History of Mathematics
March 27, 2007
It is no wonder that George Polya was a revered by so many in the mathematical world, he touched too many lives, and was too prolific a mathematician to not garner such respect. As a researcher, his broad interests led him to affect mathematics in a large number of subjects. His students regarded Polya as a master teacher and his teaching career at Stanford lasted well passed his retirement from the university in 1953. However, his most lasting contribution to mathematics may very well be to the teaching of mathematics itself. With his book How to Solve It, Polya opened a new world in the teaching of mathematics, one where problem solving took center stage. His contributions to the teaching of mathematics are extremely long lasting, as he directly affected teachers of mathematics with his teachings at Stanford and indirectly with his writings in the How to Solve it and other similar books. From here we shall take a look at the life of George Polya, and chronicle his travels from his native Hungary to Palo Alto. Much time will be devoted to the affects his life had on others along the way, through his mathematical research and his affect on the teaching of mathematics.
Biographical Sketch on George Polya
George Polya was born to Anna Deutsch and Jakab Polya in Budapest, Hungary, on December 13, 1887. Polya’s parents had an interesting history themselves, five years prior to the birth of George, the family was known as Pollak. However, Jakab a lawyer changed his name from the Jewish sounding Pollak to Polya in hopes of securing a university post. Hungary, after gaining its independence in 1867 wished to establish a more Hungarian identity, so to follow suit the family name was changed from Pollak to Polya. Jakab did gain that post shortly before his death in 1897, though it was unknown whether the name change had any bearing on his gaining the position. Polya’s father died at a young age and seemed to have worked himself to death. While working for an insurance company in Budapest it was not unusual for Jakab to act coldly towards his family to focus on his work and interests. “Jakab would arrive at the Generali at eight in the morning and work until one in the afternoon, then return at two in the afternoon and work until six, After dinner, which he required his family to eat in silence, Jakab locked himself in his office at home and, fortified with plenty of black coffee and cigars, devoted himself to his true interest; the study of history, economics and finance.’ (Taylor 8) The one luxury he did afford his family was sending them for a week in the country while he labored at his job in Budapest. George was ten years old at the time of his fathers’ death. Anna was left with five children after Jakab passed away, George’s older brother Jeno, a medical student, his two older sisters Ilona and Flora and Georges younger brother Laslo. George was not to be the only Polya that had a long lasting affect on his chosen profession, as Jeno became a gifted surgeon and has a form stomach surgery named in his honor. Georges’ youngest brother Laslo may very well have been the most talented of the Polya children, however he died in World War I before having the chance the display his talents.
A look at the education of George Polya yields some surprises. In fact in Gymnasium his marks in mathematics were average at best. The subjects that Polya he pursued at that time were languages, such as Greek, Latin German and Hungarian. Beyond languages his favorite subjects were Biology and Literature. Though he went on to have a far reaching interest in mathematics over his lifetime, he showed no particular interest in the subject while in Gymnasium. Upon entering the University of Budapest in 1905, he studied law, at the encouragement of his mother. His study of the subject lasted but a year, as he thought law to be boring. Instead he pursued his favorite subjects from Gymnasium, earning a teaching certificate in languages and literature. Polya never used that certificate, instead choosing to pursue a study in philosophy. It is under philoshphy instructor Bernat Alexander that his interest in math began to flourish, as Alexander encouraged him to study math and physics to better understand philosophy. His being forced to learn mathematics led to this comment by Polya. “ I am not good enough for physics and I am too good for philosophy. Mathematics is in between.” ( Alexanderson 10) Polya found his place in the academic world and after returning from the University of Vienna in 1911, Polya was awarded his doctorate in mathematics by the University of Budapest. His doctorate was earned essentially without an advisor as he studied a problem in geometric probability, a subject no one else seemed to have an interest in.
During his education the two most influential mathematicians on Polya were Lipot Fejer and Adolf Hurwitz. Though the men were of contrasting personal styles mathematically they operated in very similar manners, passing on their penchant for details to their student Polya. Fejer, a professor at the University of Budapest was an outgoing man known for his Bohemian attire and wild gestures. He also had a passion for music and art. Mathematically, Fejer worked a problem until he understood it to the most minute detail, a practice that rubbed off on Polya. “Polya remembered that when Fejer found an idea he tended it with care; he tried to perfect it, simplify it, free it from the unessential until the idea became absolutely clear. He eventually produced a work of art, perhaps not of large dimension, but highly polished. Polya developed this same attitude towrd mathematics, probably because of his association with Fejer”, stated Harold Taylor. ( Taylor 20) After working with Fejer, Polya was lead to give up his work on physics and philosophy and concentrated directly on mathematics. By contrast, Adolf Hurwitz was reserved, quiet and modest. Hurwitz was never one too appear in public as eccentric or Bohemian. It was mathematically that the two were extremely similar, “ And as a wirter Hurwitz was like Fejer as well. Hurwitz, too, tended his ideas with care, until he arrived at the simplest attainable expression, transparently clear and devoid of superfluous ornament. So concise and weighty was his writing that Felix Klein referred to him as an Aphoristician.” (Taylor 26) These two mathematicians gave Polya his style, he too would work a problem until it was absolutely clear. It was this style that created the effective teacher that Polya was to become, allowing hime to explain all the steps along the way, thus aiding in the discovery process.
Polya did spend some time in Gottingen in 1912, doing some postdoctoral work under Hilbert and Klein. After that work he was offered a post at the University of Frankfurt, a post that he was unable to accept. In the summer of 1913, while riding a train back to Gottingen another passenger on the train insulted Polya, calling him a “Bloody Jew”. He punched the insulter, who complained to the police. “ While Polya was not afraid of arrest and incarceration, he was sickened by the political atmosphere in Gottingen and in all of Germany. He turned down the appointment at the University of Frankfurt and went to Paris instead.” ( Taylor 26) This was blessing in disquise, as after a short stint in Paris, Polya accepted a post in Zurich at the ETH, the Swiss Federal Institute of Technology. It was in Zurich that the personal and professional life of Polya truly began.
Polya loved the city of Zurich and much good happened to him during his stay in the city. While teaching at the ETH Polya enjoyed some of the most productive years of his life, he published at a furious pace. His papers covered the gamut from polling systems to probability and analysis. It was in 1923 while in Zurich Polya began work with is friend Gabor Szego on their ground breaking work, Problems and Theorems from Analysis. The most important part of the work was the fact that they organized the problems not by type, but rather based on how their solutions were attained. Of the work Polya stated, “ We were interested in the same kind of questions and topics; but one of us knew more about one topic and the other, more about some other topic. It was a fine colloberation. The book Aufgaben und Lehrsatze aus der Analysis, the result of our cooperation, is my best work and also the best work of Gabor Szego.” ( Taylor 25) It was also at this time that Polya met his wife, Stella Weber. Stella was staying with her father, Robert Weber, a physics professor at the University of Neuchatel, in the same complex in which Polya lived. Zurich was not without its hardships though, the pace of Polya’s work concerned Stella who feared that Polya was following his fathers footsteps. It also became apparent towards the end that the Polya’s were going to be forced to leave. It was during the 1930’s that the Nazi party were establishing themselves in Europe and Polya’s Jewish ancestry would have made him a prime target. It was in 1940 Polya and Stella left Europe, with Polya teaching at Brown and Smith College before accepting an appointment as associate professor at Stanford in 1942.
Gabor Szego, Polyas old friend, who had become the executive head of the math department before Polya left Zurich, issued Polya’s invitation to Stanford. The fact that Polya was only offered an associate professors job, while he had been a full professor at the ETH, may have been seen as a slight by some, however, Polya being a man of great integrity never brought up the issue. Though it does seem that Szego forgot the help his mentor gave him by seeking him out to collaborate on Problems and Theorems from Analysis. Polya was made full professor in 1952, after Berkeley made an attempt to lure him to thir campus, though no matter what rank he was given, Polya was productive in his years at Stanford. He published How to Solve It and IsoperimetricInequalities in Mathematical Physics prior to his retirements in 1953. Following his retirement, Polya was made a Professor Emeritus and began to pursue a new twist in his career, educator of educators. He directed his efforts to teaching secondary teachers how to be more productive, through his conferences and newsletters. He also created a test for California high school students, aimed at creating excitement for math and identifying start student. The test was based on a contest in Budapest while Polya was in gymnasium. During this time it was unfair to call Polya retired, rather he shifted focus from research to education, a subject he attacked with the same vigor he gave every other problem he had ever encountered.
Before looking at the contributions of Polya made to mathematics we should look at his legacy as a person. During his long life he maintained a positive attitude even under the toughest of circumstances. “ His attitude towards life was enviable. He was seldom depressed or discouraged, and when students came up to him distressed about a test or a dissertation, he would cheer them up and sometimes even deliver a stern lecture on self-pity.”(Taylor 129) Through his half century of teaching he never lost his passion, though he never worked from notes, he would often review his lecture following the class, recounting why he would answer a student in a given manner. Children also loved Polya, and later in his life would follow him around, enjoying the games and teaching he provided them. Lastly, friends were precious to him, even after losing his sight; he could recognize the voice of a friend. Even if it were not for the work he had done in his life, Polya sure would have been remembered for the wonderful man he was. Georege Polya died on September 7, 1985, due to complications of stroke. At his memorial service M,M. Schiffer spoke these kind words about Polya, “ A good and gentle man left us. But his work will continue to affect our science, and we will always remember him with love and admiration.” (Taylor 120)
The Research of George Polya
As the interests of George Polya were too broad, we will only be able to touch on the highlights of Polya’s research career. To underscore the far reaching interests of Polya one should look at what he produced while in Zurich, where he published at a furious pace. “While in Zurich his output of mathematics was very large and wide ranging. For example he published papers on series, number theory, combinatorics and voting systems. The following year, in addition to papers on these topics, he published on astronomy and probability.” (O’ Connor) Much of this breadth came from his interest in problem sovling. He would address a given problem in hope of learning how to address the problem, then move on to the next. While Polya’s style allowed him to be a prolific researcher, not everybody approved. Polya passed on this idea in a story about G.H. Hardy
In working with Hardy, I once had an idea of which he approved. But afterwords I did not work sufficiently hard to carry out that idea, and Hardy disapproved. He
did not tell me, of course, yet it came it out whe he visited a zoological park with
Marcel Riesz. In cage there was a bear. The cage had a gate, and on the gate there
was a a lock. The bear sniffed at the lock, hit it with his paw, the growled a little,
turned around and walked away. “He is like Polya,” said Hardy. “He has excellent
idead, but he does not carry them out. (Taylor 45)
What this story does not illustrate is how Polya could make even the toughest idea easy to comprehend, making his research that much more powerful.
One piece of research he completed in Zurich was the Random Walk problem, an idea that grew out of his love of walking in the woods surrounding Zurich While on one of his many walks Polya ran into a student and his fiancée several times on the many paths in the Zurich woods. It was Polya’s belief that these meeting could be modeled, and lead to a study in probability. He simplified the paths to be square blocks and starting at an intersection of two blocks moved a point around. That point could be moved in one of four directions, all with equal probability. If you did this over and over, Polya proved that the point had a probability of one of returning to the starting point. He also showed this was true one a one-dimensional city, with the point having two choices in where it could go. “ A possible interpretation of this would be the fate of a gambler betting on heads and tails in repeated tosses of a fair coin. After so many tosses, or so many moves along the line, the final position of the moving point would correspond to the total take (or total loss) of the gambler’ (Alexanderson 16) When studied in dimensions three or greater, the result was not true as the walk was taken out to infinity, there were simply to many choices for the point to be sure that it would return to to its starting point ‘Polya was the first to consider the random walk problem and the one to give it its name. In a manner characteristic to Polya, that of giving the problem a concrete interpretation, he said, “all roads truly do lead to Rome.” Many mathematicians have since given their attention to more complicated versions of this problem.’( Taylor 40) the development of the random walk problem showed how Polya could relate math to nearly any everyday experience.
Polya also had an interest in Crystallography that in turn affected the career of artist M.C. Escher. In a paper written in 1924, Polya looked at the plane symmetries used by the Moors to decorate the Alhambra. In classifying these groups he used tilings to illustrate the groups. These tiling interested Escher, and he carefully copied the designs described by Polya.. Escher would correspond regularly concerning the geometry of these groups, and even gave Polya some art in thanks for his help. “Polya was proud that his work with symmetry groups had been invaluable to this artist that was later to attain fame. However, he was saddened by the fact that he did not arrive in the United States with the drawings.” ( Taylor 59)
Another area in which Polya had interest was the Reimann Hypothesis. This is a problem that has neve been solved, however rumor does arise every so often that a proof has been written. Polyas work on this problem included looking at zeros of functions, and those zeros after differentiation. While his work on the problem was significant, what is more interesting were the dreams he had concerning the solving of the Riemann Hypothesis. “ A note that he had made to himself on the morning of February 17, 1977, indicated the he had dreamed the previous night that he had proved the Riemann Hypothesis about the zeros of the zeta function, and that the proof was quite simple. He had dreamed that the proof relied on a formula that had not appeared in the dream, and that it took about two pages to write out the proof.” ( Taylor 42) These dreams did happen often when working on large amounts of mathematics. At times that ideas that occurred to him in these dreams did solve certain problems.