For as long as we can reliably reach into the past, we can find its development intimately connected with the development of the whole of our civilization. For as long as we have a record of man’s curiosity and his quest for understanding, we find mathematics cultivated and cherished, practiced and taught. Throughout the ages it has stood as an ultimate in rational thought and as a monument to man’s desire to probe the workings of his own mind.


The urge to understand and to create mathematics has always been remarkable, considering that those who have devoted their lives to the service of this aloof and elusive mistress could expect neither great material rewards nor widespread fame.

A champion of nuance, Kac challenges Bertrand Russell’s famous assertion that “mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.” Kac finds it to be a “dull and rather miserable picture of mathematics” and yet “not wholly wrong,” “just hopelessly incomplete and one-sided.” Instead, he considers what grants mathematics its richness as a mode of illuminating reality:

I am reminded of something Balthazaar van der Pol, a great Dutch scientist and engineer who was also a fine musician, remarked to me about the music of Bach. “It is great,” he said, “because it is inevitable and yet surprising.” I have often thought about this lovely epigram in connection with mathematics… The inevitability is, in many cases, provided by logic alone, but the element of surprise must come from an insight outside the rigid confines of logic.

It warrants noting that the altogether marvelous Enigmas of Chance is part of a series of scientists’ autobiographies funded by the Alfred P. Sloan Foundation, which has probably done more for science and its social life than any other entity in the past half-century. Complement it with this beautiful love letter to mathematics by the pioneering 19th-century English mathematician James Joseph Sylvester, the illustrated life of the eccentric mathematical genius Paul Erdos, and the great English mathematician John Horton Conway on tinkering, thinkering, and the art of being a professional nonunderstander.


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.”


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.


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.


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 interconnectedness 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.


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