胡戈·施泰因豪斯Hugo Steinhaus

胡戈·施泰因豪斯出生于加利西亚的一个波兰犹太知识分子家庭。他出生的城镇亚斯沃位于加利西亚，大约在克拉科夫和伦贝格（也称Lwów，现称利沃夫）之间（尽管离克拉科夫比离Lwów稍近一些）。加利西亚在1772年瓜分波兰时被并入奥地利。然而，当施泰因豪斯在亚斯沃出生时，奥地利已将该地区命名为加利西亚和洛多梅里亚王国，并赋予其很大程度的行政自治权。施泰因豪斯的父母是Bogusław 施泰因豪斯（1854-1933）和Ewelina Lipszyc（1855-1948）。Bogusław经营一家店铺，从事烟草和烟草制品贸易，于1880年与Ewelina结婚。他们有四个孩子：Felicja 施泰因豪斯（1881-1964）；施泰因豪斯（1887-1972），即本传记的主人公；Irena Steinhaus（1889-1982）；以及Olga Maria Steinhaus（1892-1962）。他们在亚斯沃的房子曾是一座修道院的一部分，家族于1853年购得。同一建筑群的一部分是施泰因豪斯的祖父Józef创办的一家杂货店，出售葡萄酒、酒精饮料和利口酒。施泰因豪斯这样描述他的父亲[39]：-

父亲很容易发笑，热爱生活，热爱他的家和家人、美食美酒、好伙伴、他的田地和花园、他的砖厂以及马厩里的马。他认为贪婪和野心、过度的热情和攀附社会地位，与懒惰、挥霍、无能以及随便与任何人交往一样，都是幸福的障碍。他享受好天气、一座建造精良的建筑，或花园里一棵美丽的苹果树，胜过一大笔银行存款。

施泰因豪斯的叔叔Ignacy 施泰因豪斯（1860-1928）是一位重要人物，是奥地利议会中的政治家。他的性格与施泰因豪斯的父亲截然不同，在施泰因豪斯年轻时发挥了很大作用。

当施泰因豪斯七岁仍未开始上学时，他的父亲放弃了店铺，开始从事建筑行业。不久之后，施泰因豪斯在县立学校开始上学，学校位于一座监狱上方的楼层。九岁时，他进入当地的文理中学，开始学习拉丁语和德语。他写道[39]：-

在学校里，学习被视为一件严肃的事情。历史给我带来了相当大的困难，因为我无法死记硬背事实，也想不出更好的方法来掌握这门学科。

他十二岁时染上了百日咳，被送到波罗的海沿岸科沃布热格与姑母同住。第一次见到大海是一次奇妙的经历。他回到亚斯沃的家中，但小腿上出现了大片红斑，不得不卧床近三个月；他非常无聊。他的父母雇了一位保姆照看孩子们，在两次尝试均未成功后，他们雇了一位来自里昂的法国姑娘，她做得很好，教了孩子们许多东西。

施泰因豪斯从初级中学毕业升入高级中学，此后不久他的大姐结婚了。他开始考虑应该从事什么职业，起初想从军，但他的外祖父是一位坚定的和平主义者，很快说服施泰因豪斯放弃了那个想法。他在[39]中解释了他如何决定成为一名数学家：——

我父亲的熟人中有位年轻工程师叫Ludwik Silberman，他除了在理工学院取得工程学资格外，还完成了大学哲学系的数学课程。当我第一次有机会问他可以在哪里学到更多数学时，他告诉我大学就是学数学的地方。就在那时我做出了决定。

当然，施泰因豪斯在学校学习数学，但他决定通过自学来学更多。他读了Placyd Dziwinski为Lwów理工学院学生编写的Lectures on Mathematics、Franz Vendt关于微分与积分的书，以及Adam von Burg为维也纳大学学生编写的分析书。此时他已临近1905年5月参加Matura考试的时间。他参加了数学、物理、波兰语、德语、拉丁语、希腊语和历史的笔试。由于这些笔试卷标准很高，他免去了数学、物理和历史的口试，但其他科目有口试。此时已被Lwów大学录取的施泰因豪斯，在开始大学学习前有五个月的假期。他去塔特拉山进行了一次探险。

在Lwów的住宿安排很容易，因为他的表兄Dyk已在那里学习法律。施泰因豪斯与Dyk一起住在Dyk的寡母租的公寓里。他在[39]中写道：——

给我们讲授数学的是Józef Puzyna，他曾在柏林师从拉扎勒斯·福克斯——太可惜他没有更早去那里学习，那时卡尔·魏尔斯特拉斯正在柏林！……我还修了哲学和社会科学的课程。

施泰因豪斯在利沃夫求学期间，1906年春天，柏林夏洛滕堡理工学院的画法几何学教授Stanisław Jolles来到了利沃夫。Jolles并非来访问大学，而是陪同他的妻子，她正前往她拥有的附近工厂出差。Jolles教授的妻子认识施泰因豪斯表兄的母亲，便与丈夫一同拜访了Dyk和施泰因豪斯居住的公寓。听说施泰因豪斯想成为数学家，Jolles坚决地对他说：“年轻人，收拾行囊去哥廷根吧。”1905-06学年末，施泰因豪斯回到Jasło的父母家中，向他们讲述了Jolles的“命令”。施泰因豪斯的父亲并不热衷让儿子去哥廷根，但母亲全力支持这个想法。于是很容易地安排他从1906年秋季起在哥廷根继续数学学习。

施泰因豪斯及时赶在学期开始前，经弗罗茨瓦夫、柏林、哈勒和艾兴贝格前往哥廷根。大学给了他一份寄宿公寓的名单，他在大学附近租了一个带膳食的房间。当时大学里大约有2000名学生，其中约400名是外国人。他在哥廷根大学学习了五年数学，受到了一群令人惊叹的强大数学家的影响，包括费利克斯·伯恩斯坦、康斯坦丁·卡拉西奥多里、理查·科朗特、古斯塔夫·赫格洛茨、大卫·希尔伯特、菲利克斯·克莱因、保罗·克伯、埃德蒙·朗道（尽管他是在施泰因豪斯到哥廷根三年后才到达的）、卡尔·龙格、奥托·特普利茨和恩斯特·策梅洛。他的社交生活围绕着去国家咖啡馆，每周去两次市政游泳池。他的学术生活则以数学图书馆为中心。

尽管他在利沃夫曾将哲学作为副科学习，但他决定不在哥廷根选修哲学课程，而是选择应用数学作为副科[39]：-

这包括力学、解析几何、图解统计学、数值分析和大地测量学等主题。因此，我勤奋地在Prinzenstrasse的应用数学研究所上课，绘制图纸，并在Emil Wiechert的指导下，用经纬仪在城市街道上测量多边形。

施泰因豪斯在大卫·希尔伯特的指导下攻读博士学位。1911年，他以题为Neue Anwendungen des Dirichlet'schen Prinzips Ⓣ（约翰·彼得·古斯塔夫·勒热纳·狄利克雷原理的新应用）的学位论文获得博士学位，成绩优异。获得博士学位后[55]：-

他回到加利西亚，由于没有学术职位，他在1911年至1914年期间作为“私人学者”（用施泰因豪斯自己的话）在Jasło和克拉科夫之间旅行。那时，他发表了几部作品，打网球，并在维斯瓦河上划船。他还游历了意大利和法国。

对施泰因豪斯研究方向的主要影响，相当令人意外地，不是哥廷根的任何主要数学人物，而是来自昂利·勒贝格的影响。施泰因豪斯在完成博士学位后，大约在1912年研读了昂利·勒贝格的两部主要著作Leçons sur l'intégration et la recherche des fonctions primitives Ⓣ（积分与求原函数讲义）（1904年）和Leçons sur les séries trigonmétriques Ⓣ（三角级数讲义）（1906年）。

1914年，第一次世界大战爆发，施泰因豪斯随家人迁居维也纳。他在波兰军团服役，作为第一军团炮兵团的炮组指挥官参加了沃里尼亚战役。退役后，他住在克拉科夫。他在[41]中讲述了尽管有战争，1916年在克拉科夫的普兰蒂公园散步仍是安全的：-

在一次这样的散步中，我无意中听到“勒贝格测度”这几个字。我走近公园的长椅，向两位年轻的数学学徒做了自我介绍。他们告诉我，他们还有一位名叫Witold Wilkosz的同伴，他们对他赞不绝口。这两个年轻人是斯特凡·巴拿赫和欧顿·尼科迪姆。从那时起，我们定期见面，并且……我们决定成立一个数学学会。

斯特凡·巴拿赫和欧顿·尼科迪姆在这张长椅上的雕像照片见THIS LINK。

施泰因豪斯提议的数学学会最初是作为克拉科夫数学学会成立的，战争结束后不久，它变成了波兰数学会。施泰因豪斯在[41]中描述了新数学学会的起步，这段文字相当多地讲述了他在当时克拉科夫的生活：-

作为这个想法的发起人，我腾出我的房间供开会使用，作为准备工作的第一步，我把一块油布黑板钉在墙上。当寄宿公寓的法国女经理看到我所做的事时，她吓坏了——房东会怎么说？我让她平静下来，提醒她这栋楼的主人的是我叔叔的姐夫，她原谅了我的越轨行为。然而，我犯了一个错误。L先生摆出一副传统、顽固的房东姿态，对黑板本应服务的崇高目标无动于衷。学会扩大了——这是波兰此类事物的第一缕曙光。

也在这段时间，施泰因豪斯开始与斯特凡·巴拿赫合作，他们的第一部合作作品Sur la convergence en moyenne de séries de Fourier Ⓣ（关于约瑟夫·傅里叶级数的平均收敛）（1918年）于1916年完成。他与Stefania Szmosz（1896-1983）结婚，她是铁路工程师Marek Szmosz及其妻子Pepka Grünwald的女儿。Stefania于1896年5月2日出生在克拉科夫。施泰因豪斯和Stefania 施泰因豪斯有一个女儿Lidka Joanna 施泰因豪斯（1919-2000），她于1919年4月6日出生在利沃夫。

施泰因豪斯 在利沃夫的扬·卡齐米日大学（即此前的伦贝格大学）担任助理，并于1920年前后被提升为编外教授。此时 斯特凡·巴拿赫 已在利沃夫任职，该校的重要性迅速增长。马克·卡茨 是 施泰因豪斯 20世纪30年代在利沃夫的学生，他描述了 昂利·勒贝格 的工作对利沃夫学派的影响：-

昂利·勒贝格对卢布林学派的影响非常直接。该学派由施泰因豪斯和斯特凡·巴拿赫创立，主要集中于泛函分析及其多样应用、正交级数的一般理论以及概率论。毫无疑问，如果没有对昂利·勒贝格测度和积分的基本理解，这些理论都不会达到今天的显著地位。另一方面，昂利·勒贝格测度和积分的思想在卢布林得到了最引人注目和富有成果的应用。

直到1941年，施泰因豪斯都是卢布林学派的主要人物。1923年，他在Fundamenta Mathematicae上发表了基于测度论的掷硬币理论的第一个严格论述。1925年，他第一个定义并讨论了博弈论中的策略概念。施泰因豪斯于1927年与斯特凡·巴拿赫发表了他们的第二篇联合论文Sur le principe de la condensation des singularités Ⓣ（论奇点凝聚原理）。1929年，他与斯特凡·巴拿赫一起创办了一份新期刊Studia Mathematica，施泰因豪斯和斯特凡·巴拿赫成为首批编辑。编辑方针是：-

……专注于泛函分析及相关主题的研究。

施泰因豪斯参与的另一个重要出版事业始于1931年，是Mathematical Monographs的一个新系列。该系列由来自卢布林的施泰因豪斯和斯特凡·巴拿赫以及来自华沙的Knaster、卡齐米日·库拉托夫斯基、斯特凡·马祖尔凯维奇和瓦茨瓦夫·谢尔宾斯基担任编辑。对该系列的重要贡献是施泰因豪斯于1937年与斯特凡·喀茨马茨合著的一卷，The theory of orthogonal series。关于这本书的更多信息，见THIS LINK。

施泰因豪斯最著名的是他1937年写的书Mathematical Snapshots。马克·卡茨在[17]中写道：-

……要理解和欣赏施泰因豪斯的数学风格，就必须阅读（或者更确切地说，观看）一些快照。……旨在吸引“儿童中的科学家和科学家中的儿童”……它表达——并不总是明确地，有时甚至无意识地——施泰因豪斯所认为数学是什么以及应该是什么。对施泰因豪斯来说，数学是现实和生活的镜子，就像诗歌是一面镜子一样，他喜欢“玩弄”数字、集合和曲线，就像诗人玩弄词语、短语和声音一样。

关于Mathematical Snapshots的更多信息，见THIS LINK。

Stark [38]描述了施泰因豪斯在利沃夫的讲座：-

我们班由施泰因豪斯教授指导。这是一个非常大的班，分析课有超过220名学生挤在一个狭小且通风不良的教室里，有的站在过道上，有的坐在窗台上。……他的身影高高地坐在讲台上，旁边是一块五英尺乘五英尺的小黑板，俯瞰着拥挤的房间。……尽管施泰因豪斯在备课上很用心，但这些讲座对普通学生来说还是太难了。

利沃夫学派的数学家们在利沃夫的咖啡馆里做了大量的数学研究。苏格兰咖啡馆总体上最受数学家们欢迎，但施泰因豪斯除外，据斯坦尼斯瓦夫·乌拉姆记载：-

……他通常光顾一家更雅致的茶馆，那里号称拥有波兰最好的糕点。

这是位于Akademicka街22号的Ludwik Zalewski糖果店。然而，正是在苏格兰咖啡馆里，著名的Scottish Book诞生了，其中包含了在那里工作的数学家们提出的未解决问题。施泰因豪斯有时也会在苏格兰咖啡馆与同事们相聚，他为这本书贡献了十个问题，包括1941年5月31日写下的最后一个问题，那是在纳粹军队进入该镇前几天。

你可以在 THIS LINK 上看到更多关于苏格兰咖啡馆的信息。

1938年战争阴云密布时，施泰因豪斯提议授予昂利·勒贝格利沃夫荣誉学位。施泰因豪斯对马克·卡茨开玩笑说[17]：-

留下这样的记录也不错：第一位博士生是斯特凡·巴拿赫，最后一位是昂利·勒贝格。

昂利·勒贝格获得学位后的招待会在苏格兰咖啡馆举行，但只有十五位数学家出席，这表明由于政治局势，利沃夫的数学学派已大幅萎缩。他在[39]中写道：-

1941年6月22日星期日凌晨三点，我被一大群飞机的嗡嗡声吵醒。我望向窗外，在黎明的灰光中看到粉红色的天空，整个天空中不时爆发出白色的烟团。这些是苏联炮台爆炸的炮弹。在下面的街道上，我看到一队苏联士兵列队跑过——而且没有唱歌！这一切足以让我确信战争确实爆发了。我叫醒妻子，告诉她这个可怕的消息。当然，我们一刻也不怀疑德国人会占领利沃夫。

犹太人很快陷入极大的危险之中：-

……在苏联人撤离前，最近几天被捕的所有人都已被即决处决。……德国人立即发布新闻稿，声称受害者中没有犹太人，凶手是犹太人……在几天之内，约有3000名犹太人在Lwów及其周边地区丧生。……德国人没有参与这场屠杀，他们认为普通民众对犹太人自发采取的处置方式，有助于向世界证实民族社会主义关于犹太人对难以言说的罪行负有罪责的论点。

在被党卫军成员审讯和殴打之后，施泰因豪斯和他的妻子于1941年7月4日从他们在Lwów的家中离开，穿过他们在后花园篱笆上弄出的一个洞。这是他们待在家中的最后一天，因为从那时起，他们在战争岁月中一直躲避纳粹，遭受极大困苦，大多数时候忍饥挨饿，但施泰因豪斯始终在思考数学[17]：-

……即便在那时，他那敏锐而不安分的头脑仍在思考大量想法和计划。

他们不断在Lwów的不同房屋之间迁移，有时分开，一些朋友帮助了他们，但另一些人则拒绝相助。他们持有声称自己是希腊东正教徒的假出生证明，于1941年11月离开Lwów，前往Rudno。他们之所以幸存下来，是因为他们非常小心地掩盖行踪，并留下假线索。1942年7月11日，他们再次迁移，这次到了Stróze。阅读施泰因豪斯在[39]中的战时经历，可以生动地了解他和其他犹太人在此时所遭受的恐怖。施泰因豪斯有时能够私下授课，但这带来了额外的危险。这种恐怖直到战争结束才终止。

1945年，施泰因豪斯前往Kraków，并在那里接受了Wrocław大学的一个教授职位，在那里他承担了组织数学与自然科学系的任务。在随后的几年里，他多次访问美国的大学，包括Notre Dame。[17]中的马克·卡茨写道：-

……正是他，也许比任何其他个人都更多地帮助波兰数学从第二次世界大战使其沦为的灰烬中崛起，达到它如今所拥有的新的力量和受尊重的地位。

第二次世界大战结束后，Scottish Book似乎由施泰因豪斯在战争期间保存了下来，他将其寄给了美国的斯坦尼斯瓦夫·乌拉姆。这本书由斯坦尼斯瓦夫·乌拉姆翻译成英文并出版。现于弗罗茨瓦夫大学的施泰因豪斯决定，Scottish Book的传统太好了，不能就此终结。1946年，他将这一传统扩展到弗罗茨瓦夫，开始了New Scottish Book。

最后让我们考察一些上文未提及的施泰因豪斯的数学贡献。1944年，施泰因豪斯提出了将蛋糕分成$n$份的问题，使其既成比例（每个人都满意自己的一份）又无嫉妒（每个人都满意没有人得到超过公平份额）。对于$n = 2$，这个问题是平凡的，一个人切蛋糕，另一个人选择自己的一份。施泰因豪斯为$n = 3$找到了一个成比例但无嫉妒的解决方案。对于$n = 3$的施泰因豪斯问题，一个无嫉妒的解决方案于1962年由约翰·康威和弗瑞兹·约翰 Selfridge独立找到。对于一般的$n$，这个问题由Steven Brams和Alan Taylor于1995年解决。

施泰因豪斯的著作目录，见[22]，包含170篇文章。他在泛函分析方面做了重要工作，但他自己称他在这一领域最伟大的发现是斯特凡·巴拿赫。施泰因豪斯的一些早期工作是关于三角级数的。他第一个给出了一些例子，这些例子将导致该主题的显著进展。他给出了一个三角级数的例子，该级数在每一点都发散，但其系数趋于零。他还给出了一个三角级数的例子，该级数在一个区间收敛，但在第二个区间发散。

正如我们上面所指出的，施泰因豪斯的其他贡献涉及正交级数、概率论、实函数及其应用。特别是，他与独立函数理论相关联，这源于他在概率论方面的工作，并且他是第一个精确阐述“独立”和“均匀分布”概念的人。他在Probability, verisimilitude, credibility（波兰语）（1954）中给出了托马斯·贝叶斯原理的应用。他早些时候在撰写Quality control by sampling （a plea for Bayes' rule）（1951）时为托马斯·贝叶斯原理进行了辩护。Bernard Koopman在一篇评论中写道：-

作者简要考察了现代统计质量控制方法及其通过序贯分析改进的一些例子，并给出了两种解释：依据托马斯·贝叶斯的先验概率概念，以及依据当前否定这种概念的时尚。他展示了前一种方法的一些优越性，并强调他的观点，即第二种方法只是在表面上避免了托马斯·贝叶斯的概念。最后，他试图回答一个对假设待估计参数α具有均匀先验分布的著名反对意见（即反对意见认为，$\alpha^{2}$或$\alpha$的某个其他函数β将不会均匀分布）。他的回答是，对于质量控制，$\alpha$只是一件事，即总体中合格品的比例；这个变量（显然按照惯例）必须被视为均匀分布，而不是$\beta$。

他对博弈论做出了贡献，例如在与Stanisław Trybuła合作的论文Measurement by successive comparison（波兰语）（1958）中。

除了他的著名著作Mathematical Snapshots外，他还写了备受赞誉的One Hundred Problemsin elementary mathematics。关于这些书的信息，见THIS LINK。

施泰因豪斯在支持波兰数学方面发挥了重要作用[55]：-

他分别于1937-1938年和1946-1948年担任波兰数学会的利沃夫分会和该学会的弗罗茨瓦夫分会主席，同时也是该学会的副主席。他是波兰学术院（1945年当选）的通讯院士。他还是弗罗茨瓦夫科学学会的创始人之一，该学会成立于1946年，他担任秘书长直至1947年，并于1956-1958年担任主席。他还参与了国家数学研究所的组织工作（成立于1947-1948年，后转变为波兰科学院数学研究所），担任副主任直至1952年，后来担任自然与经济应用系主任。他曾是波兰科学院人体测量学委员会主席。自1951年起，他是华沙科学学会的正式成员，自1951年起，是波兰科学院的正式成员。他参与了弗罗茨瓦夫期刊《数学讨论会》（自1947年出版）的创办，1948年，他重新创办了原在利沃夫的《数学研究》，并于1953年在弗罗茨瓦夫创办了《数学应用》（现名为《数学应用》）。

他获得了许多奖项，我们仅列举少数几个。Polish Mathematical Society授予他斯特凡·巴拿赫奖（1946年）和斯特凡·马祖尔凯维奇奖（1951年）。他获得了波兰学术院奖（1948年）、一级国家奖（1951年）、弗罗茨瓦夫市奖（1960年）和Alfred Jurzykowski奖。他获得了华沙大学、波兹南大学和弗罗茨瓦夫大学以及弗罗茨瓦夫医学院的名誉博士学位。他被授予波兰复兴勋章军官十字、带星指挥官十字（1957年）、一级劳动旗帜勋章（1959年）以及其他几个奖项。

施泰因豪斯于1972年2月在弗罗茨瓦夫去世，享年85岁。他被安葬在圣家堂区公墓；他的墓碑上写着：“在精神与物质之间，是数学。”

Hugo Steinhaus was born in Galicia into a Polish family of Jewish intellectuals. The town of his birth, Jasło, was in Galicia, about half way between Kraków and Lemberg (also known as Lwów but now called Lviv) (although a bit nearer Kraków than Lwów). Galicia was attached to Austria in the 1772 partition of Poland. However, by the time Steinhaus was born in Jasło, Austria had named the region the Kingdom of Galicia and Lodomeria and given it a large degree of administrative autonomy. Hugo's parents were Bogusław Steinhaus (1854-1933) and Ewelina Lipszyc (1855-1948). Bogusław, who had a store and traded in tobacco and tobacco products, married Ewelina in 1880. They had four children: Felicja Steinhaus (1881-1964); Hugo Steinhaus (1887-1972), the subject of this biography; Irena Steinhaus (1889-1982); and Olga Maria Steinhaus (1892-1962). Their house in Jasło had been part of a monastery which the family had acquired in 1853. Part of the same complex was a grocery store with wines, alcoholic drinks and liqueurs established by Hugo's grandfather Józef. Hugo described his father as follows [39]:-

Father laughed readily, loved life, loved his home and family, good food and drink, good company, his fields and gardens, his brickworks, and the horses in his stable. He believed that greed and ambition, excessive zeal and social climbing are just as much obstacles to happiness as idleness, prodigality, incompetence, and readiness to associate with just anybody. He enjoyed good weather, a well-constructed building, or a beautiful apple tree in the garden more than a big bank account.

Steinhaus's uncle, Ignacy Steinhaus (1860-1928), was an important person being a politician in the Austrian parliament. He had a very different personality to Hugo's father, and played a large part in his young years.

When Hugo was seven years old and still had not started school, his father gave up his store and began working in the building trade. Soon after Hugo began his schooling in the county school, situated on an upper floor above a jail. At age nine he entered the local Gymnasium and began learning Latin and German. He wrote [39]:-

Learning was treated as a serious business in the school. History caused me considerable difficulties because I could not learn facts by heart and couldn't think of a better way of mastering the subject.

When he was twelve years old he contracted whooping cough and was sent to live with his aunt in Kołobrzeg on the Baltic coast. Seeing the sea or the first time was an amazing experience. He returned home to Jasło but developed large red blotches on his calf and had to stay in bed for nearly three months; he was very bored. His parents employed a nanny to look after their children and, after two attempts which were not successful, they employed a French girl from Lyon who did very well for the children and taught them much.

Hugo graduated from the Junior Gymnasium to the Senior Gymnasium and soon after this his eldest sister got married. He began to consider what career he should follow and at first thought of an army career, but his maternal grandfather was an ardent pacifist who soon persuaded Hugo to give up that idea. He explains in [39] how he decided to become a mathematician:-

Among my Father's acquaintances was a young engineer named Ludwik Silberman, who, in addition to qualifying in engineering at the Polytechnic, had also completed the mathematics programme of the philosophical faculty of the university. When, at the first opportunity, I asked him where one might learn more mathematics, he told me that the university was the place for it. It was then that I made my decision.

Of course, Steinhaus was studying mathematics at school but he decided to learn more working on his own. He read Placyd Dziwinski's Lectures on Mathematics aimed at students at Lwów Polytechnic, Franz Vendt's book on differential and integral calculus, and Adam von Burg's book on analysis aimed at students at the University of Vienna. He was now approaching the time to take the Matura examinations in May 1905. He sat written examinations in mathematics, physics, Polish, German, Latin, Greek, and history. Because these written papers were of a high standard, he avoided oral examinations in mathematics, physics and history but had orals in the other topics. Now accepted by Lwów University, Steinhaus had five months vacation before beginning his university studies. He went on an expedition to the Tatra Mountains.

Accommodation in Lwów proved easy to arrange since his cousin Dyk was already there studying law. Steinhaus lived with Dyk in the flat rented by Dyk's widowed mother. He writes in [39]:-

We were lectured on mathematics by Józef Puzyna who had been a student of Fuchs in Berlin - too bad he hadn't studied there earlier when Weierstrass was in Berlin! ... I also took courses on philosophy and social sciences.

Steinhaus was enjoying his studies in Lwów when in spring 1906, Stanisław Jolles, professor of descriptive geometry at the Charlottenburg Polytechnic in Berlin, arrived in Lwów. Jolles had not come to visit the University but rather he was accompanying his wife who was on a business trip to factories she owned nearby. Professor Jolles' wife knew the mother of Hugo Steinhaus's cousin and came on a visit with her husband to the flat where Dyk and Hugo were living. On hearing that Steinhaus wanted to be a mathematician, Jolles said forcibly to him, "Young man, pack your trunk and go to Göttingen." At the end of the academic year 1905-06, Steinhaus returned to his parent's home in Jasło and told them about Jolles' "order." Steinhaus's father was not keen that his son should go to Göttingen but his mother fully supported the idea. It was easily arranged that he would continue his mathematical studies at Göttingen beginning in the autumn of 1906.

In good time for the start of the term, Steinhaus travelled to Göttingen via Wrocław, Berlin, Halle and Eichenberg. The university supplied him with a list of boarding houses and he took a room with board close to the University. At this time there were around 2000 students at the university; about 400 of whom were foreign. He spent five years studying mathematics at the University of Göttingen where he was influenced by an amazingly strong group of mathematicians including Felix Bernstein, Carathéodory, Courant, Herglotz, Hilbert, Klein, Koebe, Edmund Landau (although he only arrived in Göttingen after Steinhaus had been there three years), Runge, Toeplitz, and Zermelo. His social life was around visits to the café National and he went twice a week to the municipal swimming pool. His academic life centred round the mathematics library.

Although he had studied philosophy as a minor subject at Lwów, he decided not to take philosophy courses at Göttingen but rather to take applied mathematics as his minor subject [39]:-

This included such topics as mechanics, analytic geometry, graphical statistics, numerical analysis, and geodesy. Thus I diligently attended classes at the Institute of Applied Mathematics on Prinzenstrasse, made drawings, and, under Emil Wiechert's guidance, took measurements of polygons on the city streets with a theodolite.

For his doctorate Steinhaus studied under Hilbert's supervision. He was awarded his doctorate, with distinction, for a dissertation Neue Anwendungen des Dirichlet'schen Prinzips Ⓣ in 1911. After the award of his doctorate [55]:-

He returned to Galicia, and being without an academic post, he spent the period of 1911-1914 as a "private scholar" (to use Steinhaus's own term) travelling between Jasło and Kraków. At that time, he published a few works, played tennis and rowed on the Vistula river. He also travelled to Italy and France.

The main influence on the direction that Steinhaus's research would take was, rather surprisingly, none of the major mathematical figures at Göttingen but rather the influence came from Lebesgue. Steinhaus studied Lebesgue's two major books Leçons sur l'intégration et la recherche des fonctions primitives Ⓣ (1904) and Leçons sur les séries trigonmétriques Ⓣ (1906) around 1912 after completing his doctorate.

In 1914 World War I broke out and Steinhaus went with his family to live in Vienna. He undertook military service in the Polish Legion and was involved in the Volhynia campaign as a gun crew commander in the 1st Regiment of Legionary Artillery. Following his discharge, he lived in Kraków. He relates in [41] how, despite the war, it was safe to walk in the Planty Gardens in Kraków in 1916:-

During one such walk I overheard the words "Lebesgue measure". I approached the park bench and introduced myself to the two young apprentices of mathematics. They told me they had another companion by the name of Witold Wilkosz, whom they extravagantly praised. The youngsters were Stefan Banach and Otto Nikodym. From then on we would meet on a regular basis, and ... we decided to establish a mathematical society.

A picture of statues of Banach and Nikodym on this bench is at THIS LINK.

The mathematical society which Steinhaus proposed was started as the Mathematical Society of Kraków and, shortly after the war ended, it became the Polish Mathematical Society. Steinhaus described the beginnings of the new mathematical society in [41] in a passage which tells us quite a lot about his life in Kraków at the time:-

As initiator of the idea, I made my room available for meetings and, as the first step in preparations, nailed an oilcloth blackboard to the wall. When the French manager of the boarding house saw what I had done, she was terrified - what was the proprietor going to say? I calmed her down reminding her that the owner of the building was my uncle's brother-in-law, and she forgave my transgression. However, I had made a mistake. Mr L took the position of a traditional, hard-nosed landlord and was unmoved by the lofty goal the blackboard was supposed to serve. The society expanded - it was the first ray of light of this kind in Poland.

Also at this time Steinhaus started a collaboration with Banach and their first joint work Sur la convergence en moyenne de séries de Fourier Ⓣ (1918) was completed in 1916. He married Stefania Szmosz (1896-1983), the daughter of the railway engineer Marek Szmosz and his wife Pepka Grünwald. Stefania had been born in Kraków on 2 May 1896. Hugo and Stefania Steinhaus had a daughter Lidka Joanna Steinhaus (1919-2000) who was born in Lwów on 6 April 1919.

Steinhaus took up an appointment as an assistant at the Jan Kazimierz University in Lwów (as Lemberg University had become) and, around 1920, he was promoted to Extraordinary Professor. Banach was by this time on the staff at Lwów and the school rapidly grew in importance. Kac, who was a student of Steinhaus in Lwów during the 1930s, described the influence of Lebesgue's work on the Lwów school:-

The influence of Lebesgue on the Lwów school was very direct. The school, founded ... by Steinhaus and Banach, concentrated mainly on functional analysis and its diverse applications, the general theory of orthogonal series, and probability theory. There is no doubt that none of these theories would have achieved today's level of prominence without an essential understanding of the Lebesgue measure and integral. On the other hand, the ideas of Lebesgue measure and integral found their most striking and fruitful applications there in Lwów.

Steinhaus was the main figure in the Lwów School up till 1941. In 1923 he published in Fundamenta Mathematicae the first rigorous account of the theory of tossing coins based on measure theory. In 1925 he was the first to define and discuss the concept of strategy in game theory. Steinhaus published his second joint paper with Banach in 1927 Sur le principe de la condensation des singularités Ⓣ. In 1929, together with Banach, he started a new journal Studia Mathematica and Steinhaus and Banach became the first editors. The editorial policy was:-

... to focus on research in functional analysis and related topics.

Another important publishing venture in which Steinhaus was involved, begun in 1931, was a new series of Mathematical Monographs. The series was set up under the editorship of Steinhaus and Banach from Lwów and Knaster, Kuratowski, Mazurkiewicz, and Sierpinski from Warsaw. An important contribution to the series was a volume written by Steinhaus jointly with Kaczmarz in 1937, The theory of orthogonal series. For more information about this book, see THIS LINK.

Steinhaus is best known for his book Mathematical Snapshots written in 1937. Kac, writing in [17] says:-

... to understand and appreciate Steinhaus's mathematical style, one must read (or rather look at) snapshots. ... designed to appeal to "the scientist in the child and the child in the scientist" ... it expresses, not always explicitly and at times even unconsciously, what Steinhaus thought mathematics is and should be. To Steinhaus mathematics was a mirror of reality and life much in the same way as poetry is a mirror, and he liked to "play" with numbers, sets, and curves, the way a poet plays with words, phrases, and sounds.

For much more information about Mathematical Snapshots, see THIS LINK.

Stark [38] describes Steinhaus lectures in Lwów:-

My class was guided by Professor Steinhaus. It was a very big class, and the analysis lecture was attended by over 220 students squeezed into a smallish and poorly ventilated lecture room, standing in the aisles, and sitting on the window sills. ... His figure, perched high on the podium by a small five by five foot blackboard dominated the crowded room. ... despite Steinhaus's attention to preparation, the lectures were too difficult for the average student.

The mathematicians of the Lwów school did a great deal of mathematical research in the cafés of Lwów. The Scottish Café was the most popular with the mathematicians in general but not with Steinhaus who (according to Ulam):-

... usually frequented a more genteel tea shop that boasted the best pastry in Poland.

This was Ludwik Zalewski's Confectionery at 22 Akademicka Street. It was in the Scottish Café, however, that the famous Scottish Book consisting of open questions posed by the mathematicians working there came into being. Steinhaus, who sometimes joined his colleagues in the Scottish Café, contributed ten problems to the book, including the final one written on 31 May 1941 only days before the Nazi troops entered the town.

You can see more about the Scottish Café at THIS LINK.

When the prospect of war was looming in 1938, Steinhaus proposed Lebesgue for an honorary degree from Lwów. Steinhaus joked to Kac that [17]:-

It will not be a bad record to leave behind, to have had Banach as the first and Lebesgue as the last doctoral candidate.

The reception for Lebesgue, after the award of his degree, was held in the Scottish Café but only fifteen mathematicians attended, showing that the school of mathematics in Lwów had shrunk considerably due to the political situation. He writes in [39]:-

At three in the morning on Sunday 22 June 1941, I was awoken by the droning of a great fleet of planes. I looked out of the window and saw in the grey light of early dawn a pink sky over the whole of which white puffs of smoke were bursting intermittently. These were exploding shells from Soviet batteries. Down in the street I saw a squadron of Soviet soldiers running by in formation - and not singing! All this was more than enough to convince me that war had indeed broken out. I woke my wife to tell her the ghastly news. Of course, we didn't doubt for an instant that the Germans would take Lwów.

Jews were soon in great danger:-

... all those arrested in the last few days had been summarily executed before the Soviets' departure. ... The Germans immediately issued a press release stating that there were no Jews among the victims, and the murderers had been Jews ... Within the space of a few days some 3000 Jews had perished in and around Lwów. ... The Germans took no part in the massacre, considering that the spontaneity of the treatment meted out to the Jews by ordinary people served to confirm before the world the nationalist-socialist thesis of Jewish guilt for unspeakable crimes.

After being interrogated and assaulted by SS men, Steinhaus and his wife left their home in Lwów through a hole they had made in their back garden fence on 4 July 1941. This was their last day at their home for they spent the war years from then hiding from the Nazis, suffering great hardships, going hungry most of the time but Steinhaus was always thinking about mathematics [17]:-

... even then his sharp restless mind was at work on a multitude of ideas and projects.

Continually moving to different houses in Lwów, sometimes separating, they were helped by some friends but others refused. With fake birth certificates claiming they were Greek Orthodox, they left Lwów in November 1941 and went to Rudno. They survived since they were very careful to cover their tracks, and leave false leads. On 11 July 1942 they moved again, this time to Stróze. Reading Steinhaus's wartime experiences in [39] gives a vivid account of the terror he, and other Jews, suffered at this time. Steinhaus was able to give private teaching at times, but this carried extra danger. The terror only ended when the war ended.

In 1945 Steinhaus went to Kraków and while there accepted an offer of a professorship at the University of Wrocław where he had the task of organising a Department of Mathematics and Natural Science. Over the following years he made many visits to universities in the United States including Notre Dame. Kac in [17] writes:-

... it was he who, perhaps more than any other individual, helped to raise Polish mathematics from the ashes to which it had been reduced by the Second World War to the position of new strength and respect which it now occupies.

After the end of World War II the Scottish Book, which seems to have been preserved through the war by Steinhaus, was sent by him to Ulam in the United States. The book was translated into English by Ulam and published. Steinhaus, now in the University of Wrocław, decided that the tradition of the Scottish Book was too good to end. In 1946 he extended the tradition to Wrocław starting the New Scottish Book.

Let us finally examine some of Steinhaus's mathematical contributions which we have not mentioned above. In 1944 Steinhaus proposed the problem of dividing a cake into $n$ pieces so that it is proportional (each person is satisfied with their share) and envy free (each person is satisfied nobody is receiving more than a fair share). For $n = 2$ the problem is trivial, one person cuts the cake, the other chooses their piece. Steinhaus found a proportional but not envy free solution for $n = 3$. An envy free solution to Steinhaus's problem for $n = 3$ was found in 1962 by John H Conway and, independently, by John Selfridge. For general $n$ the problem was solved by Steven Brams and Alan Taylor in 1995.

Steinhaus's bibliography, see [22], contains 170 articles. He did important work on functional analysis, but he himself described his greatest discovery in this area as Stefan Banach. Some of Steinhaus's early work was on trigonometric series. He was the first to give some examples which would lead to marked progress in the subject. He gave an example of a trigonometric series which diverged at every point, yet its coefficients tended to zero. He also gave an example of a trigonometric series which converged in one interval but diverged in a second interval.

As we have noted above, other contributions by Steinhaus were on orthogonal series, probability theory, real functions and their applications. In particular he is associated with the theory of independent functions, arising from his work in probability theory, and he was the first to make precise the concepts of "independent" and "uniformly distributed". He gave applications of Bayes' principle in Probability, verisimilitude, credibility (Polish) (1954). He had earlier defended Bayes' principle writing Quality control by sampling (a plea for Bayes' rule) (1951). Bernard Koopman writes in a review:-

The author examines briefly some examples of modern statistical methods of quality control and their refinements by sequential analysis, and gives two accounts of them: in terms of the Bayes' notion of a priori probability, and in terms of the present fashion of repudiating such a notion. He shows some superiorities of the former approach, and emphasizes his view that the second avoids Bayes' notion in appearance only. He concludes by attempting an answer to a well-known objection to the assumption of uniform a priori distribution of the parameter α to be estimated (the objection, namely, that $\alpha^{2}$ or some other function β of $\alpha$ will then not be uniformly distributed). His answer is that for quality control $\alpha$ is just one thing, the proportion of good items in the population; this is the variable which (apparently by convention) must be taken as uniformly distributed, not $\beta$.

He made contributions to the theory of games, for example in the paper Measurement by successive comparison (Polish) (1958) written in collaboration with Stanisław Trybuła.

In addition to his famous book Mathematical Snapshots he also wrote the highly acclaimed One Hundred Problems in elementary mathematics. For information on these books, see THIS LINK.

Steinhaus played a major role in supporting mathematics in Poland [55]:-

He was the president of the Lwów Branch of the Polish Mathematical Society and the Wrocław Branch of that society in the 1937-1938 and 1946-1948 terms respectively, and also the vice-president of the society. He was a corresponding member of the Polish Academy of Learning (elected in 1945). He was also among the founders of the Wrocław Scientific Society, which was established in 1946, its secretary general until 1947, and its president in 1956-1958. He also took part in organising the State Mathematical Institute (established in 1947-1948, later transformed into the Institute of Mathematics of the Polish Academy of Sciences), was its vice-director until 1952, and later the head of the Department of Natural and Economic Applications. He was the chairman of the Commission of Anthropometry of the Polish Academy of Sciences. Since 1951, he was an ordinary member of the Warsaw Scientific Society, and since 1951, a full member of the Polish Academy of Sciences. He participated in the foundation of the Wrocław-based periodical "Colloquium Mathematicum" (published since 1947), in 1948, he re-established the previously Lwów-based "Studia Mathematica", and in 1953, he founded "Zastosowania Matematyki" in Wrocław (now titled "Applicationes Mathematicae").

He received many awards and we list only a few. The Polish Mathematical Society awarded him the Stefan Banach Award (1946) and the Stefan Mazurkiewicz Award (1951). He received the Award of the Polish Academy of Learning (1948), the First Degree State Award (1951), the City of Wrocław Award (1960), and the Alfred Jurzykowski Award. He received honorary doctorates from the universities of Warsaw, Poznan and Wrocław, and from the Wrocław Medical Academy. He was awarded the Officer's Cross of the Order of Polonia Restituta, the Commander's Cross with Star (1957), the Order of the Work Flag First Class (1959) and several others.

Steinhaus died in Wrocław aged 85 in February 1972. He was buried in the Holy Family Parish Cemetery; his headstone reads, "Between spirit and matter, there is mathematics."