罗莎·培特Rózsa Péter

罗莎·培特的原名是Rósa Politzer，但在20世纪30年代，她像许多其他匈牙利人一样，将自己的德式名字改为匈牙利式名字。在本传记中，我们选择使用培特这个名字。Rósa的父亲是律师Gusztav Politzer博士，母亲是Irma Klein。这是一个犹太家庭，Rósa有两个兄弟，一个比她大，另一个比她小。她成长于国家发生巨大变革的时期。当她出生在布达佩斯时，这座城市是奥匈帝国的双首都之一。然而，在她成长过程中发生了戏剧性的变化。第一次世界大战期间，匈牙利与同盟国结盟，遭受了食物短缺，许多男子被杀，生活极其艰难。Rósa就读于Maria Terezia女子学校，于1922年毕业。这所学校位于布达佩斯的Andrassy街，是一所为富裕人家设立的六年制学校，为那些能负担得起的人提供了优质教育。到她毕业时，第一次世界大战已经结束，匈牙利签署了特里亚农条约。匈牙利的领土缩减到战前的三分之一，并被迫支付巨额赔款。

培特在布达佩斯的Pázmány 培特大学（1950年更名为厄特沃什·罗兰大学）学习。遵照父亲的意愿，她报名攻读化学学位。她的哥哥已经在学医，培特的父亲认为如果培特学习化学，她可以有效地与哥哥合作。在化学学习期间，她还参加了费耶尔的数学讲座，通过这些讲座她发现了自己对数学的热爱。然后她改变了课程，跟随费耶尔和屈尔沙克·约瑟夫学习数学。另一个对培特产生重要影响的人是拉斯洛·卡尔马，他是Pázmány 培特大学的同学。培特在[12]中写道：-

当我开始上大学时，我仍然非常怀疑自己是否足够优秀，能够学习数学。后来一位同事对我说了决定性的话：不是我配得上从事数学，而是数学配得上让人去从事。

1927年毕业后，通过了成为文理中学数学教师的资格考试，培特不得不将就做些零工。接下来的几年极其艰难，1929年世界小麦价格灾难性下跌，导致了大萧条。外国债权人催收贷款，匈牙利无法偿还。通货紧缩随之而来，许多中产阶级人士被解雇。大学毕业生打扫街道，普遍不满情绪蔓延。培特没有可能获得永久教职，靠辅导数学和私人授课谋生。她进行博士研究，发表论文，1932年在苏黎世国际数学家大会上作了报告，并于1935年以优异成绩获得博士学位。两年后，她成为新创办的Journal of Symbolic Logic的编辑。然而，1939年法西斯政府通过反犹法律，禁止犹太人教书，她的就业状况变得更加糟糕。

在战争年代，培特致力于写一本书。该书于1943年出版，但许多副本在布达佩斯轰炸中被毁。我们将在下面描述这本书的内容。纳粹从1944年3月起占领布达佩斯，1944年11月，政府下令在布达佩斯建立犹太人隔离区，前犹太区的几个街区被栅栏和墙围起来。它被武装警卫完全切断，阻止人们出去和食物进入。培特被送往隔离区，但在1945年1月，苏联军队控制了布达佩斯，尽管纳粹计划屠杀隔离区中的犹太人，但他们获救了。虽然培特幸存下来，但她的兄弟和许多朋友在战争中丧生。她的第一个职位是在布达佩斯教师培训学院，于1945年获得，距毕业十八年。1955年学院关闭后，她成为厄特沃什·罗兰大学的教授，并一直担任此职直到1975年退休。

她的第一个研究课题是数论，但当她发现自己的结果已被罗伯特·丹尼·卡迈克尔和伦纳德·尤金·迪克森证明后，便灰心丧气。有一段时间，培特如此沮丧，以至于远离了数学。她写诗和翻译诗歌，但在1930年前后，她的朋友拉斯洛·卡尔马鼓励她重返数学。他建议培特研究库尔特·弗雷德里希·哥德尔关于不完性的工作[10]：

在不知道库尔特·弗雷德里希·哥德尔如何在其里程碑式论文中证明各种结果的情况下，她能够设计出自己不同的证明。这一经历不仅恢复了培特的自信，还将她的研究引向了她从此以后所遵循的方向。

在一系列论文中，她成为递归函数理论的创始人之一。这一切发生得异常迅速，因为1932年她在苏黎世国际数学家大会上提交了论文Rekursive Funktionen Ⓣ（递归函数）。请注意，这篇论文以她的原名Rósa Politzer发表。她随后的所有论文都以她大约1932年采用的匈牙利语名字培特发表。她还在1936年奥斯陆国际数学家大会上提交了一篇论文，题为Über rekursive Funktionen der zweite Stufe Ⓣ（论递归函数，第二阶段）。

Walter Felscher 在给我的私人通信 [EFR] 中，描述了 培特 在递归函数论方面工作的背景：-

递归函数是在 1920 年代由 大卫·希尔伯特 学派发明的，但关于它们并没有证明多少东西。库尔特·弗雷德里希·哥德尔 发展了 雅克·埃尔布朗 的思想，在其 1933-34 年普林斯顿讲座中定义了更一般的“一般”递归函数（威廉·阿克曼 的函数属于此类）；不久之后，旧函数获得了“原始递归”的名称，而一般函数则失去了这个形容词。
从 1934 年开始的一系列文章中，培特 发展了关于原始递归函数的各种深刻定理，其中大多数具有明确的算法内容。我钦佩这项工作，可以说她赤手空拳地将原始递归函数理论锻造成形。
[另一方面，是 Kleene 在听了 库尔特·弗雷德里希·哥德尔 的讲座后，发展了一般（包括部分）递归函数理论；这是一个远比计算更概念化的领域。]
1951 年，培特 将当时已知的内容，包括她自己的工作，收集在《Rekursive Funtionen》一书中。英译本直到 1967 年才出现。这是第一本专门讨论这个主题的书，但是
(1) 此前在 大卫·希尔伯特-保罗·贝尔奈斯 (1934-1939) 中已有关于此事的详尽章节，其中引用了 培特 的一些工作，并且
(2) 英语世界没有读她的书，而是读了 克林 1952 年的书。

在第二次世界大战期间，当她无法担任任何教职时，培特 创作了迷人的书 Playing with Infinity。首次出版是匈牙利语，由 约翰·克米尼 评论，他写道 [7]：-

这本书是对现代数学思想的通俗叙述。作者表示，她的目的是触及那部分一直想了解现代数学是什么样子，但认为它太难理解而无法理解的广大人群。她试图在不牺牲严谨性的情况下，尽可能清晰地描绘尽可能多的高级概念。……作者的幽默使每一页都令人愉悦。……作者似乎在严谨性和清晰性之间找到了完美的折衷。

关于 约翰·克米尼 评论的较长摘录，见 THIS LINK

培特 的书被翻译成英文，并有多个版本被评论。鲁宾·古德斯坦 写道 [4]：-

这无疑是我见过的面向大众的最好的数学书。作者既是一位极具创造力的数学家，又是一位经验丰富的幼儿教师，这种愉快的结合，加上清晰阐述的天赋，造就了一本令人愉悦的书……

Philip Peak 写道[11]：-

这是一本令人愉悦的书，数学家和外行人都能从中获益。

关于这些英译本评论的更长的摘录，见THIS LINK。

培特的重要著作Rekursive FuntionenⓉ（递归函数）（1951）已在上文提及。克林在一篇评论中写道[8]：-

从1932年开始，培特发表了一系列论文，考察各种特殊形式的递归之间的关系，并表明可以通过逐次更高类型的递归来定义新函数，这些论文使她成为递归函数特殊理论的主要贡献者。……在撰写这本书时，培特女士完成了一项相当可观的工作；而要更进一步，将构成一项更加巨大的工作，并且需要要么一本大得多的书，要么一种更紧凑的风格。

关于这本重要著作的这篇书评的更长摘录以及其他书评的摘录，见THIS LINK。

培特 在递归函数方面的工作当然没有以这本书结束。例如，1959 年，她在 1959 年 9 月于华沙举行的国际研讨会上提交了一篇重要论文 Über die Verallgemeinerung der Theorie der rekursiven Funktionen für abstrakte Mengen geeigneter Struktur als Definitionsbereiche Ⓣ（关于以具有适当结构的抽象量作为域的递归函数论的推广）。这篇论文分两部分发表，1961 年和 1962 年总共 68 页。在这篇论文中，培特 考察了由给定字母表形成的词，并发展了这类词的理论，推广了 朱塞佩·皮亚诺 对整数研究的方法。

从20世纪50年代中期起，培特将递归函数论应用于计算机。1976年，她的最后一本书就是关于这个主题的Recursive Functions in Computer Theory。原文为匈牙利文，英译本于1981年出版。让我们引用培特本人关于她在递归函数方面的工作如何变得对计算机科学重要的说法[12]：——

我想赢得那些认为数学有用但枯燥乏味、单调无趣——一种必要之恶——的人的心。我自己就在一个为数学内部目的而创造的领域中工作。这就是所谓“递归函数”的理论——我做梦也没想到这个理论还能实际应用。而今天呢？我那本关于递归函数的书是在苏联出版的第二本匈牙利数学书，恰恰是因为其主题对计算机理论已变得不可或缺这一实际理由。

除了她的研究贡献和那些向非数学家展示数学之乐的杰出著作之外，培特还热衷于努力改善匈牙利的数学教育。当然，她花了很多时间教中小学生，但她也编写学校教科书并致力于改进课程。引用[12]中的话似乎是恰当的，在那里我们可以很好地看到作为学校教师的培特：——

没有任何其他领域能像数学那样提供发现的喜悦，这也许是人类最大的喜悦。我过去教过的学生总是对此有共鸣，所以我也从他们那里学到了很多。例如，我从未想过在一个十二岁女孩的班上谈论欧几里得算法，但我的学生们引导我这样做了。我想讲述这堂课。

我们当时忙的是：我给出两个数，学生们算出它们的最大公约数。对于较小的数，这进行得很快。渐渐地，我给出越来越大的数，这样学生们就会感到困难，并希望有一个程序。我以为这个程序会是分解成素数。他们仍然轻松地算出了60和48的最大公约数：“十二！”但一个女孩说：“嗯，那不过就是60和48的差。” “那是巧合，”我说，并想继续下去。但他们不让我继续：“请给我们举出不是这样的数。” “好吧。60和36的最大公约数也是12，而它们的差是24。”又一次打断：“这里差是最大公约数的两倍。” “好吧，如果这能让你们都满意，那实际上并非巧合：两个数的差总是能被它们的所有公约数整除。它们的和也是如此。”当然，这需要完整地陈述，但说完之后，我真的想继续下去。然而，我还是做不到。一个女孩问：“难道他们不能仅凭这一点就发现一个求最大公约数的程序吗？”他们当然能！但那正是欧几里得算法背后的基本思想！于是我放弃了我的计划，走上了我的学生们引导我走的路。

培特因其杰出贡献获得了许多荣誉。1951年，匈牙利政府授予她科苏特奖，1953年鲍耶数学会授予她马诺·贝凯奖，1970年获得国家银质奖，1973年获得国家金质奖。1973年，她当选为匈牙利科学院院士，是第一位当选该科学院院士的女性。

至于她在数学之外的兴趣，除了我们已经讨论过的对诗歌的兴趣外，我们还应该提到电影和戏剧艺术。她于1977年死于癌症。

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 Eötvös Loránd 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 Eötvös Loránd 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.
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.

During World War II, when she had been unable to take any teaching positions, Péter has produced the charming book 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.

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.

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.

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.