Math even mathematicians don’t understand
On November 4 I published an article in the Ideas section about Shinichi Mochizuki, a mathematician who claims to have proved the ABC conjecture, one of the great unsolved problems in math. The only catch is that his proposed proof is written in mathematics so complex that literally no one in the world can evaluate its accuracy. Long, unintelligible tracts are not uncommon in mathematics and normally the math community chooses simply to ignore them — but in this case Mochizuki is so highly regarded that experts around the world have decided to puzzle it out, which could take years.
A few weeks later I received an email from a friend of a 90-year-old mathematician named Henry Pogorzelski, an emeritus professor at the University of Maine. The email explained that for the last half-century, Pogorzelski has toiled at a proof of the legendary Goldbach Conjecture and after decades of effort he believes he has it, though his work runs thousands upon thousands of pages, and no mathematicians can understand it or are even willing to invest the time to try to. Pogorzelski’s friend explained that he hoped I might write a story that would stir some interest in the professor’s work.
Though Pogorzelski is 50 years older and less internationally noted than Mochizuki, their careers have some surface similarities. After promising starts -- early in his career Pogorzelski worked under the famed Andre Weil at the Institute for Advanced Study— they devoted themselves to solving big “named” problems in mathematics. (One difference is that by the time he embarked on solving ABC, Mochizuki had already solved enough hard problems to build up considerable credibility with his peers; Pogorzelski had no similar track record at the time he embarked on Goldbach.)
Following the Institute for Advanced Study, Pogorzelski took a job at the University of Maine in order to have a quiet, out-of-the-way place to focus on the Goldbach Conjecture. He had already been toiling at a proof for 25 years when in 1988 the Bangor Daily News ran a story on him.
Pogorzelski’s situation then was much as it is now: He thought he had a proof but his writing was so voluminous and strange that he couldn’t entreat anyone to take it seriously. The Daily News article emphasized the immense personal toll this monomaniacal obsession had taken on Pogorzelski, who’d neglected his wife, by then deceased, and was not in contact with his only child. “I really thought I could get it in Maine,” he told the newspaper. “I worked day and night, neglected my family, gambled everything away on the problem. It pains me. I thought my family understood that I was doing it all for them but they did not.”
Last week a box from Pogorzelski arrived at my doorstep. It weighed about twenty pounds and contained nine volumes titled, “Transtheoretic Foundations of Mathematics,” along with a note from Pogorzelski’s current wife, Maha: “Pogo says his proofs are in his books. Good luck. ENJOY for the holiday.”
It is beyond me even to attempt to read Pogorzelski’s mathematics, but the preface to his volumes is accessible. There he explains that sometime in the early 1990s he came to understand that the approach to Goldbach he’d taken for the first three decades of his work was flawed and he set out on a new course. He realized he’d never complete this new project before his death so he decided to start the Research Institute for Mathematics (Maine) in order to take on graduate students whom he could train as “disciples” to carry on his work. In prose that sounds like a dispatch from a lost civilization, he concludes:
“After a number of years of lecturing and consecutively watering down my material, dreaming of passing on my Goldbach program to candidates. I was rather shattered to find that my disciples drowned in my Goldbach waterfalls. In consequences, I collapsed a bit from exhaustion for a while. Then I regained the confidence to carry on my program alone without having to rely on disciples. Eventually, a solution to Goldbach unfolded to me. I published the result in the third volume, IC, titled Goldbach Conjecture, Series I on Natural Numbers. Five years later (2002), I improved upon my solution and sent it to the Annals of Mathematics, which they acknowledged on Mach 12, 2002. And I never heard from them again.”
And so, a decade later he reached out to me. Since receiving Pogorzelski’s work I have been in touch with a number of professional mathematicians. Some remembered hearing his name years ago, others had never heard of him, and without exception all were skeptical that he’s achieved what he thinks he’s achieved. Not having read his work, they cited several a priori reasons for doubting its accuracy, including Pogorzelski’s lack of demonstrated achievement in mathematics and foundational issues with what they take to be Pogorzelski’s outside-the-box approach to the problem.
Several of them, too, articulated the same ruthless truth about their profession: that mathematicians who make big claims are obligated both to be right and to make themselves understood. Whether Pogorzelski has succeeded on the former point we may never know; he has certainly not, though, managed to achieve the latter.