# Rózsa Péter

### Quick Info

Budapest, Hungary

Budapest, Hungary

**Rózsa Péter**was a Hungarian mathematician and logician. She became a founder of recursive function theory.

### Biography

**Rózsa Péter**'s original name was Rósa Politzer but in the 1930s she, like many other Hungarians, changed her German style name to a Hungarian one. We choose to use the name Péter throughout this biography. Rósa's father was the lawyer Dr Gusztav Politzer and her mother was Irma Klein. It was a Jewish family and Rósa had two brothers, one older than her and the other younger. She grew up in a period in which great changes took place in the country. When she was born in Budapest, the city was one of the twin capitals of the Austro-Hungarian empire. However, there were dramatic changes as she was growing up. Allied with the Central Powers during World War I, Hungary suffered food shortages, many men were killed, and life was extremely difficult. Rósa attended Maria Terezia Girls' School, graduating in 1922. This school, on Andrassy Street in Budapest, was a six-year school for well-off people and provided an excellent education for those who could afford it. By the time that she graduated, World War I was over and Hungary had signed the Treaty of Trianon. Hungary was reduced to a third of its pre-war size and forced to pay a large indemnity.

Rózsa Péter studied at Pázmány Péter University (renamed Loránd Eötvös University in 1950) in Budapest. Following her father's wishes, she enrolled for a degree in chemistry. Her older brother was already studying medicine and Rózsa's father thought that if Rózsa studied chemistry she could usefully collaborate with her brother. During her chemistry studies she also attended lectures in mathematics given by Lipót Fejér, and through these lectures she discovered her love of mathematics. Then she changed her course and studied mathematics with Lipót Fejér and Jozef Kürschak. Another one to have an important influence on Rózsa Péter was László Kalmár who was a fellow student at Pázmány Péter University. Péter wrote in [12]:-

When I began my college education, I still had many doubts about whether I was good enough for mathematics. Then a colleague said the decisive words to me: it is not that I am worthy to occupy myself with mathematics, but rather that mathematics is worthy for one to occupy oneself with.After graduating in 1927, passing the examinations to qualify as a Gymnasium teacher of mathematics, Péter had to settle for odd jobs. The next few years were extremely difficult with the disastrous fall in world wheat prices in 1929 leading to the Great Depression. Foreign creditors called in their loans which Hungary couldn't repay. Deflation followed with many middle class people being dismissed from their jobs. University graduates cleaned the streets and there was widespread discontent. Péter had no possibility of getting a permanent teaching position and earned a living tutoring mathematics and giving private lessons. She undertook research for a doctorate, publishing papers, gave an address to the International Congress of Mathematicians in Zürich in 1932, and was awarded her doctorate with distinction in 1935. Two years later she became an editor of the recently founded

*Journal of Symbolic Logic*. However, her employment position became even worse in 1939 when the Fascist government passed anti-Semitic laws and Jews were not allowed to teach.

During the war years Péter worked on writing a book. It was published in 1943 but many copies were destroyed in the bombing of Budapest. We will describe the contents of this book below. The Nazis occupied Budapest from March 1944 and in November 1944 the government decreed that a Jewish ghetto be made in Budapest and several blocks of the former Jewish quarter was surrounded by a fence and wall. It was completely cut off by armed guards who prevented people getting out and food getting in. Péter was sent to the ghetto but in January 1945 the Soviet army took control of Budapest and, although the Nazis had planned to massacre the Jews in the ghetto, they were saved. Although Péter survived, her brother and many of her friends died during the war. Her first post, at the Budapest Teachers Training College, was obtained in 1945, eighteen years after graduating. When the College closed in 1955 she became a professor at Loránd Eötvös University and remained in this post until she retired in 1975.

Her first research topic was number theory but she became discouraged on finding that her results had already been proved by Robert Carmichael and L E Dickson. For a while Péter was so discouraged that she turned away from mathematics. She wrote and translated poetry but around 1930 she was encouraged to return to mathematics by her friend László Kalmár. He suggested Péter examine Kurt Gödel's work on incompleteness [10]:-

Without knowing how Gödel had proved various results in his landmark paper, she was able to devise her own, different proofs. This experience not only restored Péter's self-confidence, but it also pointed her research in the direction she would follow from then on.In a series of papers she became a founder of recursive function theory. All this happened in a remarkably short time for in 1932 she presented her paper

*Rekursive Funktionen*Ⓣ to the International Congress of Mathematicians in Zürich. Note that this paper appears under her original name of Rósa Politzer. All her subsequent papers appeared under the Hungarian version of her name, Rózsa Péter, that she adopted around 1932. She also presented a paper to the International Congress of Mathematicians in Oslo in 1936 entitled

*Über rekursive Funktionen der zweite Stufe*Ⓣ.

Walter Felscher, in a personal communication to me [EFR], described the context of Péter's work on recursive function theory:-

Recursive functions were invented during the 1920s in the Hilbert school, but nothing much was proved about them. Developing ideas of Herbrand, Gödel defined the more general 'general' recursive functions (to which Ackermann's function belongs) in his Princeton lectures 1933-34; soon after, the old functions received the name 'primitive recursive', and the general ones lost their adjective.During World War II, when she had been unable to take any teaching positions, Péter has produced the charming book

In a series of articles, beginning in 1934, Péter developed various deep theorems about primitive recursive functions, most of them with an explicit algorithmic content. I admire this work, and it may well be said that she forged, with her bare hands, the theory of primitive recursive functions into existence.

[On the other side, it was Kleene who, having attended Gödel's lectures, developed the theory of general (including partial) recursive functions; this is a much more conceptual than computational area.]

In 1951 Péter collected what was known by then, including her own work, in the book "Rekursive Funtionen". An English translation appeared only in 1967. It was the first book devoted exclusively to this topic, but

(1) there had been extensive chapters on this matter earlier in Hilbert-Bernays (1934-1939) where some of Péter's work was quoted, and

(2) the English speaking world did not read her book but read, instead, Kleene's book of 1952.

*Playing with Infinity*. The first publication was in Hungarian and was reviewed by John Kemeny who writes [7]:-

This book is a popular account of modern mathematical ideas. The author states that her purpose is to reach that very large section of the population which always wanted to find out what modern mathematics was like, but thought that it was too difficult to understand. She attempts to give a clear picture of as many advanced concepts as possible without sacrificing rigour. ... The author's humour makes every page enjoyable. ... The author seems to have found a perfect compromise between rigour and clarity.For a longer extract from the review by Kemeny see THIS LINK

Péter's book was translated into English and several editions were reviewed. Reuben Goodstein writes [4]:-

This is easily the best book on mathematics for everyman that I have ever seen. The author is both a highly creative mathematician and an experienced teacher of young children, and this happy combination, allied to a gift for lucid exposition has produced a delightful book ...Philip Peak writes [11]:-

This is a delightful book, and the mathematician as well as the layperson could profit from reading it.For longer extracts of these reviews of the English translations see THIS LINK.

Péter's important book

*Rekursive Funtionen*Ⓣ (1951) has already been mentioned above. Stephen Kleene writes in a review [8]:-

Beginning in 1932, Rózsa Péter has published a series of papers, examining the relationship of various special forms of recursion, and showing the definability of new functions by successively higher types of recursion, which establish her as the leading contributor to the special theory of recursive functions. ... In writing this book Ms Péter has carried out a considerable undertaking; and to go further would have constituted a still greater one, and required either a much larger book or a more compact style.For a longer extract of this review and extracts of other reviews of this important book see THIS LINK.

Péter's work on recursive functions certainly didn't end with this book. For example, in 1959 she presented a major paper

*Über die Verallgemeinerung der Theorie der rekursiven Funktionen für abstrakte Mengen geeigneter Struktur als Definitionsbereiche*Ⓣ to the International Symposium in Warsaw in September 1959. This was published in two parts totalling 68 pages in 1961 and 1962. In this paper Péter looks at words formed from a given alphabet of letters and develops a theory for such words generalising Peano's approach to a study of the integers.

From the mid 1950s Péter applied recursive function theory to computers. In 1976 her last book was on this topic

*Recursive Functions in Computer Theory*. The text was Hungarian with an English translation appearing in 1981. Let us quote from Péter herself about how her work on recursive functions became important in computer science [12]:-

I would like to win over those who consider mathematics useful, but colourless and dry - a necessary evil. I myself work in a field that was created for purposes internal to mathematics. This is the theory of the so-called 'recursive functions' - I would not have dreamed that this theory could also be applied practically. And today? My book on recursive functions was the second Hungarian mathematical book to be published in the Soviet Union, and precisely on the practical grounds that its subject matter has become indispensable to the theory of computers.In addition to her research contributions and her outstanding books showing non-mathematicians the joy of mathematics, Péter was enthusiastic in trying to improve mathematical education in Hungary. She had, of course, spent much time teaching school children but she also wrote school textbooks and worked on improving the curriculum. It seems appropriate to quote from [12] where we get an excellent picture of Péter as a school teacher:-

No other field can offer, to such an extent as mathematics, the joy of discovery, which is perhaps the greatest human joy. The schoolchildren that I have taught in the past were always attuned to this, and so I have also learned much from them. It never would have occurred to me, for instance, to talk about the Euclidean Algorithm in a class with twelve-year-old girls, but my students led me to do it. I would like to recount this lesson.Péter received many honours for her outstanding contributions. She was awarded the Kossuth Prize by the Hungarian Government in 1951, the Manó Beke Prize by the Janos Bolyai Mathematical Society in 1953, the Silver State Prize in 1970, and the Gold State Prize in 1973. She was elected to the Hungarian Academy of Sciences in 1973, being the first woman to be elected to the Academy.

What we were busy with was that I would name two numbers, and the students would figure out their greatest common divisor. For small numbers this went quickly. Gradually, I named larger and larger numbers so that the students would experience difficulty and would want to have a procedure. I thought that the procedure would be factorization into primes. They had still easily figured out the greatest common divisor of 60 and 48: "Twelve!" But a girl remarked: "Well, that's just the same as the difference of 60 and 48." "That's a coincidence," I said and wanted to go on. But they would not let me go on: "Please name us numbers where it isn't like that." "Fine. Sixty and 36 also have 12 as their greatest common divisor, and their difference is 24." Another interruption: "Here the difference is twice as big as the greatest common divisor." "All right, if this will satisfy all of you, it is in fact no coincidence: the difference of two numbers is always divisible by all their common divisors. And so is their sum." Certainly that needed to be stated in full, but having done so, I really did want to move on. However, I still could not do that. A girl asked: "Couldn't they discover a procedure to find the greatest common divisor just from that?" They certainly could! But that is precisely the basic idea behind the Euclidean Algorithm! So I abandoned my plan and went the way that my students led me.

As to her interests outside mathematics, as well as her interest in poetry which we have already discussed, we should mention the cinematic and dramatic arts. She died from cancer in 1977.

### References (show)

- D J Albers, G L Alexanderson and C Reid, Rozsa Peter 1905-1977, in
*More Mathematical People*(Harcourt Brace Jovanovich, 1990), 149. - H Andréka, Rósa Péter, in
*L S Grinstein and P J Campbell, Women of Mathematics*(Greenwood, Westport, Conn., 1987), 171-174. - A Császár, Rózsa Péter: February 17, 1905-February 16, 1977 (Hungarian),
*Mat. Lapok***25**(3-4) (1977), 257-258. - R L Goodstein, Review: Playing with Infinity, by Rozsa Peter,
*The Mathematical Gazette***46**(356) (1962), 157. - M Holt, Review: Playing with Infinity, by Rozsa Peter,
*Mathematics in School***6**(3) (1977), 35. - L Kalmár, R Péter's Work in the Theory of Recursive Functions, in
*Les Fonctions Recursives et Leur Applications*(1969). - J G Kemeny, Review: Playing with the Infinite. Mathematics for Outsiders, by Rózsa Péter,
*The Journal of Symbolic Logic***13**(3) (1948), 141-142. - S Kleene, Review: Rekursive Funktionen, by Rózsa Péter,
*Bull. Amer. Math. Soc.***58**(1952), 270-272. - S M Moite, Rozsa Peter, in R Young (ed.),
*Notable Mathematicians*(Gale Research, 1998), 394-395. - E Morris and L Harkleroad, Rósa Péter : Recursive Function Theory's Founding Mother,
*The Mathematical Intelligencer***12**(1) (1990), 59-61. - P Peak, Review: Playing with Infinity: Mathematical Explorations and Excursions, by Rozsa Peter,
*The Mathematics Teacher***70**(3) (1977), 282. - R Péter, Mathematics is beautiful,
*The Mathematical Intelligencer***12**(1) (1990), 60-64. - R M Robinson, Review: Rekursive Funktionen, by Rózsa Péter,
*The Journal of Symbolic Logic***16**(4) (1951), 280-282. - R M Robinson, Review: Rekursive Funktionen (2nd edition), by Rózsa Péter,
*The Journal of Symbolic Logic***23**(3) (1958), 362-363. - Rózsa Péter and Recursion, in
*History of Math*(Key Curriculum Press, 1996), 197-198. - Rózsa Péter (1905-1977),
*Ann. Univ. Sci. Budapest Eötvös Sect. Math.***20**(1977), 3. - I Ruzsa and J Urbán, In memoriam Rózsa Péter (Hungarian),
*Mat. Lapok***26**(3-4) (1978), 125-137. - I Tamassy, Interview with Rozsa Peter,
*Modern Logic***4**(3) (1994), 277-280.

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Written by J J O'Connor and E F Robertson

Last Update March 2014

Last Update March 2014