Monday, September 24, 2012

Quiet Please: Mathematicians Hard At Work

Pure Sciences

Hungarian mathematician Paul Erdős, although an atheist, spoke of an imaginary book, in which God has written down all the most beautiful mathematical proofs. When Erdős wanted to express particular appreciation of a proof, he would exclaim "This one's from the Book!". This viewpoint expresses the idea that mathematics, as the intrinsically true foundation on which the laws of our universe are built, is a natural candidate for what has been personified as God by different religious mystics.
John Francis, Philosophy of Mathematics (2008), p. 51

Paul Erdős [1913–1996]: At a student seminar in Budapest in the Fall of 1992. "It is not enough to be in the right place at the right time. You should also have an open mind at the right time."
Source: Wikipedia

When I was young I was fairly good at mathematics, a branch of pure science, but not good enough to become a mathematician. My abilities were limited to the level of math needed to solve engineering equations, which included basic calculus, partial differential equations, boundary value problems (BVP) and finite elemental analysis (FEA).

I was of the generation, the last cohort, to have learned the use of a slide rule in high school; this instrument, along with trigonometry tables, helped to do calculations, including multiplication, division, square roots, trigonometric and exponential functions. By the time I had reached university and the engineering school, we were all using scientific calculators made by Hewlett-Packard (HPs), starting with the HP-35, then considered the best model for our purposes. (We soon evolved to later models like the HP-45, HP-55 and HP-65.)

The calculators were a great help; and we soon were wondering how we had lived without them. While we could use scientific calculators, programmable ones were not allowed during examinations. So, we solved all of our problems with pen and paper. Needless to say, we worked hard on the problems assigned to us, all mathematical-based, but we often did our work in groups, making the time pass away in good camaraderie and competition to see who could solve the problem the fastest; it was both challenging and exhilarating.

After a long week, on Fridays we would go to the university pub for a few beers and a couple of game of pool, and then take our heavy briefcases full of books home for a weekend full of problem solving. Yes, it's also true that some of us in engineering school wore thick glasses—not me— and had pocket protectors, which also held various coloured pens, mechanical pencils, small screwdrivers and small rulers. (Even then, in the late 1970s and early 1980s, many of the top students were Asians, a point that will soon become important.)

Since engineering is essentially a discipline dedicated to problem solving, that is the application of mathematics to solving real-world problems, like energy consumption, heat transfer and mechanical systems design, in my younger years I often wondered of what use was pure mathematics and how mathematical prodigies came about. One of the twentieth century's greatest mathematicians was Paul Erdős, a Jew from Budapest, Hungary, a nation that produced Leó Szilárd, John von Neumann and Edward Teller—all phenomenal scientists whose contributions to math and physics are well known and well regarded.

And, yet, Erdős might have surpassed them all in his love, devotion and contribution to pure mathematics. In My Brain is Open: The Mathematical Journey of Paul Erdős (1998), Bruce Schechter, who holds a PhD from M.I.T., writes:
Paul Erdős's brain, when open, was one of the wonders of the world, an Ali Baba's cave, glittering with mathematical treasures, gems of the most intricate cut and surpassing beauty. Unlike Ali Baba's cave, which was hidden behind a huge stone in a remote desert, Erdős and his brain were in perpetual motion. He moved between mathematical meetings, universities, and corporate think tanks, logging hundreds of thousands of miles. "Another roof, another proof," as he liked to say. "Want to meet Erdős?" mathematicians would ask. "Just stay here and wait. He'll show up." Along the way, in borrowed offices, guest bedrooms, and airplane cabins, Erdős wrote in excess of 1,500 papers, books, and articles, more than any mathematician who ever lived. Among them are some of the great classics of the twentieth century, papers that opened up entire new fields and became the obsession and inspiration of generations of mathematicians. (14)
For most of his productive life as a mathematician, Erdős lived as a peripatetic: he had no place on earth to call home; no wife or children; no family in the traditional sense; and no permanent job. And, yet, he was productive in the truest sense of the word, that few today can emulate. The man who loved proof and conjecture had all that he needed to do mathematics: paper, pencil and his brain, always open. Any mathematician in the world might hear a knock on the front door and open it, Schechter writes, "to find a short, frail man wearing thick eyeglasses and rumpled suit, carrying a suitcase containing all his belongings in one hand and a bag full of papers in the other, who would announce, 'My brain is open.' "

By age four, Erdős was a mathematical prodigy, Schecter writes, but this is not necessarily a precondition for greatness:
Not all mathematicians start life as prodigies. Many, including some of the greatest, led fairly ordinary childhoods and one day stumble over a fascinating problem, or book, or meet an inspirational teacher and become incurably hooked. Others, like Erdős, or the greatest mathematical prodigy of all time, Kark Friedrich Gauss, seem to have been born with memories of the ideal Platinic realm of numbers that only need refreshing. (26)
Mathematicians have often been looked at as eccentric individuals; and there is some truth to that claim, but it should not at all be given any undue notice or importance. Many mathematicians never marry and raise children, but this does not mean they dislike children or the traditions of marriage. Such conventions never enter their calculus; their desires and passions, so to speak, lay elsewhere, in the world of proof and conjecture, Schechter writes:
The activities of mathematics are obscure to outsiders. One frequent guess people hazard is that mathematicians spend their days thinking about numbers. Many, but by no means all, do. More generally—a phrase of which mathematicians are especiually fond—mathematicians investigate the properties and relationships of "mathematical objects." (30)
One of the questions often asked is why some cultures are better at math than others. Apart from particular mathematical prodigies, a rarity, the ability to figure out mathematical problems is not based on evolution. and certainly less so for higher-level math, Schechter says:
"On evolutionary grounds it would be surprising if children were mentally equipped for school mathematics,"  Steven Pinker writes [in his book How The Mind Works]. The reason is that modern mathematics, developed for modern education, is too recent to have had the time to encode on our DNA. School mathematics, a product of cultural evolution, required the development of such tools as language, reading, and writing to expand the abilities of the unaided human mind. Learning mathematics beyond the first intuitive steps is, therefore, largely a matter of hard work. "Without the esteem for hard-won mathematical skills that is common in other cultures," Pinker says, "the mastery is unlikely to blossom." (29)
In other words, cultures and societies that place an importance on hard work will do well mathematically, and elsewhere. Such was the case in Hungary when Erdős was young. The United States, a beacon for immigrants, was fortunate to have attracted a number of great scientists before and after the Second World War; it still continues to do so, but perhaps less than previously. One measure of performance is the international standardized test of 15-year-olds administered every three years in 65 OECD nations, which includes the U.S., Canada and South Korea. In general, Asian and Nordic nations placed near the top; Canada placed sixth after China, South Korea and Finland, and the U.S. ranked 25th, below the OECD average. [see here, here here for more on the latest 2009 results.]

These tests show precisely what they are meant to measure: how one nation of high-school kids compares to another, a snapshot in time. It doesn't show which nation has any mathematical prodigies, since these are always rare, but a general sense of how well Grade 9 students fare on math tests. Generally, the Asians, led by China, are doing well in preparing their students for a scientific career; the students put in long days at school and at home studying, learning and integrating knowledge; leisure time is secondary. Simply put, they work hard, perhaps harder than others.

This is to their credit, since math is essential to science and scientific innovation; the U.S. doesn't require child prodigies as much as it requires a sense, a firm belief if you will, that math is important, more important than developing and playing video games. It can rely on immigration, to some degree, to attract top mathematicians and scientists, but it also has to develop an idea that hard work is the key to success. This is not always apparent.