Beauty in Times of Horror
And when the Lamb opened the third seal, I heard the third living creature saying, “Come.” I looked, and behold, a black horse; and he who sat on it had a pair of scales in his hand. And I heard something like a voice in the center of the four living creatures saying, “A quart of wheat for a denarius, and three quarts of barley for a denarius; but do not damage the oil and the wine.” – Revelation 6:5-6
Evgeny Yakovlevich Remez was a young professor of Mathematics at the Pedagogical Institute in Kiev when Famine, the third horse of the Apocalypse, rode across Ukraine, bringing despair, horror, and death. It was the early 1930s, and the agricultural collectivization imposed by Stalin depleted the country’s reserves of grain, choking its capability to feed its population. In just a couple of years, the situation became so dire that it triggered one of the worst human-made famine tragedies ever seen in this planet. This event, the Holodomor, which in Ukrainian means “death by hunger”, is undoubtedly one of the darkest pages in the long book of nightmares written in the 20th century.
The biography of Remez focuses almost entirely on his scientific career, so it is difficult to know how he lived and survived those years of dread. The ghastly pictures from Ukraine that survive to these days show images of skeletal children with large, swollen stomachs, bewildered peasants contemplating desolated fields, and abandoned bodies lying on the streets. People died from hunger while eating, and kids stopped breathing in the middle of their lessons. Madness drove parents to commit filicide; couples would kill each other fighting for the ears of wheat. Lynching and torturing of alleged thieves became common. And in the end, when there was nothing more to eat, hopeless and starving, people descended to Dante’s ninth circle of Hell, surrendering to that most desperate act of all: “fasting had more power than grief.”
It was during these grim days that Remez developed the algorithm that now bears his name, one that I became intimately familiar with during my studies at Oxford. Remez published his algorithm in a series of three papers in 1934, so I suspect most of the work was done the year before, precisely when the famine hit the worst, taking the lives of almost two million people. Although the Remez algorithm is very well known in my mathematical community and a mandatory topic in books from Approximation Theory, I have never read the observation that this beautiful creation of mathematics might have been forged in the midst of the most unimaginable catastrophe. Making this connection was as surprising and confusing as having found out that a magnificent idea was developed in Hiroshima in September of 1945 or the west coast of Indonesia in January of 2005.
The Remez algorithm computes the best polynomial approximation, that is, the polynomial among those of a fixed degree that minimizes the deviation in the supremum norm from a given continuous function on a given interval. It is elegant and efficient, a truly original result in what is known as Constructive Approximation Theory.
I spent many months going through every single nut and bolt of the machinery and became obsessed about pushing it to the limits: when the algorithm has been used to calculate best polynomial approximations in degrees of dozens, I wanted to do them in the thousands. I read the original manuscripts in French and every single paper published in the topic by all the great numerical analysts of 1950s and 60s when the Remez algorithm went through a bit of hype. I managed to get and run rustic implementations in ALGOL and FORTRAN, ancient languages that made me feel I was reading Sanskrit or Old Aramaic.
I made countless errors when I developed my version of the algorithm and went through the great pains of writing my first academic paper on my discoveries. When I submitted it for the first time for publication in a specialized journal, it got rejected and I felt my whole world came crashing down. My supervisor, a warrior of a thousand academic battles, knew better and with his guidance and lots of coffee I re-wrote it, submitted it again, and this time it got accepted with enthusiasm. My research took me to new directions, more modern and topical problems perhaps, so I never really went back to the work of Remez, but even after many years I still remember fondly all the joys and frustrations that his algorithm gave me.
It was only recently when I got a copy of “Red Famine: Stalin’s War on Ukraine” by Anne Applebaum, an authoritative book on the Holodomor, that I realized how much I knew about the idea but how little about the man. Did Remez suffer the calamities of the famine, or was he somehow shielded from them? What about his family and friends? Was he worried about his life and his future? Did he ever consider leaving Ukraine? And above all, how did he manage to find inspiration and creativity when everything around him was collapsing?
I haven’t been lucky in finding more details of the Remez’s life, so I will probably never know the answer to any of these questions unless someone can reconstruct the life of this great mathematician from scattered correspondence and notebooks left in some forgotten basement in Kiev. I have the hypothesis that if he suffered in any way the same horror that his countrymen were going through, it might have been that his creation in the abstract world of mathematics transported him away from sorrow, engaged in some form of suspension of disbelieve. But perhaps what made him prevail was something less spiritual and more mundane, like his commitment to teaching and researching at the University, for example, or just the dull routine of showing up in the office every day, trying to do the best he could.
There is only one footnote left in this story: I know for sure that Remez did not develop his algorithm in isolation, but in fact he was assisted by three other mathematicians of whom we only know their surnames and, in an intriguing twist, that the three of them were women. Mathematics, like many other disciplines, carries the shame of having been closed to women for centuries, something that only in the last few decades started to change. I was fascinated to find three women in the early 1930s doing the very same type of mathematics that I love so much, but the importance of their work deserves their own post in this blog.