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Figures of Thought: Krista Tippett Reads Howard Nemerov’s Mathematical-Existential Poem About the Interconnectedness of the Universe

A splendid song of praise for the elemental truth at the heart of all art, science, and nature.

Figures of Thought: Krista Tippett Reads Howard Nemerov’s Mathematical-Existential Poem About the Interconnectedness of the Universe

“A leaf of grass is no less than the journey work of the stars,” Walt Whitman wrote in one of his most beautiful poems in the middle of the nineteenth century, just as humanity was coming awake to the glorious interconnectedness of nature — to the awareness, in the immortal words of the great naturalist John Muir, that “when we try to pick out anything by itself, we find it hitched to everything else in the universe.”

A century later, Albert Einstein recounted his takeaway from the childhood epiphany that made him want to be a scientist: “Something deeply hidden had to be behind things.” Virginia Woolf, in her account of the epiphany in which she understood she was an artist — one of the most beautiful and penetrating passages in all of literature — articulated a kindred sentiment: “Behind the cotton wool is hidden a pattern… the whole world is a work of art… there is no Shakespeare… no Beethoven… no God; we are the words; we are the music; we are the thing itself.”

This interleaved thing-itselfness of existence, hidden in plain sight, is what two-time U.S. Poet Laureate Howard Nemerov (February 29, 1920–July 5, 1991) takes up, two centuries after William Blake saw the universe in a grain of sand, in a spare masterpiece of image and insight, found in his altogether wondrous Collected Poems (public library), winner of both the Pulitzer Prize and the National Book Award.

Howard Nemerov

On Being creator and Becoming Wise author Krista Tippett brought the poem to life at the third annual Universe in Verse, with a lovely prefatory meditation on the role of poetry — ancient, somehow forgotten in our culture, newly rediscovered — as sustenance and salve for the tenderest, truest, most vital parts of our being.

FIGURES OF THOUGHT
by Howard Nemerov

To lay the logarithmic spiral on
Sea-shell and leaf alike, and see it fit,
To watch the same idea work itself out
In the fighter pilot’s steepening, tightening turn
Onto his target, setting up the kill,
And in the flight of certain wall-eyed bugs
Who cannot see to fly straight into death
But have to cast their sidelong glance at it
And come but cranking to the candle’s flame —

How secret that is, and how privileged
One feels to find the same necessity
Ciphered in forms diverse and otherwise
Without kinship — that is the beautiful
In Nature as in art, not obvious,
Not inaccessible, but just between.

It may diminish some our dry delight
To wonder if everything we are and do
Lies subject to some little law like that;
Hidden in nature, but not deeply so.

For more science-celebrating splendor from The Universe in Verse, savor astrophysicist Janna Levin reading “A Brave and Startling Truth” by Maya Angelou and “Planetarium” by Adrienne Rich; poet Sarah Kay reading from “Song of Myself” by Walt Whitman; Regina Spektor reading “Theories of Everything” by the astronomer and poet Rebecca Elson; Amanda Palmer reading “Hubble Photographs: After Sappho” by Adrienne Rich; and Neil Gaiman’s original tributes-in-verse to women in science, environmental founding mother Rachel Carson, and astronomer Arthur Eddington, who confirmed Einstein’s relativity in the wake of a World War that had lost sight of our shared belonging and common cosmic spring.

BP

A Pioneering Case for the Value of Citizen Science from the Polymathic Astronomer John Herschel

“There is scarcely any well-informed person, who, if he has but the will, has not also the power to add something essential to the general stock of knowledge.”

A Pioneering Case for the Value of Citizen Science from the Polymathic Astronomer John Herschel

“It is always difficult to teach the man of the people that natural phenomena belong as much to him as to scientific people,” the trailblazing astronomer Maria Mitchell wrote as she led the first-ever professional female eclipse expedition in 1878. The sentiment presages the importance of what we today call “citizen science,” radical and countercultural in an era when science was enshrined in the pompous pantheon of the academy, whose gates were shut and padlocked to “the man of the people,” to women, and to all but privileged white men.

Two decades earlier, Mitchell had traveled to Europe as America’s first true scientific celebrity to meet, among other dignitaries of the Old World, one such man — but one of far-reaching vision and kindness, who used his privilege to broaden the spectrum of possibility for the less privileged: the polymathic astronomer John Herschel (March 7, 1792–May 11, 1871), co-founder of the venerable Royal Astronomical Society, son of Uranus discoverer William Herschel, and nephew of Caroline Herschel, the world’s first professional woman astronomer, who had introduced him to astronomy as a boy.

Several years before he coined the word photography, Herschel became the first prominent scientist to argue in a public forum that the lifeblood of science — data collection and the systematic observation of natural phenomena — should be the welcome task of ordinary people from all walks of life, united by a passionate curiosity about how the universe works.

John Herschel (artist unknown)

In 1831, the newly knighted Herschel published A Preliminary Discourse on the Study of Natural Philosophy as part of the fourteenth volume of the bestselling Lardner’s Cabinet Cyclopædia (large chunks of which were composed by Frankenstein author Mary Shelley). Later cited in Lorraine Daston and Elizabeth Lunbeck’s altogether excellent book Histories of Scientific Observation (public library), it was a visionary work, outlining the methods of scientific investigation by clarifying the relationship between theory and observation. But perhaps its most visionary aspect was Herschel’s insistence that observation should be a network triumph belonging to all of humanity — a pioneering case for the value of citizen science. He writes:

To avail ourselves as far as possible of the advantages which a division of labour may afford for the collection of facts, by the industry and activity which the general diffusion of information, in the present age, brings into exercise, is an object of great importance. There is scarcely any well-informed person, who, if he has but the will, has not also the power to add something essential to the general stock of knowledge, if he will only observe regularly and methodically some particular class of facts which may most excite his attention, or which his situation may best enable him to study with effect.

Diversity of snowflake shapes from a 19th-century French science textbook. Available as a print.

Pointing to meteorology and geology as the sciences best poised to benefit from distributed data collection by citizen scientists, Herschel adds:

There is no branch of science whatever in which, at least, if useful and sensible queries were distinctly proposed, an immense mass of valuable information might not be collected from those who, in their various lines of life, at home or abroad, stationary or in travel, would gladly avail themselves of opportunities of being useful.

Herschel goes on to outline the process by which such citizen science would be conducted: “skeleton forms” of survey questions circulated far and wide, asking “distinct and pertinent questions, admitting of short and definite answers,” then transmitted to “a common centre” for processing — a sort of human internet feeding into a paper-stack server. (Lest we forget, Maria Mitchell herself was employed as a “computer” — the term we used to use for the humans who performed the work now performed by machines we have named after them.)

Couple with a wonderful 1957 treatise on the art of observation and why genius lies in the selection of what is worth observing, then revisit Maria Mitchell on how to find your calling.

BP

Pioneering Mathematician G.H. Hardy on How to Find Your Purpose and What Is Most Worth Aspiring for

“If a man has any genuine talent he should be ready to make almost any sacrifice in order to cultivate it to the full.”

Pioneering Mathematician G.H. Hardy on How to Find Your Purpose and What Is Most Worth Aspiring for

“Resign yourself to the lifelong sadness that comes from never ­being satisfied,” Zadie Smith counseled in the tenth of her ten rules of writing — a tenet that applies with equally devastating precision to every realm of creative endeavor, be it poetry or mathematics. Bertrand Russell addressed this Faustian bargain of ambition in his 1950 Nobel Prize acceptance speech about the four desires motivating all human behavior: “Man differs from other animals in one very important respect, and that is that he has some desires which are, so to speak, infinite, which can never be fully gratified, and which would keep him restless even in Paradise. The boa constrictor, when he has had an adequate meal, goes to sleep, and does not wake until he needs another meal. Human beings, for the most part, are not like this.”

Ten years earlier, the English mathematician and number theory pioneer G.H. Hardy (February 7, 1877–December 1, 1947) — an admirer of Russell’s — examined the nature of this elemental human restlessness in his altogether fascinating 1940 book-length essay A Mathematician’s Apology (public library).

G.H. Hardy

In considering the value of mathematics as a field of study and “the proper justification of a mathematician’s life,” Hardy offers a broader meditation on how we find our sense of purpose and arrive at our vocation. Addressing “readers who are full, or have in the past been full, of a proper spirit of ambition,” Hardy writes in an era when every woman was colloquially “man”:

A man who is always asking “Is what I do worth while?” and “Am I the right person to do it?” will always be ineffective himself and a discouragement to others. He must shut his eyes a little and think a little more of his subject and himself than they deserve. This is not too difficult: it is harder not to make his subject and himself ridiculous by shutting his eyes too tightly.

[…]

A man who sets out to justify his existence and his activities has to distinguish two different questions. The first is whether the work which he does is worth doing; and the second is why he does it, whatever its value may be. The first question is often very difficult, and the answer very discouraging, but most people will find the second easy enough even then. Their answers, if they are honest, will usually take one or other of two forms; and the second form is a merely a humbler variation of the first, which is the only answer we need consider seriously.

Most people, Hardy argues, answer the first question by pointing to a natural aptitude that led them to a vocation predicated on that particular aptitude — the lawyer became a lawyer because she naturally excels at eloquent counter-argument, the cricketer a cricketer because he has a natural gift for cricket. In what may sound like an ungenerous sentiment but is indeed statistically accurate, Hardy adds:

I am not suggesting that this is a defence which can be made by most people, since most people can do nothing at all well. But it is impregnable when it can be made without absurdity, as it can by a substantial minority: perhaps five or even ten percent of men can do something rather well. It is a tiny minority who can do something really well, and the number of men who can do two things well is negligible. If a man has any genuine talent he should be ready to make almost any sacrifice in order to cultivate it to the full.

Illustration by artist Hugh Lieber from Human Values and Science, Art and Mathematics by mathematician Lillian Lieber

But while talent exists in varying degrees within each field of endeavor, Hardy notes that the fields themselves occupy a hierarchy of value — different activities offer different degrees of benefit to society. And yet most people, he argues, choose their occupation not on the basis of its absolute value but on the basis of their greatest natural aptitude relative to their other abilities. (Not to do so, after all, renders one the faintly smoking chimney in Van Gogh’s famous lament about unrealized talent: “Someone has a great fire in his soul and nobody ever comes to warm themselves at it, and passers-by see nothing but a little smoke at the top of the chimney.”) Hardy writes:

I would rather be a novelist or a painter than a statesman of similar rank; and there are many roads to fame which most of us would reject as actively pernicious. Yet it is seldom that such differences of value will turn the scale in a man’s choice of a career, which will almost always be dictated by the limitations of his natural abilities. Poetry is more valuable than cricket, but [the champion cricketer Don] Bradman [whose test batting average is considered the greatest achievement of any sportsman] would be a fool if he sacrificed his cricket in order to write second-rate minor poetry (and I suppose that it is unlikely that he could do better). If the cricket were a little less supreme, and the poetry better, then the choice might be more difficult… It is fortunate that such dilemmas are so seldom.

Presaging the ominous twenty-first-century trend of talented mathematicians and physicists swallowed by Silicon Valley for lucrative jobs ranging from the uninspired to the downright pernicious, Hardy adds:

If a man is in any sense a real mathematician, then it is a hundred to one that his mathematics will be far better than anything else he can do, and that he would be silly if he surrendered any decent opportunity of exercising his one talent in order to do undistinguished work in other fields. Such a sacrifice could be justified only by economic necessity or age.

[…]

Every young mathematician of real talent whom I have known has been faithful to mathematics, and not from lack of ambition but from abundance of it; they have all recognized that there, if anywhere, lay the road to a life of any distinction.

Ambition, he argues, has been the motive force behind nearly everything we value as a civilization — every significant breakthrough in art and science, “all substantial contributions to human happiness.” (George Orwell, too, pointed to personal ambition as the first of the four universal motives of great writers.) But while various ambitions can possess us, ranging from the vain and greedy to the most elevated and idealistic, Hardy points to one as the crowning achievement of the purposeful life:

Ambition is a noble passion which may legitimately take many forms… but the noblest ambition is that of leaving behind something of permanent value.

In the remainder of A Mathematician’s Apology, Hardy goes on to explore the particular aspects of mathematics that make it a pursuit of permanent value. Complement this particular portion with Dostoyevsky on the difference between artistic ambition and the ego, David Foster Wallace on the double-edged sword of ambition, and Georgia O’Keeffe on setting priorities for success.

BP

How a Hungarian Teenager Revolutionized Mathematics and Equipped Einstein with the Building Blocks of Relativity

“I have created a new universe from nothing.”

How a Hungarian Teenager Revolutionized Mathematics and Equipped Einstein with the Building Blocks of Relativity

“Euclid alone has looked on Beauty bare,” Edna St. Vincent Millay wrote in her lovely ode to how the father of geometry transformed the way we see and comprehend the world. But although the ancient Alexandrian mathematician provided humanity’s only framework for understanding space for centuries to come, shaping both science and art, his beautiful system was wormed by one ineluctable flaw: Euclid’s famous fifth postulate, known as the parallel postulate — which states that through any one point not belonging to a particular line, only one other line can be drawn that would be parallel to the first, and the two lines, however infinitely they may be extended into space, will remain parallel forever — is not a logical consequence of his other axioms.

This troubled Euclid. He spent the remainder of his life trying to prove the fifth postulate mathematically, and failing. Generations of mathematicians did the same for the next two thousand years. It even stumped Gauss, considered by many the greatest mathematician of all time. It took a Hungarian teenager to solve the ancient quandary.

In 1820, more than two millennia after Euclid’s death, the seventeen-year-old János Bolyai (December 15, 1802–January 27, 1860) told his father — the mathematician Wolfgang Bolyai, who had introduced his son to the enchantment of mathematics four years earlier — about his obsession with the parallel postulate.

János Bolyai, with graphics from Oliver Byrne’s illustrations for The Elements of Euclid

In the exchange that followed, recounted in George E. Martin’s classic 1975 primer The Foundations of Geometry and the Non-Euclidean Plane (public library), Bolyai Senior responded with the opposite of encouragement, writing to his son:

Don’t waste an hour on that problem. Instead of reward, it will poison your whole life. The world’s greatest geometers have pondered the problem for hundreds of years and not proved the parallel postulate without a new axiom. I believe that I myself have investigated all the possible ideas.

But the young man persisted. On November 3, 1823, the twenty-one-year-old mathematical maverick wrote to his father while serving as an artillery officer in the Hungarian army:

I have resolved to publish a work on the theory of parallels as soon as I have arranged the material and my circumstances allow it. I have almost been overwhelmed by them, and it would be the cause of constant regret if they were lost. When you see them, my dear father, you too will understand. At present I can say nothing except this: I have created a new universe from nothing. All that I have sent to you till now is but a house of cards in comparison with a tower. I’m fully persuaded that this will bring me honor, as if I had already completed the discovery.

The discovery in which he exults is one of humanity’s most groundbreaking insights into the nature of reality: Bolyai had laid the foundation of non-Euclidean geometry — a wholly novel way of apprehending space, which describes everything from the shape of a calla lily blossom to the growth pattern of a coral reef, and which would become a centerpiece of relativity; without it, Einstein couldn’t have revolutionized our understanding of the universe with his notion of spacetime, the curvature of which is a supreme embodiment of non-Euclidean geometry.

Illustration by Hugh Lieber from Human Values and Science, Art and Mathematics by Lillian Lieber, bridging the revolutionary discovery of non-Euclidean geometry with concepts of democracy and social justice.

Impressed by his son’s tenacity and swayed by the significance of the breakthrough, Wolfgang pivoted 180 degrees and now urged his son to publish his findings as soon as possible in order to ensure priority of discovery:

If you have really succeeded in the question, it is right that no time be lost in making it public, for two reasons: first, because ideas pass easily from one to another, who can anticipate its publication, and secondly, [because] there is some truth in the fact that many things have an epoch, in which they are discovered at the same time in several places, just as the violets appear on every side in spring.

These were words of remarkable prescience. When János’s paper, completed in 1829 and published as an appendix to a book of his father’s in 1832, reached Gauss — an old friend of Wolfgang Bolyai’s — the great German mathematician was astonished. He responded that he couldn’t praise János’s work, for it would mean praising himself — the young mathematician’s breakthrough, from the central questions he had tackled to the path he had pursued in answering them to the results he had obtained, coincided “almost entirely” with what had been occupying Gauss’s own mind for more than thirty years, though he had resolved never to publish these meditations in his lifetime. With the selfless graciousness of a true scientist, who sets aside all personal ego and celebrates any triumph of knowledge, Gauss wrote to János’s father:

So far as my own work is concerned, of which up till now I have put little on paper, my intention was not to let it be published during my lifetime… On the other hand it was my idea to write down all this later so that at least it should not perish with me. It is therefore a pleasant surprise for me that I am spared this trouble, and I am very glad that it is just the son of my old friend who anticipates me in such a remarkable manner.

But the young Bolyai’s elation at having “created a new universe from nothing” was swiftly grounded when he realized that a third mathematician — Nikolai Lobachevsky in Russia — had preceded both him and Gauss in publishing a paper outlining the selfsame ideas. Lest we forget how information traveled in the pre-Internet era, it took Bolyai sixteen years to learn of Lobachevsky’s book. Once he read it, he reconciled himself to the loss of priority by rooting his ego in the animating principle of science, which he recorded in an uncommonly poetic and profound meditation in his notebook:

The nature of real truth of course cannot be but one and the same [in Hungary] as in Kamchatka and on the Moon, or, to be brief, anywhere in the world; and what one finite, sensible being discovers, can also not impossibly be discovered by another.

The discovery at which these three finite, sensible beings had arrived simultaneously and independently forever changed not only mathematics but our fundamental grasp of nature. For a fine complement, see mathematician Lillian Lieber’s 1961 masterpiece drawing on the non-Euclidean revolution to illustrate the building blocks of moral values like democracy and social justice, then revisit physicist Alan Lightman on the shared psychology of creative breakthrough in art and science.

BP

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