米海罗·皮特罗维克Mihailo Petrović

皮特罗维克 以绰号 皮特罗维克（渔夫）闻名，因此他的名字常写作 米海罗·皮特罗维克。他的父亲尼科迪姆 皮特罗维克 是贝尔格莱德的神学教授，母亲米利察·拉扎雷维奇是副执事诺维察·拉扎雷维奇的女儿。尼科迪姆在儿子 皮特罗维克 大约五岁时去世，他由母亲和外祖父诺维察·拉扎雷维奇抚养长大。皮特罗维克 就读于贝尔格莱德第一文理中学，1885年从该校高中毕业。他的数学才能给老师们留下了深刻印象，并获得了各种奖项。此时他也是一位热情的河钓爱好者。在外祖父诺维察·拉扎雷维奇的鼓励下，他花了许多时间在萨瓦河和多瑙河上钓鱼 [19]：-

他于1882年开始跟随加什帕尔师傅当学徒，当时他是高中三年级学生。

钓鱼并不是 皮特罗维克 在学校时唯一的爱好，他还非常喜爱音乐，并学会了拉小提琴。由于他在 文理中学 最喜欢的科目是数学，离开学校后，他进入了大学校哲学系数学与自然科学部——该部于1905年成为贝尔格莱德大学——希望专攻数学，但修读的是包含物理、化学、地质学、生物学、心理学和哲学以及数学的普通理科课程。他在贝尔格莱德的数学教授是迪米特里耶·内希奇（1836-1904），他对塞尔维亚的数学教育产生了重大影响，编写了许多教科书，他还是塞尔维亚科学院的创始人之一，并担任院长。皮特罗维克 于1889年从贝尔格莱德大学校获得第一个学位，并决定前往巴黎继续深造数学。

皮特罗维克 前往巴黎，到达后便准备参加巴黎高等师范学校的入学考试。他于1890年参加考试，在获得极高分数后，开始在索邦大学听课学习。他受教于当时世界上一些顶尖的数学家，包括 儒勒·昂利·庞加莱、让·加斯东·达布、保尔·阿佩尔、于乐·达奈希、夏尔·埃尔米特 和 保罗·潘勒韦。如同在文理中学时一样，他继续追随自己对音乐的热爱，在巴黎高等师范学校期间，他建立了一个“音乐演奏社团”。他于1892年毕业，获得数学学位，即数学科学学士学位，并继续深造，于1893年被授予物理学学位，即物理科学学士学位。他被列为同代人中最优秀的学生之一，并因此受邀参加法兰西共和国总统举行的招待会。他当时已经在 夏尔·埃尔米特 和 埃米尔·皮卡 的指导下进行博士研究，并于1894年6月提交了他的主论文 Sur les zéros et les infinis des intégrales des équations différentielles algébriques Ⓣ（《代数微分方程积分的零点与奇点》）。他的第二篇论文题为 Propositions données par la Faculté Ⓣ（《院系提出的题目》）。答辩委员会由 夏尔·埃尔米特、埃米尔·皮卡 和 保罗·潘勒韦 组成，夏尔·埃尔米特 担任委员会主席。他在论文上的名字以法语形式给出为 Michel Petrowitch，他将主论文献给 于乐·达奈希 和 保罗·潘勒韦。皮特罗维克 的引言如下：-

在这项工作中，我处理了一些关于直接研究代数微分方程积分的零点、无穷大、极大值、极小值等的问题，并将研究中得到的结果应用于积分的研究，使自己站在函数论的一般观点上。

他的第二篇论文是 On recent results on the principle of least action，并由 让·加斯东·达布 批准。

获得博士学位，即数学科学博士学位后，他回到了贝尔格莱德。他在贝尔格莱德高等学校的教授是迪米特里耶·内希奇，我们上文提到过他，内希奇退休的时间几乎与皮特罗维克回国的时间相同。教授职位空缺后，皮特罗维克提出了申请，但他并不是唯一合格的候选人，因为彼得·武基切维奇，一位稍年长的数学家，于1894年获得柏林大学博士学位，也是一个强有力的候选人。竞争非常激烈，但皮特罗维克以多一票的优势被任命，而武基切维奇则成为了一名学校教师，再也没有进行任何研究。皮特罗维克非常清楚这一点，后来他说（见[12]）：-

如果我没有在申请高等学校的教授职位时多获得那一票，我永远不会将数学作为我的职业。我会生活在塞尔维亚的河流上，不是住在船上，而是住在小艇上。

1894年被任命为高等学校的数学科学教授后，他在那里继续度过了余下的职业生涯。他作为数学家获得了极大的认可，但他仍然对钓鱼充满热情，并且在1895年，他通过了钓鱼大师的考试，获得了建立他的第一家专业钓鱼公司的权利。他尤其为自己的钓鱼成就感到自豪，因为在[15]中：-

……皮特罗维克的研究……在打字机上方的墙上只挂着一张证书。那是一封大师的信，用粗糙、不熟练的笔迹写成，出自一位河流钓鱼大师之手，由委员会主席签署，证明皮特罗维克，一位教授，已经通过了河流钓鱼大师的考试，现在获得了钓鱼行业的认证，以下签名的委员会成员承认并确认这一点。

音乐在他的生活中也继续扮演着重要角色，并且在1896年1月，他创立了一个由大约十几名演奏者组成的管弦乐队。这个自称“Suz”的团体演奏城市民间音乐，并在塞尔维亚的许多城市演出。

钓鱼和音乐在皮特罗维克的生活中占据如此大的部分，人们不禁要问，他是否还有足够的时间来履行他作为数学教授的职责。然而，看看他的数学产出，很快就会发现他在这方面肯定没有懈怠，因为他发表了三百多篇数学论文。他还写了三本书：An Analysis of Number Intervals，对区间数值分析做出了重要贡献；Elliptic Functions，至今仍被塞尔维亚的学生阅读；以及Mathematical Spectres，这是一部高度原创的作品，更接近数论和密码学而非分析。

在19世纪90年代，他开发了用于求解一阶常微分方程的水力计算机。被称为水力积分器，它在他的论文Sur un procédé d'intégration graphique des équations différentielles Ⓣ（关于微分方程的图形积分方法）中描述，于1897年发表在Comptes rendus de l'Académie des Sciences de Paris上。它被翻译成塞尔维亚语，并于次年作为Hydraulic integrator出版。这项发明在1900年巴黎世界博览会上展出，他在那里获得了金牌。

自1925年以来，Paris Academy of Sciences一直在委托一系列综述当前数学问题的专著，该系列标题为Mémorial des sciences mathématiques Ⓣ（数学科学纪念文集）。皮特罗维克被要求撰写Intégration qualitative des équations différentielles Ⓣ（微分方程的定性积分），该文于1931年作为该系列的第48号出版。

关于皮特罗维克文章的开头，见THIS LINK。

他研究的另一个主题，将他的数学兴趣与其他兴趣结合起来，是现象学。他出版了Elements of mathematical phenomenology（1911年）、Méchanismes communs aux Phénomènes disparates Ⓣ（不同现象的常见机制）（1921年）和Phenomenological mapping（1933年）。他在这一领域的工作在[20]中有所描述：-

为了将现象转化为数学形式，他发展并阐述了一种现象学指代方法：当“现象的机制”被确立后，它就在多维空间中用一个“象征点”来表示，然后其机制以经典力学中描述现象机制的方式被数学地描述。他在1933年出版的专著⟦E1⟧中阐述了这一方法。然而，他并未止步于自然现象和社会中的某些现象，还投入了大量精力来确认其方法在文学中的普遍性。他为完成这一目标所收集的材料于1967年出版在⟦E2⟧一书中。

皮特罗维克在多次国际数学家大会上展示了他的数学工作：1900年8月在巴黎，1908年4月在罗马，1912年8月在英格兰剑桥，1920年9月在斯特拉斯堡，以及1924年8月在加拿大多伦多。在每一次国际数学家大会上，他都是唯一的塞尔维亚参会者，而到多伦多大会时，塞尔维亚已是南斯拉夫的一部分，他是唯一的南斯拉夫参会者。

除了是参加重大国际数学会议的唯一塞尔维亚数学家之外，皮特罗维克还是塞尔维亚数学教学发展的领导者。从1912年到1941年，他是唯一在贝尔格莱德大学指导从事数学研究的博士生的人。

我们从皮特罗维克参加国际数学家大会可以看出，他愿意旅行。事实上，他喜欢到世界偏远地区旅行，如北极和南极。1931年夏天，他从法国前往格陵兰，几乎一直到达北极。次年夏天，他横渡大西洋，到达马尾藻海、加勒比群岛、安的列斯群岛和百慕大。1933年夏天，他前往拉布拉多和纽芬兰，次年夏天则经大西洋航行至南极。1935年夏天的亮点是印度洋之旅，而1939年夏天他进行了一次较短的旅行，前往亚速尔群岛。此时他已年过70，但1939年第二次世界大战的爆发无疑使进一步的长途旅行成为不可能。

像许多数学家一样，战争和战争威胁使他们的国家招募他们从事密码学工作。皮特罗维克以记录他所做的一切而著称，但毫不奇怪，密码学是一个例外。然而，我们确实知道一些他的贡献，如11中所记载：-

皮特罗维克在密码学和加密领域的成就在南斯拉夫王国总参谋部情报处密码科的15卷文件中得到记录，标题为“密码学——加密训练学校”，以及24卷题为“系统”（编码）的文件中。在这些文件中，记录着皮特罗维克的工作及其成果可见于：(a) 加密方法；(b) “破解”密码的方法；(c) 加密技术和破解已编码信息的训练。

国际声誉带来了国际认可。皮特罗维克被选为Serbian Academy of Sciences and Arts、Yugoslav Academy of Sciences and Arts、Czech Academy of Sciences、克拉科夫的波兰科学院、Warsaw Scientific Society和罗马尼亚科学院的成员。他还是许多数学学会的成员，包括法国数学会、Mathematical Circle of Palermo、Romanian Mathematical Society和Union of Czech Mathematicians。

他于1938年退休，次年获得贝尔格莱德大学荣誉博士学位。向大学解释他为何值得这一荣誉的提案中写道：-

皮特罗维克创建了南斯拉夫第一所数学学派，通过这一创造性努力，他将贝尔格莱德大学的数学教学提升到了当代世界一流学派的水平。我们的学院、大学、我们的国家以及整个国家都应对皮特罗维克致以最高的赞誉。

同样在1939年，南斯拉夫授予他一等圣萨瓦勋章。

关于他一生中众多活动的某些信息，我们参考[12]，其中一些我们已在上文提及：-

皮特罗维克拥有丰富、有趣且不拘一格的一生。要完全列举，更不用说详细描述皮特罗维克所从事的所有事情，是很困难的。除了对数学各个领域的兴趣外，皮特罗维克还出现在许多其他往往意想不到的领域。他撰写法律并起草国际协议，但也是一位发明家，拥有成功且已实施的专利。皮特罗维克被许多人认为是塞尔维亚最哲学的人物之一，即数学现象学。他的写作风格优美有趣，他的一些小说是塞尔维亚青年文学中最受欢迎的作品之一。他撰写学术论文并研究其他自然科学，主要是天文学、相对论和化学。他创建了一套编码系统，并是塞尔维亚和南斯拉夫军队的主要密码学家。他拉小提琴并指挥乐队“Suz”，直到第二次世界大战爆发前，该乐队一直是贝尔格莱德波西米亚生活的主要场所之一。最后，他是一位伟大而热情的渔夫，也是一位伟大的世界旅行家和航海家，航行过南北海域。一位伟大的数学家和世界旅行家静静地去世了，梦想着一次新的伟大海洋之旅。

皮特罗维克于1943年6月8日在贝尔格莱德Kosančićev venac街22号的家中去世。事实上，战争一定加速了他的死亡。德国飞机从1941年4月6日开始轰炸贝尔格莱德，德国地面部队在同一天越过南斯拉夫边境。南斯拉夫于4月17日投降。皮特罗维克被征召为陆军中校，尽管当时他已73岁，他立即被占领的德国军队逮捕，在监狱中度过了一年。他被释放，可能是因为他的健康状况，并获准返回他的公寓。一到那里，他就只是坐在房间里写作，直到终点来临。

最后，我们引用[6]中关于皮特罗维克个性的内容：-

与皮特罗维克同时代的人常常见证，他是一个诚实、正直，尤其是极其谦逊的人。

Mihailo Petrović was known by the nickname Alas, the fisherman, so his name often appears as Mihailo Petrović Alas. His father, Nikodim Petrović, was the theology professor at Belgrade, and his mother Milica Lazarević was the daughter of the archdeacon Novica Lazarević. Nikodim died when his son Mihailo was around five years old and he was brought up by his mother and grandfather Novica Lazarević. Petrović attended the First Belgrade Gymnasium, graduating from that High School in 1885. He had impressed his teachers with his mathematical ability and had won various prizes. Also by this time he was an enthusiastic river fisherman. Encouraged by his grandfather Novica Lazarević, he had spent many hours fishing on the Sava and the Danube rivers [19]:-

He began apprenticing with Master Gašpar in 1882, as a third-year student of the High School.

Fishing was not Petrović's only hobby while at school, for he also was very musical and learnt to play the violin. Since his favourite subject at the Gymnasium was mathematics, after leaving the school he enrolled in the Department of Mathematics and Natural Sciences of the Faculty of Philosophy in the Grand School, which in 1905 became the University of Belgrade, wanting to specialise in mathematics but taking a general science course which included physics, chemistry, geology, biology, psychology and philosophy as well as mathematics. His mathematics professor at Belgrade was Dimitrije Nešić (1836-1904) who was a major influence on mathematical education in Serbia writing many textbooks, and he was also a founder of the Serbian Academy of Sciences, serving as its president. Petrović graduated with his first degree from the Belgrade Grand School in 1889 and decided to pursue further studies in mathematics in Paris.

Petrović travelled to Paris and once there he prepared to take the entrance examinations for the École Normale Supérieure. He took the examination in 1890 and, after the award of very high marks, began his studies attending lectures at the Sorbonne. He was taught by some of the leading mathematicians in the world including Henri Poincaré, Jean Gaston Darboux, Paul Émile Appell, Jules Tannery, Charles Hermite and Paul Painlevé. As in the Gymnasium, he continued to follow his love of music and, while at the École Normale Supérieure, he established a "music playing society". He graduated in 1892 with a degree in mathematics, the Licence es sciences mathématiques, and continued his studies, being awarded a degree in physics, the Licence ès Sciences Physiques, in 1893. He was named as one of the best students of his generation and as such was invited to a reception given by the President of the French Republic. He was already undertaking research for his doctorate advised by Charles Hermite and Émile Picard, and submitted his main thesis Sur les zéros et les infinis des intégrales des équations différentielles algébriques Ⓣ in June 1894. His second thesis was entitled Propositions données par la Faculté Ⓣ. The examining committee was Charles Hermite, Émile Picard and Paul Painlevé, with Hermite the president of the committee. His name on the thesis is given in a French form as Michel Petrowitch and he dedicated his main thesis to Jules Tannery and Paul Painlevé. Petrović begins his introduction as follows:-

In this work, I deal with some questions concerning the direct study of zeros, infinities, maxima, minima, etc. of integrals of algebraic differential equations, and I apply the results found in the study of the integrals by placing myself in the point of view of the general theory of functions.

His second thesis was On recent results on the principle of least action and it was approved by Gaston Darboux.

After the award of his doctorate, the Docteur es Sciences Mathématiques, he returned to Belgrade. His professor at Belgrade Grand School had been Dimitrije Nešić, who we mentioned above, and Nešić retired at almost the same time as Petrović returned. The professorship becoming vacant, Petrović applied but he was not the only well-qualified candidate since Petar Vukićević, a slightly older mathematician who had been awarded a doctorate by the University of Berlin in 1894, was also a strong candidate. The competition was very close but Petrović was appointed having received one more vote that Vukićević. What a difference that one vote made, for Vukićević became a school teacher and never did any further research. Petrović was very aware of this and later said (see [12]):-

If I had not obtained that one additional vote for my application for a Grand School professor, I would have never pursued mathematics as my profession. I would have lived on Serbian rivers, not on a boat, but on a dinghy.

Appointed to the Grand School in 1894 as professor for mathematical sciences, he continued there for the rest of his career. He had gained great recognition as a mathematician but he still had a passionate love of fishing and, in 1895, he passed the examination for a master fisherman and earned the right to establish his first professional fishing company. He was proud particularly of his fishing achievements for in [15]:-

... Mihailo Petrović's study ... on the wall above the typewriter stood only one diploma. It was a master's letter, written in rough, unskilled handwriting of a river master fisherman, and signed by the president of the commission, that Mihailo Petrović, a professor, has passed the master's exam for a master river fishermen, and is now certified in the fisherman trade, which the undersigned members of commission recognise and confirm.

Music also continued to play a big part in his life and, in January 1896, he founded an orchestra consisting of about a dozen players. This group, which called themselves "Suz", played urban folk music and performed in many cities throughout Serbia.

With fishing and music playing such a large part in Petrović's life, one wonders if he had much time left to carry out his job as a professor of mathematics. A look at his mathematical output, however, soon shows that he certainly did not slack in this for he published over three hundred mathematics papers. He also wrote three books: An Analysis of Number Intervals which makes an important contribution to interval numerical analysis; Elliptic Functions which is still read by students in Serbia today; and Mathematical Spectres which is a highly original work closer to number theory and cryptography than to analysis.

In the 1890's he developed the hydraulic computer for the solution of first order ordinary differential equations. Called the hydraulic integrator, it was described in his paper Sur un procédé d'intégration graphique des équations différentielles Ⓣ published in Comptes rendus de l'Académie des Sciences de Paris in 1897. It was translated into Serbian and published as Hydraulic integrator in the following year. The invention was displayed at the World Exposition in Paris in 1900 where he received a gold medal.

Since 1925 the Paris Academy of Sciences had been commissioning a series of monographs reviewing current mathematical problems, the series having the title Mémorial des sciences mathématiques Ⓣ. Petrović was asked to write Intégration qualitative des équations différentielles Ⓣ which appeared as No 48 in the series in 1931.

For the beginning of Petrović's article, see THIS LINK.

Another topic on which he worked, which combined his mathematical interest with other interests, was phenomenology. He published Elements of mathematical phenomenology (1911), Méchanismes communs aux Phénomènes disparates Ⓣ (1921), and Phenomenological mapping (1933). His work in this area is described in [20]:-

In order to cast phenomena into a mathematical form, he developed and elaborated a method of phenomenological referencing: when "the mechanism of the phenomena" is established, it is represented by a "figurative point" in a multidimensional space and then its mechanism mathematically described in a way mechanisms of phenomena are described in classical mechanics. He expounded this method in the monograph 'Phenomenological Mapping' published in 1933. However, he did not stop at phenomena in nature and some phenomena in society, but also invested a great deal of effort to confirm the universality of his method in literature as well. The material he gathered for the accomplishment of this goal was published in 1967 in the book 'Metaphors and Allegories'.

Petrović presented his mathematical work at several International Congresses of Mathematicians: in Paris in August 1900, in Rome in April 1908, in Cambridge, England in August 1912, in Strasbourg in September 1920, and in Toronto, Canada in August 1924. At each of these International Congresses of Mathematicians he was the only participant from Serbia, and by the time of the Toronto congress, Serbia was part of Yugoslavia and he was the only participant from Yugoslavia.

As well as being the only Serbian mathematician taking part in the major international mathematics conferences, Petrović was the leader for developing the teaching of mathematics in Serbia. He was the only person who supervised Ph.D. students undertaking mathematical research at the University of Belgrade from 1912 to 1941.

We have seen from Petrović's attendance at International Congresses of Mathematicians that he was prepared to travel. In fact he loved travelling to remote parts of the world such as the Arctic and the Antarctic. In the summer of 1931 he went from France to Greenland and almost all the way to the North Pole. In the following summer he crossed the Atlantic, to the Sargasso Sea, the Caribbean Islands, Antilles, and Bermuda. The summer of 1933 saw him travel to Labrador and Newfoundland, and the following summer involved sailing via the Atlantic ocean to the Antarctic. The summer of 1935 had a trip to the Indian Ocean as its highlight while in the summer of 1939 he made a shorter trip, going to the Azores. By this time he was over 70 but it was the outbreak of World War II in 1939 which certainly made further major travels impossible.

Like many mathematicians, both war and the threat of war, saw their country recruit them for work in cryptography. Petrović is remarkable for documenting all the things he did but, not surprisingly, cryptography is an exception. We do know something of his contributions, however, as recorded in [11]:-

Petrović's achievements in the field of cryptography and ciphering are documented in 15 volumes of the Department for Ciphering of the Intelligence Service of the General Staff Head Office of the Kingdom of Yugoslavia, under the title "Cryptography - school for training in ciphering" as well as in 24 volumes entitled "The System" (of coding). In these documents, it was recorded that Petrović's work and results of that work are to be found in: (a) ciphering methods; (b) methods for "cracking" codes and (c) training in ciphering techniques and cracking of messages that had been coded.

International fame brought international recognition. Petrović was elected to the Serbian Academy of Sciences and Arts, the Yugoslav Academy of Sciences and Arts, the Czech Academy of Sciences, the Polish Academy of Sciences in Kraków, the Warsaw Scientific Society, and the Romanian Academy. He was also a member of many mathematical societies including the French Mathematical Society, the Mathematical Circle of Palermo, the Romanian Mathematical Society, and the Union of Czech Mathematicians.

He retired in 1938 and he received an honorary doctorate from the University of Belgrade in the following year. The proposal made to the University explaining why he deserved this honour states:-

M Petrović has created the Mathematical School, the first in Yugoslavia, and with this creative endeavour he elevated the teaching of mathematics at the University of Belgrade to the level of contemporary world-class schools. Our Faculty, University, our state and this entire country owe the highest accolade to Mihailo Petrović.

Also in 1939 Yugoslavia awarded him the Decoration of St Sava of the first degree.

We refer to [12] for some information about the many activities that filled his life, some of which we have mentioned above:-

Mihailo Petrović had a rich, interesting and unconventional life. It is difficult to fully enumerate, let alone describe in detail, all the things Petrović engaged in. In addition to his interest in various areas of mathematics, Petrović appears in many other, often unexpected fields. He wrote laws and drafted international agreements, but was also an inventor, holding successful and executed patents. Petrović is considered by many to be one of Serbia's most philosophy, namely mathematical phenomenology. His style of writing was beautiful and interesting and some of his novels are among favourite Serbian pieces of youth literature. He wrote academic papers in and studied other natural sciences, primarily astronomy, theory of relativity and chemistry. He created a coding system and was the main cryptographer of the Serbian and Yugoslav Army. He played violin and conducted the musical band "Suz" which up to the onset of the Second World War provided one of the main loci of bohemian life of Belgrade. Finally, he was a great and passionate fisherman, and a great world traveller and seafarer who sailed northern and southern seas. A great mathematician and world traveller died quietly, dreaming about a new and great ocean journey.

Mihailo Petrović died in Belgrade on 8 June 1943 at his home in 22 Kosančićev venac street. In fact the war must have hastened his death. German planes bombed Belgrade beginning on 6 April 1941 and German ground forces crossed the Yugoslav border on the same day. Yugoslavia surrendered on 17 April. Petrović had been drafted as a Lieutenant Colonel in the army, although he was 73 years old at the time, and he was immediately arrested by the occupying German forces spending one year in prison. He was released, probably because of the state of his health, and allowed to return to his flat. Once there he just sat in his room and wrote until the end came.

Finally we quote from [6] regarding Petrović's personality:-

Contemporaries of Petrović often witnessed that he was an honest, honourable, and above all an extremely modest man.