Wednesday, March 6, 2013

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




-->
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?

-->
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.

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 = C3 ,

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 20th century mathematics to solve a seventeenth century theorem




9 comments:

  1. When I read the first proofs of a poetry collection I had spent five years working on ('To Sing Away the Darkest Days. Poems Re-imagined from Yiddish Folksongs') I also began to cry. I think all the explanations above have validity. The contact with parents, the immersion into Yiddishkeit, the release in completion of a determined work.

    Norbert

    ReplyDelete
  2. "We now know that he was mistaken."

    Is this a slip?

    If so what does it say?

    ReplyDelete
  3. Fermat was mistaken in the belief that he had proved the theorem. His conjecture/guess of course turned out to be correct

    ReplyDelete
  4. Larry’s question piqued my interest and I took the opportunity to view the full video to confirm some thoughts … with a surprise for me.

    To the question “Why does the mathematician… cry? I am in agreement with ……

    “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.”

    A brief observation: because of the enormity of the accomplishment for Andrew as well as mathematicians from the entire globe and the investment of, for all intents and purposes, a person’s entire life to that point his emotions seemed quite subdued. The moment he moved to tears was when he was recounting where he had resigned himself to ‘putting aside’ his work because of the flaw in the 1993 version of the formulation of his proof. Coming this close to not being successful in his 30+ year task which formed the core of his life, I believe, was an additional reason for his becoming overwhelmed. I believe his awe at the ‘beauty and elegance’ of the final, corrected solution was also a major contributor to his emotions.


    A bit longer observation.

    Dimensions of Andrew’s approach, I believe, made the final success possible:

    • Being a massively successful connector of the dots – throughout his life, Andrew drew on seemingly disparate and ‘audacious’ (to his colleges) areas of mathematics – there simply were not any rational, logical connections between much of what he brought into his ‘working set’ – Andrew’s effort was the convergence of over 300 years of mathematician’s work from around the globe. His test for what should be material to be considered was "broad, mathematical concepts ...." he intentionally kept his mind open and receptive ... if I might, he had a keenly developed "evenly hovering attention".

    • Used the only computing ‘environment’ capable of handling the task – the conscious but more importantly the unconscious capabilities of the human mind.
    - A key (for me) is that Andrew never used computers, he 'doodles and made handwritten notes’. ALL the material, that he was gathering for 30 years was in his head.
    - For whatever reason, Andrew used his unconscious very effectively as part of his work regimen. In the full video he describes his walks by the lake as an opportunity for the 'subconscious to work on it'.
    - I also believe his young children's presence in his life was key. As he shared w/ his spouse – “I only have time for my family and my ‘problem’. Devoting time to his small children provided a clean conscious break on a frequent, regular basis that allowed the work to continue with his unconscious and avoid the trap of more and more ‘heads down’ conscious focus.
    - Andrew makes reference to multiple ‘aha’ moments (when the unconscious speaks forcefully to us). During his 7-years of conscious focus when he would arrive at an unknown he says "sometimes you have to figure something out entirely new ... it's a mystery where it comes from". The second video snippet where he becomes emotional is after the 6/'93 conference and several months of trying to resolve the flaw in that formulation. At that moment, he had "the most important moment of my working life, what appeared to me suddenly and unexpectedly was this incredible revelation …. so beautiful, so elegant .... I stared in disbelief for 20 minutes”.

    I believe that thoughts of the theorem’s proof were just below the surface for decades and he actually worked on the problem since he was 10 until he was 42 …… over 3 decades of intensely personal and highly meaningful work for him. People have much stronger emotions after days worth of effort, much less decades.

    My surprise was the degree that, I believe, Andrew’s unconscious played in the work.

    I would speculate that he is working on another 'Big Problem' .... in even more rigorous secrecy.

    Bob.

    ReplyDelete
    Replies
    1. Thanks for these thoughts Bob. I love your idea that his connection to his children was a conduit for his problem solving. Perhaps not simply as a diversion or break, but because it helped him stay in contact with this own childhood excitement upon encountering the theorem for the first time.

      Delete
  5. When the first edition of my book "Adult Life" was published and I attended a reception my editor threw for me at a conference, I cried when I saw it. I hadn't experienced the awe of the universe. My parents didn't have high school educations, so it wasn't about joining their world. The whole Oedipal explanation doesn't work so well for women. I had worked on this first draft for about three years, but I had been told when I first showed some sample chapters to an editor that I couldn't write and I shouldn't show the manuscript to anyone else. Ever. Now, it turned out that he had someone else under contract to write a book on adult development, but I didn't know that at the time. I also know that the tears were a combination of relief (Wow! I actually did something that I didn't think I could do) and joy (it's so beautiful and it's really good). I'm not too sure why this seems mystifying. Recalling an experience of triumph over great odds and producing a thing of beauty seem like good reasons to cry to me. But, then, I'm a crier.

    ReplyDelete
  6. Judy the question you raise, as others did, is whether or not I am making much ado about nothing. I do note in the post that crying after a great accomplishment is not uncommon. I favor the idea that Wiles had an uncommon experience, which is seeing a level of beauty and coherence in the universe which few of us have access to; that is why I likened it to a religious experience. So I guess I am privileging his tears over our own.

    I raised the issue of oedipal dynamics because Wiles himself notes that he had the special experience of solving a problem he first encountered when he was a child. The question is why should such a problem, as opposed to those we might encounter in adulthood, have a special character. If you can't take in the oedipal language, could you take in the idea that children experience a certain excitement in imagining themselves as adults which is s reflection in part of their innocence about what the adult world actually entails. Most of us dismiss these early fantasies (e.g "I want to be a policeman/fireman,") but Wiles held on to his ("i want to solve this theorem.") So he recovered his childhood excitement but through the route of adult work - seven years of postponed gratification.

    ReplyDelete
  7. ADRIAN FRONDA HAS ASKED ME TO POST THIS COMMENT ON HIS BEHALF (LH)

    In the past, more than one famous French mathematician has given lectures to the public at large. The assumption of these lectures was that the public is naive and not-knowing, but willing to learn. What I mean below by public is an audience who does not want to learn; it rather wants to get enough "facts" to become opinionated. Similarly, a barbarian is not merely a person lacking knowledge but a person lacking the will to learn.
    I presume Andrew Wiles looks away and dismisses with his hand the presence of the barbarian journalist with camera on behalf of a barbarian public out of pudicity. All mathematicians participating to the BBC programme were aware that they can create at most an illusion of a crude and brash understanding in barbarians, with a clearly occult tinge as can be seen in the comments following the BBChorizon video. The BBChorizon producers misspelt the word "theorem" even in the title. I assume that speaking with a knowledgeable individual Andrew Wiles would not have turned his face away.

    Publicity is perceived as rape by mathematicians but they are too civilised to defend themselves, ISPSO folk is very happy to become more public but this is not the case with mathematicians. Besides, ISPSO, like psychoanalysis, belongs to a specific culture, whereas mathematics is the same across cultures, what is important and beautiful in mathematics, the thrust of mathematics, is always the same, in all cultures. To my knowledge, in no culture have mathematicians ever sought publicity. They are happy to be among themselves and do mathematics. On the other hand, they are happy to share their knowledge with whomever wants to learn.

    ReplyDelete
  8. Why are you hung up over his emotional response, as though to suggest having a response is an oddity, or a sign of weakness, or a character flaw? Maybe it is your flaw.

    ReplyDelete