Biographical Paper:

George Pólya

by Leah Grant

For partial completion of

MATH 4010

Dr. Cherowitzo
15 March 2005

The Great Teacher

I came very late to mathematics. [A]s I … learned something of it, I thought: Well it is so, I see, the proof seems to be conclusive, but how can people find such results? My difficulty in understanding mathematics: How was it discovered? [1]

Considered as one of the most influential contributors to mathematical problem solving,[2] George (György) Pólya supplied numerous donations to our current study and understanding of several different areas in mathematics.[3] Throughout his lifetime he produced notable work in real and complex analysis, combinatorics, probability, number theory, and algebra, among other fields.[4]

Born to Anna Deutsch and Jakab Pollák in Budapest on December 13th, 1887, young George showed no particular interest in the study for which he would eventually become so well known.[5] He later attributes this indifference to the poor instruction of “despicable teachers,”[6] and dedicates much of his time to developing effective teaching methods. In doing so he was one of the few individuals in his research field who made significant contributions both to mathematical study and its instruction.[7]

Life and Times

One of five children, George’s early childhood was marked with adversity. His Jewish parents changed their last name five years before George was born in hopes that Polish-born Jakab would be granted a position at the University of Budapest[8] (an institution very Magyar[9]-biased in its hiring policies). The family also converted to Catholicism, as they felt it another necessary measure to avoid persecution.[10]

George’s father died several years after having finally procured his university title, leaving forty-four year old Anna and his five children to fend for themselves.[11] George’s two older sisters, Ilona and Flóra, went to work to support the family, and to allow Jëno (his older brother) to pursue medical studies. It is interesting to note that Jëno Pólya became quite a distinguished surgeon, as renowned in his profession as George would later become in the world of mathematics.[12]

George attended school at the Markó Street Gymnasium,[13] where he excelled impressively in most subjects (but only “satisfactor[ily]” in math). Once he graduated from the Gymnasium, he was able to attend the University of Budapest with monetary aid from his brother,[14] studying law to appeal to his mother’s suggestion. Yet George quickly bored of this subject, moving on to study literature and languages, particularly Latin and Hungarian. He became interested in philosophy, and also began studying physics and mathematics at the advocacy of his philosophy professor Bérnat Alexander.[15] In becoming familiar with these subjects, George laughingly remarked that he was “…not good enough for physics and … to good for philosophy. Mathematics is in between.”[16]

And So Began the Love Affair

Under the instruction of some very celebrated professors (including Lóránd Baron von Eötvös and Lipót Fejér), George excelled in his study of math and science.[17] He went on to graduate school and finished his doctoral studies, save for the completion of his dissertation, in 1910.[18]

George spent the next few years at the University of Vienna, earning money by tutoring the son of a baron. Apparently this youth lacked any semblance of mathematical ability, so George spent a lot of time formulating some problem solving techniques that his young student could understand and apply.[19] In this venture, George developed his (later famous) view that problem solving skills were not necessarily inherent traits, but should be learned and taught.[20]

In 1912 George was granted his doctorate in mathematics, having independently examined “a problem in the theory of geometric probability… essentially without supervision.”[21] He continued academic study at Göttingen, hobnobbing with various and distinguished mathematicians including David Hilbert, Felix Klein, Richard Courant, Herman Weyl, Erich Hecke, and Otto Toeplitz.[22]

Unfortunately, George was compelled to leave Göttingen; interestingly enough there are several quite different explanations surrounding this event. Probably the most reliable is one related by George in a letter to his friend in 1921. He describes,

On Christmas 1913 I travelled [sic] by train from Zürich to Frankfurt and at that time I had a verbal exchange – about my basket that had fallen down – with a young man who sat across from me in the train compartment. I was in an overexcited state of mind and I provoked him. When he did not respond to my provocation, I boxed his ear. Later on it turned out that the young man was the son of a certain Geheimrat; he was a student, of all things, in Göttingen. After some misunderstandings I was told to leave by the Senate of the University…[23]

Life Goes On

In 1913 George met a man by the name of Gábor Szegö, who had also been a student of Lipót Fejér. They became friends, and collaborated throughout their careers to assemble proofs and various analysis problems. When one needed guidance he would often solicit the help of the other; and their friendship grew throughout the next few years.[24]

George was offered a position in Frankfurt a year after meeting Szegö, but decided first to visit a university in Paris. While staying there, he learned that Adolf Hurwitz had secured him a position at the Eidgenössische Technische Hochschule in Zürich (Swiss Federal Institute of Technology) where Hurwitz was chair.[25] This was excellent for George, for he could serve there as Privatdozent[26] and gain his habilitation.[27] He accepted the offer and left for his new post in Zürich.[28]

While this was a very sought-after opportunity for career advancement, the political situation in Zürich was one in which George wanted no part. World War I had commenced, and fortunately enough for his pacifist political alignment, a convenient soccer injury initially kept the young scholar out of army service.[29] However, as the war continued his country desperately needed soldiers, and required that he return to Hungry to fight. Eventually George’s refusal to cooperate with the army recruiters resulted in his having to flee Hungary and take up residence in Switzerland; he did not return to his homeland for nearly five and a half decades.[30]

While in Switzerland George married Stella Vera Weber, the daughter of a Swiss physics professor at the University of Neuchâtel.[31] They never had any children,[32] but George kept busy publishing numerous works in different areas of mathematics and science. In 1918, the same year he married, and for several years thereafter he continuously presented papers on number theory, astronomy, combinatorics, probability, series, and voting methods. During this same time he also did innovative work in advancing the analysis of integral functions.[33]

George and Stella enjoyed frequent walks near their home in Switzerland, and it was after one of these that George formulated his aptly named concept of “random walks.” Upon meeting the same couple six different times during one afternoon stroll, George began to wonder whether steps chosen in random directions would ever result in one reaching the same place he left off. After several experiments, he was able to prove that if the subject walked long enough, his hypothesis was true.[34]

By 1920, George had achieved the position of extraordinary professor at the Swiss Federal Institute, and continued to collaborate with Szegö on various problems and proofs. Shortly after George received a grant to study in England, their first collaborative work in two volumes Aufgaben und Lehrsätze aus der Analysis was published with great reception.[35]

This was only one of many publications that George was involved in; he wrote 13 books and scores of papers throughout his lifetime. After Stella and he moved to the U.S. due to Nazism in Germany,[36] George published probably the most famous of his titles, a book called How To Solve It. Never having been out of print since its publication, the text has sold over 1,000,000 copies; it has been translated into 17 different languages [37] and is still sold in many bookstores. [38]

Of his more lighthearted works, How To Solve It contains numerous and witty quotes, while also providing ingenious guidelines for problem solving. His method involves four steps:

1.  Understanding the problem,

2.  Devising a plan,

3.  Carrying out the plan, and

4.  Looking back,

which he summarizes throughout the book.

George also supplies interesting learning techniques, such as this mnemonic for remembering the first fourteen digits of pi (the number of letters in each word corresponds to the numerical value of its corresponding digit):

“How I need a drink, alcoholic of course, after the heavy chapters involving quantum mechanics!”[39]

Critics hailed the book, claiming that How To Solve It distinguished “…a line of demarcation between two eras, problem solving before and after Pólya.”[40]

Luckily for mathematics, however, not all of George’s work is directed towards the layman. For what has become known as the Hilbert-Pólya conjecture, George worked with David Hilbert to produce a proof of the Riemann hypothesis involving spectral theory.[41] He collaborated with other colleagues to develop the Burnside-Pólya Counting and “necklace” theorems, which examine all possible permutations of different numbers of colored objects. Pólya’s Enumeration Theorem involves combinatorics and generating functions, and is often combined with his aforementioned counting theorem.[42]

George proposed, titled, and proved the concept of two-dimensional “random walks;” he also studied limit laws and even named the Central Limit Theorem. He contributed some fundamental insights to our present knowledge of probability theory, Fourier transforms, stable distributions, roots of arbitrary polynomials, random variables in exchangeable sequences, and the theorem of continuity for moments.[43] Much of his work in these capacities is highly specialized to its field, and can be truly appreciated only by those aptly acquainted.[44]

However, applications still trickle down even to elementary levels. In fact, a great deal of George’s mathematical contributions have been “…elaborated on by other mathematicians and have become the foundations of important branches of mathematics.”[45] Many of the results that George found in different areas, especially probability theory, even appear today in textbooks as fundamental concepts or exercises. It might probably be very difficult to find a probability student who has not encountered his work. [46]

After moving to the United States, George took up residence in Palo Alto, California. He worked at Brown University and Smith College for a few years before securing a long-standing position at Stanford. It was here that he more deeply developed his concept of heuristic learning, which he felt was necessary to truly understand problem solving.[47] The approach encourages the study of “…methods and rules of discovery and invention … it’s purpose is to discover the solution of the present problem.”[48] Over the course of his instructional career at Stanford, this technique of “heuristics” served as a basis for his teaching philosophy.[49]

George officially retired from Stanford University in 1953, yet strove to continue his involvement with the mathematics world.[50] He periodically taught courses as a “Professor Emeritus” at Stanford until 1978 (he was 91 years old!).

George died on September 7, 1985 in Palo Alto – exactly a month after his friend Szegö.[51] He is survived by his wife of 71 years,[52] and a legend of brilliance and innovation. For his numerous and valuable mathematical contributions, George was awarded honorary memberships to the Hungarian Academy, the mathematical Association of Great Britain, the London Mathematical Society, and the Academie des Sciences in Paris. He was selected for positions in the National Academy of Sciences in the United States, the Académie Internationale de Philosophie des Sciences de Bruxelles, the American Academy of Arts and Sciences, and the California Mathematics Council, among others.[53]

Considering George’s extensive and significant influence in so many areas of mathematical study, these honors come as no surprise. His triumph through initial difficulty and consequent rise to prominence serves as a remarkable testimony to the value of hard work and persistence… mixed with a small measure of genius. We would all do well to follow George Pólya’s advice, both in mathematics and in life…

“If there is a problem you can’t solve, then there is an easier problem you can solve: find it.”

References

Albers, D. J., and Gerald L. Alexanderson (eds.), Mathematical People: Profiles and Interviews (Boston, 1985).

Pólya, George. Collected Papers. 4 vols. (Cambridge, Massachusetts; London, England: MIT Press. (1974 –1984).

Boas, R. P. “George Pólya.” Retrieved March 15 from http://books.nap.edu/books/030-0401988/gifmid/339.gif

Braden, Bart. “Polya's Geometric Picture of Complex Contour Integrals.”
Mathematics Magazine. Vol. 60, No. 5 © Dec. 1987. Mathematical Association of America. pp. 321-327

Pólya, George. How to Solve It. (Princeton University Press, 1957). 2nd ed.

“A Letter by Professor Polya.” The American Mathematical Monthly. Vol. 80, No. 1. © Jan. 1973 Mathematical Association of America. pp. 73-74

Dale, Nell, and John Lewis. “Problem Solving and Algorithm Design.” Chapter 6, Computer Science Illuminated. (Jones and Bartlett, 2004). 2nd ed.

Taylor, Peter. “George Pólya (1887 – 1985).” AMT Publishing. (2000).

Frank, Tibor. George Polya and the Heuristic Tradition. “Fascination with Genius in Central Europe.” Polanyiana, Vol. 6, No. 2. (1997).

Motter, A. “George Polya.”(no date). Retrieved March 10, 2005, from http://www.math.wichita.edu/history/men/polya.html.

Long, C.T., and D. W. DeTemple. Mathematical Reasoning for Elementary Teachers. Reading MA: Addison Wesley. (1996).

Alexanderson, Gerald L. The Random Walks of George Pólya. Washington, MAA. (2000).

Schoenfeld, A.H. “Pólya, problem solving, and education,” Mathematics Magazine. Vol. 60, No. 5. (1987).

Dahlke, Karl. “Polya’s Enumeration Theorem.” (2005) Retrieved 12 March 2005 from http://www.mathreference.com/cmb-gf.pet.html.

Titles Authored/Coauthored