瓦茨瓦夫·谢尔宾斯基Wacław Sierpiński

瓦茨瓦夫·谢尔宾斯基的父亲是一名医生。他在华沙上学，他的第一位数学老师很快发现了他的数学天赋。这是俄罗斯占领波兰的时期，对于有天赋的谢尔宾斯基来说，在波兰接受教育是一段艰难的时期。俄罗斯人在1869年至1874年间对所有中学实施的全面改革中，将他们的语言和文化强加给波兰人。俄罗斯的目的是让波兰的文盲率尽可能高，因此他们阻碍学习，学生人数下降。

尽管困难重重，谢尔宾斯基于1899年进入华沙大学数学与物理系。更准确地说，应将其描述为沙皇大学，因为这是该大学的官方名称，它于1869年成为一所俄罗斯大学。大学的讲座全部用俄语进行，教职员也全是俄罗斯人。因此，首先吸引谢尔宾斯基的是他的老师之一格奥尔吉·沃罗诺伊这位俄罗斯数学家的著作，这并不奇怪。

1903年，数学与物理系设立了一项奖项，奖励学生就格奥尔吉·沃罗诺伊对数论的贡献所写的最佳论文。谢尔宾斯基因其学位论文在竞赛中荣获金奖。他描述了这些事件（见[12]）：-

……我因一篇关于数论的竞赛论文而被大学授予金奖。这是我的第一部科学著作。它被华沙大学《消息报》接受发表。然而，次年发生了罢工，以抵制波兰的俄罗斯学校，我不想让我的第一部作品以俄语印刷，因此我将其从华沙《消息报》撤稿。这就是为什么它直到1907年才在萨穆埃尔·迪克施泰因出版的数学杂志《数学与物理著作》上印刷。

从华沙大学毕业后五十年，谢尔宾斯基回顾了他在俄罗斯占领时期作为波兰人攻读学位时所遇到的问题：-

……我们不得不每年参加俄语讲座。……每个学生都把在该科目中取得最差成绩视为荣誉。……我一个问题也没回答……我得到了不及格的分数。……我通过了所有考试，然后讲师建议我参加补考，否则我将无法获得数学科学候选人的学位。……我拒绝了他，说这将是本校首例：在所有科目中成绩优异、学位论文被接受并获得金奖的人，却因为俄语一门分数较低而无法获得数学科学候选人的学位，而只能获得较低的学位，即‘真正学生’的学位（奇怪的是，较低学位就是这么叫的）。

谢尔宾斯基很幸运，因为讲师将他的俄语课程分数改为‘良好’，这样他就能获得学位了。正如他所说：-

这位警察是有人性的。

谢尔宾斯基在1904年所写的获奖论文中的结果是对一个关于格点的著名问题的重要贡献。假设$R(r)$表示包含在以$O$为圆心、$r$为半径的圆内的点$(m, n), m, n$ ∈ $\mathbb{Z}$的数目。存在一个常数$C$和一个数$k$，使得

$| R(r) - \pi r^{2}| < Cr^{k}$。

设$d$为$k$的最小值。卡尔·弗里德里希·高斯在1837年证明了$d ≤ 1$。谢尔宾斯基的主要贡献是表明可以将该不等式改进为$d ≤ \large\frac{2}{3}\normalsize$。1913年，埃德蒙·朗道简化了谢尔宾斯基的证明，并将这一结果描述为深刻。

让我们暂时离题，讨论一些进一步的工作，这些工作源于谢尔宾斯基关于通常被称为“卡尔·弗里德里希·高斯圆问题”的这一结果。1915年，戈弗雷·哈罗德·哈代和埃德蒙·朗道证明了$d > \large\frac{1}{2}\normalsize$，而1923年van der 约翰内斯·范德科皮特证明了$d < \large\frac{2}{3}\normalsize$。次年，利特尔伍德和阿诺德·华菲斯证明了$d ≤ \large\frac{37}{56}\normalsize$，次年这一结果被改进为$d ≤ \large\frac{163}{247}\normalsize \approx 0.6599$。1932年伊万·维诺格拉多夫和1934年爱德华·查尔斯·蒂奇马什做了轻微的进一步改进。目前已知的最佳结果（迄今为止）是$d ≤ \large\frac{131}{208}\normalsize \approx 0.6298$。

谢尔宾斯基于1904年毕业，并在华沙一所女子学校担任了一段时间的数学和物理教师。然而，当学校因罢工而关闭时，谢尔宾斯基决定去克拉科夫攻读博士学位。在克拉科夫的雅盖隆大学，他听了斯塔尼斯瓦夫·萨伦巴的数学讲座，此外还学习天文学和哲学。他获得了博士学位，并于1908年被任命到伦贝格大学（该大学位于现在的乌克兰利沃夫）。

事实上，正是在1907年，谢尔宾斯基第一次对集合论产生了兴趣。事情发生在他遇到一个定理时，该定理指出平面上的点可以用单个坐标来指定。他写信给当时在哥廷根的塔德乌什·巴那齐耶维茨，询问他这样的结果怎么可能。他收到了一个词的回复“格奥尔格·康托尔”。谢尔宾斯基开始研究集合论，并在1909年首次开设了完全致力于集合论的讲座课程。

在他的一生中，谢尔宾斯基保持了令人难以置信的研究论文和书籍产出。在1908年至1914年期间，当他在伦贝格大学任教时，除了许多研究论文外，他还出版了三本书。这些书是The theory of 无理数 numbers（1910年）、Outline of Set Theory（1912年）和The theory of numbers（1912年）。

1912年，他引入了Sierpiński curve，它描述了一条穿过正方形每个内点的闭合路径。曲线的长度是无穷大，而它所包围的面积是正方形的$\large\frac{5}{12}\normalsize$。

你可以在THIS LINK看到这条曲线构造的一些阶段。

大约在这个时候，他引入了现在称为Sierpiński triangle或Sierpiński gasket的东西。它是一种分形曲线的例子，是四十年前引入的Cantor set的二维类似物。

你可以在THIS LINK看到这个集合构造的一些阶段。

1914年第一次世界大战爆发时，谢尔宾斯基和他的家人恰好在俄罗斯。此时奥地利和俄罗斯政府试图利用波兰问题作为政治武器。谢尔宾斯基被拘留在维亚特卡。然而德米特里·叶戈罗夫和尼古拉·卢津听说他被拘留，并安排允许他去莫斯科。谢尔宾斯基在莫斯科度过了战争余下的岁月，与尼古拉·卢津一起工作。他们一起开始了分析集的研究。1916年，在莫斯科期间，谢尔宾斯基给出了第一个绝对正规数的例子，即无论以何种进制书写，其数字出现频率都相等的数。埃米尔·博雷尔证明了这样的数存在，但谢尔宾斯基是第一个给出例子的人。

1918年第一次世界大战结束时，谢尔宾斯基回到了伦贝格。然而在伦贝格再次接受任命后不久，他获得了华沙大学的职位，他接受了。1919年，他被提升为华沙的教授，并在那里度过了余生。

1920年，谢尔宾斯基与他以前的学生斯特凡·马祖尔凯维奇一起创办了重要的数学期刊Fundamenta Mathematicae。谢尔宾斯基编辑了这本专门发表集合论论文的期刊。

从这一时期起，谢尔宾斯基主要工作在集合论领域，但也研究point set topology和实变函数。在集合论中，他对选择公理和连续统假设做出了重要贡献。

谢尔宾斯基继续与尼古拉·卢津合作研究解析集和射影集。他在实变函数方面的工作包括关于函数级数、函数可微性和勒内-路易·贝尔分类的结果。

谢尔宾斯基还深度参与了波兰数学的发展。他于1921年当选为波兰科学院会士，同年被任命为华沙大学学院院长。1928年，他成为华沙科学学会的副主席，并于同年当选为波兰数学会的主席。

1939年，随着第二次世界大战的到来，华沙的生活发生了剧变。谢尔宾斯基继续在“地下华沙大学”工作，而他的正式工作是在华沙议会办公室做职员。他的出版物继续发表，因为他设法将论文寄往意大利。这些论文每一篇都以这样的话结尾：——

这些定理的证明将发表在Fundamenta Mathematicae上

每个人都明白这意味着“波兰将会幸存”。

1944年起义后，纳粹烧毁了他的房子，毁掉了他的藏书和个人信件。谢尔宾斯基在1945年于克拉科夫的雅盖隆大学所做的一次演讲中谈到了战争中的悲剧事件（见[13]）。他谈到了在战争中死去的他的学生们：-

1941年7月，我最年长的学生之一Stanisław Ruziewicz被杀害。他是利沃夫扬·卡齐米日大学的退休教授……一位杰出的数学家和优秀的教师。1943年，我最杰出的学生之一斯坦尼斯拉夫·萨克斯被杀害。他是华沙大学的助理教授，世界上积分理论领域的顶尖专家之一……1942年，我的另一位学生阿道夫•林登鲍姆被杀害。他是华沙大学的助理教授，集合论方面著作的杰出作者。

在列举了像尤利乌什·绍德尔这样在战争中被杀害的同事，以及像萨穆埃尔·迪克施泰因和斯塔尼斯瓦夫·萨伦巴这样因战争而死的其他人之后，谢尔宾斯基继续说道：-

因此，在我们学术学校中授课的数学家有超过一半被杀害。这对波兰数学是一个巨大损失，波兰数学在集合论和拓扑学等一些领域正发展良好……除了令人痛惜的个人损失之外，波兰数学还因战争期间德国的野蛮行径而遭受了物质损失。他们烧毁了华沙大学图书馆，其中藏有数千卷书籍、杂志、数学书籍以及不同作者数学著作的数千份抽印本。几乎所有各期的⟦E1⟧（32卷）和十卷⟦E2⟧都被完全烧毁。华沙大学所有四位数学教授的私人藏书，以及他们在战争期间撰写的相当多手稿和手册也被烧毁。

谢尔宾斯基是数量惊人的724篇论文和50本书的作者。他于1960年从华沙大学教授职位上退休，但他继续在波兰科学院主持数论讨论班，直到1967年。他还继续他的编辑工作，担任他于1958年开始的Acta Arithmetica的主编，以及Rendiconti del Circolo Matematico di Palermo, Compositio Mathematica和Zentralblatt für Mathematik的编委会成员。

他获得了如此多的荣誉，以至于不可能在此一一列举。我们列出一些。他获得了扬·卡齐米日大学（利沃夫，即利沃夫的伦贝格大学后来为人所知的名字）（1929年）以及圣马可大学、埃隆·拉格斯·利马大学（1930年）、阿姆斯特丹大学（1931年）、塔尔塔大学（1931年）、索菲亚大学（1939年）、布拉格大学（1947年）、弗罗茨瓦夫大学（1947年）、勒克瑙大学（1949年）和莫斯科罗蒙诺索夫大学（1967年）的荣誉学位。

他于1931年当选为利马地理学会会员，1934年当选为列日皇家科学学会会员，1936年当选为保加利亚科学院会员，1939年当选为利马国家科学院院士，1939年当选为那不勒斯皇家科学学会会员，1947年当选为罗马Accademia dei Lincei院士，1950年当选为德国科学院院士，1959年当选为美国艺术与科学院院士，1960年当选为巴黎科学院院士，1961年当选为荷兰皇家科学院院士，1961年当选为布鲁塞尔科学院院士，1964年当选为伦敦数学会院士，1965年当选为罗马尼亚科学院院士，1967年当选为宗座科学院院士。

谢尔宾斯基的学生罗特凯维奇在[12]中写道：-

谢尔宾斯基身体极好，性格开朗。……他能在任何条件下工作。……他不喜欢别人对他的论文作任何修改。当有人提出修改意见时，他就在论文中加一行：‘某先生指出……’他富有创造力，喜欢创造性的数学。他是波兰最伟大、最多产的数学家。

Wacław Sierpiński's father was a doctor. He attended school in Warsaw where his talent for mathematics was quickly spotted by his first mathematics teacher. This was a period of Russian occupation of Poland and it was a difficult time for the gifted Sierpiński to be educated in Poland. The Russians had forced their language and culture on the Poles in sweeping changes to all secondary schools implemented between 1869 and 1874. The Russian aim was to keep illiteracy in Poland as high as possible, so they discouraged learning and the number of students fell.

Despite the difficulties, Sierpiński entered the Department of Mathematics and Physics of the University of Warsaw in 1899. It would be more accurate to describe it as the Czar's University since this was the official name of the University which had become a Russian university in 1869. The lectures at the University were all in Russian and the staff were entirely Russian. It is not surprising therefore that it would be the work of a Russian mathematician, one of his teachers Voronoy, that first attracted Sierpiński.

In 1903 the Department of Mathematics and Physics offered a prize for the best essay from a student on Voronoy's contribution to number theory. Sierpiński was awarded the gold medal in the competition for his dissertation. He described the events (see [12]):-

... I was awarded a gold medal by the university for work in a competition on the theory of numbers. It was my first scientific work. It was accepted for publication in the 'Izvestia' of Warsaw University. However, in the following year there was a strike to produce a boycott of Russian Schools in Poland and I did not want to have my first work printed in the Russian language and that is why I had it withdrawn from print in Warsaw's 'Izvestia'. That is why it was not printed until 1907 in the mathematical magazine 'The works of Mathematics and Physics' published by Samuel Dickstein.

Fifty years after he graduated from the University of Warsaw Sierpiński looked back at the problems that he had as a Pole taking his degree at the time of the Russian occupation:-

... we had to attend a yearly lecture on the Russian language. ... Each of the students made it a point of honour to have the worst results in that subject. ... I did not answer a single question ... and I got an unsatisfactory mark. ... I passed all my examinations, then the lector suggested I should take a repeat examination, otherwise I would not be able to obtain the degree of a candidate for mathematical science. ... I refused him saying that this would be the first case at our University that someone having excellent marks in all subjects, having the dissertation accepted and a gold medal, would not obtain the degree of a candidate for mathematical science, but a lower degree, the degree of a 'real student' (strangely that was what the lower degree was called) because of one lower mark in the Russian language.

Sierpiński was lucky for the lector changed the mark on his Russian language course to 'good' so that he could take his degree. As he says:-

The policeman was human.

The results in the prize essay that Sierpiński wrote in 1904 were a major contribution to a famous problem on lattice points. Suppose $R(r)$ denotes the number of points $(m, n), m, n$ ∈ $\mathbb{Z}$ contained in a circle centre $O$, radius $r$. There exists a constant $C$ and a number $k$ with

$| R(r) - \pi r^{2}| < Cr^{k}$.

Let $d$ be the minimal value of $k$. Gauss proved in 1837 that $d ≤ 1$. Sierpiński's major contribution was to show that it was possible to improve the inequality to $d ≤ \large\frac{2}{3}\normalsize$. In 1913 Edmund Landau shortened Sierpiński's proof and described the result as profound.

Let us digress for a moment to discuss some further work which flowed from this result of Sierpiński on what is often called the 'Gauss circle problem'. In 1915 Hardy and Landau proved that $d > \large\frac{1}{2}\normalsize$, while in 1923 van der Corput proved that $d < \large\frac{2}{3}\normalsize$. The following year Littlewood and Walfisz proved that $d ≤ \large\frac{37}{56}\normalsize$, this being improved to $d ≤ \large\frac{163}{247}\normalsize \approx 0.6599$ the following year. Slight further improvements were made by Vinogradov in 1932 and Titchmarsh in 1934. The best result (so far) known is $d ≤ \large\frac{131}{208}\normalsize \approx 0.6298$.

Sierpiński graduated in 1904 and worked for a while as a school teacher of mathematics and physics in a girls' school in Warsaw. However, when the school closed because of a strike, Sierpiński decided to go to Kraków to study for his doctorate. At the Jagiellonian University in Kraków he attended lectures by Zaremba on mathematics, studying in addition astronomy and philosophy. He received his doctorate and was appointed to the University of Lemberg (which is in what is now Lviv in Ukraine) in 1908.

In fact it was in 1907 that Sierpiński first became interested in set theory. It happened when he came across a theorem which stated that points in the plane could be specified with a single coordinate. He wrote to Banachiewicz, who was at Göttingen at the time, asking him how such a result was possible. He received a one word reply 'Cantor'. Sierpiński began to study set theory and in 1909 he gave the first ever lecture course devoted entirely to set theory.

Throughout his life Sierpiński maintained an incredible output of research papers and books. During the years 1908 to 1914, when he taught at the University of Lemberg, he published three books in addition to many research papers. These books were The theory of irrational numbers (1910), Outline of Set Theory (1912) and The theory of numbers (1912).

1n 1912 he introduced the Sierpiński curve which describes a closed path which passes through every interior point of a square. The length of the curve is infinity, while the area enclosed by it is $\large\frac{5}{12}\normalsize$ that of the square.

You can see some of the stages of the construction of this curve at THIS LINK.

About this time he introduced what is now called the Sierpiński triangle or Sierpiński gasket. It is an example of a fractal curve which is a two-dimensional analogue of the Cantor set which had been introduced forty years before.

You can see some of the stages of the construction of this set at THIS LINK.

When World War I began in 1914, Sierpiński and his family happened to be in Russia. At this time the governments of Austria and Russia tried to use the Polish question as a political weapon. Sierpiński was interned in Viatka. However Egorov and Luzin heard that he had been interned and arranged for him to be allowed to go to Moscow. Sierpiński spent the rest of the war years in Moscow working with Luzin. Together they began the study of analytic sets. In 1916, during his time in Moscow, Sierpiński gave the first example of an absolutely normal number, that is a number whose digits occur with equal frequency in whichever base it is written. Borel had proved such numbers exist but Sierpiński was the first to give an example.

When World War I ended in 1918, Sierpiński returned to Lemberg. However shortly after taking up his appointment again in Lemberg he was offered a post at the University of Warsaw which he accepted. In 1919 he was promoted to professor at Warsaw and he spent the rest of his life there.

In 1920 Sierpiński, together with his former student Mazurkiewicz, founded the important mathematics journal Fundamenta Mathematicae. Sierpiński edited the journal which specialised in papers on set theory.

From this period Sierpiński worked mostly in the area of set theory but also on point set topology and functions of a real variable. In set theory he made important contributions to the axiom of choice and to the continuum hypothesis.

Sierpiński continued to collaborate with Luzin on investigations of analytic and projective sets. His work on functions of a real variable include results on functional series, differentiability of functions and Baire's classification.

Sierpiński was also highly involved with the development of mathematics in Poland. He had been honoured with election to the Polish Academy in 1921 and he was made dean of the faculty at the University of Warsaw in the same year. In 1928 he became vice-chairman of the Warsaw Scientific Society and, in the same year was elected chairman of the Polish Mathematical Society.

In 1939 life in Warsaw changed dramatically with the advent of World War II. Sierpiński continued working in the 'Underground Warsaw University' while his official job was a clerk in the council offices in Warsaw. His publications continued since he managed to send papers to Italy. Each of these papers ended with the words:-

The proofs of these theorems will appear in the publication of Fundamenta Mathematicae

which everyone understood meant 'Poland will survive'.

After the uprising of 1944 the Nazis burned his house destroying his library and personal letters. Sierpiński spoke of the tragic events of the war during a lecture he gave at the Jagiellonian University in Kraków in 1945 (see [13]). He spoke of his students who had died in the war:-

In July 1941 one of my oldest students Stanisław Ruziewicz was murdered. He was a retired professor of Jan Kazimierz University in Lviv ... an outstanding mathematician and an excellent teacher. In 1943 one of my most distinguished students Stanisław Saks was murdered. He was an assistant professor at Warsaw University, one of the leading experts in the world in the theory of the integral... In 1942 another student of mine, Adolf Lindenbaum was murdered. He was an assistant professor at Warsaw University and a distinguished author of works on set theory.

After listing colleagues who were murdered in the war such as Schauder and others who died as a result of the war such as Dickstein and Zaremba, Sierpiński continued:-

Thus more than half of the mathematicians who lectured in our academic schools were killed. It was a great loss for Polish mathematics which was developing favourably in some fields such as set theory and topology ... In addition to the lamented personal losses Polish mathematics suffered because of German barbarity during the war, it also suffered material losses. They burned down Warsaw University Library which contained several thousand volumes, magazines, mathematical books and thousands of reprints of mathematical works by different authors. Nearly all the editions of Fundamenta Mathematicae (32 volumes) and ten volumes of Mathematical Monograph were completely burned. Private libraries of all the four professors of mathematics from Warsaw University and also quite a number of manuscripts of their works and handbooks written during the war were burnt too.

Sierpiński was the author of the incredible number of 724 papers and 50 books. He retired in 1960 as professor at the University of Warsaw but he continued to give a seminar on the theory of numbers at the Polish Academy of Sciences up to 1967. He also continued his editorial work, as editor-in-chief of Acta Arithmetica which he began in 1958, and as an editorial board member of Rendiconti del Circolo Matematico di Palermo, Compositio Mathematica and Zentralblatt für Mathematik.

He received so many honours that it would be impossible to mention them all here. We list a few. He was awarded honorary degrees from Jan Kazimierz University, Lwów (as the University of Lemberg in Lviv became known) (1929) and from the universities of St Marks, Lima (1930), Amsterdam (1931), Tarta (1931), Sofia (1939), Prague (1947), Wrocław (1947), Lucknow (1949), and Lomonosov University of Moscow (1967).

He was elected to the Geographic Society of Lima (1931), the Royal Scientific Society of Liège (1934), the Bulgarian Academy of Sciences (1936), the National Academy of Lima (1939), the Royal Society of Sciences of Naples (1939), the Accademia dei Lincei of Rome (1947), the German Academy of Science (1950), the American Academy of Arts and Sciences (1959), the Paris Academy (1960), the Royal Dutch Academy (1961), the Academy of Science of Brussels (1961), the London Mathematical Society (1964), the Romanian Academy of Sciences (1965) and the Pontifical Academy of Sciences (1967).

Rotkiewicz, who was a student of Sierpiński's wrote in [12]:-

Sierpiński had exceptionally good health and a cheerful nature. ... He could work under any conditions. ... He did not like any corrections to his papers. When someone suggested a correction he added a line to it: 'Mr X remarked that ...' He was a creative mind and liked creative mathematics. He was the greatest and most productive of Polish mathematicians.