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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.”

By Maria Popova

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

Three Balls of Wool: An Illustrated Celebration of Nonconformity and the Courage to Remake Society’s Givens

A poignant and hope-giving allegory based on the true story of a refugee family.

By Maria Popova

Three Balls of Wool: An Illustrated Celebration of Nonconformity and the Courage to Remake Society’s Givens

It may be an elemental property of human nature to fantasize about utopias — a fantasy all the more alluring the more dystopian one’s actual society is. But the inescapable fallacy of the fantasy is that while a utopia promises universal flourishing for everyone, not everyone has the same criteria for flourishing. Homogeneity, as Zadie Smith observed in her superb essay on optimism and despair, is no guarantor of a just and equitable society. We all dream different utopias — something Margaret Mead and James Baldwin tussled with in their fantastic forgotten conversation about the problematic metaphor of the melting pot. An ideal society is not one that seeks to level the differences into a flat universality but one that welcomes them into a glorious topography of diverse human flourishing.

That is what Portuguese writer Henriqueta Cristina and Brazilian artist Yara Kono explore in Three Balls of Wool (Can Change The World) (public library) — an unusual and poignant picture-book about the meaning of freedom and human dignity, published in partnership with Amnesty International.

The story, reminiscent of Umberto Eco’s lovely vintage semiotic children’s books on humanitarian subjects, is told through the perspective of an eight-year-old girl and her refugee family, exiled from their home country under threats of imprisonment and a sense of political gloom, which the young girl can’t quite understand, though she intuits their gravity through the deep lines furrowing her parents’ foreheads.

The family arrives in a new country that at first shines with the promise of a better life, clean and orderly. “Here there are no poor people and all the children go to school,” the girl’s mother tells her. But soon the seeming utopia unravels into a tyranny of uniformity. At school, all the children wear sweaters in one of the three permitted colors — solid gray, solid green, or solid orange — and all the buildings look like “a bunch of gray shoeboxes staked one on top of the other.”

One day, the mother launches a quiet rebellion against the tyranny of homogeneity and conformity — an embodiment of artist Ben Shahn’s insistence that “without the nonconformist, any society of whatever degree of perfection must fall into decay.” It starts with a grey sweater she unravels into a ball of yarn, then an orange one, then a green one. Out of these three balls of wool, she begins knitting sweaters of all stripes and patterns, remixing the solid givens into previously unimagined possibilities.

Soon, the little girl and her brothers are clad in countercultural sweaters that become the awe of the neighborhood.

People begin gathering at the local park each Sunday, unraveling their old solid sweaters and knitting variegated new ones, transfiguring the same three colors of yarn into every imaginable combination of shape and pattern — a testament to Kierkegaard’s assertion that “the more a person limits himself, the more resourceful he becomes.”

By springtime, the entire city is an explosion of combinatorial color.

The story doesn’t reveal what country the refugees left, nor what country they arrived in, but through the semiotic description of their new home I sense familiar echoes of my own childhood in Communist Bulgaria. The afterword confirms my intuition — the book is based on the true story of a Portuguese family who fled from Portugal’s right-wing dictatorship at the peak of the Cold War in the 1960s, then lived as refugees in Algeria and Romania before finally settling in Czechoslovakia. They returned home to Portugal shortly before the end of the Carnation Revolution — a peaceful uprising that began on April 25, 1974 and ended the half-century dictatorship. “Today, Portugal has a democratically elected government and every child goes to school,” Cristina writes in the closing of the afterword.

Printed on the last spread of the book is the Universal Declaration of Human Rights, adopted by the United Nations General Assembly in 1948, before Portugal was a member. Its first article reads:

All human beings are born free and equal in rights. Endowed with reason and conscience, we should act towards one another in a spirit of kindness and community.

Its thirtieth and final article reads:

This declaration does not give any government, group, or person the right to infringe upon the freedoms of any other.

Three Balls of Wool, which is absolutely lovely both as a picture-book and as a symbolic cultural message, comes from Enchanted Lion Books — the Brooklyn-based independent powerhouse that has brought us treasures like Cry, Heart, But Never Break, The Lion and the Bird, The Paper-Flower Tree, and Bertolt. Complement it with Albert Camus on what it really means to be a rebel and Ben Shahn on nonconformity and the creative spirit.

Illustrations courtesy of Enchanted Lion Books; photographs by Maria Popova

BP

Neither Victims Nor Executioners: Albert Camus on the Antidote to Violence

“If he who bases his hopes on human nature is a fool, he who gives up in the face of circumstances is a coward.”

By Maria Popova

Neither Victims Nor Executioners: Albert Camus on the Antidote to Violence

“Progress is never permanent, will always be threatened, must be redoubled, restated and reimagined if it is to survive,” Zadie Smith wrote in her spectacular essay on optimism and despair. Seventy years earlier, just after the close of World War II, another genius of the times addressed this predicament and its attendant question of what reimagining progress looks like as we behold the future from the precarious platform of the present.

In November of 1946, a decade before he became the second-youngest laureate of the Nobel Prize in Literature, awarded him for work that “with clear-sighted earnestness illuminates the problems of the human conscience in our times,” Albert Camus (November 7, 1913–January 4, 1960) wrote a series of short essays for the journal of the French Resistance, which were later published as Neither Victims nor Executioners: An Ethic Superior to Murder (public library).

Albert Camus, 1946 (Photograph: Cecil Beaton)

At the outset of the war, Camus considered what it means to have strength of character in difficult times. Now, at its grim end, he examines what it takes to maintain that strength and make it a heritage to future generations in the aftermath of extreme moral weakness at the level of a whole civilization. He begins his final installment in the series, titled “Toward Sociability,” with an inquiry into the vital interplay of reason and emotion in inciting the movements that bring about social change:

Yes, we must raise our voices. Up to this point, I have re­frained from appealing to emotion. We are being torn apart by a logic of History which we have elaborated in every detail — a net which threatens to strangle us. It is not emotion which can cut through the web of a logic which has gone to irrational lengths, but only reason which can meet logic on its own ground. But I should not want to leave the impression, in con­cluding, that any program for the future can get along without our powers of love and indignation. I am well aware that it takes a powerful prime mover to get men into motion and that it is hard to throw one’s self into a struggle whose objectives are so modest and where hope has only a rational basis — and hardly even that. But the problem is not how to carry men away; it is essential, on the contrary, that they not be carried away but rather that they be made to understand clearly what they are doing.

In a sentiment he would later ferment into the assertion that “real generosity toward the future lies in giving all to the present,” Camus counters the dangerous notion that the present must be made a sacrificial offering to some abstract ideal future:

To save what can be saved so as to open up some kind of future — that is the prime mover, the passion and the sacrifice that is required. It demands only that we reflect and then decide, clearly, whether humanity’s lot must be made still more miserable in order to achieve far-off and shadowy ends, whether we should accept a world bristling with arms where brother kills brother; or whether, on the contrary, we should avoid bloodshed and misery as much as possible so that we give a chance for survival to later generations better equipped than we are.

A decade before he reached a warm hand across the Iron Curtain of the Cold War in his beautiful letter of cross-cultural solidarity to Boris Pasternak, Camus writes:

We are asked to love or to hate such and such a country and such and such a people. But some of us feel too strongly our common humanity to make such a choice. Those who really love the Russian people, in gratitude for what they have never ceased to be — that world leaven which Tolstoy and Gorky speak of — do not wish for them success in power-poli­tics, but rather want to spare them, after the ordeals of the past, a new and even more terrible bloodletting. So, too, with the American people, and with the peoples of unhappy Eu­rope. This is the kind of elementary truth we are liable to forget amidst the furious passions of our time.

Art by Eugenio Carmi from The Three Astronauts, Umberto Eco’s vintage semiotic children’s book about world peace

Five years before Hannah Arendt considered loneliness as the common ground for terror and how dictators use isolation as a weapon of oppression in her own postmortem of WWII, Camus adds:

Yes, it is fear and silence and the spiritual isolation they cause that must be fought today. And it is sociability (“le dialogue”) and the universal intercommunication of men that must be defended. Slavery, injustice and lies destroy this inter­course and forbid this sociability; and so we must reject them.

Exactly forty years later, in his magnificent Nobel Peace Prize acceptance speech, Holocaust survivor Elie Wiesel would echo Camus and insist that “silence encourages the tormentor, never the tormented.”

Half a century after Tolstoy asserted in his forgotten correspondence with Gandhi about violence and human nature that “love is the only way to rescue humanity from its ills,” Camus argues that framing the violence of the present as a necessary evil of what will eventually be “History” is a grave disservice to posterity:

Modern nations are driven by powerful forces along the roads of power and domination. I will not say that these forces should be furthered or that they should be obstructed. They hardly need our help and, for the moment, they laugh at at­tempts to hinder them. They will, then, continue. But I will ask only this simple question: what if these forces wind up in a dead end, what if that logic of History on which so many now rely turns out to be a will o’ the wisp? What if, despite two or three world wars, despite the sacrifice of several generations and a whole system of values, our grandchildren — supposing they survive — find themselves no closer to a world society? It may well be that the survivors of such an experience will be too weak to understand their own sufferings. Since these forces are working themselves out and since it is inevitable that they con­tinue to do so, there is no reason why some of us should not take on the job of keeping alive through the apocalyptic histor­ical vista that stretches before us, a modest thoughtfulness which, without pretending to solve everything, will constantly be prepared to give some human meaning to everyday life. The essential thing is that people should carefully weigh the price they must pay.

A year after he contemplated the three antidotes to the absurdity of life, Camus concludes the essay with a mighty reminder that we choose our future with our actions in the present — and that violence is a choice, as is its counterpoint:

All I ask is that, in the midst of a murderous world, we agree to reflect on murder and to make a choice. After that, we can distinguish those who accept the conse­quences of being murderers themselves or the accomplices of murderers, and those who refuse to do so with all their force and being. Since this terrible dividing line does actually exist, it will be a gain if it be clearly marked. Over the expanse of five continents throughout the coming years an endless strug­gle is going to be pursued between violence and friendly per­suasion, a struggle in which, granted, the former has a thou­sand times the chances of success than has the latter. But I have always held that, if he who bases his hopes on human nature is a fool, he who gives up in the face of circumstances is a coward. And henceforth, the only honorable course will be to stake everything on a formidable gamble: that words are more powerful than munitions.

Complement Neither Victims nor Executioners with Camus on happiness, despair, and the love of life, consciousness, the most important question of existence, and what it really means to be a rebel, then revisit Rebecca Solnit on resisting the defeatism of easy despair and Toni Morrison on the artist’s task through turbulent times.

BP

The Continuous Thread of Revelation: Eudora Welty on Writing, Time, and Embracing the Nonlinearity of How We Become Who We Are

“Greater than scene… is situation. Greater than situation is implication. Greater than all of these is a single, entire human being, who will never be confined in any frame.”

By Maria Popova

The Continuous Thread of Revelation: Eudora Welty on Writing, Time, and Embracing the Nonlinearity of How We Become Who We Are

To be human is to unfold in time but remain discontinuous. We are living non sequiturs seeking artificial cohesion through the revisions our memory, that capricious seamstress, performs in threading the stories we tell ourselves about who we are. It is, after all, nothing but a supreme feat of storytelling to draw a continuous thread between one’s childhood self and one’s present-day self, since hardly anything makes these two entities “the same person” — not their height, not their social stature, not their beliefs, not their circle of friends, not even the very cells in their bodies. Somewhere in the lacuna between the experiencing self and the remembering self, we create ourselves in what is literally a matter of making sense — of craftsmanship — for, as Oliver Sacks so poignantly observed, it is narrative that holds our identity together.

But while this self-composition is native to the human experience, there is a subset of humans who have elected the transmutation of discontinuity into cohesion as their life’s work and have made temporality the raw material of their craft: writers. The essence of that craftsmanship is what Pulitzer-winning author Eudora Welty (April 13, 1909–July 23, 2001) explores in a passage from One Writer’s Beginnings (public library) — her three-part memoir adapted from the inaugural Massey Lectures she delivered at Harvard in 1983, shortly after she was awarded the nation’s highest civilian honor, the Presidential Medal of Freedom, and exactly half a century after The New Yorker rejected her brilliant job application.

Eudora Welty

Welty writes:

The events in our lives happen in a sequence in time, but in their significance to ourselves they find their own order, a timetable not necessarily — perhaps not possibly — chronological. The time as we know it subjectively is often the chronology that stories and novels follow: it is the continuous thread of revelation.

Drawing on one of her short stories, whose protagonist holds up her fingers to frame what she is about to paint before she beings painting it, Welty reflects on the evolution of her own understanding of writing and selfhood — a beautiful counterpoint to today’s fashionable fragmentation of the wholeness of personhood into sub-identities:

The frame through which I viewed the world changed too, with time. Greater than scene, I came to see, is situation. Greater than situation is implication. Greater than all of these is a single, entire human being, who will never be confined in any frame.

With an eye to the retrospective pattern-recognition by which we wrest our personhood from our experience — an existential act on which Joan Didion had some magnificent advice — Welty adds:

Writing a story or a novel is one way of discovering sequence in experience, of stumbling upon cause and effect in the happenings of a writer’s own life. This has been the case with me. Connections slowly emerge. Like distant landmarks you are approaching, cause and effect begin to align themselves, draw closer together. Experiences too indefinite of outline in themselves to be recognized for themselves connect and are identified as a larger shape. And suddenly a light is thrown back, as when your train makes a curve, showing that there has been a mountain of meaning rising behind you on the way you’ve come, is rising there still, proven now through retrospect.

Complement this particular passage of Welty’s altogether fantastic One Writer’s Beginnings with anthropologist Mary Catherine Bateson on the nonlinearity of how we become who we are and with more life-earned insight into the craft of writing from Susan Sontag, James Baldwin, Gabriel García Márquez, Ernest Hemingway, Zadie Smith, T.S. Eliot, and other titans of literature, then revisit Welty on friendship, the difficult art on seeing one another, and a rare recording of her reading her comic and quietly heartbreaking masterpiece “Why I Live at the P.O.”

BP

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