I read in today’s New York Times this obituary: Goro Shimura, 89, Mathematician With Broad Impact, Is Dead. The man is justifiably famous. As the article notes, Shimura was “a mathematician whose insights provided the foundation for the proof of Fermat’s Last Theorem and led to tools widely used in modern cryptography.” Upon reading this, I was immediately plunged into a sea of memories, for I once had a friend who spent hours daily reflecting on that one-time diamond of mathematical pursuit, Fermat’s Last Theorem.
It was once again 1962. The year before, I had left a well-paying job as a research associate in high strength steel at the Republic Steel Research Center near Cleveland, Ohio in order to become a poor graduate student in mathematics at Ohio State University. I was still pursuing what my path taken would be. I thought that path might be a math researcher and professor.
I settled into a rented room in an old three-story Victorian mansion at 1368 Neil Avenue and began a series of courses in the real numbers, integral calculus, and topology. I was soon to discover that I was in way over my head, but I was fortunate to form a friendship with Tom Brieter, who became my regular study partner, along with another graduate student, Don Boner. Both of these good fellows went on to have careers teaching university level mathematics, while I found my way eventually to a different academic profession, philosophy of education.
Back in those ancient times most college students, even graduate students, didn’t think in terms of loans to finance their education. The smarter ones got teaching stipends. Others, like myself, lived very frugally, perhaps on savings or low-paying part-time jobs. There was a cafeteria on campus called the Pomerene Refectory that was quite popular as an eating, meeting and greeting place. They served two meals a day, lunch and dinner, and at both times there was a special offering that could be purchased for 55¢. That is what I lived on for the better part of two years.
I met many interesting and intriguing people at my low-cost food venue at Ohio State, but I believe the most genuinely eccentric person (also interesting and intriguing) that I met was one Herb Dowdy. He was given to almost stereotypical mathematical cogitation—hand-on-forehead, sometimes standing Plato-like, oblivious to surroundings, wandering around in halls and walking into walls, scribbling occasionally in notebooks or on a found scrap of paper or napkin. The object of his devout contemplation, as he would freely volunteer to curious bystanders, was a proof of Fermat’s Last Theorem. [Ed. note. That name is French and pronounced “fir-mah.”]
Now Fermat was a 17th Century mathematician of significant achievement in the developing fields of analytic geometry, number theory, and integral calculus. Suffice it to say that Descartes, Newton and Leibniz, later reknowned for their own amazing mathematical achievements, all profited from the work of Fermat. Today, he is most often remembered for his conjecture that
If you have read this far, but find this statement off-putting, don’t quit yet, for I shall endeavor to make Fermat’s Conjecture more meaningful.
But first, though I have completely lost touch with Herb Dowdy, I remember him vividly. I want to sketch a few more details on the canvas of his portrait. Herb was taller than six feet, fairly skinny, with just a hint of a beginning “pot,” but no one would ever guess by looking at him that he was an incredibly voracious and voluminous eater. When we went through the line at Pomerene Refectory, we used amply sized trays. Mine always had a beverage and a couple of dishes: usually a small casserole and a side dish of a vegetable. What Herb did was to eat between two and three times a normal-sized tray. His tray would be literally crowded to overflowing with casseroes, dishes, plates, and beverages, and usually at least one of these containers was piled on top of another one or was balanced between two other containers or was teetering over the edge of the tray. Still, somehow, Herb always managed to pay for his food and to get it to table, where he consumed it all with a vengeance.
Returning to Fermat’s Last Theorem, lots of us took Euclidean Geometry in high school. In my high school, the 11 grade geometry course was dreaded by many students who were in the college preparatory strain, but not particularly gifted in math. I did not have that reservation, though the course was sometimes a challenge for me.
One of the crowning glories of Euclidean Geometry is the Pythagorean Theorem, a statement of a mathematical relationship that was known to the ancient Greeks. They figured it out with pictures like the one to the upper left: the sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c). In our current mathematical notation this amounts to
where c represents the length of the hypotenuse and a and b the lengths of the right triangle’s other two sides.
There are actually an infinite number of sets of integers that satisfy this Pythagorean relationship. For example, a = 3, b = 4, and c = 5 works. 3 squared is 9, 4 squared is 16, 9 + 16 = 25, and 25 is 5 squared. Another trio that works which you can verify is a = 5, b = 12, and c = 13. However, what Fermat conjectured, and claimed to have discovered a proof for, was that there was NO trio of integers that would satisfy an equation of the form
or for that matter for any equation of the form, where n is any power greater than 2.
That’s quite a large claim. According to Wikipedia, “around 1637, Fermat wrote his Last Theorem, along with his famous apology, in the margin of his copy of the Arithmetica next to Diophantus’s sum-of-squares problem:
It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.”
Alas, Fermat never wrote down his demonstration, and centuries of tantalized mathematicians were motivated to seek the missing proof. Eventually, they did prove him right.
Herb Dowdy had another eccentric but memorable quality. He seemed to have no warning that he was going to have to relieve himself of all that food which he had consumed. On more than one occasion, he would interrupt a meal, conversation or deep mathematical mediation by a headlong rush to the nearest men’s room. Once I heard in the distance, “Out of the way!” followed by the sound of a door banging shut. Perhaps 20 minutes later, he would saunter back to his former location—without a clue that this was anything but an ordinary event—and resume whatever activity it was in which he had been engaged.
Science and mathematics are cooperative endeavors. Newton is famous for his quote “If I have seen further than others, it is by standing upon the shoulders of giants.” But it is not only giants that contribute to the advance of science, it is people who spend their careers working out a small detail in a complex discipline. And so it was with the work of Goro Shimura. Around 1955 he worked with another mathematician, Yutaka Taniyama, to develop the Taniyama-Shimura Conjecture: every eliptic curve is modular. Many other mathematicians worked together and alone for decades to develop this idea, and eventually it was understood that proving the Taniyama-Shimura conjecture, or a limited form of it, was the mathematical equivalent of proving Fermat’s Last Theorem. This research is far too complex to characterize in a blog like this.
However, it fell to one mathematician, Andrew Wiles of Britain, to make the final, very difficult step in a long string of mathematical discoveries. The Wikipedia article on Fermat’s Last Theorem states:
Andrew Wiles, an English mathematician with a childhood fascination with Fermat’s Last Theorem, and who had worked on elliptic curves, decided to commit himself to accomplishing the second half: proving a special case of the modularity theorem (then known as the Taniyama–Shimura conjecture) for semistable elliptic curves. … Wiles worked on that task for six years in near-total secrecy, covering up his efforts by releasing prior work in small segments as separate papers and confiding only in his wife.
Wiles presented his results for peer review in June of 1993 at the Isaac Newton Institute for Mathematical Science, but the review pointed out an error in his proof. Wiles, then at times quite discouraged, continued to work on the problem, abandoning one approach and returning to an earlier one. The May 1995 issue of the Annals of Mathematics contained two articles by Wiles, establishing the proof of the 358 year old Fermat’s Last Theorem.
I wonder how many thousands of mathematicians, including my eccentric friend, Herb Dowdy, puzzled endlessly about this famous mathematical quandry. Herb had a beautiful and statuesque African-American fiancee named Dora, who eventually moved in with him and may have married him. Here I am 67 years later, remembering him, even down to his crazy sense of humor. Apparently, he was legendary for having the most messy apartment ever seen on the OSU campus. I asked him about that once, and he replied, “Well, I do live on Chittenden Avenue—shit in den!” He laughed uproariously at that remark. I think you do have to be a little eccentric to work so hard on such an abstract problem.
However, our deceased hero, whom Kenneth Chang in the New York Times honored today with an obituary, had the most regular of work habits.
He wrote almost all of his mathematical papers alone. Dr. Sarnak of Princeton recalled visiting Dr. Shimura’s house and seeing two desks in his office. In the morning, Dr. Shimura would work at one, exploring new ideas. In the afternoon, he would work at the second, polishing papers for publication. Once he made a breakthrough and finished a draft of a paper at the morning desk, he would place it in a drawer in the second desk and not return to it for about a year.
Goro Shimura got to savor his role in the centuries-long proving of Fermat’s Last Theorem for 25 years, even though he little realized his contribution back in 1958, when his colleague, Yutaka Taniyama, committed suicide. In this troubled time of political and climate upheaval, it is good to remember Goro Shimura. New York Times, you got it right once again. Keep up the good work.