Mathematics and the Character of Tragedy

Rebecca Goldstein

There is a tension between mathematics and the narrative arts. I am speaking

here of an uneasiness that goes much deeper than the hatred of math so often confessed to

(even with a perverse sort of pride) by literary types, or the symmetrical inclination

among a good many mathematical types to dismiss storytelling as so much fluff. I am

alluding rather to an argued strain between mathematics and the narrative arts that comes

with the finest intellectual pedigree, reaching back to the fifth-century B.C.E. Its first and

finest expression occurred in Greece, not so surprising since this is the location for the

first and the finest expression of a dismaying number of fundamental ideas and themes

and preoccupations of Western culture. It was Plato who best expressed— who veritably

embodied—the tension between the narrative arts and mathematics.

Of course, there were no novelists in Plato’s day. The closest thing to novelists

were the dramatists and epic poets. Mastery of the poetic legacy was largely what

counted as wisdom. Sprinkling one’s oratory with passages from Homer and Hesiod

yielded an authority almost tantamount to proof. Though Plato certainly rejected the

probative value of poetic quotations, he nevertheless continually availed himself of them

throughout his writings, though always in the way of illustrative flourishes, never as

substantive elements of argument.

Plato clearly loved them both, both mathematics and poetry. But he approved of

mathematics, and heartily, if conflictedly, disapproved of poetry. Engraved above the

entrance to his Academy, the first European university, was the admonition: Oudeis

ageometretos eiseto. Let none ignorant of geometry enter. This is an expression of high

approval indeed, and the symbolism could not have been more perfect, since mathematics

was, for Plato, the very gateway for all future knowledge. Mathematics ushers one into

the realm of abstraction and universality, grasped only through pure reason. Mathematics

is the threshold we cross to pass into the ideal, the truly real.

In the Allegory of the Cave—the best piece of storytelling in all of Western

philosophy—he presents us with a sort of epistemological Pilgrim’s Progress,

deliberately setting up an alternative myth as a rival to the poet’s tales of the immortal

gods. His new myth is presented in a few paragraphs of the Republic but its scope is as

large as any myth offered up in the millennia since. It presents not only a vision of the

nature of reality, but of our cognitive relationship to reality. Plato is attempting to tell us

what knowledge is and how it is possible for the likes of us. Even more largely, it is a

view about how we can save ourselves by coming to know the real. All reference to the

gods is stricken in his narrative; whether we happen to find favor with that capricious

mob is not the contingency that makes the difference between a life well-lived or not. We

need never stand passive before them, nor anything else, again. Instead the difference

between a life worth living and one that is not is a difference that we ourselves make,

relying only on our capacity to reason out objective truth. Reason gives us the world as it

really is and arriving at the world is the one activity worth doing, bringing us the only

solace that we can honestly attain.

The allegory presents a prisoner chained with others so that he, like them, stares

only at the back wall of the Cave, over which plays flickering configurations of shadows

that he and the other prisoners regard as the sum total of reality. They study the shadows,

keen on discerning any underlying patterns. Some among them, shadow-scientists, seem

to excel at predicting events in the random sequence and they are honored as thinkers.

This level of confusion is called by Plato eikrasia, the lowest rung of consciousness.

One prisoner is freed and sees that the shadows are only shadows, cast by

puppets being toted on sticks by men. Only then does he understand the nature of the

entities studied in the pseudo-science of his former colleagues, knows the shadows for

what they are by ascending to the next level where he can see their explanation.

The light that creates the shadows comes from a small fire burning within the

Cave. The fire stands for the personal ego in Plato’s allegory. (I am engaging in some

interpretation here. An allegory obviously calls for such a response.) A chained prisoner

staring at the flickering shadows is looking at how the world would be if he were really

as important in it as he unreflectively feels himself to be—as we all unreflectively feel

ourselves to be, which is why Plato solemnly intones that this strange situation, being

chained so that one’s scope of vision is restricted to shadows that one cannot identify as

shadows, is one that universally holds: “It is a strange picture, he said, and a strange sort

of prisoner. Like us, I replied” (vii. 514).

In the Timeus, Plato’s creation myth, he has the Demiurge delegating to a lesser

god the doling out of human emotions. This lesser god, in his inexperienced zeal, infuses

too much of the “necessary and dreadful” emotions needed for survival. It is necessary

that each person confer special significance on himself, be interested in himself as he is in

nothing else. But we do tend to go overboard in this direction, Plato is telling us, and to

objectify into false metaphysical schemes what is merely a contingent fact: that one

happens to be who one happens to be, and, being that thing, one wants very much that

that thing that one is to particularly and objectively matter: to matter largely, on a scale to

flatter one’s own sense of one’s importance. To be imprisoned in such a viewpoint

(many people’s personal religiosity comes to mind) is to be at level of the chained

prisoner.

The prisoner, once he is free of his chains, can turn his head and take in others.

This is progress, but still the whole inside of the Cave is limited to the view from the

distinctly human perspective. It is still dark with subjectivity—even if the viewpoint is

broadened beyond that of the single individual. Plato has in mind at this stage the sort of

undergraduate-beloved specious enlightenment that comes with recognizing that one’s

own point of view is but one’s own, others having others, and therefore—here comes the

specious part—there is nothing at all that makes one point of view superior to any other.

This sort of relativism, despite its postmodern tone, is as old as philosophy itself,

represented in Plato’s own day by such sophists as Protagoras. Plato deplored relativism.

It is only one step above eikrasia, closer to it than to the next stage of the ascension.

Now the prisoner is taken to the opening of the Cave. He steps out, with

difficulty, into the sunlight. He doesn’t look yet at the sun; he can’t. He must get to the

sight of it by degrees. With this motion outside of the Cave he takes the big step, going

from the dark and narrow confines of points of view thrown up by the fire of human

subjectivity that casts its weak and illusion-making light within the Cave, into the broad

and bright Out There of the objectively real. One can only get Out There through the

exercise of reason, the only means one has of overcoming ego-induced illusions. Reason

acquaints one, according to Plato, with a new level of reality—the objects of pure

abstraction—existing precisely as reason deduces them to be, unobscured by the

emotional smoke of the ego-fueled perspective, seeing the world as it wishes it to be for

such purposes as meet its own narrow needs.

The prisoner is at first too stunned by the dazzle of light to take in the objects

around him and instead looks at their reflections in water. These reflections, Plato tells

us, represent mathematics. Mathematics is the training ground for the faculty of reason.

Mathematics is the first encounter with rigorous rationality, the first acquaintance with

the objects we know to exist because we deduce that they must exist. The self’s desires—

the major source of illusions—have no role to play in the deductions of pure reason. The

world to which one ascends is perfectly objective, appearing to all as it appears to each.

For Plato, there are objects of abstraction beyond the mathematical, but

mathematics is our entry into the realm of abstraction. For those who have no head for

mathematics, there is simply no hope of attaining the sights of true reality. They will

never make their way further into the bright and broad reality beyond the Cave. No

wonder that he inscribed that warning to the ageometretos above the entrance to his

Academy.

Mathematics, then, plays a special role in Plato’s epistemological myth.

Interestingly enough, to the extent that a Platonic metaphysics—a commitment to the real

existence of abstract entities—still survives in contemporary thinking it is almost

exclusively restricted to mathematical Platonism, the view to which many

mathematicians appear to subscribe. For example. G.H. Hardy, in A Mathematician’s

Apology, unapologetically confesses his core Platonism:

I believe that mathematical reality lies outside of us, that our function is to

discover or observe it, and that the theorems which we prove, and which we

describe grandiloquently as our “creations,” are simply our notes of our

observations. This view has been held, in one form or another, by many

philosophers of high reputation from Plato onwards, and I shall use the language

which is natural to a man who holds it . . .

This realistic view is much more plausible of mathematical than of physical

reality, because mathematical objects are so much more what they seem. A chair

or a star is not in the least what it seems to be; the more we think of it, the fuzzier

its outlines become in the haze of sensation which surrounds it; but “2” or “317”

has nothing to do with sensation, and its properties stand out the more closely we

scrutinize it. It may be that modern physics fits best into some framework of

idealistic philosopher—I do not believe it, but there are eminent physicists who

say so. Pure mathematics, on the other hand, seems to me a rock on which all

idealism founders. 317 is a prime, not because we think so, or because our minds

are shaped in one way or another, but because it is so. Because mathematical

reality is built that way.

In any case, whether or not mathematicians are disposed to think well of

Platonism, Plato was certainly disposed to think well of mathematics. Mathematics

presents us with the model of objective reason, of thought processes that have been

purified of all the smoky soot of subjectivity, of how things merely appear to us, these

appearances conditioned by our own particular place in the world, and transports us out

into the wide impersonal open of the truly real, the necessarily existent, the pure and

perfect and certain.

So math is good, according to Plato. Very very good. What of the narrative arts?

Not so good, according to Plato.

Not only do the arts in general belong to the ego-warped, emotionally clouded,

narrowly particular, shifting quasi-reality of mere subjectivity, presenting the world as it

appears to be from the merely human point of view, or worse, from the point of view of

particular idiosyncratic “artistic” human beings, rarely the most stable or virtuous among

us. Plato emphasized the cognitively empty status of the arts by identifying them with the

flickering shadows dancing their meaningless dance on the back wall of the Cave. The

arts, of all the modes of thinking, are the furthest removed from salubrious objectivity.

All the flaws from which the sooty world within the Cave suffer are accentuated in art.

To take the world as it is given to us in art is to remove oneself even further from

knowledge of the real. Art, as beautiful as it may be, in the end has nothing to teach us.

Yes, as beautiful as if may be. The beauty of art is worth remarking on, since

Plato does not undervalue beauty, as any mathematically inclined metaphysician would

not. Beauty, he believes, is vital to us in helping us see our way clear to objectivity.

Without our susceptibility to beauty there would be no hope for us at all. It is only

because we fall in love with beautiful things, including truth, that our attention can be

wrested away from our default object of fascination: ourselves. Beauty then is part of our

cognitive salvation. So there is a bit of a mystery as to why the arts, dedicated as they are

to ideal beauty, are dismissed as inhabiting the absolute lowest sphere of human thought.

And of all the arts he singles out the narrative arts for special abuse. In that very

same dialogue, the Republic, in which the Allegory of the Cave is presented, he—

regretfully and with many terms of affection—gives the poets the boot. The outrage

occurs toward the end of Book X. Socrates, in an attempt to answer the large question of

what justice consists in, has been constructing a utopian state, down to the smallest detail,

even prescribing who shall mate with whom, and how it shall be brought about, and how

the issue of those matings shall be educated, according to their innate cognitive abilities.

In his micro-management he must, of course, consider the issue of the arts, which in

ancient days, as in ours, formed such a substantive part of shared communal sensibility.

He allows for certain kinds of music, mainly martial, to stir up the right sort of stuff for

battle, and for simple forms of visual art—small and uninspired art, not the sort to make

us swoon with aesthetic bliss. It sounds as if, for Plato, safe art is pretty bad art.

But there is to be no place at all for the narrative poets. Were one of these gifted

fellows to wander into the reason-sanctioned city he would be told that there are none at

all like him in residence—ambiguous words of praise. The poet will be anointed with

myrrh, his head wreathed in laurel, and then he will be respectfully but firmly

accompanied to the nearest border and bidden farewell.

Why did Plato feel he had to go to such lengths to de-poetize his rational city? Is

there no safe narrative art at all? No, there isn’t, not according to Plato. The better, the

narrative artist is, the more he moves us, the more he undermines the rational life that

Plato espouses.

The answer as to why Plato took such drastic measures is not straightforward at

all, and I am doing a good deal of extrapolation and reconstruction—not to speak of

condensation—here. There are various answers, though they interweave with one

another in a reinforcing sort of way.

The simplest answer as to why Plato, poet-lover though he was, felt compelled to

take such drastic measures in his imaginative city, is to concentrate on Plato’s reality-

chauvinism, so to speak, which inclined him to harbor little respect for the products of

imagination. Reality, for him, consists in the abstractions of pure reason, the world

outside the Cave. Art, though it may reflect abstractions and universal, is made up out of

particulars, and particulars were of no particular interest to Plato. Even worse, art does

not present real particulars but imitations of particulars, placing art at an even further

remove from the real.

But this observation, often presented as the sum and substance of Plato’s

argument against the artists, does not begin to do justice to the complexity of Plato’s

conflicted attitude toward art. And it does not explain why the narrative arts, of all the

mimetic arts, were singled out for specially abusive treatment.

Another possible answer centers on Plato’s view of beauty. Beauty is just too

potent a power for Plato to allow it to be utilized by any but the rigorously rational, those

who put truth above all. Artists will go after a beautiful effect whether it is edifying or

not, and this is dangerous because we are helpless before beauty, ready to believe

anything, or anyone, if it, or he, stirs our aesthetic sense. It is true that were we not

disposed to love the beautiful we would all remain staring at the shadows of the back wall

of the Cave, enchanted by our own personal vision of the way the world would look if we

were the most important thing in it. Our love of beauty breaks the heavy daze of our self-

enchantment, turns the vectors of our attention around so that they are pointing away

from us and we can take in something of the world beside ourselves. But because it is so

potent, our love of beauty can be misused, making us fall in love with what is neither true

nor good. The philosopher alone knows how—like the mathematician that he must also

be—to use beauty responsibly, as a guide for the truth.

This is all true, though it is also true that all the arts—music and painting and

sculpture, as well as the narrative arts— are guilty of slavishly worshiping beauty,

detached from its truth-seeking rationale. No answer has yet been offered as to why Plato

singles out the narrative artists alone for banishment. Why can’t the poets of his day, and

by extension the novelists of today, undergo Platonic rehabilitation so that the wares of

their creativity can pass muster in the rational city?

Can’t we novelists be good for the soul?

No, we can’t, not according to Plato, even though we too have the power of

pulling an individual out of his little ego-warped worldview, by making him participate in

other points of view, inhabit them as if they were his own. But the subjective point of

view is not a good thing in itself, so pulling people out of their own, so that they see the

world from others’ viewpoints, is not much progress. Multiplying what has no value in

itself doesn’t amount too much.

But more seriously still, the narrative arts are profoundly and irreconcilably at

odds with the Platonic project of rational salvation, of saving ourselves by becoming

rational. Saving ourselves from what? From being human, with all of the dreadful and

necessary emotions—and tragedy—that this entails. The project of rationality as Plato

conceived of it was to be nothing less than delivering us from the tragic dimension

altogether, by bringing us to that glaring sunlit world where not a shadow of the personal

perspective can survive. Within this brave sunlit world there can be no storytelling—deep

participation in the narrative of people’s lives. These narratives all conjure up the

emotions of subjective attachments that get blessedly burned away in the sun’s bright

glare.

In order for narrative art to move us, to yank out us out of the own enchained

personal visions by awakening us to a deep emotional participation in other narratives,

making us momentarily forgetful of who precisely we are, we must take it for granted

that the individual matters in incalculably significant proportions, that his diminishments

are diminishments of the world itself. And, too, good literature makes us feel what a

pitiful thing it is to be human—all of our terrible longings and the indifference with

which the world treats them, not least our longing to live forever. We are infinite only in

our longings—which means that we long to be gods.

Good art offers no false consolations. From Plato’s day to ours, art tells us not

only that we are not gods, but that it is a tragic shame that we are not, and that there is

nothing that we can do to make it better. As the literary critic George Steiner beautifully

put it:

Tragic drama tells us that the spheres of reason, order, and justice are terribly

limited and that no progress in our science or technical resources will enlarge

their relevance. Outside and within man is l'autre, the "otherness" of the world.

Call it what you will: a hidden or malevolent God, blind fate, the solicitations of

hell, or the brute fury of our animal blood. it waits for us in ambush at the

crossroads. It mocks us and destroys us. In certain rare instances, it leads us after

destruction to some incomprehensible repose.

Our essential pitifulness is impressed on us most intensely when the protagonists

of a tragedy are cast in the heroic mold, with strengths that make them stand out in vivid

bas relief from hoi polloi. Their very giftedness, and the ambition for transcendence that

these gifts engender, is what typically brings them down. It is not that were they to have

stayed nearer to the earth, crouching on its surface, they would have been able to avoid

tempting tragedy altogether; after all. none of us can. But what they could have avoided

were the spectacular tragedies that make their stories worth the telling—stories like that

of Icarus, who soared and plunged, or of Marsyas, the musical mortal who dared to

challenge Apollo to sing music more beautiful than his. Marsyas lost to the god, of

course, and was flayed alive for his hubris, crying out to the god in his agony: “Why do

you pull myself from myself?” This is a particularly apt cry of agony for a tragic hero;

the heroic attempt is, after all, an attempt to pull one’s self from one’s self, and to fail,

and to suffer the consequences.

This whole tragic way of looking at things is what Plato is keen to deny. And this

is why literature, from his day to ours, is at odds with his ambitious project of rationality,

and why he was compelled to send the laurel-wreathed poets packing. Plato would like

to save us from our fragility, which means that he would like to save us from being

ourselves—infinite only in our longings to be other than what we necessarily are, to pull

ourselves from our selves. Plato, too, is asking us to be heroic, to pull ourselves from

ourselves, but he is asking that we complete the job, become the impersonal point of

view, submerge our own particularity into the fullness of objectivity. The more our own

identity merges with the impersonal world of pure abstraction into which our love of

beauty carries us, entering it through the portal of mathematics, the more indifferent we

are to the vicissitudes of our own personal fortune, which is the only salvation that we

can know. We render our particular selves invulnerable by ceasing to be, or at least

caring particularly about, those particular selves.

Poetry’s beauty is then particularly dangerous, according to Plato, for it points us

precisely where we should not go, toward deep participation in the heartbreaking

poignancy of our frail and mortal lives. And, too, the plight of the hero, singled out for

suffering precisely because he is heroic, could not go further against what Plato is urging

of us, viz. to be heroically rational—his tale of the Cave is precisely such an exhortation,

and one for which, too, he foresees, in some sense, a tragic end: The epistemological

pilgrim having emerged into the glorious refulgence of reality will feel compelled to go

back and try to enlighten the others. But back down in the darkness of the Cave, his eyes

still filled with the sunlight of impersonal Truth, he will appear the fool, and the others

are almost guaranteed not to understand him, and to mock him, and if he still insists on

these truths that they cannot begin to guess at he will begin to seriously annoy them.

Helpless down in their darkness, precisely because he is enlightened, he will be

vulnerable to their low cunning and “if they catch him they will kill him” ( vii. 517).

A tragic denouement, in a certain, sense, but not in Plato’s sense. What Plato is

trying to do is change our frame of reference so that these are no longer seen as tragic

tales, all these stories that seek to move us by recounting the demise of one particular

person, no matter how poignantly heroic. Who is he, who is anyone, compared with the

truth? Plato’s project of rationality rests in excising the tragic dimension altogether from

our conception. And this is why he says in the Phaedo that to philosophize is to prepare

to die.

Plato then loves and approves of mathematics, but he only loves— and against his

better reason—the narrative arts. He doesn’t approve of serious—that is, tragic—

storytelling. He can’t approve of it, since it is designed to make us feel what a pitiful

thing it is to be human—even for the greatest among us, the heroes, who can seem, in

their audacious powers, to transcend the limits of finitude, but who, in the end, don’t,

because none of us can.

But still Plato—how I love his conflicted soul—can’t help loving poetry. He

betrays his better philosophical instinct by sneaking poetry in wherever he can. And of

course he is the greatest literary artist among the philosophers, wielding the tools of the

literary trade—dialogue, metaphor, allegory—with enviable artistry.

Even after he gets through banishing the poets he confesses his love for their art

and holds out the hope that someone will convince him that his anti-poetical stance isn’t

necessary:

But in case we are charged with a certain harshness and lack of sophistication,

let’s also tell poetry that there is an ancient argument between it and philosophy . .

. Nonetheless, if . . . poetry has any argument to bring forward that proves it ought

to have a place in a well-governed city, we at least would be glad admit it, for we

are ourselves very susceptible to its charms. (Book X 607)

What I am going to offer is nothing like the sort of poetry-justifying argument that

Plato is looking for. In fact, if anything what I will have to say will only strengthen

Plato’s claim that the narrative arts stand irreconcilably opposed to the project of

rationality, as Plato conceives of it. I wouldn’t want to read novels if they didn’t. I

wouldn’t want to write them.

Still, I confess I take Plato’s argument seriously. I feel his banishment of those of

my ilk with a certain admiring sense of justification. I always hear Plato’s voice in my

head and I’m always trying to answer him. Plato’s right that narrative art must be

poignant. It must pierce us through and through. It must devastate us with the terrible

knowledge of our fragility—even the most god-gifted among us. If the project of

rationality is to make ourselves immune from devastation, then Plato’s logic is faultless:

fiction is no damn good for us. I’m not going to try to answer him here on that score.

Mostly, I try to answer him by writing fiction.

And often this fiction that I write features mathematicians as major characters,

which is ironic, given whom I’m trying to answer here. Why is this? What draws me,

again and again to mathematicians as characters?

There is something about being a mathematician that crystallizes, at least for me,

some of the very features of the human predicament—the ways in which we partake of

the infinite and the ways in which we don’t, the ways in which we are caught between

the gods and our mortality— from which Plato sought to rescue us by way of

mathematical reasoning itself. That is to say, there is something that I find essentially

tragic—in the noblest and most moving sense of that word—about being a

mathematician. If to be a tragic figure is to be a finite mortal who strains toward the

infinite, who resists the mortal limitations and tries to gain entry into the sphere of the

limitless and eternal, the necessary and ideal, well then, isn’t that the job description of

the mathematician? Only the mathematician can lay claim to intimacy with the infinite,

proving theorems about its true nature, revealing that there are infinities beyond infinity,

an ascending order of uncountable domains containing more things than could have been

dreamed of in any Horatian philosophy.

“I wish I could describe it to you, could capture it in words. But I never could, not

even if I had the verbal facility which I lack. I wish I were able to describe the

beauty and excitement when it’s working, when I’m seeing it.

“Like the past few hours?”

“Like the past few hours. It’s not always like that, of course. Sometimes I wonder

round and round in circles, going over the same ground, getting lost, sometimes

for hours, or days, or even weeks. But I know that if I immerse myself in it long

enough, things will clarify, simplify. I can count on that. When it happens, it

happens fast. Boom ba boom ba boom! One thing after the other, taking the

breath away. And then you know I feel like I’m walking out in some remote

corner of space, where no mortal’s ever been, all alone with something beautiful.”

This is a character, Noam Himmel, speaking from my first novel The Mind-Body

Problem, trying to explain to his wife of a few weeks, whom he has largely ignored on

their honeymoon, the exhilaration of his work. There is something heroic about such

ascents into infinity, ascents impossible for all but a handful of people (Noam, when he

was twelve, had discovered numbers he dubbed the “supernaturals”). Noam’s wife

listens to his passionate description of his solitary work, and feels chastened for having

resented being left so completely on her own on their ostensible honeymoon. “You’re a

god,” she gushes to him. He has the modesty of knowing the history of mathematics, and

demurs:

“Some are gods. Archimedes, Newton, Gauss, they were gods.”

“A minor deity, then?”

”No, not that either. Euclid, Descartes, Fermat, Euler, Lagrange, Riemann,

Cantor, Poincaré, Hilbert, and Gödel—they’re the minor deities.”

“A demigod, then, the offspring of a god and mortal?”

”Okay.” He laughed. “Maybe I’m a demigod in the pantheon of math.”

There is no tragic hero without intimations of divinity, and, too, there is no tragic

hero without those intimations bringing him to his ruin. Mathematicians, despite their

pilgrimages to the infinite, are finite. In fact, they are typically all too finite, often

extravagantly flawed as human beings, vulnerable not only in the ordinary ways that

plague us all, but subject to their own extraordinary afflictions and mortifications. In

almost all narrative art—including not only novels but plays and movies as well—which

feature mathematical characters, these special vulnerabilities enter prominently into the

plot, making ultimately for a tragic tale.

There are three particular vulnerabilities to which mathematicians, at least in their

fictional appearances, are portrayed as being peculiarly vulnerable, rendering them tragic

figures.

The first of these is the classic hubris of the hero. Plato placed mathematics at the

very portals of the extreme objectivity which would cure us of the “dreadful and

necessary emotions” required for survival. To emerge, by way of pure reason, into the

Out There was to be on one’s way to saving one’s self by transcending the self. But just

because you have transcendent mathematical talent doesn’t mean that you’ve signed on

for the whole Platonic project of transcendence.

Mathematicians, just like artists, just like novelists, are often astonishing in their

single-mindedly commitment to the personal I, intent not only on knowing the truth for

its own sake, but on being known as the one who first knows the truth. Someone else may

get there first, and then all—all!— will be lost. Pure mathematicians tend not to be so

pure. Plato’s project of rationality was to save us from being hostages to fate; the only

thing that we can truly control is the progress of our own understanding, but that is the

only thing that we need to control in order to live a life worth living. But a

mathematician’s personal ambition in regard to impersonal truth makes him more

hostage than most. The difference between winning and losing is not entirely within

one’s control and this can produce extraordinary agony, worthy of the narrative artist.

So for example, in Uncle Petros and Goldbach’s Conjecure, the eponymous

Uncle Petros does not mention to anyone when he has derived an important result which

he regards as valuable primarily as a stepping stone to the big one, proving Goldbach’s

Conjecture. He’s afraid that someone else will use his result and get to the proof that all

even numbers are the sum of two primes before him. So he keeps mum and is punished

on the grand tragic scale, for someone else of course proves the intermediate step, an

important result in its own right, and Uncle Petros, his obsession with glory now laced

with the bitter poison of feeling himself the victim of misfortune, descends into further

tragedy, following the path of the classical hero, made to suffer for his wanton hubris:

Reminiscing for my benefit, Uncle Petros marked this decision as a turning point

in his life. From then onwards, he said, difficulties began to pile upon difficulties.

By withholding publication of his first truly important contribution to

mathematics, he had placed himself under double time-pressure. In addition to the

constant gnawing anxiety of days and weeks and months and years passing

without his having achieved the desired final goal, he now also had to worry that

someone might arrive at his discovery independently and steal his glory.

William Boyd’s Brazzaville Beach also has a tragic mathematician, John

Clearwater, whom, as in The Mind-Body Problem, we see through the eyes of his non-

mathematical wife:

“Do you remember,” he said, “when you came to see me, after I’d moved out?”

“A Sunday?”

“Yes. And I said that the work was going well, that I was very close to a breaking

through? I had worked out this set?”

“The Clearwater set.”

His laugh was dry. “Well, I was beaten to it. Someone got there first.”

He wrote a formula in the condescension of the car window.

“Unreal,” he said. “Dazzling. But someone else thought it up.”

“You know it means nothing to me.”

“It’s so simple. And that’s why it’s beautiful. If you knew, if you just knew what

you could achieve with that. If you knew what it implied.”

She looked at the innocuous figures, bleeding now, mysterious and unknowable.

For the first time in ages she felt that old envy of him.

“You see how much better I am,” he laughed, not quite so calm.

“I can look at that,” he rubbed it out with a swipe of his fist, “and not burst into

tears.”

Brazzaville Beach’s John Clearwater, as well as Uncle Petros, bring me to the

second affliction to which the tragic mathematicians of fiction are presented as being

painfully susceptible. It is a delicate subject to touch upon: madness. Why do you pull

myself from myself, Marsyas had cried out, in the name of all heroes, to the god.

Madness is perhaps the cruelest way in which this agony can be realized.

T he current play—and now movie—“Proof,” not to speak of the academy-award-

winning movie “A Beautiful Mind” also dwelled on the madness of mathematicians. “A

Beautiful Mind, loosely based on the biography of that same title by Sylvia Nasser,

features Russell Crowe playing John Nash—of course a real mathematician, one of the

original developers of game theory, who lost decades of his life to schizophrenia. There

are others—already mentioned in the roll call of the pantheon of mathematics that my

character Noam Himmel recited—who similarly lost themselves to madness. There is

Georg Cantor, who more than any other mathematician could be described as achieving

intimacy with infinity, and who died in an insane asylum, and there is Kurt Gödel who

ended up starving himself to death under the paranoid delusion that he was being

poisoned. One of Gödel’s mathematics professors, Phillip Fürtwangler, once remarked in

regard to Gödel’s paranoia: “Is his illness a consequence of proving the non-provability

or is his illness necessary for such an occupation?”

In Uncle Petros, the narrator nephew catches a glimpse of Gödel and decides to

forswear his own mathematical ambitions:

Yet, hopeful as I still was in my ignorance about my abilities, it wasn’t

professional failure that I feared. It all started with the sorry sight of the father of

the Incompleteness Theorem padded with layers of warm clothing, of the great

Kurt Gödel as a pathetic , deranged old man sipping his hot water in total isolation

in the lounge of the Institute for Advanced Study.

A few paragraphs later he remarks:

Sammy’s theory of hubris had haunted me ever since I’d heard it, and after my

brief review of mathematical history I embraced it wholeheartedly. His words

about the dangers of coming too close to Truth in its absolute form kept echoing

in my mind. The proverbial ‘mad mathematician’ was more fact than fancy. I

came increasingly to view the great practitioners of the Queen of Sciences as

moths drawn towards an inhuman kind of light, brilliant but scorching and harsh.

There have been spectacular specimens of the mad mathematician—I mean in real

life, not only in literature—the very prominence of the individuals in question tending to

exaggerate perhaps the appearance of a strong correlation between the two mental

anomalies, mathematical talent and insanity. We don’t really know what the precise

statistics are correlating the continuous variable mathematical talent with the many forms

of madness, which must also be distinguished from one another if empirical hypotheses

are to be tested. (The British psychologist, Simon Barrett-Cohen, who studies autism,

has theorized that the mathematically inclined have an increased inclination toward the

mental problems that fall in the continuum spreading from mild Asperger Syndrome to

full-fledged autism, and has set up a longitudinal study to test whether offspring of two

mathematical parents are at an increased risk of falling within the autism spectrum.)

But no matter what the true statistical situation should prove to be correlating

mathematicians and madness, the mathematicians of narrative art tend to be seriously

unhinged. Why is it that when narrative artists create mathematicians they so often create

them going mad? True, to outsiders the mental life of the mathematician is so

inaccessible and weird, that it can suggest, in and of itself, the idea of madness . (The

inaccessibility is also what keeps many narrative artists from even thinking of creating

mathematical characters, or sometimes, when they do, causes them to get the inner life

risibly wrong.) That fixed concentration on questions of no practical applicability that

can’t be communicated to regular people can in itself seem like lunacy. (When the play

“Proof” moved from Off-Broadway to Broadway, the producers held a day of panel

discussions on matters mathematical at the Courant Institute for purposes of publicity. I

was on a panel with David Auburn, the author of “Proof,” which revolves around the

theme of mathematics and madness. Auburn, at one point in the panel discussion, burst

out, partly in jest but with some sincerity as well, with the remark that the day at Courant

had convinced him that all mathematicians are howlingly mad.)

But at a deeper level, there is something in the figure of the brilliant

mathematician who is losing his hold on reality that comes close to the heart of tragic

poetry. Here is a person with extraordinary powers—in the case of the mathematician,

mental powers—being reduced to below the level of the ordinary, powerless in the way

that before he was powerful. It is impossible not to think of Lear. The mad

mathematician, no matter what extraordinary sights he has glimpsed of the infinite real,

is more a chained prisoner than anyone, staring helpless at the shadows thrown up by his

own mind. This is the essence of tragedy. It can also be comic, and at the very same time

that it is tragic (if the work is only comedy, devoid of the tragedy, then it is merely cruel),

and such a blending is itself a complicated literary device for portraying the human

condition. Just because we’re absurd doesn’t mean we’re not also tragic.

There is a third interesting peculiarity of mathematicians that often finds its way

into plotlines, yet another susceptibility to remind us of the inescapable finitude of even

those among us who have the greatest claims to intimacy with the infinite. I am speaking

now about getting older, not pleasant for any of us, but of particular significance to the

mathematician. Mathematical talent, while often emerging prodigiously young, also gives

out heartbreakingly young. Perhaps only the aging athlete and the fading femme fatale

feel the progression of time as devastatingly as the mathematician.

G.H. Hardy’s A Mathematician’s Apology is, in addition to everything else, one

of the saddest testaments I have ever read, the very conditions of its composition

contributing to the moving quality of the book. Hardy had lost his mathematical powers,

had attempted suicide and failed, and was convinced by his friend C. P Snow, the

physicist and novelist, to write a book for non-mathematicians, describing what it is like

to be a mathematician. Hardy produced a little book that is eloquent and interesting, and

whose almost every line—even when he is speaking of proofs and of Platonism—speaks

of loss. This is its opening paragraph:

It is a melancholy experience for a professional mathematician to find himself

writing about mathematics. The function of a mathematician is to do something,

to prove new theorems, to add to mathematics, and not to talk about what he or

other mathematicians have done. Statesmen despise publicists, painters despise

art-critics, and physiologists, physicists, or mathematicians have usually similar

feelings; there is no scorn more profound, or on the whole more justifiable, than

that of the men who make for the men who explain. Exposition, criticism,

appreciation is words for second-rate minds.

A few pages later he speaks more directly about again:

I had better say something here about this question of age, since it is

particularly important for mathematicians. No mathematician should ever allow

himself to forget that mathematics, more than any other art of science, is a young

man’s game . . . Galois died at twenty-one, Abel at twenty-seven, Ramanujan at

thirty-three, Riemann at forty. There have been men who have done great work a

good deal later: Gauss’s great memoir on differential geometry was published

when he was fifty (though he had the fundamental ideas ten years before). I do not

know an instance of a major mathematical advance initiated by a man past fifty. If

a man of mature age loses interest in and abandons mathematics, the loss is not

likely to be very serious either for mathematics or for himself.

It was, by the way, Hardy’s elegiac memoir that caused me to become a novelist. I

read it while I was a graduate student of philosophy and was so moved by its underlying

pathos that, almost despite myself, it suggested the plot of a novel to me, and

mathematicians have continued to people my fiction ever since.

But let’s not end by speaking of diminution and loss but of power and grandeur.

Mathematicians qualify, at least in my book, as tragic heroes not only because they can

be tragic but also heroic, not least of all in their susceptibility to beauty. The

mathematician and the artist share a well-trained instinct for beauty. The mathematician

and the artist can both, in the service of their muses, even manage to subdue their

respective raging egos in order to submit themselves to the chastening aesthetic dictates.

So it is on this note that I will end, quoting from one of my own short stories, entitled

“Strange Attractors”:

Suddenly there’s a great stamping noise outside of her door, and Oren

Glube comes bursting in, his normally pale face flushed as if with fever.

“Come! Now! This second, Phoebe!”

Her only thought is of fire, of which she’s always been frightened: The

Institut des Hautes Etudes Scientifiques is in flames, she thinks.

She rushes to the door, and Oren grabs her hand and is pulling her down

the hall with him, stomping loudly in his big black rainboots.

Antoine’s taken Phoebe’s other hand, and they run for the main door of

the building; all of the mathematicians have come from their offices and are

running for the door.

Only the secretaries sit calmly at their desks, typing and making their

jokes about the “inmates of Bures.”

“They’re overexciting themselves again,” says Suzette. “We’ll have our hands full

trying to calm them down this evening.”

And outside the mathematicians all stand gathered together on the wet

lawn, staring up into the western sky, where there’s a rare double rainbow

stretching itself:

The colors of the primary arc are intense and pure; and beneath is the

secondary rainbow, with its paler inversion of the spectrum.

And all of the mathematicians are standing together in silence; on every

face the same look of transfixed bliss.