Why does Andrew Wiles, the mathematician, cry?

This post is different from the ones I typically write, focused as they are on current events in business and society. Instead, I wrote this post in response to a BBC video documentary posted on Youtube, which I found moving and perplexing. I am interested in readers' responses to these videos. 

In 1995 Andrew Wiles, a British mathematician, proved a theorem, known as Fermat’s last theorem, which had stumped mathematicians for three and a half centuries. In proving it, he opened up a new vista in the abstruse world of topology, highlighting unexpected links between different domains of mathematical inquiry. At the end of this blog post I describe the theorem, and what it posits, for those who are interested. In the body of this post,  I include two video segments that I extracted from a BBC documentary on Youtube, describing his discovery. (Also, here is the link to the full BBC video: https://www.youtube.com/watch?v=Hkz45Ivr12k)

If you click on this first video below, you will note immediately that Wiles begins to cry as he describes his moment of discovery, likening it to the sudden illumination of a dark room,  Feeling somewhat embarrassed he then turns away from the camera.

                                                                First video

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To recapitulate he says,  “At the beginning of September I was sitting at this desk, when suddenly, totally unexpectedly, I had this incredible revelation.. It was the most important moment in my working life…(he is tearing)..  nothing I ever do again.. (he is overcome by feeling)..  I’m sorry.. (he turns away from the camera). 

My question is, why does he cry? 

I have asked many of my colleagues this question. Here are four possibilities.

Explanation 1: Wiles is paradoxically moved by his sense of loss just as he recounts his glorious achievement.  As he begins to say, “nothing I ever do again..,’’ the listener can fill in the missing words, “will ever match my experience of that moment.” It is as if as all of his subsequent work will pale in comparison to his moment of illumination. How will he be able to take this prospective work seriously?

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Explanation 2: It is a common experience that people cry at the happy ending of an otherwise sad movie. Why should this be? One conception is that while watching the movie, people steel themselves against feeling overwhelmed by the movie’s sadness.  But upon experiencing the happy ending, they can now acknowledge their sadness. In effect they now feel that it is safe to cry.

A great undertaking, shot through with frustration, creates this same dynamic. One has to bear up under the tension and frustration, lest these feelings overwhelm one’s ability to do good work. Only upon reaching the goal, is it safe to acknowledge the overwhelming impact the frustration has had on one’s emotional life.  Indeed, Wiles had circulated an earlier version of the proof a year earlier, which proved faulty. After another year of encountering dead ends, he was prepared to relinquish his entire effort. This suggests that his illumination came at a moment of despair.

Now in the video, Wiles notes that the BBC reporter is interviewing him as he is sitting at the same desk where he made his discovery and experienced his revelation. Perhaps the stimulus of the desk, as a cue, triggers his reliving the tension and release associated with seven years of strenuous effort and frustration that led up to his moment of illumination.

Explanation 3: Wiles had a kind of religious experience through which he came into touch with the profound and unexpected unity of mathematics. He was both the agent of, and witness to, this profound unity.  Indeed, in the documentary Wiles describes the experience of a paradox. At the moment he was ready to finally give up, he realized that the reason his effort, based on work by the mathematician Flach, had failed, provided just the explanation for why an even an earlier attempt based on a theory by Iwasawa was at first unsuccessful. But paradoxically this revelation showed him how Iwasawa’s theory could now provide a way forward! I think that paradoxes intensify the experience of having uncovered hidden connections. When a contradiction leads to a truth, two distinct arguments, are now seen as two sides of the same coin. 

But why should someone cry at such a revelation? Drawing on the second explanation, perhaps we cry at the revelation of an unexpected unity, because the discovery makes it safe for us to acknowledge the day-to-day alienation we all feel as finite beings in a completely mysterious universe. After all, few of us know why the universe exists and why any one of us was born. Normally, we keep this feeling under wraps because it is too difficult to bear. But when we experience unexpectedly our link to this universe by witnessing its coherence, when in effect it reveals its secrets to us, it is safe for us to acknowledge the repressed pain of our existential separation. Hence we cry.

This explanation has the merit of connecting us to one of the greatest puzzles of our culture. How is that mathematics, a human invention, should give us a language for describing the universe, an object that stands outside of culture and is of course indifferent to human striving. How strange that a product of our mind is connected in this way to the most basic elements of matter! It is no accident in this sense that Pythagoras, the founder of a school of mathematics in ancient Greece ,was also a mystic.

Explanation 4: Recall that Freud, argued that religious experiences, stimulated by what a colleague of his called the “oceanic feeling,” move us emotionally because they help us relive the infant’s sense of unity when at the mother’s breast. Freud’s explanation has that distinctive character that makes many of his ideas appear ironic. In other words, he is saying that profound religious experiences, which at first glance appear to express a great cultural achievement, are “nothing but” the feelings of an infant whose psychological life is after all primitive.

Perhaps Wiles too experienced this oceanic feeling, but because his was not directly a religious experience, based for example on a belief in a God or a spirit to whom he could then be permanently connected, he recognized that his experience was as temporary and fleeting as it was profound. This sense of both having and then losing moved him to tears. He did not simply lose the prospect of doing work in the future that could engage him, as our first explanation suggests, Instead he lost his connection to the universe. This experience may be similar to losing a great love, though in Wiles case, the loss was inevitable and meaningful, while in the case of a lost loved one, the loss may seem arbitrary and cruel. 

Freud’s conception drew me to another segment of the BBC documentary (click on the video segment below)
                                                                   Video two

As Wiles notes, as a ten-year-old boy he was fascinated by Fermat’s last theorem. He says in the video,  “there is no other problem that will mean the same to me. I had this very rare privilege of being able to pursue.. in my adult life what had been my childhood dream. I.. know that it’s a rare privilege, but if one can do this, it is more rewarding than anything I can imagine.”

In a way, consistent with Freud’s conception, Wiles, invokes an early experience, though in this case, of childhood, not infancy. But why does Wiles insist that it is such a rare privilege to solve a problem first encountered when one was a child?  One hypothesis is that Wiles is describing the experience of a certain kind of childhood innocence. What does that innocence consist of? A ten-year-old boy experiences human culture, as itself all encompassing and self justifying. Not yet touched by fears of death, the despair that accompanies the belief that the universe is indifferent to our happiness, or that human striving ends in tragedy, the child anticipates gaining freedom and power by joining in the work of creating culture. In this sense his innocence protects him from feelings of alienation. Indeed, this is the source of his innocence.

In this sense, Wiles is expressing the positive side of what Freud called the Oedipus complex. This is the moment when the child identifies with his elders who promise to turn over to him their work of creating culture, if he is willing to bear the discipline of educating and preparing himself.  The child relinquishes the infant’s bliss at the mother’s breast for the prospect of gaining freedom and power by joining father’s world. This prospect for a ten year may very well be enthralling. After all, children express this when they excitedly talk about what they will be “when they grow up.” Wiles is saying that in solving Fermat’s last theorem he recaptured the innocence of the oedipal child, the belief that father’s world is its own justification and offers unlimited opportunities. 

This suggests in addition, that one reason a person like Wiles is able to soldier on for seven years to achieve something glorious, despite the psychic pain he experiences, is that he retains in his memory the fantasy of a  child's delight in joining father's world as a free and powerful person. In effect, the two video segments are joined, and not simply by my selection of them. Rather,  the second gives an account of how Wiles was able to tolerate the pain of working despite the frustration he faced, the first what happened when at last the frustration was lifted.

If this explanation has merit it also poses some questions about Freud’s conception of religious experience. Freud argued that religious experience reproduced what is now called the “pre-oedipal” child’s experience, when mother was everything. But if a religious experience opens up a path for useful work, for engaging in the work of building culture, rather than just passively accepting God’s love, perhaps it too provokes the child’s excitement of joining the father’s world, where at least in the western tradition, God is also a father. My brother Norbert told me that the composer Handel wrote that he “saw heaven” upon writing the Messiah in just 24 days. It is likely that Wiles and Handel had the same emotional experience.

This line of argument also leads us to the questions that my colleague Howard Schwartz has raised about post-modernism. What happens if and when we come to believe that human culture is intrinsically toxic or destructive, that the Western tradition for example, which gave rise to the glory of modern mathematics, is at its core, flawed.  Does this suggest that our children will no longer have the privilege of experiencing innocence, and that we will lose interest in preparing them to take on the work of building civilization? Howard speaks of an “anti-oedipal” culture, in which the father is seen as an illegitimate authority figure, blocking our return, as pre-oedipal infants, to our mother. His argument suggests that we are at risk of creating an infantilized culture, or what others have called a culture of narcissism. What is the evidence for this?

I am interested in people’s thoughts about all the questions raised in this post.

Fermat's last theorem 

In the seventeenth century Fermat, suggested that the formula known as the Pythagorean theorem and shown as  

A²+B² = C² , or when sounded out., “A squared plus B squared equals C squared,”

is true only when you square the numbers, or raise them to the power of two. In this case you are stating the schoolboy’s formula that in a right triangle,  the sum of the squares of the length of each side is equal to the square of the length of the hypotenuse.

A  triangle that fits this formula is when one side is 3, one is 4 and the hypotenuse is 5 

We have in this case

 3²+4² = 5² or  9+16=25

 

Fermat said that if you try to find three numbers that fit the formula

A³+B³ = C^{3 ,}

In other words, the numbers are cubed, you will never find them. The same goes for raising numbers to the powers of 4 and so on. Fermat wrote in his notebook that he had found a wonderful proof of this theorem but alas, the margins of his notebook were too narrow to permit him to write it down! We now know that he was mistaken. It took late 20^th century mathematics to solve a seventeenth century theorem

The Newtown shooting and the NRA

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This blog post does not take a position on whether the National Rifle Association (NRA) or gun control advocates are right or wrong in their responses to the shootings in Newtown. 

(Note to my non-U.S. friends. On December 20, a young man, Adam Lanza, entered a school building in suburban Connecticut and killed twenty school children, kindergartners among them, and six adults, including the principal and the school psychologist.  He used a semi-automatic rifle and first killed his mother at home before entering the school building.  The shooting has revived a debate in the United States about whether or not the federal government should control gun sales). 

Instead, I suggest that there is always value in stepping away from such debates and asking how a particular debate resonates with wider trends and issues. For those who favor more gun control, as I do, it is nonetheless important to understand what the NRA might represent, both for us and against us, as participants in the wider cultural questions that affect us all.  This approach is linked to the hypothesis that, in social systems, we polarize our differences into camps of opinion when we cannot reconcile conflicting goals we face together. 

I was drawn in this regard to the press conference the NRA gave fully a week after the shooting. Its leaders, particular its executive vice president, Wayne LaPierre, came to the press conference, “loaded for bear,” as the hunter’s expression goes. He attacked the media for its anticipated critique of the NRA. He proposed that the best way to protect school children, was not to control the sale of guns and the licensing of gun holders, but instead, to place armed guards in every school building. There was a brutal logic to his argument. 

“How have our nation's priorities gotten so far out of order, he asked, “think about it. We care about our money, so we protect our banks with armed guards. American airports, office buildings, power plants, courthouses -- even sports stadiums -- are all protected by armed security. We care about the President, so we protect him with armed Secret Service agents. Members of Congress work in offices surrounded by armed Capitol Police officers. Yet when it comes to the most beloved, innocent and vulnerable members of the American family -- our children -- we as a society leave them utterly defenseless, and the monsters and predators of this world know it and exploit it. That must change now!”

I can readily imagine how some parents, under the press of fear and anxiety, might feel reassured if an armed officer or civilian volunteer was deployed to protect their children while they were in school. In this sense, the NRA’s position is not as outrageous as some of its critics have suggested.

However, what I found telling was the NRA’s failure to consider what it means to provide children with a psychological sense of safety. Armed guns will protect their physical security, but they threaten to bring into the child’s view and experience the belief and fear that the world is dangerous, filled with bad people who can harm them. Why else have armed guards in the school building in the first place? In other words, what the NRA did not recognize is our shared hope that we can protect a child’s innocence; that is, an experience of the world as a loving place, where strangers are friendly, authority is dependable, optimism is realistic, curiosity is rewarded, and of course armed guards are unnecessary.

But having noted this lapse, I can also ask, whether or not we adults, as bearers of our culture, any longer believe that children are entitled to their innocence. There is a sizeable literature, think of Neal Postman’s prescient, “The Disappearance of Childhood,” published in 1982, which argues that we have created a social world that undermines the experience of childhood innocence. Children are told to be wary of adults who touch them, lest they be abusers, and they are exposed to sexual stimuli as well as fantasies of violence throughout their childhood. In our anxiety for their future we overschedule their lives with programmed activities, and test preparation sessions, so that they experience little spontaneous play. Many children are also exposed to our psychological conflicts when, as parents, we divorce one another and then fight over their custody. The television series, “The Wire,” a story of Baltimore as a decaying postindustrial city, and widely regarded as realistic, highlights how some African-American children are far too soon introduced to the world of adult violence, ineptitude and corruption.

The list could go on, but do we care?  In others word are we committed to sustaining the experience of childhood innocence? One reason we might not be is because as adults we no longer feel psychologically, financially or physically safe ourselves. As a result we are pulled toward protecting ourselves rather than our children.  The psychologist David Bakan, argues in The Slaughter of Innocents,” that throughout history adults abandoned infants when faced with food shortages. Children in other words, potentially distract us from our own struggle for survival.  While in the developed world we have ample food, the anxieties associated with securing an adult role that confers dignity, purpose and a livelihood, have grown substantially. This may be one reason why birth rates are falling throughout much of the West. Perhaps the NRA's conviction that monsters threaten us and our children, represents, in an exaggerated form, a widely held belief that the world has become more dangerous for adults.  

If this is true, one question then is how one makes sense of danger? I want to suggest that the NRA’s philosophy, or perhaps theology, is based on the idea of an evil presence in the world, which, if and when acknowledged, makes innocence seem delusional. Harlon Carter, who helped “overthrow” the NRA’s “old guard” in 1977,  attacked it using the discourse of good and evil. “The latest news release from the NRA,” he said, "embraces a disastrous concept, that evil is imputed to the sale and delivery, the possession of a certain kind of firearm, entirely apart from the good or evil intent of the man who uses it.” This discourse of evil is also why LaPierre referred in the press conference to “predators and monsters,” rather than, for example, to mentally disturbed people. In this sense, the discourse of evil stands in contrast to the discourse of disease and health. The idea that Adam Lanza, was mentally disturbed, in other words, he had a diseased mind, can be usefully challenged by the idea that he, or much more likely, his mother and first victim, was evil.

Does the idea of evil have standing? Scott Peck, the psychiatrist and religious thinker, wrote a widely read book, “The People of the Lie,” based on the idea that evil people are hidden, their impacts insidious, and that through their failure to tolerate imperfection in themselves and others, they often drive others, particularly children, into acts of desperation. The book, published in 1978, in this sense might be read as a précis of the Newtown shooting.

The idea of evil is not compatible with the belief, that if social conditions are right, fair and just, people can be perfected. This belief has been the basis for much modern social policy as well as the foundation for tolerance and for a commitment to pluralism. My own sense however is that idea of evil remains unsettled within us, we are so to speak “bedeviled” by it, as we contemplate not simply the Holocaust, but more recently, the slaughters in Rwanda, in Bosnia, and the attack on women in parts of the Muslim world. Certainly, the psychoanalytic conception of character holds that loving and destructive feelings are comingled in our most intimate relationships. This is one basis for domestic violence. In other words, one hypothesis is that we are dishonest when we project our own struggle with the idea of evil onto people and groups, like the NRA, whose view of it we then label as extreme.

The belief that evil is real and insidious can certainly give rise to the conviction that we must defend ourselves against evil at all costs. But one question is why we can’t rely on the state. the government, to protect us against evil. Certainly, in the NRA’s worldview, the state cannot be counted on, and in fact may become the enemy. This is why its leaders put such great store on one interpretation of the U.S. constitution’s second amendment; namely that it protects the right of people, “to keep and bear arms.”

It is tempting to dismiss this as paranoid thinking, but surely one trend in the wider world is the apparent decline of states and the rise of non-state actors, such as terrorist organizations and criminal networks fully capable of attacking and defeating police forces and armies. In the U.S. we need only look southward to our neighbor, Mexico, to envision a scenario of how criminal gangs, wealthy and armed, might defeat the state. Moreover the conspiratorial idea that the United Nations is the first step toward global domination by hidden powers, an idea that attracts some NRA members, bears a family resemblance to the idea that the global corporation, which under certain conditions can equip private armies, is growing more powerful than the states that regulate them. Truth be told, in the United States, outlaws have often successfully challenged, defeated and corrupted the state. Think of outlaws on the western frontier, the Mafia in Chicago, whose leaders undermined judges and policemen with bribes and threats, or local urban police who, in decaying industrial cities, abandoned high crime areas to criminals.

I am reminded here – of all things! – of J.D. Salinger’s classic novel, “The Catcher in the Rye.”  It shows how young people lose their innocence, and potentially their sanity, when they come to grips with adult hypocrisy, or what Holden Caulfield, its hero, called its “phoniness.” I think the book had such enormous resonance for two reasons,  First,  it raised the question of whether we could sustain the innocence of childhood long enough in the life of each child so that children could become optimistic adults, even as they had to confront, in their adolescence, the dirty secrets of the adult world. Second, it had resonance because, published in 1949, it was anticipatory, foreshadowing young people's loss of trust in the adult world some fifteen years later.  Some thirty years after its publication, Neil Postman argued in his own book that the idea of childhood innocence in the United States, had a hundred year run, from 1850 to 1950, suggesting, once again, that an artist’s antenna, in this case Salinger's, picks up signals from the future.

“The Catcher in the Rye,” is both the book’s title and Holden Caulfield’s fantasy of his life’s work. As he tells his younger sister Phoebe, “Anyway, I keep picturing all these little kids playing some game in this big field of rye and all. Thousands of little kids, and nobody’s around- nobody big, I mean – except me. And I’m standing on the edge of some crazy cliff. What I have to do, I have to catch everybody if they start to go over the cliff- I mean they’re running and they don’t look where they’re going I have to come out from somewhere and catch them. That’s all I do all day. I’d just be the catcher in the rye and all. I know it’s crazy, but that’s the only thing I’d really like to be.”

Perhaps the Newtown shooting and the NRA’s response both raise questions we all face. Who, if anyone, will do the work of “child-catching” in a post-industrial world, how should they do it and how do we help them?