论坛 › 数学哲学&悖论&本体论 › [转帖]崇高的人格与光辉的数学成就——希尔伯特的数学生涯

zhcf 发表于 2006-7-18 17:26

[转帖]崇高的人格与光辉的数学成就——希尔伯特的数学生涯

<p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">崇高的人格与光辉的数学成就——希尔伯特的数学生涯</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">&nbsp;</font></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">摘要：本文叙述了希尔伯特的生平，列举并论述了他的数学研究成果，探</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">讨希尔伯特对后世数学发展的巨大影响。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">&nbsp;</font></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">关键词：《几何基础》</span><font face="Times New Roman"> </font><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">公理化</span><font face="Times New Roman"> </font><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">哥廷根</span><font face="Times New Roman"> </font><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">“</span><span lang="EN-US"><font face="Times New Roman">23</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">个数学问题”</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">&nbsp;</font></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">&nbsp;</font></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">前</span><font face="Times New Roman"> </font><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">言</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">&nbsp;</font></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">作为一个时代数学界的领袖，德国人民伟大的儿子，当大卫·希尔伯特</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">1943</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年在哥廷根与世长辞时，人们开始回顾他所留下的精神印记和正在</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">消失在地平线下的那个数学时代，似乎感到希尔伯特的时代比起以往和</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">以后贯穿着更完美的平衡——精通单个具体问题和形成一般抽象概念之</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">间的平衡。</span><font face="Times New Roman"> </font></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">我们称之为希尔伯特的时代，正因为是他，希尔伯特，通过自己的工作</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">做出了巨大的贡献，开创了二十世纪初那个数学大发展的时代。而后继</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">者们所走的道路，也几乎都可以追溯到他的推动。希尔伯特是推动着一</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">个时代的数学的人，“在以后的时代里我们还没有找到可以达到与他相</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">比的崇高形象”（赫尔曼·外尔语）。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">&nbsp;</font></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">&nbsp;</font></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">大卫·希尔伯特对数学的贡献是巨大的和多方面的，研究领域涉及代数</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">不变式，代数数域，几何基础，变分法，积分方程，无穷维空间，物理</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">学和数学基础等。当然，他在数学领域所做出的最具影响的贡献还是著</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">名的几何基础和“</span><span lang="EN-US"><font face="Times New Roman">23</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">个数学问题”，它们贯穿整个</span><span lang="EN-US"><font face="Times New Roman">20</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">世纪的数学乃至现</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">在，影响之深远是我们所无法估量的。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">另一方面，希尔伯特的崇高人格更加为人称道的。许多在数学发展中</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">起了相当大作用的年轻数学家，都曾在</span><span lang="EN-US"><font face="Times New Roman">1900</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">至</span><span lang="EN-US"><font face="Times New Roman">1914</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年间侨居哥廷根，师</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">从希尔伯特。而他的问题、观点和数学研究方法的影响更远远超过直接受</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">他教导所鼓舞的那些人的范围。希尔波特的政治人格同样崇高，是“独一</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">无二地没有国家和种族偏见的人”，他反对沙文主义，并且主张“科学无</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">国界”，在政治上始终站在自由和民主的一方。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">我们研究希尔伯特的数学思想和数学成就，以及产生这种成就的源泉，</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">就要领略他伟大的人格，追溯他的人生轨迹。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">&nbsp;</font></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">一、</span><font face="Times New Roman"> </font><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">生平与为人</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">&nbsp;</font></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">1861</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年春的一天，奥托·希尔伯特和他夫人玛丽亚的遗传基因偶然地</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">结合，孕育了一个非同寻常的天才人物；</span><span lang="EN-US"><font face="Times New Roman">1862</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年</span><span lang="EN-US"><font face="Times New Roman">1</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">只</span><span lang="EN-US"><font face="Times New Roman">23</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">日下午一点钟，</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">他们的第一个孩子降生在靠近东普鲁士首府哥尼斯堡的韦洛。父母给他起</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">了个名字叫大卫。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">希尔伯特和德国的国家主义几乎同时诞生。他来到人世前的几个月，</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">已故普鲁士国王的兄弟到哥尼斯堡进行了一次传统的朝拜。在那座古老的</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">城堡里，他带上了王冠。东普鲁士首都建于公元</span><span lang="EN-US"><font face="Times New Roman">13</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">世纪中叶，是条顿族骑</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">士修筑的城堡。市内有七座各具特色的大桥，横跨普累格尔河。其中有五</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">座把河岸同河中的克亲芳福岛相连接。这些桥可不简单，哥尼期堡因此而</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">载入了数学史：桥的配置能引出一个数学问题，牵涉著名的拓扑学基础，</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">之前的一个世纪被欧拉</span><span lang="EN-US"><font face="Times New Roman">(Euler)</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">所解决。哥尼斯堡大教堂在克余芳福岛，近</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">旁是一所古老的大学，还有哥尼斯堡最伟大的居民伊曼努尔·康德</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">(1mmanud Kant)</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">的墓地。像哥尼斯堡所有的孩子一样，大卫的成长也深受</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">康德言论的抚育。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">1880</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年秋天，</span><span lang="EN-US"><font face="Times New Roman">18</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">岁的希尔伯特进人家乡的哥尼斯堡大学，他不顾当法</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">官的父亲希望他子承父业的愿望，毫不犹豫地进了哲学系学习数学（当时</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">的大学，数学还设在哲学系内）．希尔伯特发现当时的大学生活要多自由有</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">多自由．意想不到的自由，使许多年轻人把大学第一年的宝贵时光都花费</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">在学生互助会的传统活动饮酒和斗剑上。然而对希尔伯特来说，大学生活</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">的更加迷人之处却在于他终于能自由地把全部精力给予数学了．</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">许多德国学生有从一个大学到另一个大学周游的习惯，希尔伯特却不</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">同，一直在家乡求学，正是在嘉兴的大学里，他攀上了学术界的最初几级</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">台阶，成为大学讲师，在适当的时候升为副教授。</span><span lang="EN-US"><font face="Times New Roman">1895</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年在费力克斯·克</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">莱因的建议下，被授予哥廷根的正教授的职位，这是他的一流代数家的声</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">誉已经建立起来了。而哥廷根由于有希尔伯特和稍后闵可夫斯基的加盟，</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">一下子成为了世界数学的中心。从这时一直到去世，希尔伯特一直在哥廷</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">根直至</span><span lang="EN-US"><font face="Times New Roman">1930</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年退休。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">&nbsp;</font></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">希尔伯特与三个人的关系很值得关注。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">一是他与克罗内克不免充满矛盾的态度。希尔伯特是康托尔的一般集</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">合论的早期一个少数的拥护者之一。而克罗内克正是康托尔的死敌。在希</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">尔伯特看来，克罗内克的数学就是普洛克鲁斯的梯床，这位老数学家利用</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">自己的全市和声望，压制那些不符合自己数学思想的其它声音。克罗内克</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">坚持定理的证明必须通过整数明显构造出来。然而另一方面。他又依赖克</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">罗内克，因为在希尔伯特的代数时期，克罗内克的工作的重要性是毋庸讳</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">言的。晚年的希尔伯特在这一方面的矛盾其实更加尖锐，与布劳维尔的直</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">觉主义的论战，其实是在与克罗内克的鬼魂的论战。希尔伯特一方面同克</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">罗内克斗争事实上又在另一方面追随他：他须沿着严格的直觉主义的路线</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">来思考，以求保护非直觉主义的数学。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">另两位是阿道尔·胡尔维茨和闵可夫斯基，前者是希尔伯特的老师、</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">朋友和前任，后者则在青年时就成为希尔伯特挚友并且成为希尔伯特前半</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">生最忠实的数学伙伴。</span><span lang="EN-US"><font face="Times New Roman">1902</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年希尔伯特和闵可夫斯基在哥廷根重新聚首，</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">这之后的十年，直至闵可夫斯基逝世，数学领域因二人的共同工作经历了</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">一段伟大而光辉的时期。希尔伯特后来这样谈到他的朋友和他们共同工作</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">的这段时期：“我们的科学，我们爱它超过一切，它把我们联系在一起。在</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">我们看来，它好像鲜花盛开的花园。在花园中，有许多踏平的路径可以使</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">我们从容的左右环顾，毫不费力的尽情享受，特别是有趣味相投的游伴在</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">身旁。但是我们也喜欢寻求隐秘的小径。发现许多美观的新景，当我们向</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">对方指出来，我们就更加快乐。”这也不仅证明他们基于共同的科学兴趣的</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">友谊是如此的深厚，而且我们似乎由这几句话听到希尔伯特这位吹笛人所</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">吹的甜蜜的芦笛声，它诱惑许多老鼠跟着他投入数学的深河（赫尔曼·外</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">尔语）。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">&nbsp;</font></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">这篇文章的主旨只是想简略谈一下希尔伯特个性中的个性方面，因此</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">笔者并不准备过多涉及他对人们生活中的态度，像社会及政治、艺术、宗</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">教道德和规范、家庭、友谊、爱情等等方面，也更加没有必要指出在的人</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">格光环下的某些阴影。然而不可忽略的是以上我们所谈到的他的同行和更</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">多的环境因素。像哥廷根那样的小镇中的大学，特别是处于</span><span lang="EN-US"><font face="Times New Roman">1914</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年以前美</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">好平静的日子里，正是发展理论科学的最有利场所。教授们的崇高的科学</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">地位，以及大学城中一切事情都和大学密切相关，这在当今的中国几乎是</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">一种不可想象的气氛。此外，一旦一帮充满求知欲望的学生围绕着希尔伯</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">特，不被教学杂务打扰而专门从事研究，彼此之间相互激励，又怎能不产</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">生丰富的数学硕果。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">当然，这其中更加不能忽略的是希尔伯特的个人魅力。一个希尔伯特</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">的学生回忆说：“我去听希尔伯特开的课，课程讲的是数的概念和化圆为方。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">他讲的内容一直钻进我的脑子里。新世界的门向我敞开了。我在他的班上</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">听课没多久就在我年青的心里下了决心，我必须用一切方法去阅读和研究</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">他所写的一切。……这之后几个月使我一生中最幸福的几个月，经历了我</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">们共同分担的疑虑和失败的岁月之后，它的光辉仍抚慰着我的心灵。”</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">一个数学家对于他所处时代的推动并不直接和他科学研究工作的分量</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">成比例。希尔伯特的数学工作博大精深，然而他的影响并不完全来自这些</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">工作。同样是哥廷根历史上领军人物的高斯，其数学成就甚至要高过希尔</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">伯特，但是他对同时代的人的激励却很少，不仅没有形成学派，甚至给某</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">些年青数学家的前途带来了毁灭性的灾难。高斯与小波尔约和阿贝尔的纠</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">葛中，高斯都未能表现出一个数学领袖应有的风范。这或许是个人的天性</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">所决定。尽管很多的创造性的天才习惯于孤独与默默无闻，但是希尔伯特</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">充满生活热情的天性使他选择了另一种方式。他喜欢和其他人交往，尤其</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">是同年青科学家交往，并在交往中发展自己，也给对方带来启发。正如他</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">从胡尔维茨那儿学到东西，在年青时不顾世俗的偏见和闵可夫斯基结成了</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">终生的友谊。他也在环绕哥廷根的树林中长时期漫步或雨天在有顶的花园</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">中走来走去时，把科学传授给自己的学生，至少是那些他深感兴趣的学生。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">他的乐观主义，他的热情，他对科学的崇高价值的不可动摇的信念以及他</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">坚信对于简单明了的问题能够求出简单明了的答案的理性的力量，都具有</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">不可抗拒的感染力。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">他憎恶假装冷淡的势利态度与游手好闲甚至玩世不恭的犬儒主义，他</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">对人对事总是采取直截了当的态度。即使这样。在他周围总是充满快乐和</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">欢笑。他惊人的勤奋。“天才就是勤奋”是他的座右铭。而最卓著的是他伟</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">大的启示性力量，有时甚至使平庸的智利提高到远远超过你所期待的水平</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">而取得惊人的成就。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">&nbsp;</font></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">二、</span><font face="Times New Roman"> </font><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">数学工作</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">&nbsp;</font></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">希尔伯特的数学工作涉及的领域非常广泛而且成就巨大。他的工作可</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">以清楚的分为不同的时期，每一时期他都几乎集中于一组特殊的问题，当</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">他全神贯注于微分方程时，微分防城似乎就是一切，放弃一个题目，他就</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">永远的离开，转向另外的题目。他就是以这样特殊的方式造就他的广博。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">一般的，将他的工作氛围五个主要时期：</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">1</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、不变式理论</span><font face="Times New Roman"> </font><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">1885</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">－</span><span lang="EN-US"><font face="Times New Roman">1893</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">2</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、代数数域理论</span><font face="Times New Roman"> </font><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">1893</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">－</span><span lang="EN-US"><font face="Times New Roman">1898</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">3</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、基础论</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">a</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、几何基础（</span><span lang="EN-US"><font face="Times New Roman">1898</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">－</span><span lang="EN-US"><font face="Times New Roman">1902</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">b</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、一般数学基础（</span><span lang="EN-US"><font face="Times New Roman">1922</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">－</span><span lang="EN-US"><font face="Times New Roman">1930</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">4</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、积分方程（</span><span lang="EN-US"><font face="Times New Roman">1902</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">－</span><span lang="EN-US"><font face="Times New Roman">1912</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">5</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、物理学（</span><span lang="EN-US"><font face="Times New Roman">1910</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">－</span><span lang="EN-US"><font face="Times New Roman">1922</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">尽管以上前</span><span lang="EN-US"><font face="Times New Roman">4</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">个时期的每个时期的数学成就都足以使希尔伯特位列一</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">流数学家，但论及对整个数学的发展真正举足轻重的还是他的公理化理论</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">和“</span><span lang="EN-US"><font face="Times New Roman">23</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">个数学问题”。后者虽然不是一项严格意义上的数学成果，但它的</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">影响却不比任何一个理论的成果影响小。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">&nbsp;</font></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（一）、《几何基础》与公理化理论</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">&nbsp;</font></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">1</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、《几何基础》</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">希尔伯特在名著《几何基础》中第一次给出了自然、简明、全面、严</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">格的公理系统，提出了形式公理法，这是公理学上的里程碑。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">在欧几里德的实质公理法中，所讨论的对象是在所列举的公理也是不</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">完备的。公理系统以前早就已知的，而这些对象（指基本概念）的定义仅</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">限于直觉描述而不是逻辑定义。因而在定理的证明过程中并非为严格的逻</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">辑推演，往往诉诸“直观”和“经验”。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">希尔伯特还注意到，要研究数学的逻辑推理，要考察哪种推理过程可</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">以实现，哪种过程不能实现，与作为前提的诸命题和作为结论的命题的具</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">体含义无关，只与其逻辑构成形式有关，例如，由“如果</span><span lang="EN-US"><font face="Times New Roman">A</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">则</span><span lang="EN-US"><font face="Times New Roman">B</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">”与“如</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">果</span><span lang="EN-US"><font face="Times New Roman">B</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">则</span><span lang="EN-US"><font face="Times New Roman">C</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">”可以推出“如果</span><span lang="EN-US"><font face="Times New Roman">A</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">则</span><span lang="EN-US"><font face="Times New Roman">C</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">”，这种推理是常见的、正确的和可以</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">实现的。并且这种推理与命题</span><span lang="EN-US"><font face="Times New Roman">A</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、</span><span lang="EN-US"><font face="Times New Roman">B</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、</span><span lang="EN-US"><font face="Times New Roman">C</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">的具体含义无关。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">因此，希尔伯特在研究欧式几何的基础上，放弃了《几何原本》里公</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">理系统的直观显然性，而强调逻辑结构，给出了由五组公理构成的公理系</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">统。这便是希尔伯特《几何基础》的主要内容，也就是所谓的形式公理法。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">在希尔伯特的公理系统中，把“点、线、面”作为一组抽象元素，把</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">“在…之上”，“在…中间”，“合同于…”作为一组抽象的关系，这是六个</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">基本概念。然后把他们的基本属性用五组公理形式陈述出来，也就是说与</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">欧式的实质公理法不同，不是把所讨论的对象作直接的定义，做明显的直</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">觉描述。而是把它们之间的基本关系的根本性质用公理来刻画。从而把所</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">讨论的对象整体的公理形式做“隐”的规定，所以，作为抽象元素的点、</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">线、面和作为抽象关系的“在…之上”，“在…中间”，“合同于…”完全不</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">必与直觉关系下的点、线、面和他们之间的关系发生任何联系。这六个基</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">本概念，我们只知道他们是适合上述五组公理的元素和关系，换言之。这</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">些公理就是在其中出现的概念的定义。同时公理自身就是自己的证明。应</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">用纯粹逻辑推理来建立几何是，所需要的一切都包含在公理之中了。“</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">当我们把这些基本概念看成某一具体领域的元素的关系时，若公理成</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">了该领域的真命题，就称给公理系统一个解释，或者说给出公理系统的一</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">个模型。举一例如下：</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">考虑非空的集合</span><span lang="EN-US"><font face="Times New Roman">S</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">＝</span><span lang="EN-US"><font face="Times New Roman">{x, y, z,</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">…</span><span lang="EN-US"><font face="Times New Roman">}</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，</span><span lang="EN-US"><font face="Times New Roman">S</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">上有一抽象关系</span><span lang="EN-US"><font face="Times New Roman">R</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">：“在…之前”，</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">简记为“</span><span lang="EN-US"><font face="Times New Roman">&lt;</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">”。这里</span><span lang="EN-US"><font face="Times New Roman">x</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，</span><span lang="EN-US"><font face="Times New Roman"> y</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，</span><span lang="EN-US"><font face="Times New Roman"> z</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">……作为抽象的元素，“</span><span lang="EN-US"><font face="Times New Roman">&lt;</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">”作为抽象的关系，</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">还并未说明由什么具体含义。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">我们列举两条关于它们的公理：</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">1</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、任何元素不再自身之前，即关系“</span><span lang="EN-US"><font face="Times New Roman">&lt;</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">”不自反。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">2</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、如果</span><span lang="EN-US"><font face="Times New Roman">x&lt;y</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，</span><span lang="EN-US"><font face="Times New Roman">y&lt;z</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，则</span><span lang="EN-US"><font face="Times New Roman">x&lt;z,</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，即关系“</span><span lang="EN-US"><font face="Times New Roman">&lt;</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">”是传递的。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">不难看出公理系统</span><span lang="EN-US"><font face="Times New Roman">{1,2}</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">是有模型的。例如，当</span><span lang="EN-US"><font face="Times New Roman">x, y, z</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">指的是人，而人</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">与人之间有个年龄的大于关系，如果把“</span><span lang="EN-US"><font face="Times New Roman">x&lt;y</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">”解释作“</span><span lang="EN-US"><font face="Times New Roman">x</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">大于</span><span lang="EN-US"><font face="Times New Roman">y</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">”，那么</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">公理系统</span><span lang="EN-US"><font face="Times New Roman">{1</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，</span><span lang="EN-US"><font face="Times New Roman">2}</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">的一个解释就是：人的集合和人之间的年龄大于关系。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">再如，把</span><span lang="EN-US"><font face="Times New Roman">x, y, z</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">看成整数，则</span><span lang="EN-US"><font face="Times New Roman">S</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">就成了整数集</span><span lang="EN-US"><font face="Times New Roman">Z</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，而把“</span><span lang="EN-US"><font face="Times New Roman">&lt;</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">”看成“</span><span lang="EN-US"><font face="Times New Roman">Z </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">上的小于关系”，那么整数集</span><span lang="EN-US"><font face="Times New Roman">Z</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">和他们之间的小于关系就是公理系统</span><span lang="EN-US"><font face="Times New Roman">{1,2} </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">的又一解释。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">&nbsp;</font></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">2</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、公理系统的系统特征</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">A</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、</span><font face="Times New Roman"> </font><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">系统的无矛盾性</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">希尔伯特用形式公理法研究初等几何的逻辑结构时，首先提出了公理</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">系统的协调性，即无矛盾性。也就是基于它的公理系统作逻辑演绎时不会</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">推出互相矛盾的命题来。否则这个公理系统就不能反应“真”、“假”，因而</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">也就没有意义了。希尔伯特认为公理系统的协调性是解决悖论问题的方法。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">为了给出协调性的证明，希尔伯特创立了证明论和有穷方法。这对数学思</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">想的发展，对数理逻辑和数学基础的研究有很大的促进作用和深远的影响。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">以协调性为中心论题的证明论已发展成为数理逻辑的西大分之之一。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">B</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、系统的独立性</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">在一个公理系统中，若一个公理</span><span lang="EN-US"><font face="Times New Roman">A</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">不能从其他公理对出，则称</span><span lang="EN-US"><font face="Times New Roman">A</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">对于</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">其它公理是独立的。公理系统应该尽可能的精简。然而一条独立的公理就</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">不能轻易删除，否则它所包含的内容不能有其它公理推出，系统也就具有</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">缺陷。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">C</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、系统的完备性</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">从公理系统出发借助于逻辑规则可以推演出一个数学分支的全部真命</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">题，即为公理的完备性。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">&nbsp;</font></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">公理化方法的出现和研究可以上溯到古希腊时期。在希尔伯特的《几</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">何基础》问世之后，公理化方法则成为数学研究中的主要研究方法。几乎</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">现在所有的数学分支都经历过公理法的分析和讨论。可以讲，没有希尔伯</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">特形式公理法的数学是难以想象的。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">&nbsp;</font></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（二）、希尔伯特的</span><span lang="EN-US"><font face="Times New Roman">23</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">个数学问题</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">&nbsp;</font></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">在</span><span lang="EN-US"><font face="Times New Roman">1900</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年巴黎国际数学家代表大会上，希尔伯特发表了题为《数学问</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">题》的著名讲演。他根据过去特别是十九世纪数学研究的成果和发展趋势，</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">提出了</span><span lang="EN-US"><font face="Times New Roman">23</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">个最重要的数学问题。这</span><span lang="EN-US"><font face="Times New Roman">23</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">个问题通称“希尔伯特问题”</span><font face="Times New Roman"> </font><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，后</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">来成为许多数学家力图攻克的难关，对现代数学的研究和发展产生了深刻</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">的影响，并起了积极的推动作用，希尔伯特问题中有些现已得到圆满解决，</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">有些至今仍未解决。他在讲演中所阐发的想信每个数学问题都可以解决的</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">信念，对于数学工作者是一种巨大的鼓舞。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">&nbsp;</font></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">希尔伯特的</span><span lang="EN-US"><font face="Times New Roman">23</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">个问题分属四大块：第</span><span lang="EN-US"><font face="Times New Roman">1</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">到第</span><span lang="EN-US"><font face="Times New Roman">6</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">问题是数学基础问题；第</span><span lang="EN-US"><font face="Times New Roman">7 </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">到第</span><span lang="EN-US"><font face="Times New Roman">12</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">问题是数论问题；第</span><span lang="EN-US"><font face="Times New Roman">13</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">到第</span><span lang="EN-US"><font face="Times New Roman">18</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">问题属于代数和几何问题；第</span><span lang="EN-US"><font face="Times New Roman">19 </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">到第</span><span lang="EN-US"><font face="Times New Roman">23</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">问题属于数学分析。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">1</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）康托的连续统基数问题。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">1874</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年，康托猜测在可数集基数和实数集基数之间没有别的基数，即</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">著名的连续统假设。</span><span lang="EN-US"><font face="Times New Roman">1938</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年，侨居美国的奥地利数理逻辑学家哥德尔证明</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">连续统假设与</span><span lang="EN-US"><font face="Times New Roman">ZF</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">集合论公理系统的无矛盾性。</span><span lang="EN-US"><font face="Times New Roman">1963</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年，美国数学家科思</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">P.Choen</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）证明连续统假设与</span><span lang="EN-US"><font face="Times New Roman">ZF</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">公理彼此独立。因而，连续统假设不能</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">用</span><span lang="EN-US"><font face="Times New Roman">ZF</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">公理加以证明。在这个意义下，问题已获解决。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">2</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）算术公理系统的无矛盾性。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">欧氏几何的无矛盾性可以归结为算术公理的无矛盾性。希尔伯特曾提</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">出用形式主义计划的证明论方法加以证明，哥德尔</span><span lang="EN-US"><font face="Times New Roman">1931</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年发表不完备性定</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">理作出否定。根茨（</span><span lang="EN-US"><font face="Times New Roman">G.Gentaen</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，</span><span lang="EN-US"><font face="Times New Roman">1909-1945</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）</span><span lang="EN-US"><font face="Times New Roman">1936</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年使用超限归纳法证明</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">了算术公理系统的无矛盾性。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">3</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）只根据合同公理证明等底等高的两个四面体有相等之体积是不可</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">能的。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">问题的意思是：存在两个登高等底的四面体，它们不可能分解为有限</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">个小四面体，使这两组四面体彼此全等德思（</span><span lang="EN-US"><font face="Times New Roman">M.Dehn</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）</span><span lang="EN-US"><font face="Times New Roman">1900</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年已解决。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">4</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）两点间以直线为距离最短线问题。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">此问题提的一般。满足此性质的几何很多，因而需要加以某些限制条</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">件。</span><span lang="EN-US"><font face="Times New Roman">1973</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年，苏联数学家波格列洛夫（</span><span lang="EN-US"><font face="Times New Roman">Pogleov</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）宣布，在对称距离情况</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">下，问题获解决。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">5</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）拓扑学成为李群的条件（拓扑群）。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">这一个问题简称连续群的解析性，即是否每一个局部欧氏群都一定是</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">李群。</span><span lang="EN-US"><font face="Times New Roman">1952</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年，由格里森（</span><span lang="EN-US"><font face="Times New Roman">Gleason</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）、蒙哥马利（</span><span lang="EN-US"><font face="Times New Roman">Montgomery</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）、齐宾</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">Zippin</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）共同解决。</span><span lang="EN-US"><font face="Times New Roman">1953</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年，日本的山迈英彦已得到完全肯定的结果。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">6</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）对数学起重要作用的物理学的公理化。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">1933</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年，苏联数学家柯尔莫哥洛夫将概率论公理化。后来，在量子力</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">学、量子场论方面取得成功。但对物理学各个分支能否全盘公理化，很多</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">人有怀疑。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">7</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）某些数的超越性的证明。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">需证：如果α是代数数，β是无理数的代数数，那么αβ一定是超越</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">数或至少是无理数（例如，</span><span lang="EN-US"><font face="Times New Roman">2</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">√</span><span lang="EN-US"><font face="Times New Roman">2</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">和</span><span lang="EN-US"><font face="Times New Roman">e</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">π）。苏联的盖尔封特（</span><span lang="EN-US"><font face="Times New Roman">Gelfond</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）</span><span lang="EN-US"><font face="Times New Roman">1929 </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年、德国的施奈德（</span><span lang="EN-US"><font face="Times New Roman">Schneider</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）及西格尔（</span><span lang="EN-US"><font face="Times New Roman">Siegel</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）</span><span lang="EN-US"><font face="Times New Roman">1935</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年分别独立地证</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">明了其正确性。但超越数理论还远未完成。目前，确定所给的数是否超越</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">数，尚无统一的方法。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">8</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）素数分布问题，尤其对黎曼猜想、哥德巴赫猜想和孪生素共问题。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">素数是一个很古老的研究领域。希尔伯特在此提到黎曼（</span><span lang="EN-US"><font face="Times New Roman">Riemann</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">猜想、哥德巴赫（</span><span lang="EN-US"><font face="Times New Roman">Goldbach</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）猜想以及孪生素数问题。黎曼猜想至今未解</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">决。哥德巴赫猜想和孪生素数问题目前也未最终解决，其最佳结果均属中</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">国数学家陈景润。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">9</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）一般互反律在任意数域中的证明。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">1921</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年由日本的高木贞治，</span><span lang="EN-US"><font face="Times New Roman">1927</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年由德国的阿廷（</span><span lang="EN-US"><font face="Times New Roman">E.Artin</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）各自给以</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">基本解决。而类域理论至今还在发展之中。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">10</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）能否通过有限步骤来判定不定方程是否存在有理整数解？</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">求出一个整数系数方程的整数根，称为丢番图（约</span><span lang="EN-US"><font face="Times New Roman">210-290</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，古希腊</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">数学家）方程可解。</span><span lang="EN-US"><font face="Times New Roman">1950</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年前后，美国数学家戴维斯（</span><span lang="EN-US"><font face="Times New Roman">Davis</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）、普特南</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">Putnan</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）、罗宾逊（</span><span lang="EN-US"><font face="Times New Roman">Robinson</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）等取得关键性突破。</span><span lang="EN-US"><font face="Times New Roman">1970</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年，巴克尔</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">Baker</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）、费罗斯（</span><span lang="EN-US"><font face="Times New Roman">Philos</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）对含两个未知数的方程取得肯定结论。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">1970</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年。苏联数学家马蒂塞维奇最终证明：在一般情况答案是否定的。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">尽管得出了否定的结果，却产生了一系列很有价值的副产品，其中不少和</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">计算机科学有密切联系。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">11</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）一般代数数域内的二次型论。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">德国数学家哈塞（</span><span lang="EN-US"><font face="Times New Roman">Hasse</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）和西格尔（</span><span lang="EN-US"><font face="Times New Roman">Siegel</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）在</span><span lang="EN-US"><font face="Times New Roman">20</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年代获重要结果。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">60</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年代，法国数学家魏依（</span><span lang="EN-US"><font face="Times New Roman">A.Weil</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）取得了新进展。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">12</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）类域的构成问题。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">即将阿贝尔域上的克罗内克定理推广到任意的代数有理域上去。此问</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">题仅有一些零星结果，离彻底解决还很远。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">13</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）一般七次代数方程以二变量连续函数之组合求解的不可能性。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">七次方程</span><span lang="EN-US"><font face="Times New Roman">x7+ax3+bx2+cx+1=0</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">的根依赖于</span><span lang="EN-US"><font face="Times New Roman">3</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">个参数</span><span lang="EN-US"><font face="Times New Roman">a</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、</span><span lang="EN-US"><font face="Times New Roman">b</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、</span><span lang="EN-US"><font face="Times New Roman">c</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">；</span><span lang="EN-US"><font face="Times New Roman">x=x(a,b,c)</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">这一函数能否用两变量函数表示出来？此问题已接近解决。</span><span lang="EN-US"><font face="Times New Roman">1957</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年，苏联</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">数学家阿诺尔德（</span><span lang="EN-US"><font face="Times New Roman">Arnold</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）证明了任一在［</span><span lang="EN-US"><font face="Times New Roman">0</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，</span><span lang="EN-US"><font face="Times New Roman">1</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">］上连续的实函数</span><span lang="EN-US"><font face="Times New Roman">f(x1</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">x2</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，</span><span lang="EN-US"><font face="Times New Roman">x3)</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">可写成形式∑</span><span lang="EN-US"><font face="Times New Roman">hi(</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">ξ</span><span lang="EN-US"><font face="Times New Roman">i(x1,x2),x3)(i=1--9)</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，这里</span><span lang="EN-US"><font face="Times New Roman">hi</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">和ξ</span><span lang="EN-US"><font face="Times New Roman">i</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">为连续实函数。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">柯尔莫哥洛夫证明</span><span lang="EN-US"><font face="Times New Roman">f(x1</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，</span><span lang="EN-US"><font face="Times New Roman">x2</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，</span><span lang="EN-US"><font face="Times New Roman">x3)</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">可写成形式∑</span><span lang="EN-US"><font face="Times New Roman">hi(</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">ξ</span><span lang="EN-US"><font face="Times New Roman">i1(x1)+</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">ξ</span><span lang="EN-US"><font face="Times New Roman">i2(x2)+</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">ξ</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">i3(x3))(i=1--7)</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">这里</span><span lang="EN-US"><font face="Times New Roman">hi</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">和ξ</span><span lang="EN-US"><font face="Times New Roman">i</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">为连续实函数，ξ</span><span lang="EN-US"><font face="Times New Roman">ij</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">的选取可与</span><span lang="EN-US"><font face="Times New Roman">f</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">完全无关。</span><span lang="EN-US"><font face="Times New Roman">1964 </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年，维土斯金（</span><span lang="EN-US"><font face="Times New Roman">Vituskin</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）推广到连续可微情形，对解析函数情形则未解决。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">14</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）某些完备函数系的有限的证明。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">即域</span><span lang="EN-US"><font face="Times New Roman">K</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">上的以</span><span lang="EN-US"><font face="Times New Roman">x1,x2,</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">…</span><span lang="EN-US"><font face="Times New Roman">,xn</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">为自变量的多项式</span><span lang="EN-US"><font face="Times New Roman">fi</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">i=1,</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">…，</span><span lang="EN-US"><font face="Times New Roman">m</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">），</span><span lang="EN-US"><font face="Times New Roman">R</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">为</span><span lang="EN-US"><font face="Times New Roman">K </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">［</span><span lang="EN-US"><font face="Times New Roman">X1</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，…，</span><span lang="EN-US"><font face="Times New Roman">Xm]</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">上的有理函数</span><span lang="EN-US"><font face="Times New Roman">F</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">X1</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，…，</span><span lang="EN-US"><font face="Times New Roman">Xm</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）构成的环，并且</span><span lang="EN-US"><font face="Times New Roman">F</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">f1</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，…，</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">fm</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）∈</span><span lang="EN-US"><font face="Times New Roman">K</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">试问</span><span lang="EN-US"><font face="Times New Roman">R</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">是否可由有限个元素</span><span lang="EN-US"><font face="Times New Roman">F1</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，…，</span><span lang="EN-US"><font face="Times New Roman">FN</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">的多项式</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">生成？这个与代数不变量问题有关的问题，日本数学家永田雅宜于</span><span lang="EN-US"><font face="Times New Roman">1959 </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年用漂亮的反例给出了否定的解决。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">15</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）建立代数几何学的基础。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">荷兰数学家范德瓦尔登</span><span lang="EN-US"><font face="Times New Roman">1938</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年至</span><span lang="EN-US"><font face="Times New Roman">1940</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年，魏依</span><span lang="EN-US"><font face="Times New Roman">1950</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年已解决。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">注一舒伯特（</span><span lang="EN-US"><font face="Times New Roman">Schubert</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）计数演算的严格基础。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">一个典型的问题是：在三维空间中有四条直线，问有几条直线能和这</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">四条直线都相交？舒伯特给出了一个直观的解法。希尔伯特要求将问题一</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">般化，并给以严格基础。现在已有了一些可计算的方法，它和代数几何学</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">有密切的关系。但严格的基础至今仍未建立。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">16</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）代数曲线和曲面的拓扑研究。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">此问题前半部涉及代数曲线含有闭的分枝曲线的最大数目。后半部要</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">求讨论备</span><span lang="EN-US"><font face="Times New Roman">dx/dy=Y/X</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">的极限环的最多个数</span><span lang="EN-US"><font face="Times New Roman">N</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">n</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）和相对位置，其中</span><span lang="EN-US"><font face="Times New Roman">X</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、</span><span lang="EN-US"><font face="Times New Roman">Y </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">是</span><span lang="EN-US"><font face="Times New Roman">x</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、</span><span lang="EN-US"><font face="Times New Roman">y</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">的</span><span lang="EN-US"><font face="Times New Roman">n</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">次多项式。对</span><span lang="EN-US"><font face="Times New Roman">n=2</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（即二次系统）的情况，</span><span lang="EN-US"><font face="Times New Roman">1934</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年福罗献尔</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">得到</span><span lang="EN-US"><font face="Times New Roman">N(2)</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">≥</span><span lang="EN-US"><font face="Times New Roman">1</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">；</span><span lang="EN-US"><font face="Times New Roman">1952</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年鲍廷得到</span><span lang="EN-US"><font face="Times New Roman">N(2)</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">≥</span><span lang="EN-US"><font face="Times New Roman">3</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">；</span><span lang="EN-US"><font face="Times New Roman">1955</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年苏联的波德洛夫斯基宣</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">布</span><span lang="EN-US"><font face="Times New Roman">N(2)</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">≤</span><span lang="EN-US"><font face="Times New Roman">3</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，这个曾震动一时的结果，由于其中的若干引理被否定而成疑</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">问。关于相对位置，中国数学家董金柱、叶彦谦</span><span lang="EN-US"><font face="Times New Roman">1957</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年证明了（</span><span lang="EN-US"><font face="Times New Roman">E2</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）不</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">超过两串。</span><span lang="EN-US"><font face="Times New Roman">1957</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年，中国数学家秦元勋和蒲富金具体给出了</span><span lang="EN-US"><font face="Times New Roman">n</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">＝</span><span lang="EN-US"><font face="Times New Roman">2</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">的方程</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">具有至少</span><span lang="EN-US"><font face="Times New Roman">3</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">个成串极限环的实例。</span><span lang="EN-US"><font face="Times New Roman">1978</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年，中国的史松龄在秦元勋、华罗</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">庚的指导下，与王明淑分别举出至少有</span><span lang="EN-US"><font face="Times New Roman">4</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">个极限环的具体例子。</span><span lang="EN-US"><font face="Times New Roman">1983</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年，</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">秦元勋进一步证明了二次系统最多有</span><span lang="EN-US"><font face="Times New Roman">4</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">个极限环，并且是（</span><span lang="EN-US"><font face="Times New Roman">1</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，</span><span lang="EN-US"><font face="Times New Roman">3</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）结构，</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">从而最终地解决了二次微分方程的解的结构问题，并为研究希尔伯特第</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">16</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）问题提供了新的途径。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">17</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）半正定形式的平方和表示。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">实系数有理函数</span><span lang="EN-US"><font face="Times New Roman">f(x1,</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">…，</span><span lang="EN-US"><font face="Times New Roman">xn)</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">对任意数组（</span><span lang="EN-US"><font face="Times New Roman">x1</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，…</span><span lang="EN-US"><font face="Times New Roman">,xn</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）都恒大于或等</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">于</span><span lang="EN-US"><font face="Times New Roman">0</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，确定</span><span lang="EN-US"><font face="Times New Roman">f</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">是否都能写成有理函数的平方和？</span><span lang="EN-US"><font face="Times New Roman">1927</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年阿廷已肯定地解决。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">18</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）用全等多面体构造空间。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">德国数学家比贝尔巴赫（</span><span lang="EN-US"><font face="Times New Roman">Bieberbach</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）</span><span lang="EN-US"><font face="Times New Roman">1910</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年，莱因哈特（</span><span lang="EN-US"><font face="Times New Roman">Reinhart</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">1928</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年作出部分解决。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">19</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）正则变分问题的解是否总是解析函数？</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">德国数学家伯恩斯坦（</span><span lang="EN-US"><font face="Times New Roman">Bernrtein</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，</span><span lang="EN-US"><font face="Times New Roman">1929</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）和苏联数学家彼德罗夫斯基</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">1939</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）已解决。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">20</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）研究一般边值问题。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">此问题进展迅速，己成为一个很大的数学分支。日前还在继读发展。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">21</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）具有给定奇点和单值群的</span><span lang="EN-US"><font face="Times New Roman">Fuchs</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">类的线性微分方程解的存在性</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">证明。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">此问题属线性常微分方程的大范围理论。希尔伯特本人于</span><span lang="EN-US"><font face="Times New Roman">1905</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年、勒</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">尔（</span><span lang="EN-US"><font face="Times New Roman">H.Rohrl</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）于</span><span lang="EN-US"><font face="Times New Roman">1957</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年分别得出重要结果。</span><span lang="EN-US"><font face="Times New Roman">1970</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年法国数学家德利涅</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">Deligne</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）作出了出色贡献。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">22</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）用自守函数将解析函数单值化。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">此问题涉及艰深的黎曼曲面理论，</span><span lang="EN-US"><font face="Times New Roman">1907</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年克伯（</span><span lang="EN-US"><font face="Times New Roman">P.Koebe</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）对一个变</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">量情形已解决而使问题的研究获重要突破。其它方面尚未解决。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">23</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）发展变分学方法的研究。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">这不是一个明确的数学问题。</span><span lang="EN-US"><font face="Times New Roman">20</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">世纪变分法有了很大发展。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">&nbsp;</font></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">希尔伯特认为，数学科学是一个不可分割的整体，它的生命正是在于</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">各个部分之间的联系。尽管数学知识千差万别，但是在作为整体的数学中，</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">数学家们都在使用着相同的工具，存在着概念的亲缘关系，同时，在它的</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">不同部分之间，也有大量相似之处。并且希尔伯特相信，数学理论越是向</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">前发展，它的结构就变得越加的调和一致，并且，这门科学一向相互隔绝</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">的分支之间也会显露出原先意想不到的关系。因此，随着数学的发展，它</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">的有机的特性不会丧失，只会更清楚的呈现出来。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">时至今日，希尔伯特的高度预见性已经得到了验证，他向人们指出的</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">数学方向和具体问题也被证明是极为正确的。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">&nbsp;</font></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">三、</span><font face="Times New Roman"> </font><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">希尔伯特带给我们的启发</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">&nbsp;</font></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">希尔伯特是一个伟大的数学现象</span><span lang="EN-US"><font face="Times New Roman">,</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">这个现象可以用三个方面来概括：</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">1</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、</span><font face="Times New Roman"> </font><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">数学领域中基础性和革命性的成果；</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">2</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、</span><font face="Times New Roman"> </font><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">对于数学未知的关键领域的高度敏感和预见性；</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">3</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、</span><font face="Times New Roman"> </font><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">一代数学领袖的光辉、无比高尚的人格和巨大的精神感召力。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">是什么给了他这些呢？</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">通过以上的探讨，我想可以从以下几个方面来阐述：</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">1</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、少年和青年时代生活环境的影响。哥尼斯堡是一方具有伟大文化和</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">科学传统的沃土。伟大的先辈，古老的传统，浓重的文化与科学氛围，都</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">对处于发展时期的希尔伯特产生着潜移默化的影响。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">2</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、一直处于像哥廷根这样的数学研究中心，长期有杰出的数学家共同</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">工作。相互之间的交流和关怀对于产生数学硕果具有的决定性的影响。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">3</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、对朋友真挚的热情，对数学无限的热爱，对知识的不懈追求和严谨</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">的治学态度，这些都是希尔伯特成其为希尔伯特的关键因素。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">4</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">、在数学道路上对简单性和严格性的坚持和把握。应该说，公理化理</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">论和“数学问题”都是这两个精神或者说原则的产物。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><p><font face="Times New Roman">&nbsp;</font></p></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">希尔伯特是</span><span lang="EN-US"><font face="Times New Roman">20</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">世纪真正意义上的数学大师，他的研究成果博大精深，</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">无论是生前还是身后，人们对他的评价都是那样高崇。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">由于希尔伯特的杰出贡献，德国政府授予了他“枢密顾问”的称号。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">在他六十八岁那年，哥尼斯堡市政会授予了他“荣誉市民”称号。</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">希尔伯特毕生投身于数学研究。在他去世时，德国《自然》杂志发表</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">了这样的观点：“现在世界上难得有一位数学家的工作不是以某种途径导源</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">于希尔伯特的工作。他像是数学世界的亚历山大，在整个数学版图上，留</span><span lang="EN-US"><font face="Times New Roman"> </font></span></p><span style="FONT-SIZE: 10.5pt; FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'; mso-bidi-font-size: 12.0pt; mso-font-kerning: 1.0pt; mso-bidi-font-family: 'Times New Roman'; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA;">下了他那显赫的名字。”</span>

量小非君子也 发表于 2007-4-17 15:23

<p>强人啊</p>

查看完整版本: [转帖]崇高的人格与光辉的数学成就——希尔伯特的数学生涯