On Remarkable Mathematicians: Goro Shimura is Dead

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

no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2.

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.

520px-Pythagorean.svgOne 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
pythagorean3 or for that matter for any equation of the formpythagoreann, 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.



How an Ethics Course Changed My Life

When I was an undergraduate at Marietta College, I was fortunate to take several philosophy courses with Elizabeth Steiner Maccia. Her classrooms were overflowing with students eager to learn this esoteric subject from her, and her reputation was well-established when my fellow classmate and mathematics student, Jim Murtha, said “You ought to get a course from Liz Maccia; she’s good.” I think it is fair to say that Elizabeth Maccia’s course on Ethics changed my life.

If my memory serves me right we read four original works, one each by Aristotle, Immanuel Kant, John Stuart Mill, and Friederich Nietzsche. I approached all my college courses just like I had my high school courses. I went to class, took notes, studied hard, and got generally good grades on a test. Aristotle’s Politics was interesting, but when we got to Kant, I was blown away, no, transformed.

It’s been sixty years, so I might get a detail here and there wrong, but I believe she had us read Fundamental Principles of the Metaphysics of Morals. Kant’s book was published in 1785. It is no mere coincidence that the Constitution of the United States was being written in 1787. Kant’s writings buit on the Enlightenment idea of men entering into a rational social contract to secure and protect individual freedom, and these writings influenced our governing document. To protect everyone’s freedom, we all agree to live within a legal sytem which protects freedom.

What got me so excited about the little book with the long title was that Kant was laying out the foundation for a rational and objective ethics. In it he argues that the good will—that will which has the pure intention of doing what is right (whatever that may be)— is, to a rational being, a perfectly good thing. He then sets about to establish a principle, elaborated in several forms, by which a person with a good will can live. Here is one such statement, “Act only on that maxim whereby thou canst at the same time will that it should become a universal law.”

Kant called this principle the categorical imperative. It’s also sometimes called the universalizability principle. It’s not quite the same thing as the Golden Rule: “Do onto others as you would have them do onto you.” It’s more cerebral; it envisions a community of people asking “What is my duty?” And neither one of these rules is the same as “You scratch my back and I’ll scratch yours.” That’s the kind of ethics we try to teach toddlers when they start hitting their playmates.

Well, because of Kant’s “monograph,” as Dr. Maccia referred to it, I was hooked on ethics/philosophy. I carried that thin paperback around with me day and night for the rest of that whole semester. And a decade and a half later, when I began to teach philosophy of education at the university, my general approach to ethics was neo-Kantian, as elaborated by the British philosopher, R. S. Peters, in the book Ethics and Education.

Of course, there are a lot of other approaches to ethics that merit serious consideration and have their devoted followers, and Professor E. S. Maccia introduced me to two of them.

About the time we in the USA were starting our Civil War, John Stuart Mill was beginning to develop his theory of utilitarian ethics. He proposed that happiness is the greatest good, and that, when individuals are making their moral choices, they should pick the action which will maximize happiness. He asserted that all other moral principles are derived from the Greatest Happiness Principle. His book, Utilitarianism, published in 1863, was widely circulated and read. This is by no means an uncontroversial approach to ethics, but Mill has had a huge effect on our philosophical thinking, especially during the 19th Century. One effect of such an ethics is that it forces us to think about the consequences down the line of what we do. That, of course, is both a strength and a weakness, because it’s sometimes not at all clear what the result of choosing this or that action will be.

For Kant, the good life was spelled out in following this set of rules that were designed to appeal to the rational mind. For Mill, we become calculators of consequences, maximizers of mirth. While I could understand the importance of maximizing happiness in our lives, the ideal of polishing my will until it shined held sway over my still very limited but gradually expanding life experience. Yep, I decided, I was still a neo-Kantian.

Both Friedrich Nietzsche and his ideas made it into the 20th Century. He died in 1900 at the age of 55, suffering a complete mental and physical collapse at age 44. He is famous for arguing both that the will to power, not rational action, is the primary determinant of human behavior, and also that this is as it should be. Nietzsche’s sister, Elizabeth Förster-Nietzsche, became his caretaker in the last three years of his life. During her life, Elizabeth aligned herself with German nationalism and antisemitism through her marriage to Bernhard Förster. Förster left his high school teaching job to become a leader in the German nationalist movement. It was through Elizabeth’s curation and actual alteration of Friedrich Nietzsche’s writings that they became an inspiration for and associated with German militarism and National Socialism.

Getting my mind around Friederich Nietzsche’s Beyond Good and Evil turned out to be quite a project. There were just too many challenges in his philosophy to my rather conventional Christian upbringing. I’m not going to try to do a good job of presenting his core concepts, but this quote from Wikipedia will serve as a springboard for this discussion:

In [Beyond Good and Evil, Nietzsche] exposes the deficiencies of those usually called “philosophers” and identifies the qualities of the “new philosophers”: imagination, self-assertion, danger, originality, and the “creation of values.” He then contests some of the key presuppositions of the old philosophic tradition like “self-consciousness”, “knowledge”, “truth”, and “free will”, explaining them as inventions of the moral consciousness. In their place, he offers the “will to power” as an explanation of all behavior; this ties into his “perspective of life”, which he regards as “beyond good and evil”, denying a universal morality for all human beings. Religion and the master and slave moralities feature prominently as Nietzsche re-evaluates deeply held humanistic beliefs, portraying even domination, appropriation and injury to the weak as not universally objectionable.

With Nietzsche, we don’t so much get rules for guiding our lives as we get permission to be rule-breakers. He thinks that humans are still evolving, and it is doubtful that any system of moral thought—even the system of the “Chinaman of Königsberg” (Nietzsche’s snarky characterization of Kant)—is valid for everyone and every age.  He sees these grand ethical systems as the crystallized and hand-painted neuroses and psychoses of their philosophic creators. The new humans who will evolve will not want to subject themselves to an “ethical” system that treats everyone the same, for they will not be the same as everyone else, i.e., “the herd,” they will be better, superior to most people. They may even want to dominate their inferiors, treat them as slaves, experiment with forms of control that are actually torture in disguise.

One other aspect of Nietzsche’s philosophical thinking I’d like to note before moving on is his attachment, not to philosophy, but rather to art and music as our inspiration and guide. He proposes more of an aesthetic, rather than an ethical Weltanshauung or worldview. Creativity more than duty is important to him. He notes that in Ancient Greek culture two elements existed, one represented by the god, Apollo, representing harmony, progress, clarity and logic, the other, by Dionysus, representing disorder, intoxication, emotion and ecstasy. Both perspesctives are essential for an aesthetic view of life, but the Dionysian element is essential for overcoming inhibitions and breaking boundaries.

Appalling as aspects of Nietzsche’s philosophy were and still are to me, I find great value in reflecting on his perspective. Here I was, a naïve college student still much attached to family origins and traditional Christian values, even though I had professed agnosticism at that point. Still, my very well-developed conscience told me that living an aesthetic life beyond moral control was likely to lead me into unexplored, dangerous and even sinful territory (from my own perspective). Nietzsche did not snatch Kant’s monograph from my trembling hands.

Nietzsche was visionary in imagining a future of super humans who exist beyond good and evil. (He would have said “supermen;” for Nietzsche, women existed to be possessed by their male lovers.) A defect of his vision is that he did not give us clear criteria by which we could distinguish evolved human beings from their inferior predecessors.

There are and will always be pretenders to the throne of superiority. They, of course, are absolutely convinced of their magnificence, though others may not be so impressed. Indeed, one even might make the argument that any human being who claims to be superior to the herd has just disqualified themself. And this is precisely why the very First Amendment of the U.S. Constitution, indeed, the very first words of the amendment, state “government shall make no law respecting an establishment of religion.” Human beings are zealous in the pursuit of elevating their own preconceived notions into universal law. Therefore, any human beings in government are prohibited from so doing by our Constitution.

One of the failures of Nietzsche’s vision has to do with the evolution, not just of the human race, but actually of a new dimension in our experience, cyberspace, and a new dimension of our thinking, artificial intelligence (AI). Computer scientists are now predicting that by 2030, AI will catch up with and surpass human intelligence. Far from becoming super human, it may well be that super intelligence incorporates human beings as a component in a larger system. Let’s just hope that any future evolutions of Hal think like a Kantian rather than a Nietzschian philosopher, or some of us are doomed.

So here we are on Planet Earth, the first generation of humans to live in the Anthropocene Era. In the past, massive species extinction has been a sure sign of a severe suppression of all life on Earth. And here we are in the USA, where a current political party has decided to flout due process in the interest of re-establishing a set of values that favors white males over other members of the human species. The system of checks and balances that were put in place over two centuries ago was inexorably evening the playing field, so the Republican Party went Nietzschean on us. They are beyond good and evil, in the worst sense. Their maxim is “Do anything to win, and blame any bad results on the opposing party.” They definitely are not following either a Kantian or a utilitarian ethics.

And here am I, still a believer in the social contract, still a neo-Kantian, and a Christian, to boot. What am I to do? Well, my deontic, or duty-based, ethics gives me a very clear road map for my future. I may not know how to maximize good in the USA in this era of profligate public prevarication by poseurs to the presidency. I may not know how to live “beyond good and evil.” But I sure as hell know what my duty is. I shall act accordingly.




A Mother’s Day Tale

arthurVictorine Dorval Munier Andris was the grandmother that I never met. My own mother, Ella Lorene Sullivan Andris fortunately planted and kept alive in my mind the memory of “Torienne.” I don’t know too much about her, but story telling and genealogical excavation have allowed me to reconstruct some of the story of her life. One thing is clear to me, though. If it weren’t for Grandmother Andris, I wouldn’t be here.

During the 19th Century the smallish town of Binche, Belgium was near factories where window glass was made. In those days the “small pot” method was used, and it required the services of professional glassblowers. It was a difficult and demanding, yet respected craft. A master glassblower would take up to 35 kilograms of molten glass from a pot in a nearly white-hot oven onto a heavy lead pipe. He would “start” the bubble by blowing into the pipe, which he constantly rotated and periodically returned to the blazing furnace. When the bubble was large enough, it had to be swung in a deep pit, so that the mass of glass became elongated into a cylinder. Eventually the cylinder of hot glass was swung back up onto a stone slab. It was caused to split down its length by a cold iron chisel and quickly flattened into a sheet of clear but wavy glass which could eventually be cut to fit into a window frame.

Victorine Dorval married two glassblowers. The first one, Jules Munier, died at a young age in a fall from their roof. At the time he and Victorine had one daughter, Julia. In 1895, Victorine married my grandfather, Arthur Louis Nicolas Andris. Arthur had also lost his first spouse, and brought two young sons, Arthur and Aimé, to the marriage. Things were not going well for young glassblowers in Binche at that time, and so in a few years, around 1902, Arthur packed up most of his family and travelled to Russia, where there supposedly was work to be found. There, Arthur’s mother died and had to be left for the neighbors to bury her, because the “Cossacks” were coming to the village on horses carrying torches, looking for “foreigners” who were taking jobs from the Russians. They barely made it across the border, but the Andris family escaped and returned to Binche, bringing with them infant, Louie.

In 1908, Arthur came to the U.S. and secured a job in a glass plant in Clarksburg, West Virginia. The rest of the family immigrated over the next three years, except for Arthur’s oldest daugter, Louise, who “stayed in the old country.” In 1909, first Arthur and Aimé came over, and then Torienne and her two small sons, Louie and Alphonse. Finally, Julia arrived in 1911. Things went well for a while, and Arthur, Jr. and Aimé helped papa Arthur. In 1992 I interviewed my dad, Fernand Andris, youngest surviving son of Arthur and Victorine. He had this to say about our family’s involvement in glassblowing:

“Amy (Aimé) was a strapper and Arthur was a gatherer. Dad was the blower. They had lead pipes this big (shows about 2 feet long), and they was heavy. They had a hole in the floor this wide, and it was deep. After dad got the glass (on his pipe) he would have to blow hard to get it started. A brain tumor killed my dad; blowing glass might have killed him. When he got the glass started, he’d swing it in the pit, and it would become a cylinder, maybe 8 or 9 feet long. Dad couldn’t do all the work, so Arthur would throw the thing. They had these horses and he’d swing the cylinder up on them. Alfred Bourmark (Julia’s husband) was the glass cutter. They’d cut the glass long ways, then they’d fold them over to make a big sheet. Bourmark would cut that sheet. The glass house was across the Putnam Street Bridge just beyond the College crew shed and on the left.”

Unfortunately, that work also dried up, because mass production of window glass had been successfully put into place, and the old small pot glass making technique could not compete with machine manufactured glass on either cost or speed. Fortunately, Mrs. Andris, my grandmother, was a resourceful person. Again, from my father’s recollection: “Torrienne bought 313 Greene St. around 1922 from Mrs. Morris, whose husband was dying of TB. she wanted to go back to West Virginia. She payed $900 for the (grocery) store.” From that time forward, grandmother Torienne took the reins of the family fortune and built a successful trade for her small family grocery store on downtown Greene St. Arthur, Sr. became less and less able to help and died in 1930. It is another story to tell, but Arthur, Jr. and Aimé went off on their own, while the three sons of Arthur and Victorine, Louis, Alphonse, and Fernand, remained and worked in the family business.

I have written on my genealogy website:

“In 1937 the family grocery store was hit by a flood. Various records show that the flood hit 55 feet on Jan. 23. This was nearly 20 feet above flood stage. According to my mother, it was a terrible tragedy. They had moved all the groceries and equipment up the stairs to the second floor, but the flood went to 55 feet, and was well into the second floor. When they were able to get into the building, everything was ruined. Mud was everywhere. Sacks of flour and sugar were ruined. Canned goods were rusted and the labels had come off. Mrs. Andris was devastated. Mother said that she would set for hours scrubbing rusted cans with steel wool and try to identify them for reduced sale. The anguish of the loss and the worry of financial ruin probably contributed in large part to her death on March 4, 1937.”

My mother, Lorene, told me the rest of the story more than once. My dad, Fernand and she were dating at the time. Mom loved Torienne, and they had a very good relationship, unlike the sometimes turbulent relationship she had with her own mother, Clara. Torienne had a heart attack around March 1, 1937. She was quite obese, and Fernand and Lorene half carried, half pushed his mother up the narrow stairway to the second floor of 313 Greene St., where she remained until her death a few days later. The doctor came to visit, but offered little hope for her recovery. While she was lying on her death bed, Torienne extracted from her future daughter-in-law a promise that she would marry Fernand and take care of him. “He’ll never be able to make it on his own,” Torienne said, according to my mother.

But Lorene wasn’t completely sure about this promise. Fernand had a habit of binge drinking, and he could be hard to live with when he was drunk. The problem continued into my adulthood. The night before Mrs. Andris died, my mother had a vivid dream about my grandmother. In the dream, Torienne was seated astride Lorene’s trunk, and she was pounding on her chest and screaming, “Lorette! You promised! Marry my Fernand!! Remember, you promised me to marry him!” Mom awoke in a pool of sweat, but the dream was so vivid that it convinced her to keep her oath. On August 16, 1937, Fernand and Lorene were united in marriage. Theirs was one of the last “bellings” that happened at the Lafayette Hotel. Over 500 people showed up to wish the newly minted family well and to party.

So Happy Mother’s day, Victoria Dorval Munier Andris, from the bottom of my heart. If it weren’t for you, that heart would not even exist to pay homage to you.