“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
“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).
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.
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.
“Euclid’s system,” mathematician Lillian Lieber, of whom Einstein was an admirer, wrote in her brilliant free-verse primer on mathematics and social justice, “has served for many centuries as a model for clear thinking, and has been and still is of the greatest value to the human race.” But more than a beacon of truth, Euclid was also a torchbearer of beauty. In fathering geometry with Euclid’s Elements, one of the most influential scientific texts of all time, he grounded mathematics in the real world — a groundbreaking cross-pollination of truth and beauty that shaped art through science and science through art. By giving rise to the development of perspective, Euclidean geometry invited architecture and the figurative arts into the three-dimensional world for the first time, then through them gave back to science — Galileo’s Moon drawings were so revolutionary in large part because his training in perspective allowed him to depict the topography of its mountains and craters, refuting the old notion that our satellite is a perfectly smooth orb of ethereal matter and revealing it instead to be as solid and rugged as the Earth.
Ralph Waldo Emerson grasped Euclid’s significance when he wrote in his journal:
The problem of the poet is to do the impossible… to unite the wildest freedom with the hardest precision… Dante was free imagination, all wings, yet he wrote like Euclid.
But it was another great poet who most precisely captured Euclid’s supreme and abiding contribution to humanity.
Edna St. Vincent Millay (February 22, 1892–October 19, 1950) was twenty-one when she enrolled in Vassar College as a freshman. Immersed in the strong science curriculum established there by pioneering astronomer Maria Mitchell, who had paved the way for women in science in the previous century, Millay composed one of her earliest sonnets as a tribute to Euclid’s legacy of revolutionizing how we look at the world and what we see. Geometry became for her a singular portal to truth and beauty. Her ardor for this branch of science came to color her own art and the art she cherished above all others. Years after leaving Vassar, she wrote to a friend:
Without music I should wish to die. Even poetry, Sweet Patron Muse forgive me the words, is not what music is… All that remains is Bach. I find that I never lose Bach. I don’t know why I have always loved him so. Except that he is so pure, so relentless and incorruptible, like a principle of geometry.
Euclid alone has looked on Beauty bare.
Let all who prate of Beauty hold their peace,
And lay them prone upon the earth and cease
To ponder on themselves, the while they stare
At nothing, intricately drawn nowhere
In shapes of shifting lineage; let geese
Gabble and hiss, but heroes seek release
From dusty bondage into luminous air.
O blinding hour, O holy, terrible day,
When first the shaft into his vision shone
Of light anatomized! Euclid alone
Has looked on Beauty bare. Fortunate they
Who, though once only and then but far away,
Have heard her massive sandal set on stone.
“The fact that religions through the ages have spoken in images, parables, and paradoxes means simply that there are no other ways of grasping the reality to which they refer. But that does not mean that it is not a genuine reality. And splitting this reality into an objective and a subjective side won’t get us very far.”
By Maria Popova
In the autumn of 1911, just as the dawn of quantum mechanics and Einstein’s groundbreaking theory of relativity were unsettling our understanding of existence, some of the world’s most influential physicists were summoned to Brussels for the Solvay Conference — an invitation-only gathering that would become a turning point for modern physics and our basic understanding of reality. The conference was such a towering success that it became a regular event, with twenty-five installments over the next century. The most famous was the fifth, convened in 1927 and chaired by the Dutch Nobel laureate Hendrik Lorenz, whose transformation equations had become the centerpiece of Einstein’s theory of special relativity. Of the 29 attendees that year, 17 would become Nobel laureates; Marie Curie, the sole woman since the inaugural gathering, would become the only scientist to win two Nobel Prizes in two different disciplines. (It was at the first Solvay Conference that Curie had met Einstein — the inception of a lifelong friendship in the course of which he would buoy her during a crisis with his splendid advice on how to handle haters.)
One evening during the 1927 conference, some of the younger attendees — including twenty-seven-year-old Wolfgang Pauli, who was yet to co-invent synchronicity with Carl Jung, and twenty-six-year-old Werner Heisenberg, who had just published his revolutionary uncertainty principle earlier that year — stayed up at the hotel lounge and launched into a swirling conversation at the borderline of physics and metaphysics, ignited by the young physicists’ unease about Einstein’s views on God. (Three years later, Einstein himself would traverse that borderline in his historic conversation with the Indian poet and philosopher Tagore, the first non-European to win the Nobel Prize in Literature.) They collided with the difficulty of reconciling science and religion, some adamantly insisting that the two were simply incompatible, for religion is a vestige of a pre-scientific world of superstition, while others suggesting that science can never supplant but can only complement the essential moral guidance by which theology strengthens society.
The unresolved question stayed with Heisenberg. After the conference, he recounted the conversation to quantum theory founding father and Nobel laureate Niels Bohr (October 7, 1885–November 18, 1962). Bohr surprised him with a nuanced and uncommonly insightful take on the subject, which Heisenberg recounts in Physics and Beyond: Encounters and Conversations (public library) — part of the pioneering World Perspectives series envisioned by philosopher Ruth Nanda Anshen as a canon of books by the world’s great “spiritual and intellectual leaders who possess full consciousness of the pressing problems of our time with all their implications,” with a board of editors including Robert Oppenheimer and Bohr himself.
Bohr tells Heisenberg:
We ought to remember that religion uses language in quite a different way from science. The language of religion is more closely related to the language of poetry than to the language of science. True, we are inclined to think that science deals with information about objective facts, and poetry with subjective feelings. Hence we conclude that if religion does indeed deal with objective truths, it ought to adopt the same criteria of truth as science. But I myself find the division of the world into an objective and a subjective side much too arbitrary. The fact that religions through the ages have spoken in images, parables, and paradoxes means simply that there are no other ways of grasping the reality to which they refer. But that does not mean that it is not a genuine reality. And splitting this reality into an objective and a subjective side won’t get us very far.
That is why I consider those developments in physics during the last decades which have shown how problematical such concepts as “objective” and “subjective” are, a great liberation of thought. The whole thing started with the theory of relativity. In the past, the statement that two events are simultaneous was considered an objective assertion, one that could be communicated quite simply and that was open to verification by any observer. Today we know that “simultaneity” contains a subjective element, inasmuch as two events that appear simultaneous to an observer at rest are not necessarily simultaneous to an observer in motion. However, the relativistic description is also objective inasmuch as every observer can deduce by calculation what the other observer will perceive or has perceived. For all that, we have come a long way from the classical ideal of objective descriptions.
In quantum mechanics the departure from this ideal has been even more radical. We can still use the objectifying language of classical physics to make statements about observable facts. For instance, we can say that a photographic plate has been blackened, or that cloud droplets have formed. But we can say nothing about the atoms themselves. And what predictions we base on such findings depend on the way we pose our experimental question, and here the observer has freedom of choice. Naturally, it still makes no difference whether the observer is a man, an animal, or a piece of apparatus, but it is no longer possible to make predictions without reference to the observer or the means of observation. To that extent, every physical process may be said to have objective and subjective features. The objective world of nineteenth-century science was, as we know today, an ideal, limiting case, but not the whole reality. Admittedly, even in our future encounters with reality we shall have to distinguish between the objective and the subjective side, to make a division between the two. But the location of the separation may depend on the way things are looked at; to a certain extent it can be chosen at will.
This, Bohr notes, is why the language of objectivity doesn’t belong in religious rhetoric — religion and its pluralities are best understood, and best applied to human life as an instrument of moral enrichment rather than one of dogmatic constriction, through the lens of complementarity:
The fact that different religions try to express this content in quite distinct spiritual forms is no real objection. Perhaps we ought to look upon these different forms as complementary descriptions which, though they exclude one another, are needed to convey the rich possibilities flowing from man’s relationship with the central order.
A quarter century before mathematician Lillian Lieber demonstrated how mathematical abstractions like infinity, which have no correlate in physical reality, offer an analogue for moral questions, Bohr considers whether or not the tenets of religion can similarly offer useful abstractions, even though they are not to be taken as objective truth:
In mathematics we can take our inner distance from the content of our statements. In the final analysis mathematics is a mental game that we can play or not play as we choose. Religion, on the other hand, deals with ourselves, with our life and death; its promises are meant to govern our actions and thus, at least indirectly, our very existence. We cannot just look at them impassively from the outside. Moreover, our attitude to religious questions cannot be separated from our attitude to society. Even if religion arose as the spiritual structure of a particular human society, it is arguable whether it has remained the strongest social molding force through history, or whether society, once formed, develops new spiritual structures and adapts them to its particular level of knowledge. Nowadays, the individual seems to be able to choose the spiritual framework of his thoughts and actions quite freely, and this freedom reflects the fact that the boundaries between the various cultures and societies are beginning to become more fluid. But even when an individual tries to attain the greatest possible degree of independence, he will still be swayed by the existing spiritual structures — consciously or unconsciously. For he, too, must be able to speak of life and death and the human condition to other members of the society in which he’s chosen to live; he must educate his children according to the norms of that society, fit into its life. Epistemological sophistries cannot possibly help him attain these ends. Here, too, the relationship between critical thought about the spiritual content of a given religion and action based on the deliberate acceptance of that content is complementary. And such acceptance, if consciously arrived at, fills the individual with strength of purpose, helps him to overcome doubts and, if he has to suffer, provides him with the kind of solace that only a sense of being sheltered under an all-embracing roof can grant. In that sense, religion helps to make social life more harmonious; its most important task is to remind us, in the language of pictures and parables, of the wider framework within which our life is set.