My hard feelings toward math began to be mitigated by an awareness of its enigmas. Of course, I am acquainted only with modest enigmas and those only modestly, but they introduced me to a world I had been unaware of, one that was capacious and profound but not so abstruse that I couldn’t find the means to appreciate it, even as a tourist. Numbers began to seem as alive in their being as I am in my mind (and nothing very important has happened to me, except in my mind).
The second starter mystery I encountered is that of prime numbers, those that can be divided cleanly only by themselves and by 1. The first primes are 2, 3, 5, 7, 11, 13, 17, 19, and 23. The number 1 is not prime, because, if it were, the fundamental theorem of arithmetic, which is also called the unique factorization theorem, would not hold. A counting number that is not prime is called a composite number. The fundamental theorem of arithmetic says that any composite number can be expressed as a unique product of primes (1 is not a composite number, either). The primes that express a composite number are called its factors, and composite numbers factor in only one arrangement of primes. 2 × 2 = 4. 2 × 3 = 6. 2 × 2 × 2 = 8. 3 × 3 = 9 … 2 × 3 × 3 × 37 = 666 … 29 × 31 = 899 … 2 × 2 × 2 × 5 × 5 × 5 = 1000. If 1 were allowed as prime, no series of factors would be unique. 2 × 2 = 4, but so does 2 × 2 × 1 and so on.
Prime numbers are where imaginary mathematics begins. They are an example of our discovering properties of numbers, rather than creating them. Idealism is the name of the eighteenth-and nineteenth-century discipline that believed that the mind creates what we know and that there is nothing we can know that the mind hasn’t created. G. H. Hardy writes that prime numbers seemed to him the place where Idealism failed: “317 is a prime not because we think so, or because our minds are shaped in one way rather than another,” he writes, “but because it is so, because mathematical reality is built that way.”
With prime numbers I was made aware of the conundrum of how numbers have properties that no one gave them. And that if mathematics is a human creation, how is it that our creation has attributes that we only partly comprehend and are still discovering? And how is it that our minds could create something that they can’t entirely understand or contain? Is it possible to create something useful in pedestrian ways that becomes more than it was intended to be, something that exceeds our grasp and control? (AI might prove to be, but AI is dynamic and can act on its own, whereas numbers are inert.) None of the inventors of language imagined novels and poetry, but the possibility of novels and poetry is inherent in the medium. There are no novels or poems without human beings, but there is math and there are prime numbers without us. It would be as if we had invented language and then discovered that certain words had the capacity to talk to each other when we weren’t around. In any book we opened there might be conversations taking place that were unrelated to the text. The exchanges in the dictionary might amount to a din. We would know this to be so, but we wouldn’t know why, and while we knew many, many words that could talk to each other, not all words could, and we had no way of predicting what words had this ability and no way of finding them, either. All we could do is go through all the possible combinations of the alphabet looking for them, since they seem to appear randomly, and there exists an infinite collection of them.
Becoming aware that prime numbers were lurking among ordinary numbers was for me like finding in the background of a familiar photograph a figure I had not seen before and whose presence overthrew the narrative that the photograph had appeared to tell. Prime numbers are a means of escape from the hegemony of the number line. Absent primes, one can go only farther or deeper into the number line, toward infinity in either direction, but the way has no turns. Prime numbers wander. They turn left or right completely. They are a secret society, equipped for more than measuring and commerce, the sorcerers and shamans of the mathematical realm. I am not alone in thinking so. The mathematician Henryk Iwaniec told me, “When you get something unexpected like prime numbers, it seems God given, like something mysterious being found.”
It may be beguiling to know that there is no last number, but it is an observation more about infinity than about the hidden qualities of numbers. It might provoke awe, but it is also a mere fact involving the simplest mathematical proof there is, a child can come up with it on his or her own and often does: for any last number n, there is n + 1.
PRIME NUMBERS HAVE their own taxonomy. Twin primes are two apart. Cousin primes are four apart; sexy primes are six apart, six being sex in Latin; and neighbor primes are adjacent at some greater remove. From Prime Curios!, by Chris Caldwell and G. L. Honaker, Jr., I know that an absolute prime is prime regardless of how it is arranged: 199, 919, 991. Palindromic primes are the same forward and backward—133020331; they are also called smoothly undulating primes. Tetradic primes are palindromic primes that are also prime backward and when seen in a mirror, such as 11, 101, 1881881.
A beastly prime has 666 in its center. 700666007 is a beastly palindromic prime. A depression prime is a palindromic prime whose interior numbers are the same and smaller than the numbers on the end; 75557, for example. Conversely, plateau primes have interior numbers that are the same and larger than the numbers on the ends, such as 1777771. A circular prime is prime through all its rotations: 1193, 1931, 9311, 3119. There are Cuban primes; Cullen primes; curved digit primes, which have only curved numbers—0, 6, 8, and 9; and straight-digit primes, which have only 1, 4, and 7. A prime from which you can remove numbers and still have a prime is a deletable prime, such as 1937. An emirp is prime even when you reverse its numbers: 389, 983. Invertible primes can be turned upside down and rotated: 109 becomes 601. Gigantic primes have more than 10,000 digits; holey primes have only numbers with holes (0, 4, 6, 8, and 9). There are Mersenne primes; minimal primes; naughty primes, which are made mostly from zeros, naughts; ordinary primes; Pierpont primes; snowball primes, which are prime even if you haven’t finished writing all of the number—73939133; titanic primes; Wagstaff primes; Wall-Sun-Sun primes; Wolstenholme primes; Woodall primes; and Yarborough primes, which have neither 0 nor 1.
The only even prime number is 2. Since all other primes are odd, the interval between any two successive primes has to be even, but no one knows a rule to govern this. The largest known prime exceeds by far the estimates of the number of atoms in the universe.
PRIME NUMBERS, WHICH are the subject of number theory, are the origin figures of pure mathematics—mathematics done, that is, without an interest in being useful in any practical way. Applied mathematics begins with the ability to count and measure and is procedural; pure mathematics is imaginative. That the classifications have about them a suggestion of snobbery and side-taking has a lot to do with the British mathematician G. H. Hardy, who sometimes called pure mathematicians “real mathematicians.” In “A Mathematician’s Apology,” Hardy writes, “Is not the position of an ordinary applied mathematician in some ways a little pathetic? If he wants to be useful, he must work in a humdrum way, and he cannot give full play to his fancy even when he wishes to rise to the heights. ‘Imaginary’ universes are so much more beautiful than this stupidly constructed ‘real’ one.”
The remoteness of pure mathematics is partly why Hardy’s essay is an apology, but he was also pleased at pure mathematics not being helpful in commerce and especially not in war. He defines usefulness as “knowledge which is likely, now or in the comparatively near future, to contribute to the material comfort of mankind, so that mere intellectual satisfaction is irrelevant.” He believes of pure mathematics that “the best of it may, like the best literature, continue to cause intense emotional satisfaction to thousands of people after thousands of years.”
The aloofness of pure mathematics and its reverence for thinking infused itself into physics. In April of 1969, Robert Wilson, a physicist who had worked on the Manhattan Project and was the director of the Fermi National Accelerator Laboratory, in Illinois, appeared before Congress to request money for building the accelerator, which was called the 200-BeV Synchrotron. The Synchrotron was a proton accelerator that would make it possible to observe subatomic particles, some of which were theoretical.
Wilson was questioned by Senator John Pastore, a Democrat from Rhode Island, who was sympathetic to science and was hoping for arguments he might use to persuade the accelerator’s opponents. Senator Pastore asked if the accelerator involved the security of the country.
“No, sir; I do not believe so,” Dr. Wilson said.
“Nothing at all?”
“Nothing at all.”
“It has no value in that respect?”
“It only has to do with the respect with which we regard one another, the dignity of men, our love of culture,” Wilson said. “It has to do with those things. It has nothing to do with the military,” for which he added that he was sorry.
Senator Pastore told him not to be sorry.
“I am not, but I cannot in honesty say it has any such application,” Wilson said.
Senator Pastore tried another tack. “Is there anything here that projects us in a position of being competitive with the Russians, with regard to this race?” he asked.
Very little, Wilson said. “Otherwise, it has to do with: Are we good painters, good sculptors, great poets? I mean all the things that we really venerate and honor in our country and are patriotic about. In that sense, this new knowledge has all to do with honor and country, but it has nothing to do directly with defending our country, except to help make it worth defending.”
Pure mathematics sometimes finds a practical use, but typically after so long that, according to James Newman, the use is thought to be either a coincidence or evidence that mathematics and the world’s deeper workings are mystically aligned. Hardy was not entirely correct about its aloofness, though. In Mathematics Without Apologies, a response to Hardy, Michael Harris writes that pure mathematics has figured in “radar, electronic computing, cryptography for e-commerce, and image compression, not to mention control of guided missiles, data mining, or options pricing.”
Primes, which are secretive, are also essential to keeping secrets. Much of internet commerce and finance depends on a cryptographic system called RSA, which takes the initials of the men who are said to have come up with it, in 1977—Ron Rivest, Adi Shamir, and Leonard Adleman. RSA works by means of a pair of codes called a private key and a public key. A transaction involving a credit card is scrambled using the public key, sent to a bank, then unscrambled using the private key. The public key is the result of two prime numbers multiplied together. Typically, each prime is more than a thousand digits long. (The largest known prime, which took several thousand computers nearly a month to find, has seventeen million digits.) The public key can be discovered from the private key, but the private key cannot easily be discovered from the public.
Keys are protected by a principle called the trapdoor function. Trapdoor functions are simple to calculate in one direction but hard to calculate in the other without special information, which is the trapdoor. In the case of RSA, the trapdoor involves a field called modulo math. The most familiar form of modulo math is the clockface. Ten hours from four o’clock on the clockface is two o’clock, not fourteen o’clock, except in military time; this is an example of arithmetic modulo twelve. To know the time at any number of hours into the future, you divide the hours by twelve and apply the remainder to the clockface. A version of a clockface can be composed of any number of hours, and this is what happens with RSA. The trapdoor function is a result of knowing the modulo involved in selecting the keys. Simply finding by luck or application the prime numbers involved in the key is useless without the modulo number.
Henry Cohn, a principal researcher at Microsoft, told me that “cryptographic security is based on conjectures that are unproved and seemingly very difficult to resolve. The security of RSA depends on there not being a fast algorithm to factor a number into primes.” An algorithm that would factor huge numbers into primes does not appear to exist, “but it’s by no means certain,” Cohn said.
Primes seem to many mathematicians to be distributed according to a mysterious pattern, which may or may not exist. The more elusive the pattern is, the more many mathematicians are persuaded that it exists. “When you’re trying to make a cryptosystem like RSA, you are depending on the fact that numbers have a good deal of structure,” Cohn said. He went on to say, however, that “this structure could come back to bite you, by enabling an attacker to defeat the system. Ultimately, what you’d like is a situation that looks beautifully regular and structured to the legitimate users while looking unapproachably random to hackers. This is a subtle balancing act, and lots of cryptosystems have fallen apart over the years when it turned out there was just a little too much hidden structure that could be taken advantage of.” This has not yet happened to RSA.
Occasionally a version of pure mathematics put to practical use misleads. This happened with Johannes Kepler, the German astronomer on whose three laws of planetary motion, published between 1609 and 1619, Newton based his law of gravitation, in 1687.
In a portrait of Kepler painted in 1610, when he was forty-nine, he is wearing a ruff collar, and he is looking into the middle distance, as if he were somewhere else in his mind. Maybe he is only impatient with having to sit for the painter, but it makes him seem as if he might have been difficult to talk to, unless you had something to say that interested him. Kepler is a hinge figure. The physicist Wolfgang Pauli, speaking in 1948, describes him as “a spiritual descendant of the Pythagoreans,” devoted to finding a harmony among the proportions in nature, where for him “all beauty lay,” and as a signal intermediary “between the earlier, magical-symbolical and the modern, quantitative-mathematical descriptions of nature.” Reading War and Peace I feel something like astonishment that a single intelligence contained it, and I feel something similar when I think of the version of the heavens that Kepler held in his head.
Kepler exemplifies the premodern belief that practical mysteries conceal the divine. According to Pauli, he thought that the theorems of geometry “have been in the spirit of God since eternity,” and that God was represented by the sphere. “The Father is the center,” Kepler wrote. “The Son is in the outer surface, and the Holy Ghost is in the equality of the relation between point and circumferences,” which as a model is more concise and apprehensible than anything I remember being taught in Sunday school. In the natural world Kepler saw the arrangement as embodied by the sun and the planets. He felt compelled to discover the mechanics according to which the planets revolved, since he believed that they would reveal the sacred design underpinning creation.
Kepler imagined that the planets’ orbits accorded with the forms of the platonic solids. A platonic solid is a shape in which all the faces are the same and are triangles, squares, or pentagons—that is, they have three, four, or five sides, which are all equal to one another. Fantasy games often have dice in the shapes of the platonic solids.
Much of the following material was challenging for me, although I don’t think it’s all that complicated. It requires an ability to imagine shapes from more than one vantage, though—a talent that studying mathematics made me aware that I don’t have—and also, it’s a little dense. It interests me because Kepler’s accomplishment is so enormous. The Chinese, the Indians, the Babylonians, the Egyptians, the Greeks, the Arabs, the Jewish rabbis who set up the Jewish calendar all followed movements in the sky, and so did the Mayans, and surely the Africans did, too. The Greeks saw Mercury, Venus, Mars, Jupiter, and Saturn and thought they were stars, but ones that moved differently from, and erratically compared with, the other stars—planets means wanderers in Greek. Kepler was the first to describe the mathematical laws governing celestial mechanics—the paths the planets followed and the pace at which they followed them—to find an order, that is, in something that had seemed disorderly. Carl Sagan called Kepler “the first astrophysicist and the last scientific astrologer.” If chess were played with figures from mathematical history as pieces, Kepler might be one of the pieces.
So here goes: At each vertex of a platonic solid—that is, corner where they form an angle—the same number of faces meet. A cube is a platonic solid. There are four others: the tetrahedron, the octahedron, the icosahedron, and the dodecahedron. The tetrahedron is a pyramid with three equilateral triangles meeting at each vertex. The octahedron has eight faces and four equilateral triangles meeting at each vertex. An octahedron looks like a pyramid sitting on top of an upside-down pyramid. An icosahedron has five triangles meeting at each vertex, and I can’t think of anything it looks like. The dodecahedron has three pentagons at each vertex and twenty sides, forming a pattern like the pattern on the side of a soccer ball. Euclid proved that there can be only five platonic solids, because six equilateral triangles arranged around a vertex form a circle, being 360 degrees, and therefore lie flat.
Kepler’s belief in a correspondence between the solar system and an arrangement of the platonic solids descended from Plato, who says in Timaeus that the world is made from earth, air, water, and fire, and that each of these is formed from one of the solids. Earth, being “the most immoveable,” is formed from the cube, which has the most stable base. Fire, which dissolves other elements by its sharpness, is made of the tetrahedron, which is pointy. The octahedron is assigned to air for being something like smooth. Water is made from the icosahedron, which is complex and heavy and able to crush fire and earth. The dodecahedron represents the universe itself, since it most closely approaches a sphere.
Kepler believed that each planet, according to its orbit, could be placed inside a solid that is enclosed by the next larger orbit and, like a nesting doll, is itself inside another solid. By their distances from the sun, from least to greatest, Mercury sits in an octahedron enclosed by Venus; Venus sits in an icosahedron enclosed by Earth; Earth sits inside a dodecahedron enclosed by Mars; Mars sits inside a tetrahedron enclosed by Jupiter; and Jupiter sits inside a cube enclosed by Saturn. The orbit of the inner planet is tangent to the center of the face of the solid enclosing it, meaning that it touches the face at a single point, and the orbit of the outer planet travels through the solid’s vertices. The arrangement requires that the planets’ orbits be circles, also a platonic notion. For Kepler, the planets moving in a circle demonstrated a divine pattern.
Eventually, Kepler noticed that if he drew a line from the sun to a planet and a similar line at an interval in the planet’s orbit, either of hours or days, the area enclosed is always the same no matter where the planet is, so long as the interval of time is the same. Such a circumstance could happen only if the planet travels faster when it is closer to the sun than it does when it’s farther away. And this is possible only if the orbit is an ellipse, which is not a divine form.
Kepler’s three laws of planetary motion defined the correct orbits. The first law is that the orbit of a planet is essentially an ellipse. The second is that the progress of a line drawn between a planet and the sun encloses equal areas during equal intervals of time. And the third is that the square of the period of the orbit is proportional to the cube of the semi-major axis of its orbit. According to Amie, in whose field Kepler’s work figures, this means that if you square the planet’s orbital year and divide it by the cube of the distance to the sun, the number is the same for all planets. Finis.
The more I learned about the habits of mathematicians, the more it seemed that they approached their work in much the same way that writers and artists approach theirs, which didn’t make doing math easier, but it interested me. So far as I understand it, mathematical creativity involves the same stages that creative thought involves in any discipline or art. Like novelists and musicians, mathematicians produce thought objects that have no presence in the physical world. (Anna Karenina is no more actual than a thought about Anna Karenina.) Like other artists, mathematicians also have the run of a world that others hardly or only rarely visit or don’t travel very far into. For mathematicians, though, this territory has more rules than it does for others. Also, what is different for mathematicians is that all of them agree about the contents of that world, so far as they are acquainted with it, and all of them see the same objects within it, even though the objects are notional. No one’s version, so long as it is accurate, is more correct than someone else’s. Parts of this world are densely inhabited and parts are hardly settled. Parts have been visited by only a few people, and parts are unknown like the dark places on a medieval map; somewhere among this territory would be where the proof of the Collatz conjecture resides. The known parts are ephemeral but also concrete for being true and more reliable and everlasting than any object in the physical world. Two people who do not share a language or understand a word the other is saying can do mathematics with each other, silently, like a meditation.
An imaginary world’s being infallible is very strange. This spectral quality is bewildering even to mathematicians. In text accompanying his portrait in Mathematicians: An Outer View of the Inner World, a series of portraits by the photographer Mariana Cook, the mathematician John Conway says, “It’s quite astonishing, and I still don’t understand it, despite having been a mathematician all my life. How can things be there without actually being there?”
We don’t often encounter the limits of our intelligence, but the way I struggled with algebra sometimes made me wonder if I was finding my own. At such times I felt myself to be a poorly equipped version of human possibility, sort of a discard. I was also almost daily reminded of how some things needed to be learned more slowly. Meanwhile, I was harassed by my upbringing to believe that I had to work quickly; any half-smart person could work out a problem given sufficient time. I found these attitudes difficult to combine. Sometimes I realized that I was talking to different parts of myself, and the exchange was not polite.
Occasionally I got good at operations that were hard at first. This happened with factoring, a process in algebra of simplifying expressions and with expanding, which is the opposite of factoring. The axioms of arithmetic imply that when you expand (a + b)2, for example, you get a2 + 2ab + b2 in the following way: (a + b)2 is equal to (a + b)(a + b). Each term in one parentheses multiplies the terms in the other: a × a = a2; a × b = ab; b × a = ab; b × b = b2. Combining the terms, a2 + ab + ab + b2 = a2 + 2ab + b2. In a similar way, a2 – b2, a squared number subtracted from another squared number, called a difference of squares, becomes (a – b)(a + b), which becomes a2 + ab – ab – b2, which is a2 – b2.
Simple, but I really liked it. As the formulas became more complicated, there were more steps, each of which followed from the one before it, so that in addition to finding the answer, there was the pleasure of enacting a procedure properly, plus no textbook skipped the steps. Each time I turned a page and saw more factoring, I was pleased. It was like being good at spelling and wanting to be asked more words. Accompanying my pleasure, though, was a voice saying, “Listen, Slick, this is algebra for kids. We can throw problems at you, you won’t even know what they mean.”
Sometimes I dreamed that there were numbers falling from the sky into chasms I couldn’t see the bottom of.
Aside from learning starter mathematics, I became interested in trying to understand what mathematics is. Variously, I thought, a structure, a terrain, a republic, a façade concealing an infinite kingdom, an archive that goes on forever, a library of manuscripts and artifacts penned behind a series of more and more elaborate locks and doors.
For as long as I can remember I have been drawn to the hidden life—scenes that play behind the eyes before sleep, the fleeting impressions from the periphery of one’s vision that are there and not there, reveries, anything that resides on the border of consciousness. I like it in all its guises and representations and almost anywhere it shows up, but I am especially drawn to objects and places that are underground. The bottom of the ocean or a pond or the beds of underground rivers.
I often dream that I am in a basement, or descending stairs to the subway, or walking in a cave, or swimming toward lights at the bottom of a river, or finding rooms belowground that no one else knows about and that aren’t there when I try to go back to them. The world of symbols is fairly impenetrable to me, but I am not so thick that I don’t see that the underground stands for the Unconscious. I sometimes think of the Unconscious as a series of rooms, each opening out from the next. In one of them, perhaps, a book lies open on a table. In another an old woman sits in a rocking chair while rain beats against a window. On a wall in another is writing that you can almost read, or a mural depicting a scene that turns up later in a dream or perhaps as a premonition. Or maybe not as a series of rooms but as a landscape. It has weather and there is night and day, but it is not always a landscape I recognize, and it changes all the time, as if each vista were a fragment of another one, like the planes in a Cubist painting. I am drawn to the Unconscious for the reasons I assume most people are, which is the belief that something it contains has the power to release you from torment. Or that something lost can be recovered there. Or that it is rich in inspiration and productive of inexplicable intimations and feelings.
Occasionally I hear about subterranean places in New York City, and I go visit them, such as the corridors and tunnels under Grand Central or the railroad lines along the West Side, by the Hudson River, where until a few years ago, when the railroad police began chasing them out, squatters lived in the cinder-block chambers that the railroad had built in the walls beside the tracks for its workers to use while they were constructing the tunnels. The railroad had lit them but they weren’t lit anymore, and the darkness was so complete that the men and women who inhabited them couldn’t see their hands in front of their faces.
I had a hobby of looking for tunnels that I had heard were dug by bootleggers between their basements and the rivers; tunnels in Chinatown built during the Tong Wars for fighters to disappear into; and a section of the aqueduct in the Bronx that someone told me went all the way to Manhattan. One night I persuaded a friend who had bolt cutters to cut the lock on a chain that held a gated door closed over a tunnel in Central Park on one of the transverses. The next night I came back and went through the door. There was a long vaulted tunnel of stone with a dirt floor leading to the basement of the pump house at the southern end of the Reservoir, which I explored with a flashlight for as long as I thought was safe. Not long after that I took a long subway and bus ride to the end of Brooklyn, where it plays out into marshland and sand dunes, and walked up a hill past gates that no longer worked to visit the shoebox-shaped concrete holes where surface-to-air missiles were kept during the 1950s and ’60s. It was so far away from the rest of the city and so quiet that I might as well have been in the desert listening to the wind whistle.
The more I read about mathematics, the more it seemed to me that unconscious thought was responsible for a lot of it. The French mathematician Jacques Hadamard published The Psychology of Invention in the Mathematical Field in 1945, having sent a questionnaire to a number of mathematicians and physicists asking about their methods. Hadamard had been inspired by ideas he heard described by Poincaré in a talk on mathematical invention that Poincaré had given in Paris in 1908.
Hadamard describes four stages of mathematical creation: preparation, incubation, illumination, and verification. Preparation involves the gestures made on a first encounter, a throwing of what one knows at the problem. (The artist Saul Steinberg once told me that he began each working day by setting himself a problem.) For some mathematicians the first encounter is aggressive, a frontal assault, and for others, such as the French mathematician Alain Connes, it is oblique, because Connes believes that too assertive an approach can leave a mathematician not having solved the problem but having run out of tactics.
Amie said that one of her teachers had told her, “A good problem fights back.” Beyond the obvious inference, I took this to mean that such a problem might welcome attention but diffidently. Or that it is something like hills in the distance observed through a mist. The hills are apparent but the way there is not, or even if there is a way there. An approach that appears promising might collapse immediately, but it might also fall apart after a lot of work.
Incubation describes that period when, having been rebuffed, the mathematician might put the problem aside, hoping or perhaps expecting that some other part of his or her mind would continue to work on it, a method that Einstein told Hadamard he favored. Making associations among disparate means of progress takes time. Connes says that sometimes he finds it helpful to work in a field parallel to the one occupied by his problem, one that is perhaps a little easier and has more opportunities for engagement, in which he might develop a framework for his thoughts and approach the parallel problem indirectly. Thinking in a way that isn’t linear, that is abstracted and involves a dissociated state of mind, is characteristic of what the neuroscientist Jean-Pierre Changeux describes as “a blackout of rational thought.” He says this in Conversations on Mind, Matter, and Mathematics, which is a collection of exchanges between him and Alain Connes and a book I liked so much that I read it several times.
Illumination ends incubation. Maybe the mathematician drank too much coffee, or drank coffee and isn’t used to it, something that happened to Poincaré. Maybe he or she worked to the point of exhaustion and the defenses that might be invoked against psychic intrusions are fatigued. Maybe a long walk stimulates the imagination. Or waking from a nap.
Regardless, something previously unseen seems suddenly at hand. This is a sanctified state of being. Connes says of such times, “I couldn’t keep the tears from coming to my eyes.” All that remains, Hadamard says, is verifying one’s vision while hoping that it is genuine and doesn’t collapse.
Mathematics is not a practice that one ought necessarily to expect to be better at as one gets older, but I did expect, somehow, to be better at it. I had the feeling that the limitations I had placed on myself when I was a boy were matters I had disposed of, partly by showing myself that I wasn’t inept in the adult world. I also felt that I was returning to the encounter with more tools. Many of the subversive processes that overtook me in attempting math a second time, however, were bound into my being, my nature; they were me, at least a part of me, those hesitations and reluctances and fears of incapacity and, although it pains me a bit to say this, the embarrassment I had felt at being found to be insufficient. This was a struggle I hadn’t anticipated.
Hegel somewhere remarks that the reason there are so many examples of childhood prodigies in mathematics is that mathematics doesn’t involve (as music doesn’t either) a grasp of the complexities of mature life, which need experience to make sense of them. It is a commonplace in mathematics that young mathematicians do more consequential work than older ones. This seems to be a judgment that mathematics can’t escape. I have heard a number of explanations for this, aside from the obvious one that vitality and exuberance matter. Experience can make you hesitant is one. Also, children can see patterns at an abstract level that are obscured for older people, who see the patterns embedded in a larger structure and likely a familiar one that they aren’t able to see anew. Older mathematicians have independently to learn complicated new mathematics that younger mathematicians are fluent in, from having studied them recently in school. Also, young mathematicians don’t know what to be afraid of, and so can make breakthroughs on problems that older mathematicians have given up on. Andrew Granville, a mathematician born in 1962, once told an interviewer, “To crush a great problem, it often means you have no respect for where you are told not to go.” And, finally, the absence of a collection of elders who can impede the progress of a young mathematician whose work doesn’t fit their tastes, since mathematics is largely an individual task and taste does not figure extravagantly in mathematics, although, as elsewhere, favoritism of gender and race sometimes figures in academic life.
Serious mathematics requires a single-mindedness that Hadamard defines as “a tenacious continuity of attention,” and as “a voluntary faithfulness to an idea,” phrases that reverberate poetically. A mathematician might spend a career on only a few problems. Georges-Louis Leclerc, a French mathematician of the eighteenth century, thought that genius was a capacity for great patience. Henryk Iwaniec, who was born in Poland in 1947, told me that persistence was also necessary for a mathematician and that arriving at a result was a matter of “really conquering the turf to the end.” Mathematics for Iwaniec, as for many mathematicians, is an intimate endeavor. “We are driven by emotions,” he said. “Mathematics is still a beautiful personal challenge.”
HADAMARD DIVIDES A mathematician’s methods into two categories. One consists of the time spent with a problem, beginning with selecting it. The second involves the work that goes on unconsciously. From this engagement an answer emerges, sometimes entire or else in the form of an organizing principle. Hadamard quotes a nineteenth-century mental calculator named Ferrol who writes to the German mathematician August Möbius, “It often seems to me, especially when I am alone, that I find myself in another world. Ideas of numbers seem to live. Suddenly, questions of any kind rise before my eyes with their answers.”
Furthermore, Hadamard says, there are two types of invention. One type sees a result and imagines the means of achieving it. The other knows an answer and pursues ways of using it. One type of mathematician sees things at close hand, and the other sees them from a remove that is congenial to making generalizations between or among theorems or fields.
Breakthroughs on big problems are exhilarating. In Mathematics Without Apologies, Michael Harris quotes Marcus du Sautoy, saying “Once you have experienced a buzz of cracking an unsolved problem or discovering a new mathematical concept, you spend your life trying to repeat that feeling.” For the French mathematician Marie-France Vignéras, also quoted by Harris, “it happens suddenly: one direction becomes more dense, or more luminous. To experience this intense moment is the reason why I became a mathematician.” For Bourbaki, the pseudonym of a group of French mathematicians from the middle part of the twentieth century, a form of intuition, not “popular sense-intuition, but rather a kind of direct divination,” sheds light “at one stroke in an unexpected direction,” and this “illumines with a new light the mathematical landscape.”
Mathematical investigations depend on the tools at hand, but mathematicians can also invent new tools or find new uses for established ones. As with all arts, the more one can define, or perhaps restrict, one’s intentions, the more progress is likely, although many of the problems in deep mathematics are so difficult to approach that progress is sometimes made only incrementally, over long periods of time, and often by people not necessarily working with each other or even in the same era. A mathematician at work on a deep problem might have only companions from history or, conversely, ones from his or her posterity.
A pure mathematician seeking a result is pursuing something that he or she believes exists. The work isn’t being done in the expectation of finding a surprising result that hadn’t been contemplated, although possibly this happens. Pure mathematics is not generally haphazard, though. A mathematician typically knows what he or she is pursuing—the steps necessary to approach a big problem make a mathematician more liable to think, Something might happen, than, Anything can happen.
The mathematician Gregory Chaitin once told an interviewer, though, “Science is the same idea as magic: that there are hidden things behind everyday appearances. Everyday appearance is not the real reality.”