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.
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
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
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.
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.
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.
drew me to another segment of the BBC documentary (click on the video segment below)
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.
that fits this formula is when one side is 3, one is 4 and the hypotenuse is 5
We have in this case
= 52 or 9+16=25
Fermat said that if you try to find three numbers that fit
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