& 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, 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