**Uncle Petros
& Goldbach's Conjecture:A Novel of Mathematical
Obsession**

(This column was first published in the March 29, 2001 ArtVoice of Buffalo.)

Mathematicians are very different from
the rest of us and even from other members of the scientific research
community. I state that from personal experience, having dealt with many of
them over a lifetime of work in an activity parallel to but never intersecting
theirs. Theirs - and I speak here of world class mathematicians - is an
activity so different from that of the rest of us that it is extremely
difficult to gain insights into it.

Greek author Apostolos Doxiadis has, in **Uncle Petros & Goldbach's Conjecture: A Novel of
Mathematical Obsession** (Bloomsbury, 2000), achieved the near
impossible. He gives the intelligent outsider a view of what it is like to be
on the cutting edge of mathematical research. That he does so in a charming
family story, a pleasant tale that holds together the mathematical insights is
a further achievement of high order.

Here, for example, is how the story
begins:

"Every family has its black sheep -- in ours it was Uncle Petros.

"My
father and Uncle Anargyros, his two younger brothers, made sure that my cousins
and I should inherit their opinion of him unchallenged.

"'That no-good brother of mine,
Petros, is one of life's failures,' my father would say at every
opportunity.'"

Indeed, Uncle Petros is an odd duck,
supported by but constantly complained about by his brothers. But Petros is
also a mathematician who enjoys common ground with the world-renowned 20th
century mathematicians G. H. Hardy, J. E. Littlewood, Srinivasa Ramanujan, Alan
Turing, and Kurt Godel - all of whom have roles to play in this story.

The narrator of this story - as the
title suggests, Petros' nephew - is a student who is considering a career in
mathematics. Quite naturally then, despite the criticisms of his father and his
other uncle, he turns to Uncle Petros for advice and encouragement. Petros
helps at first and the boy does well in school math but then, when his nephew
announces his ambition to enter university and study to become a mathematician,
he backs off. He challenges the youngster to solve a problem and exacts a
promise from him not to continue his focus on math if he fails. The nephew
agrees and Petros assigns him his challenge:

"'Here's the problem...I assume you already know what a prime
number is?'

"'Of course I know, Uncle. A prime is an integer greater than 1
that has no divisors other than itself and unity. For example 2, 3, 5, 7, 11,
13, and so on.'...

"First he wrote it out on a piece of paper and then he read it to
me.

"'I want you to try to demonstrate,' he said, 'that every even
number greater than 2 is the sum of two primes.'

"I considered it for a moment, fervently praying for a flash of
inspiration that would blow him away with an instant solution. As it wasn't
forthcoming, however, I just said, 'That's all?'

"Uncle Petros wagged his finger in warning: 'Ah, it's not that
simple! For every particular case you can consider, 4=2+2, 6=3+3, 8=3+5,
10=3+7, 12=7+5, 14=7+7, etc., it's obvious, although the bigger the numbers get
the more extensive the calculating. However, since there is an infinity of
evens, a case-by-case approach is not possible. You have to find a general
demonstration and this, I suspect, you may find more difficult than you
think.'"

Unknown to the student, his uncle has
assigned him the very problem on which Petros has spent a lifetime of
frustrating research. You have just read one of the easiest-to-state and
hardest-to-prove theorems in all of mathematics. This so-called Goldbach's
Conjecture (slightly restated from its original form) remains unsolved even
today after so many of the famous problems - the Five Color Map problem and
Fermat's Last Theorem, for example - have fallen at the hands of contemporary
mathematicians. Petros obviously wants to head off his nephew from following
his own tragic path.

At the end of this book the publisher
offers a million dollar reward to anyone who solves this problem and has the result
published before March 15, 2002. The details of this offer are spelled out for
any of you who are intent on setting to work.

I offer a personal experience as an
aside - but not to be taken to suggest that the publisher's challenge is
unreasonable. Early in the 20th century it was PROVED that an angle CANNOT be
trisected by geometric construction with the so-called Euclidean tools - the
straightedge and compasses that are familiar to high school geometry students.
Misunderstanding what that proof meant, people set to work trying to carry out
that construction. Because I edited a math journal for a time, I used to
receive these constructions. Most were simple and easy to refute, but one -
published by a Buffalo priest - is a full volume containing all kinds of
intricate diagrams. That is where I drew the line - joining mathematicians to
whom such nonsense is often submitted - and refused to spend hours locating the
necessary error. I am certain that that author remains convinced that he has
quite literally accomplished the impossible - something that can only be done
in the religious world.

Doxiadis, a mathematician himself (as
well as a businessman and film director) knows what the life of a committed
mathematical researcher is:

"The loneliness of the researcher doing original mathematics is
unlike any other. In a very real sense of the word, he lives in a universe that
is totally inaccessible, both to the greater public and to his immediate
environment. Even those closest to him cannot partake of his joys and his
sorrows in any significant way, since it is all but impossible for them to
understand their content.

"The
only community to which the creative mathematician can truly belong is that of
his peers; but from that Petros had willfully cut himself off. During his first
years at Munich he had submitted occasionally to the traditional academic
hospitality towards newcomers. When he accepted an invitation, however, it was
sheer agony to act with a semblance of normality, behave agreeably and make small
talk. He had constantly to curb his tendency to lose himself in
number-theoretical thoughts, and fight his frequent impulses to make a mad dash
for home and his desk, in the grip of a hunch that required immediate
attention. Fortunately, either as a result of his increasingly frequent
refusals or his obvious discomfort and awkwardness on those occasions when he
did attend social functions, invitations gradually grew fewer and fewer and in
the end, to his great relief, ceased altogether.

"I
don't need to add that he never married.... In truth, he was very much aware
that his lifestyle did not allow for the presence of another person. His
preoccupation with his research was ceaseless. Goldbach's Conjecture demanded
him whole: his body, his soul and all of his time."

But despite his failure to reach his goal, Petros remains convinced
that, with appropriate effort, he can reach it.

"Despite the fact that he hadn't yet managed to attain his goal
and prove Goldbach's Conjecture, Uncle Petros firmly believed that his goal was
attainable. Being himself Euclid's spiritual great-grandson, his trust in this
was complete. Since the Conjecture was almost certainly valid (nobody with the
exception of Ramanujan and his vague 'hunch' had ever seriously doubted this),
the proof of it existed somewhere, in some form.

"He continued with an example:

"'Suppose a friend states that he has mislaid a key somewhere in
his house and asks you to help him find it. If you believe his memory to be
faultless and you have absolute trust in his integrity, what does it mean?'

"'It means that he has indeed mislaid the key somewhere in his
house.'

"'And if he further ascertains that no one else entered the house
since?'

"'Then we can assume that it was not taken out of the house.'

*"'Ergo?'*

* *

*"'Ergo, *the key is
still there, and if we search long enough -- the house being finite -- sooner
or later we will find it.'

"My uncle applauded. 'Excellent! It is precisely this certainty
that fuelled my optimism anew. After I had recovered from my first
disappointment I got up one fine morning and said to myself: 'What the hell --
that proof is still out there, somewhere!'"

But Uncle Petros is unprepared for Godel's proof in his so-called
Incompleteness Theorem that there are mathematical propositions than can
neither be proved nor disproved and Turing's subsequent proof that you can
never determine which statements fall into this "forever undecided"
category. Returning to Petros' earlier analogy:

"The hypothetical friend who had enlisted his help in seeking a
key mislaid in his house might (or again might not, *but there was no way to
know which) *be suffering from
amnesia. It was possible that the 'lost key' had never existed in the first
place!"

This is what finally drives Petros over the edge. But it is not the end
of the story.

Despite
the trials of Uncle Petros, this is a tale told with affection and good humor.
In a cover blurb Olive Sacks accurately calls it "a very funny, tender,
charming, and irresistible novel." I recommend it to anyone who did well
in school mathematics and would like to know more about this challenging
subject. No further background is necessary.-- Gerry Rising