George Pólya and the Heuristic Tradition
Fascination with Genius in Central Europe
„How To Solve It”
Hungarian-born mathematician George Pólya (1887-1985) was one of those who channeled the Hungarian and, more broadly speaking, European school tradition into American education in a series of books and articles, starting with his 1945 book How to Solve It. In 1944 Pólya remembered the time when, at the turn of the century in Hungary,
he was a student himself, a somewhat ambitious student, eager to understand a little mathematics and physics. He listened to lectures, read books, tried to take in the solutions and facts presented, but there was a question that disturbed him again and again: ‘Yes, the solution seems to work, it appears to be correct; but how is it possible to invent such a solution? Yes, this experiment seems to work, this appears to be a fact; but how can people discover such facts? And how could I invent or discover such things by myself?‘
Pólya came from a distinguished family of academics and professionals. His father, Jakab, an eminent lawyer and economist provided the best education for his children. They included George‘s brother, Jenõ Pólya, the internationally recognized professor of surgery and honorary member of the American College of Surgeons. George Pólya first studied law, later changing to languages and literature, then philosophy and physics, to settle finally on mathematics, in which he received his Ph.D. in 1912. He was a student of Lipót Fejér, whom Pólya considered one of the key people who influenced Hungarian mathematics in a definitive way.
Pólya felt that Fejér, the competitive examination in mathematics and the mathematical journal Középiskolai Matematikai Lapok were responsible for the development of a large number of major mathematicians in Hungary.
For emancipated Jews in Hungary, who received full rights as citizens in 1867, it was the Hungarian Law 1867:XII that made it possible, among other things, to become teachers in high schools and even professors at universities. This is one of the reasons that lead to the explosion of mathematical talent in Hungary, just as happened in Prussia after the emancipation of Jews in 1812. John Horváth of the University of Maryland was one who pointed out the overwhelming majority of Jewish mathematicians in Hungary in the early part of the 20th century.
Culture in the second half of the nineteenth century became a matter of very high prestige in Hungary, where the tradition to respect scientific work started to loom large after the Austro-Hungarian Ausgleich (Compromise) in 1867 between Austria and Hungary. For sons of aspiring Jewish families, a professorship at a Budapest university or membership in the Hungarian Academy of Letters and Science promised entry into the Hungarian élite and eventual social acceptance in Hungarian high society, an acknowledged way to respectability. Pursuing scientific professions, particularly mathematics, secured a much desired social position for sons of Jewish-Hungarian families, who longed not only for emancipation, but for full equality in terms of social status and psychological comfort. Thus, in many middle class Jewish families, at least one of the sons was directed into pursuing a career in academe.
Distinguished scientists such as Manó Beke, Lipót Fejér, Mihály Fekete, Alfréd Haar, Gyula and Dénes König, Gusztáv Rados, Mór Réthy, Frigyes Riesz and Lajos Schlesinger belonged to a remarkable group of Jewish-Hungarian mathematical talents, who, after studying at major German universities, typically Göttingen or Heidelberg, became professors in Hungary‘s growing number of universities before World War I. Several of them, like Gyula König and Gusztáv Rados, even became university presidents at the Technical University of Budapest. There were several renowned scientists active in related fields, such as physicist Ferenc Wittmann, engineer Donát Bánki and others. Mathematicians were also needed outside the academoc world: just before the outbreak of World War I George Pólya was about to join one of Hungary‘s big banks, at the age of 26, with a Ph.D. in mathematics and a working knowledge of four foreign languages in which he already publisched important articles.
Despire what we know about the social conditions which nurtured and even forced out the talent of these many extraordinary scientists, how this occured still remains somewhat mysterious. Stanislaw Ulam recorded an interesting quote from John von Neumann when describing their 1938 journey to Hungary in his Advdentures of a Mathematician.
I returned to Poland by train from Lillafüred, traveling through the Carpathian foothills. ... This whole region on both sides of the Carpathian Mountains, which was part of Hungary, Czechoslovakia, and Poland, was the home of many Jews. Johnny [von Neumann] used to say that all the famous Jewish scientists, artists, and writers who emigrated from Hungary around the time of the first World War came, either directly or indirectly, from these little Carpathian communities, moving up to Budapest as their material conditions improved. The [Nobel Laureate] physicist I[sidor] I[saac] Rabi was born in that region and brought to America as an infant. Johnny used to say that it was a coincidence of some cultural factors which he could not make precise: an external pressure on the whole society of this part of Central Europe, a feeling of extreme insecurity in the individuals, and the necessity to produce the unusual or else face extinction.
An interesting fact about Jewish-Hungarian geniuses at the turn of the century was that several of them could multiply huge numbers in their head. This was true of von Kármán, von Neumann and Edward Teller. Von Neumann, in particular commanded extraordinary mathematical abilities. Nevertheless, there is no means available to prove that this prodigious biological potential was more present in Hungary at the turn of the century than elsewhere in Europe.
Similarly, heuristic thinking was also a common tradition that many other Hungarian mathematicians and scientists shared. John Von Neumann‘s brother remembered the mathematician‘s „heuristic insights” as a specific feature that evolved during his Hungarian childhood and appeared explicity in the work of the mature scientist.
Von Neumann‘s famous high school director, physics professor Sándor Mikola, made a special effort to introduce heuristic thinking in the elementary school curriculum in Hungary already in the 1900s.
Fejér drew a numer of gifted students to his circle, such as Mihály Fekete, Ottó Szász, Gábor Szegõ and, later, Paul Erdõs. His students remembered Fejér‘s lectures and seminars as „the center of their formative circle, its ideal and focal point, its very soul”. „There was harcly an intelligent, let alone a gifted, student who could exempt himself from the magic of his lectures. They could not resist imitatting his stress patterns and gestures, such was his personal impact upon them.” George Pólya remembered Fejér‘s personal charm and personal drive to have been responsible for his great impact: „F[ejér] influenced more than any other single person the development of math[ematic]‘s Hungary. ...”
In Budapest, Pólya was one of the founders, along with Károly Polányi, of the student society called Galilei Kör [Galileo Circle], where he lectured on Ernst Mach. The Galileo Circle (1908-1918) was the meeting place of radical intellectuals, mostly Jewish college students from the up and coming Budapest families of a new bourgeoisie. Members of the circle became increasingly radical and politicized. Oddly enough, the Communists of 1919 found it far too liberal, while the extremist right-wing régime of Admiral Horthy considered it simply Jewish. In a Hungary of varied totalitarian systems, the radical-liberal tradition remained unwanted. Soon however, Pólya went to Vienna where he served the academic year of 1911, after receiving his doctorate in mathematics in Budapest. In 1912-13 he wen to Göttingen, and later to Paris and Zurich, where he took an appointment at the Eidgenössiche Technische Hochschule (Swiss Federal Institute of Technology). He became full professor at the ETH in 1928.
A distinguished mathematician, Pólya drew on several decades of teaching mathematics based on new approaches to problem solving, first as a professor in Zurich, Switzerland, and later in his life at Stanford, California. It was in Zurich that Pólya and fellow Hungarian Gábor Szegõ started their long collaboration by signing a contract in 1923 to publish their much acclaimed joint collection of Aufgaben und Lehrsätze aus der Analysis. Considered a mathematical masterpiece even today, Aufgaben und Lehrsätze took several years to complete, and it continues to impress mathematicians not only with the range and depth of the problems contained in it, but also with its organization: to group the problems not by subject but by solution method was a novelty. His primary concern had always been to provide and maintain „an independence of reasoning during problem solving”, an educational goal he declared to be of paramount impoprtance when addressing the Swiss Association of Professors of Mathematics in 1931. Several of his articles on the subject preceded this lecture, probably the earliest being published in 1919. Pólya had provided a model for problem solving by the time he was in Berne, Switzerland, suggesting „a systematic collection of rules and methodological advices”, which he considered „heuristics modernized”.
Pólya was active in a number of important fields of mathematics, such as theory probability, complex analysis, combinatorics, analytic number theory, geometry, and mathematical physics. In the United States after 1940, and at Stanford as of 1942, Pólya became the highest authority on the teaching of problem solving in mathematics.
With his arrival at the United States, Pólya started a new career based on his new found interst in teaching and in heuristics. He developed several new courses such as his „Mathematical Methods in Science”, which he first offered in the Autumn 1945 Quarter at Stanford, introducing general and mathematical methods, deduction and induction, the relationship between mathematics and science, as well as the „use of physical intuition in the solution of mathematical problems”. In his popular and often repeated Mathematics 129 course on „How to Solve the Problem?” Pólya taught mathematical invention and mathematical teaching, quoting Samuel Butler:
All the inventions that the world contains
Were not by reason first found out, nor brains
But pass for theirs, who had the luck to light
Upon them by mistake or oversight.
He surveyed all aspects of a problem, general and specific, restating it in every possible way and pursued various courses that might lead to solving it. He studied several ways to prove a hypothesis or modify the plan, always focusing on finding the solution. He compiled a characteristic list of „typical questions for this course”, which indeed contained his most important learnings from a long European shcooling.
In a course on heuristics he focused on problems and solutions, using methods from classical logic to heuristic logic. Offering the course alternately as Mathematics 110 and Physical Sciences 115, he sought to attract a variety of students, including those in education, psychology and philosophy. The courses were based on Pólya‘s widely used textbook How to Solve It.
In due curse, Pólya published several other books on problem solving in mathematics such as the two-volume Mathematics and Plausible Reasoning (1954), and Mathematical Discovery, in 1965. Both became translated into many languages. Towards the end of his career his „profound influence of mathematical education” was internationally recognized.
Pólya‘s significance in general methodology seems to have been his proposition to interpret heuristics as problem solving, more specifically, the search for those elements in a given problem that may help us find the right solution. For Pólya, heuristics equaled „Enfindungskunst”, a way of inventive or imaginative power, the ability to invent new strategems of learning, and bordered not only on mathematics and philosophy but also psychology and logic. In this way a centuries-old European tradition was renewed and transplanted into the United States where Pólya had teremendous influence on subsequent generations of teachers of mathematics well into the 1970s. In 1971 the aged mathematician received an honorary degree at the University of Waterloo where he addressed the Convocation, appropriately calling for the use of „heuristic proofs”: „In a class for future mathematicians you can do something more sophisticated: You may peresent first a heuristic proof, and after that a strict proof, the main idea of which was foreshadowed by the heuristic proof. You may so do something important for your students: You may teach them to do research.” „Heuristics should be given a new goal”, Pólya argued, „that should in no way belong to the realm of the fantastic and the utopian”.
Problem wolving for Pólya was seen as „one third mathematics and two thirds common sense”. This was a tactic which he emphatically suggested for teachers of mathematics in American high schools. If the teaching of mathematics neglects this tactic, he commented, it misses two important goals: „It fails to give the right attitude to future users of mathematics, and it fails to offer an essential ingredient of general education to future non-users of mathematics.”
Throughout his career as a teacher he strongly opposed believing in what authorities profess. Teachers and principals, he argued, „should use their own experience and their own judgment.” His matter-of-fact, experience-based reasoning has been repeatedly described in books and articles. He even made two films on the teaching ot mathematics („Let Us Teach Guessing”, a prize winner at the American Film Festival in 1968; „Guessing and Proving”, based on an Open University Lecture, Reading, 1962). The most simple and straightforward summary of his ideas on teaching was presented in the preface of a course that he gave at Stanford and subsequently published in 1967. Pólya‘s description is the best introduction to heuristic thinking.
Start from something that is familiar or useful or challenging: From some connection with the world around us, from the prospect of some application, from an intuitive ide.
Don‘t be afraid of using colloquial alnguage when it is more suggestive than the conventional, precise terminology. In fact, do not introduce technical terms before the student can see the need for them.
Do not enter too early or too far into the heavy details of a proof. Give first a general idea or just the intuitive germ of the proof.
More generally, realize that the natural way to learn is to learn by stages: First, we want to see an outline of the subject, to perceive some concrete source or some possible use. Then, graudally, as soon as we can see more use and connections and interest, we take more willingly the trouble to fill in the details.
Pólya had lasting influence on a variety of thinkers in and beyond mathematics. The first curriculum recommendation of the National Council of the Teachers of Mathematics suggested that „problem solving be the focus of school mathematics in the 1980s [in the U.S.]. The 1980 NCTM Yearbook, published as Problem Solving in School Mathematics, the Mathematical Association of America‘s Compendia of Applied Problems and the new editor of the American Mathematical Monthly, P.R. Halmos, all called for more use of problems in teaching in 1980. Philosophers Imre Lakatos, who described mathematical heuristics as his main field of interest in 1957, acknowledging his debt to Pólya‘s influence, and particularly to How to Solve It, which he translated into Hungarian.
Critics, however, like mathematician Alan H. Schoenfeld, pointed out that wile Pólya‘s influence extended „far beyond the mathematics education community”, „the scientific status of Pólya‘s work on problem solving strategies has been more problematic.” Students and instructors often felt that the heuristics-based approach rarely improved the actual problem-solving performance itself. Researchers in artificial intelligence claimed that they were unable to write problem solving programs using Pólya‘s heuristics. „We suspect the strategies he describes epiphenomenal rather than real”. Recent work in cognitive science, however, has provided methods for making Pólya‘s strategies more accessible for problem solving instruction. New studies have provided clear evidence that students can significantly improve their problem-solving performance through heuristics. „It may be possible to program computer knowledge structures capable of supporting heuristic problem-solving strategies of the type Pólya described.”
The Stanford Mathematics Competition
Initiated jointly by Professors George Pólya and Gabor Szegõ, one of the most significant Hungarian contributions to the teaching of mathematics was the introduction of the Stanford Mathematics Competition for high school students. Modelled after the Eötvös Competition organized in Hungary from 1894 on, the main purpose of the competition was to discover talent, and to revive the competitive spirit of the Eötvös Competition, which Szegõ himself won in 1912. This contest was held annually for over 30 years until it was terminated in 1928. Stress was laid on inherent cognitive ability and insight rather than upon memorization and speed. Those who were able to go beyond the question posed were given additional credit. Those who were cognizant of the preponderance of Hungarian mathematicians were tempted to speculate upon the relationship between the Eötvös Prize and „the mathematical fertility of Hungary”. Winners of the Eötvös Prize have included Lipót Fejér, Theodore von Kármán, Alfréd Haar, George Pólya, Frigyes Riesz, Gabor Szegõ, and Tibor Radó.
The Stanford competition was started in 1946 and discontinued in 1965 wen the Stanford Department of Mathematics turned more towards graduate training. When first started, the Stanford Ecamination was administered to 322 participants in 60 California high schools. The las examination, in 1965, was administered to about 1200 participants in over 150 larger schools in seven states from Nevada to Montana. The Stanford Univesity Competitive Examination in Mathematics emphasized „originality and insight rather than routine competence”. Even a typical question required a high degree of ingenuity and the winning student was asked „to demonstrate research ability”.
Organizers of the competition thought of mathematics „not necessarily as an end in itself, but as an adjunct necessary to the study of any scientific subject”. It was suggested that ability in mathematical reasoning correlated with succdess in higher education in any field. Also, the discovery of singularly gifted students helped identify the originality of mind displayed by grappling with difficult problems: mathematical ability was regarded as an index of general capacity. Those responsible for the competition were firmly convinced that Dan early manifestation of mathematical ability is a definite indication of excepcional intelligence and suitability for intellectual leadeship.” Several of the winners of the Stanford competition did not go into mathematics but went on to specialize in electrical engineering (1946), physics (1947), biology (1948) and geology (1956).
It is intersting to note that by introducing Pólya‘s article about the 1953 Stanford Competitive Examination, the California Mathematics Council Bulletin found it important to make a connection between „the best interests of democracy” and the need „that our superior students be challenged by courses of appropriate content, encouraged to prograss in accordance with their capacities”. It seems as if the Competitive Examination was viewed by some as reflecting the dangerously mounting international tensions, somewhat forecasting the era of the Sputnik fears yet to come. Speaking at the National Council of Teachers of Mathematics in 1956, Gábor Szegõ articulated this opinion when declaring that „much is said in these days about the pressing need for science and engineering graduates. Our view is that the nation needs just as well good humanists, lawyers, economists, and politicla scientists in its present struggle. This is a view which can be defended, I think, in very strong terms.” (This ominous reference was dropped from a similar introduction by 1957.)
Through its long and distinguished tenure the Stanford examination proved to be a pioneer in the discovery of mathematical talent not only in California and the West Coast, but nationally. To this day, George Pólya is best remembered in the United States as one who introduced European models of competitive educational methods of problem solving in mathematics. He served as one of the several bridges that linked Central European as well as distinctly Hungarian patterns of thinking and reasoning with American achievements in problem solving. The study of Pólya may reveal some of the Central European origins of heuristic thinking in the United States.
19. G. Pólya, How to Solve It. A New Aspect of Mathematical Method (Princeton, N.J.: Princeton University Press, 1945). How to Solve it has never been out of print and has sold well over 1 million copies. It has been translated into 17 languages, probably a record for a modern mathematics book. Gerald L. Alexanderson, „Obituary. George Pólya”, Bulletin of the London Mathematical Sociaty, Vol. 19, 1987, p. 563, 603.
20. G. Pólya, „How to Solve It”, op. cit., p. vi.
21. Vilmos Milkó, Pólya Jenõ emlékezete [In memoriam Jenõ Pólya], Archivum Chirurgicum, Vol. 1, No. 1, 1948, p. 1.
22. G. Pólya, „Leopold Fejér”, Journal of the London Mathematical Society, Vol. 36., 1961. p. 501.
23. R. Hersch and V. John-Steiner, „A Visit to Hungarian Mathematics”, Ms., pp. 35-37.
I received a copy of this arcitle from Professor Gerald L. Alexanderson of the Department of Mathematics, Santa Clara University, Santa Clara, CA. John Horváth compared this explosion of Jewish talent after the Jewish emancipation to the surprising number of sons of Protestant ministers entering the mathematical profession in Hungary after World War II, „Those kids would have become Protestant ministers, just as the old ones would have become rabbis... mathematics is the kind of occupation where you sit at your desk and read. Instead of reading the Talmud, you read proofs and conjectures. It‘s really a very similar occupation”. R. Hersch and V. John-Steiner, op. cit., p. 37.
24. György Pólya to Baron Gyula Madarassy-Beck, Paris, February 23, 1914. I am grateful to Professor Gerald Alexanderson of the University of Santa Clara for showing me this document as well as his collection of Pólya documents that were to be transferred to the George Pólya Papaers, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.
25. Nobel Prize in Physics, 1944.
26. S. M. Ulam, Adventures of a Mathematician (New York: Scribner‘s, 1976), p. 11.
Cf. Tibor Fabian, „Carpathians Were a Cradle of Scientists”, Princeton, NJ. November 16, 1989, The New York Times, December 2, 1989. -- George Pólya‘s nephew John Béla Pólya had an even more surprising, though cautious proposition to make. He suggested that through George Pólya‘s mother, Anna Deutsch (1853-1939), Pólya was realted to Eugene Wigner and Edward Teller, „who are thought to have” acestry originating from the same region between the towns of Arad and Lugos in Transsylvania (the Hungary, today Roumania). Though this relationship is not yet documented and should be taken at this point merely as a piece of Pólya family legend, it is nonethelles an interesting reflexion of the strong belief in the productivity of the Jewish community in North-Eastern Hungary and Transsylvania in terms of maethematical talent. John Béla Pólya, „Notes on George Pólya‘s family”, attached to John Béla Pólya to Gerald L. Alexanderson, Greensborough, Australia, July 28, 1986. -- I am deeply grateful to Gerald L. Alexanderson of Santa Clara University, Santa Clara, CA. for his generous and highly informative support of my research on George Pólya.
27. Norman Macrae, John von Neumann (New York: Pantheon, 1992), p. 9. J. M. Rosenberg, Computer Prophets (New York: Macmillan, 1969) p. 155. ff. Edward Teller and Alan Brown, The Legacy of Hiroshima (Garden City: Doubleday, 1962) p. 160. Cf. William O. McCagg, op. cit.. 211.
28. Nicholas A. Vonneuman, John Von Neumann as Seen by His Brother(Meadowbrook, PA, 1987), p. 44.
29. Sándor Mikola, „Die heuristischen Methode in Unterricht der Mathematik der unteren Stufe”, in: E. Beke und. S. Mikola, Hrg., Abhandlungen über die Reform des mathematischen Unterrichts in Ungarn (Leipzig und Berlin: Teubner, 1911), pp. 57-73.
30. Gábor Szegõ, „[Lipót Fejér],”, MS. Gábor Szegõ Papers, SC 323, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.
31. [Lecture outline, n.d. unpublished MS] George Pólya Papers, SC 337, 87-034, Box 1, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.
32. The full history of the Galileo Circle is yet to be written. Important documents were published by Zoltán Horváth in 1971: „A Galilei Körre vonatkozó ismeretlen dokumentumok”, Századok, Vol. 105, No. 1, 1971, pp. 95-104. Cf. György Litván, Magyar gondolat--szabad gondolat [Hungarian Thought-- Free Thought] (Budapest: Mwagvetõ, 1978); cf. György Litván, „Jászi Oszkár, A magyar progresszió és a nemzet”, [Oscar Jaszi, Hungarian Progressives and the Nation] in Endre Kiss, Kristóf János Nyíri, eds., A magyar gondolkodás a századelõn [Hungarian Philosophy at the Turn of the Century], (Budapest: Kossuth, 1977). Litván pointed out that while similar social science organizations, such as Társadalomtudományi Társaság or Húszadik Század had a fair number of gentile contributors, the Galileo Circle almost exclusively drew upon progressive Jewish students. Cf. Attila Pók, A magyarországi radikális demokrata ideológia kialakulása. A „Huszadik Század” társadalomszemléléete (1900-1907) [The Rise of Democratic Radicalism in Hungary: the Social Concept of Huszadik Század (1900-1907)] (Budapest: Akadémiai Kiadó, 1990) p. 152-165.
33. Georg Pólya-Gábor Szegõ, Aufgaben und Lehrsätze aus der Analysis (Berlin: Springer, 1925, new editions: 1945, 1954, 1964, 1970-71), Vols. I-II; Translations: English, 1972-76; Bulgarian, 1973; Russian, 1978; Hungarian, 1980-81.
34. Gerald L. Alexanderson, „Obituary. George Pólya”, op. cit., pp. 562.
35. G. Pólya, „Comment chercher la solution d‘un problème de mathématiques?” L0enseignement mathématique, 30e année, 1931, Nos. 4-5-6.
36. G. Pólya, „Geometrische Darstellung einer Gedankenkette”, Schweizerische Pädagogische Zeitschrift, 1919.
37. G. Pólya, „Comment chercher”, op. cit.
38. Gerald L. Alexanderson, „Obituary. George Pólya”, Bulletin of the London Mathematical Society, Vol. 19, 1987, p. 563; on „Pólya the mathematician and teacher”, see pp. 566-570.
39. Paul Kirkpatrick, Acting Dean, School of Physical Sciences, Stanford University, Course outline, September 4, 1945, George Pólya Papaers, SC 337, 87-137, Box 2, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.
40. G. Pólya, „Elementary Mathematics from Higher Point of View”, Mathematics 129, George Pólya Papers, SC 337, 86-036, Box 1, Folder 9, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA. Cf. Samuel Butler, „Miscellaneous Thoughts”, in: The Poems of Samuel Butler, Vol. II. (Chiswick: C. Willingham, 1822), p. 281.
41. G. Pólya, „Elementary Mathematics from a Higher Point of View”, Survey of Typical Questions, George Pólya Papers, SC 337, 87-147, Box 3, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA. -- Pólya was indeed very well read and liked to show his erudition by quoting Socrates, Descartes, Leibniz, Kant, Herbert Spencer, Thomas Arnold, J. W. Goethe, Leonhard Euler and his famous colleagues, such as Albert Einstein, adn many others. George Pólya Papers, SC 337, 87-034. Box 1&3, 87-147, Box 2, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.
42. Untitled course description, nd. George Pólya Papers, SC 337, 86-036, Box 1, Folder 3, Department of Special Collekctions and University Archives, Stanford University Libraries, Stanford, CA.
43. G. Pólya Mathematics and Plausible Reasoning (Princeton, N.J.: Princeton University Press, 1954, 2nd ed. 1968), Vols. 1-2. Translations: Bulgarian, 1970; French, 1957-58; German, 1962-63; Japanese, 1959; Roumanian, 1962; Russian, 1957, 1975; Spanish, 1966; Turkish, 1966. G. Pólya, Mathematical Discovery. On Understanding, Learning and Teaching Problem Solving(New York - London - Sidney: John Wiley and Sons, 1965, Vols. 1-2; combined paperback ed. 1981), Translations: Bulgarian, 1968; French, 1967; German, 1966, 1967, 1979, 1983; Hungarian, 1969, 1979, Italian, 1970-71, 1979, 1982; Japanese, 1964; Polish, 1975; Roumanian, 1971; Russian 1970, 1976, Cf. Gerald L. Alexanderson, „Obituary, George Pólya”, op. cit. pp. 604-605.
44. A good example was the Second International Congress on Mathematical Education at the University of Exeter, England. Cf. The invitation sent to Pólya by the Chairman of the Congress, Professor Sir James Lighthill, FRS, June 24, 1971. [Cambridge] George Pólya Papers, SC 337, 87-034, Box 1, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.
45. G. Pólya, „Die Heuristik. Versuch einer vernünftigen Zielsetzung”, Der Mathematikunterricht, Heft 1/64 (Stuttgart: Ernst Klett, 1964); cf. ‘L‘Heuristique est-elle un sujet d‘étude raisonnable?‘, La méthode dans les sciences modernes, ‘Travail et Méthode‘, numéro hors série, pp. 279-285.
46. G. Pólya, „Guessing and Proving”. Adddress delivered at the Convocation of the University of Waterloo, October 29, 1971. George Pólya Papers, SC 337, 87-034, Box 1, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.
47. G. Pólya, „Die Heuristik”, op. cit., p.5
48. George Pólya, Untitled note, n.d., George Pólya Papers, SC 337, 87-034, Box 1, department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.
49. G. Pólya, „Formation, Not Only Information”, Address at the Mathematical Association of America, George Pólya Papers, SC 337, 87--034, Box 1, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.
50. George Pólya tob Robert J. Griggin, Stanford, June 12, 1962. George Pólya Papers, SC 337, 87-034, Box 2, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.
51. George Pólya to Anthony E. Mellor, Harper and Row, Stanford, March 11, 1974; Stanford University News Service, Februar 17, 1969. George Pólya Papers, SC 337, 87-034, Box 1, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.
52. George Pólya, „Preface”, MS George Pólya Papers, SC 337, 87-034, Box 1, Department of Special Collections and Unviersity Archives, Stanford University Libraries, Stanford, CA.
53. Imre Lakatos to Dr Maier (Rockefeller Foundation), Cambridge, England, May 5, 1957. George Pólya Papers, SC 337, 87-137, Box 1, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.--In turn, Pólya expressed his admiration for Lakatos‘s „Proofs and Refutations”, and recommended him as Professor of Logic at the London School of Economics and Political Science, „with special reference to the Philosophy of Mathematics”. George Pólya to Walter Adams (LSE), Stanford, CA, January 13, 1969, George Pólya Papers, SC 337, 87-034, Box 1, Department of Special Collections and University Archives, Stanford University Libraries, Stanford, CA.
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