Sunday
Oct272013

Boléro

Michael Sheehan


 

The mathematician's patterns, like the painter's or the poet's, must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way.

—G.H. Hardy

 

Allison painted long lists of prime numbers, wrapped her brush around them over and over and over again, listening to music, the sounds measured carefully into divisions of time and duration. The brush is wrapping sloppily, agitatedly around 997 as the beat starts to pulse like the blood in the vein in her wrist: 1 and 2 and 3 and. The numbers are layered with color, the painting a massive palimpsest, primes upon primes upon primes.

 

 

At twenty-seven, she earned her PhD. Her work was on primes, focused on a facet of the so-called twin prime conjecture. Allison was careful to keep her personal motivation and feelings out of her work, for fear of making it seem unserious, but she believed there was a pattern, an order, and she believed finding it could open up for us the world beyond what we yet know.

At twenty-nine she married Tim, a lawyer, serious-minded and responsible, someone she knew she could trust. Not that that was what really counted. It was not that she had planned to love him, had planned for what she felt in the early hours of their first night together. She knew, despite the regularity and order and pattern she sought to study and surround herself with, much of life was made up of the unexpected.

At thirty she became a mother for the first time, and the experience was vaguely songlike. The epidural separated her thoughts from her body, allowing the metronymic rhythm of her breathing, pushing efforts to somehow harmonize with the arrhythmic feelings that had come before, the indescribable real physical awareness that extended over minutes, hours, an eternity as she struggled to reveal an unknown, a completely new and complex potentiality, her daughter Zoë.

At thirty-one, she gained tenure, and discovered she loved teaching. It came as something of a surprise to her, but the persona she adopted in front of the amphitheater of students was comfortable. This went beyond the interactions with the students on a person-to-person level, though she did find she liked the students and being with them, despite having always been terribly shy and quiet. She loved ordering the material, translating the crystalline beauty of mathematical architecture into language anyone could understand. She loved creating a lesson, one particular aspect of math she would construct whole in the minds of her students ex nihilo, form from void. Her work on primes slowed to a stop as she focused on teaching. She had given up on her search for a pattern in the primes, but she still believed it was there, still wanted to think there was more and deeper meaning to numbers than we knew.

At thirty-three she gave birth again, this time to her son Timothy Michael Newmann— after his father, though they would call him Mike. His name was a perfect trilogy, seven letters in the first, second, and third names. There were three in Tim, seven in Timothy, and also in Allison. Zoë's name, too, she took secret pleasure in: Zoë Marie Newmann: 3, 5, 7.

At thirty-eight, she started having migraines. Though they often ended full-body, they began like white electricity in her brain. Shortly before they overtook her, she would forget things, would be unable to recall the word for something, the name of something, would even be unable to explain basic mathematical functions. She would be in class and find herself unable to explain something she had taught many times, something that would seem so strange and unfamiliar it was as if she had broken a clock and were trying to put its pieces together again, trying to make it measure time.

At forty-one, she was forced to take a sabbatical the bulk of which was devoted to frequenting the doctor. Tests were done: MRIs, CT scans, blood tests. Her doctor recommended a respected neurologist, a man who mapped brains. He was the first to suggest some of the prognosis, the arresting news—as she and Tim sat side by side in upholstered chairs in the hospital's Comfort Lounge—that the primary hemisphere of her brain had switched. She had the sickening sensation as he described this condition that her thoughts were not her own, that soon everything would change, decline, be erased, and she would be gone, some other woman in her place.

She was granted an extended leave by the university at forty-three. By forty-six, she had trouble bringing to mind the words for household objects, including at times the names of her two children. Objects existed at the edge of her cognition: she perceived them, often felt a strong sense that she knew this thing (sink, counter, rack, door, window, light, stair) but could not name it, could not quite place what it was for. She would suddenly become frozen in the middle of some ordinary act, unable to place the objects around her, to put names and rules upon the world she perceived, to bring the seizing mass of sense perceptions into some kind of shape.  

 

  2       3      5     7    11     13    17    19   23    29
 31    37    41    43   47     53    59    61   67    71
 73    79    83   89    97   101  103  107  109  113
127  131  137  139  149  151  157  163  167  173
179  181  191  193  197  199  211  223  227  229
233  239  241  251  257  263  269  271  277  281
283  293  307  311  313  317  331  337  347  349
353  359  367  373  379  383  389  397  401  409
419  421  431  433  439  443  449  457  461  463
467  479  487  491  499  503  509  521  523  541
547  557  563  569  571  577  587  593  599  601
607  613  617  619  631  641  643  647  653  659
661  673  677  683  691  701  709  719  727  733
739  743  751  757  761  769  773  787  797  809
811  821  823  827  829  839  853  857  859  863
877  881  883  887  907  911  919  929  937  941
947  953  967  971  977  983  991  997

 

The greater and more complicated the primes become, the less tangible. The numbers themselves sink into an array of meaningless figures in a visual field. She paints over them and over again in time with Schumann's Arabeske Op. 18, explosions of color cry out from the rigid order, the colors make a sfumato of porous numbers, their edges bleed one into the next. Though she thinks little of it, the painting has a sort of vatic brain-mapping imagery to it, an understanding of its own troubled making.

 


Aphasia. "Loss of ability," she looked the definition up in the dictionary, had to read it three times through, "to understand or express speech." Simple enough. Seven letters: a prime.

Painting felt vaguely like math: they were both means of expression that were nonverbal and—she could feel—more capable of achieving something true. Solving a problem in differential calculus, reaching the moment when the structural and visual elements of her painting came together and harmonized—these both resulted in similar feelings of having touched something beyond her experience. The elements of the painting came together and in an Archimedean epiphany, she could feel it was true, could feel something that was not just internal, but a sort of resonance between the world and the perceiving self.

Allison found this aesthetic bliss not only with difficult problems. She saw the simple harmonic beauty in 2+2=4, the conversions of objects (two hats plus two shoes is four items) to abstractions. Suddenly numbers could be seen to have greater universality and primary truth, where numerals are a priori. 2+2 = 4. Such depth in such simplicity. And when she was bringing her dissertation to its close, she would periodically feel the rare aesthetic beauty, that overwhelming all-body beauty, of truth—pure and real and outside her head—shining through. She would shiver with the sense, which remained ever incomplete, that she had brushed up against that which is.

 

In the Philosophical Investigations, Ludwig Wittgenstein says of language that meaning is use: There is only an object performing the role of 'cup'.

The cup holds the brushes and the cardboard box flap holds the oil paints. The flat stretch of canvas—like a sheet on a line gone taut in the wind—holds the paint itself as she applies it over the surface, sometimes inexpertly, carried away, her actions all appassionato; at other times careful, labored, showing the deliberative way she applies thought through paint to image. Her brushes are dirty, their stems covered with the scabs of old paint, the wood grain darkened in places, showing the stains from her hands, from use. They sit in an old Pizza Hut promotional plastic cup, emblazoned with the Jurassic Park logo set atop a lush jungle, a Tyrannosaurus's shadow haunting its leaf-fringed shape. The cardboard box flap she uses to hold her paint—which commingles and shifts as she uses her knife and brush to force colors together, to add them and thereby make some new thing, tertium quid—is torn from an Amazon.com order, the underside where her splayed fingers hold it still shows the smiling arrow and a remnant of the address label. The paint she scrapes together has made a landscape unto itself across the face of the cardboard, a topography of colors, old and new, the dried sediment of old paint thickly coated over by swaths of the molten new.

The canvas, this canvas, remains vague, indistinct, the signs on it not yet signifying anything. It looks like the scratch-paper she would collect in calculus exams.

Still, though, looking at it, Allison feels the twin urges to destroy and to make. She is filled with rage at her futile stabbing with the brush, her pointless mark-making and her decreasing ability to express anything, and yet, too, she has the familiar longing to see this through, like a difficult proof; it is the belief that despite all evidence to the contrary there is an immanent pattern in the emptiness.

 

 

The doctor seemed to think her painting was therapeutic. She was no longer able to work, in any real sense. Her lectures had become jumbled pages of signifiers, little signs she couldn't connect to thought. Some of the time she knew the words; most of the time the pages were merely visual displays, dense sketches in pen and pencil, a balance between line and empty space.

 

 

Allison prefers music that is very ordered when she paints, music that trusts to harmonies and develops patterns. Music that fills her mind with wordless expressions of image and object, a world rendered from sound and light, one that manages to both progress and yet be static, to repeat upon itself and yet to advance.

She is painting an ekphrastic piece for Ravel's Boléro. It is a large canvas, covered with squares, and squares within squares and more squares within those. It is impossible to break free of the squares and the deeper you look, the more there are. And yet the whole canvas has the appearance not of order but of a wild unleashed beauty, a disorder burst across the canvas. The squares are there, and within each is a smaller and smaller universe of disarray. The squares try to hold the burgeoning drips and splashes of color. The paint dances within the pattern and whirrs like atoms thrumming the world into existence.

Ravel's Piano Concerto for the Left Hand had been commissioned by Paul Wittgenstein, Ludwig's brother, who had lost his right hand while a captive during World War I. He'd been an accomplished piano player, and refused to lose what he valued most; he worked his hands in air across imaginary keys, developing the almost impossible ability to do with one hand what others needed two to do. He compulsively played soundless scores, working until his left hand could perform imaginary acts of musical creation. He repeated the motions on walls, on a table, on his left leg, and most of the time on nothing but the open space that arrayed itself in his mind into eighty-eight alternating black and white rectangles upon which he could perform music that filled the space inside him, while the world he was sitting in and the empty space where his right hand had been faded.

Her wrist aches. She is reaching an ecstasy of creative feeling now, her arms moving as the sounds drown out the world, her world—the garage in which she paints, the dried paint on her fingers and hands—the tempo's slowing meets with a greater ferocity of brushstrokes in the upper right quadrant of the picture plane, as the music drives to its finish.

 

 

She was a professor of mathematics. Today as she sits at the kitchen table, periodically distracted from what she's doing by the calls of birds and the play of light through a prism that hangs in the window above the sink, she suddenly finds herself looking at the numbers on the homework Zoë's spread out on the table without the slightest shudder of recognition. She cannot add them, nor even can she recall that is what she is meant to do with them.

 

Two plus two is four. Two plus two is four. Two plus two is four. Two plus two is four. Two plus two is four. Two plus two is four. Two plus two is four. Two plus two is four. Two plus two is four. Two plus two is four. Two plus two is four. Two plus two is four. Two plus two is four. Two plus two is four. Two plus two is four. Two plus two is four. Two plus two is four. Two plus two is four. Two plus two is four. Two plus two is four. Two plus two is four. Two plus two is four. Two plus two is four. Two plus two is four.


As an undergrad, Allison lay in bed looking out at the sliver of stars and the nighttime nothingness their light came across that she could see from the angle at which her bed met the wall's window. The distance of those stars was so immense, so impossibly difficult to conceive that thinking about the distance drove her thoughts further up inside, away from the world, into abstraction. The world itself prompted a type of disconnected pure thought that existed only in her head. Nothing more real than this: all our thoughts are all we get, there is no access to the beyond.

But looking at the stars and considering the seeming paradox of how the great expanses outside her could only illuminate the greater expanses within, Allison would think—even as a sophomore—of Gödel, who became one of a small number of mathematical heroes for a time. Of the twenty-five year old—sensitive, asocial, friendlessly shy—who conceived of the outer limits of all math and showed there is indeed more to life than is dreamt of in our philosophy. In later life, in adoration of Leibniz, Gödel tried to prove the existence of God. She would think of this man of immense faith and confused thoughts, who dreamt of truths outside of any math, proofs of how incomplete our knowing is. How alone he must have felt when he delivered his incompleteness theorem and none could understand it. When he first conceived of it, of the idea of incompleteness, what did that feel like? Did he feel the loss of certainty, the dissolution of order, or was he able to see God in the void?

 

 

Euclid had to leave the infinite continuity of parallels as a postulate, without a hard proof. It must have seemed impossible that they would collapse and suddenly intersect and fail to hold their form forever.

It seemed just as impossible the day the doctor tried to describe frontotemporal dementia. He explained that the manic burst of creativity Allison had undergone and the staggering speed of her loss of numerical expression and understanding were symptomatic and resulted from a sort of intersection of the hemispheres of her brain. She had taken for granted this was impossible, yet suddenly she found herself in the nonspace outside of Euclid's orderly world. The familiar laws ceased to hold, the world they described disappeared.

 

 

She had never considered not living long enough to see her children grow old. Mike and Zoë played in the yard while she sat on the steps of the back porch. Coruscating sunlight shone, a migraine's keen rays piercing already her periphery. Allison watched the two of them playing, wondering whether she was dying. Would she die before they had become adults? Before they had become adolescents? All the things she wished to share with them.

Fibonacci's rabbits: What a way to quantify sex. Each mature pair of rabbits creates one young pair, and each young pair matures every cycle. One young, one mature, one young and one mature, two mature and one young, etc. 1, 1, 2, 3, 5 . . . ∞.

If her children had children, and those children had children, and their children's children had children, was it possible for her to reach infinity through them? She comes close to losing consciousness, sitting there on the porch, trying to calculate how many children each of hers will have, by some order, by an unseen law, and how long it will take to reach threshold numbers: 100, 1000, beyond. She can't hold the numbers in her head long enough, and must continually start over. She wonders whether there is a pattern; she seeks order in meaningless places.

This is what her days feel like. It is as if she were standing at a board at the front of a classroom, looking at a problem there, but as she follows its contours and complications, by the time she reaches the end it does not sum to anything. Some days, it is as if she stands before the problem and by the time her eyes have reached the ending of it she has no idea what she is there to do, why she is standing before it.

The sunspots are turbid and yet vibrant, a diaphanous tumor metastasizing behind her eyes, a fading bruise of yellow, purple, red, and green. Allison looks up to see she is on her side in the grass and her children are kneeling beside her. "Momma," her daughter whispers almost inaudibly. Her little boy cries.

 

 

"A prime number is that which is measured by a unit alone."

The simplest notion of a prime number is an integer that cannot be divided, does not break down into anything smaller.

For years, she studied primes, wrote articles on them, taught the occasional upper level seminar on number theory, enunciating the history from Euclid to Goldbach, Euler to Gauss, Riemann et al.

Euclid's Elements, Book IX, Proposition 20: "Prime numbers are more than any assigned multitude of prime numbers." Furstenberg proved ∞ primes in 1955; Saidak in 2005.

She would sometimes tell her students about the near-mystical pursuit of the impossible in math. She would sketch out for them the Goldbach Conjecture and the Reimann hypothesis. These were holy grails, akin to Gödel's proof of God, not just the type of thing to stake a career on. Problems whose solutions feel bigger than the sum of all parts. That any even number greater than 2 can be the sum of two primes. Never yet proven, or disproven. And what does that mean? Its continued unresolved status, the role of primes in number theory: What do these things mean? Allison would marvel at this, was staggered by the unknowability, the boundariless state of what seemed a simple idea. Goldbach wrote this problem in the margin of a letter to Euler, who responded some days later, "I regard this as a completely certain theorem, although I cannot prove it."

These were the Olympian heights of modern math, the stuff of Millennium Prizes. Allison had never much indulged in chasing down prizes or fame; she sought pattern and wished to see primes as they really were, to know them, but she harbored no illusions.

The first and thus far only Millennium Prize to be awarded, following a Fields Medal for the same achievement, was to Grigori Perelman for his solution of the Poincaré conjecture. However, he rejected both prizes, and shortly thereafter withdrew—like the young Wittgenstein almost one hundred years earlier—amid a wide range of speculation that he had given up math altogether, was living unemployed with his mother, or that he was secretly in pursuit of other Millennium Prize problems. Allison read about him, but could not understand the paper he'd published that started it all. In an interview she read (which may have been faked), Perelman explains his rejection of the prizes, saying "I've learned how to calculate the voids. . . . Voids are everywhere. They can be calculated, and this gives us great opportunities . . . I know how to control the Universe. So tell me—why should I chase a million?"

 

 

The white floor of the empty gallery struck her as profound. The space was slightly dusty, the walls bare but showing the traces of previous shows' hangings: holes, outlines, an aura of absence pervaded the whole place. The lighting was "natural," and sound seemed to die in the walls, each noise existed as if in a vacuum, without resonance or any lingering or extension.

This was Tim's idea. He felt she should bring her art out, sell it, show it. She felt strange about the whole conversation. She understood what he was doing and what motivated it, but was too kind to tell him this. He was trying to cope, she knew, in his own way. He was acting as if her painting was a normal, natural expression of her artistic side, latent all along. He tried to treat it like a career change, a pursuit of passion, her real nature coming through. This was to willfully ignore the connection between her bursts of manic creativity and her inability to add, to think of even simple commonplace words, the doctor's account of what had happened inside her head. He was trying to encourage her in order to be, himself, encouraged. It was less about his overwhelming acceptance and support, and more about his ability to view this as something he could handle, eliminating all trace of deterioration and death.

Tim was talking with the gallery's owner, a small woman with wide hips and high cheekbones who wore ostentatious tortoiseshell glasses connected to a string around her neck, wore them, that is, in her thick nest of curls.

"They seem to be dealing with a very rigid, mechanical aesthetic, there's a sort of Jasper Johns quality to this one. I can see there is an idea of imposition of order on chaos, yes?"

Tim nodded academically, his hand on his chin in unconscious pretension. "It's incredibly beautiful. I think her mathematical ideas and her artistic passion work together, they create a sort of fluid . . . a harmony. My wife," he touched her elbow gently, "was a math professor until she started painting."

"Big change," the owner said, incongruously lowering her glasses from her hair to the tip of her nose.

"Mathematics possesses not only truth, but also supreme beauty," Allison said, quoting Bertrand Russell, her voice surprising even herself with its hollow sound, its emptiness.

"I can see that idea in your painting. Yes, definitely."

Tim went on, moving his hand in wide flourishes, sine waves and parabolas. He was translating the wild sort of beauty in her paintings, their capture of her departing mind's last wordless, disorderly expressions of something she herself no longer knew the words for.

In his lifetime, van Gogh was not successful, famously failing to sell paintings or start an aesthetic movement. Though his works now top the lists of all-time art buys, abstractly large amounts of money being exchanged for an artist's subjective impressions of experience and the world, in the time before his presumably-syphilitic deterioration led him to shoot himself in the chest, his primary buyer was his brother, Theo. She thinks of this, as Tim continues his perorations and exhortations with the gallery owner, as he imagines wordy formulations of what she has done, as he falsifies whatever is on the canvas by trying so hard to give it a name.

Gödel showed that our math is not really that which is the case. It can't describe what really is. Whatever hope she has of finding even the least bit of truth must rely, she realizes, on its inexpressibility. On the fact that, Cassandra-like, she can perceive something true but cannot tell it. Can only paint whatever beyond words she understands in the desperate hope that some other will see it, feel it, and will know it. Like a rainbow, it will be unique to each viewer, never seen the same twice. Like primes, it will be the indivisible, the infinite, the alone.

 

Two plus two is four. Two plus two is four. Two plus two is four. Two plus two is four. Two plus two is. Two plus two. Two plus. Two two four. Two two is. Two is four. Two plus is four. Two plus four. Two plus. Two two is. Plus two is four. Plus is. Plus. Four. Two. Is.

 

 

And she believed finding it could open up for us the world beyond to understand or express covered with squares and squares within squares and more squares within those that manages to both progress and yet be static to repeat upon itself and yet to advance to leave the infinite whatever beyond words like white electricity in her brain primes upon primes upon primes and the play of light distance of those stars was so immense so impossibly difficult to see God in the void his hands in air across imaginary keys I regard this as a completely certain theorem, although I cannot prove it.