大卫·希尔伯特(1862—1943),20世纪上半叶国际数学界的一位领袖人物。他于1900年在巴黎第二届国际数学家代表大会上提出的23个数学问题(史称希尔伯特问题),激发了整个数学界的想象力,此后,这些问题几乎成为检阅数学重大成就的指标。这位创造了20世纪数学史奇迹的数学家和数学思想家,就像数学世界的亚历山大,在整个数学版图上留下了他巨大显赫的名字。
希尔伯特,这个极富想象力的人却在一个极抽象的世界里如鱼得水,他的一生平淡而神奇。本书原来是为数学专业外的好奇人士而作,如今已成为科学界公认的经典读本。
康斯坦丝·瑞德,美国一名普通的数学爱好者,并非数学专业人士。她因为对数学有浓厚的兴趣而萌发了撰写数学大师希尔伯特传记的念头。这是一个极富挑战性的想法,因为,即使是专业内人士也少有人动这样的念头,希尔伯特的复杂性格和卓越贡献使大多数传记作家望而却步。所以,当瑞德为准备写作而采访那些曾与希尔伯特共事的大数家时,迎接她的不是充满疑虑和劝阻的目光,就是闭门谢客的拒绝态度。然而,当这本为普通人而写的传记于1969年面市时,却意外受到了广大专业人士的好评.
- posted on 01/12/2009
大卫·希尔伯特(David Hilbert,1862年1月23日-1943年2月14日),德国数学家,是19世纪和20世纪初最具影响力的数学家之一。希尔伯特1862年出生于哥尼斯堡,1943年在德国哥廷根逝世。他因为发明和发展了大量的思想观念(例如:不变量理论,公理化几何,希尔伯特空间)而被尊为伟大的数学家、科学家。希尔伯特和他的学生为形成量子力学和广义相对论的数学基础做出了重要的贡献。他还是证明论、数理逻辑、区分数学与元数学之差别的奠基人之一。他热忱地支持康托的集合论与无限数。他在数学上的领导地位充分体现于:1900年,在巴黎的国际数学家大会提出的一系列问题(希尔伯特的23个问题)为20世纪的许多数学研究指出方向。
希尔伯特的著作有《希尔伯特全集》《几何基础》《线性积分方程一般理论基础》等。1928年他跟威廉·阿克曼合写《理论逻辑原理》(Grundzuge der Theoretischen Logik)。
- posted on 01/12/2009
希尔伯特的23个问题
1900年,希尔伯特在巴黎的国际数学家大会上作了题为《数学问题》的演讲,提出了23道最重要的数学问题,这就是著名的希尔伯特的23个问题。希尔伯特问题对推动20世纪数学的发展起了积极的推动作用。在许多数学家努力下,希尔伯特问题中的大多数在20世纪中得到了解决。
希尔伯特问题中未能包括拓扑学、微分几何等领域,除数学物理外很少涉及应用数学,更不曾预料到电脑发展将对数学的产生重大影响。20世纪数学的发展实际上远远超出了希尔伯特所预示的范围。
希尔伯特问题中的1-6是数学基础问题,7-12是数论问题,13-18属于代数和几何问题,19-23属于数学分析。
以下列出希尔伯特的23个问题:
第一题 连续统假设 已解决。1963年美国数学家保罗·柯恩以力迫法(forcing)证明连续统假设不能由ZFC推导。也就是说,连续统假设成立与否无法由ZFC确定。
第二题 算术公理之相容性 已解决。库尔特·哥德尔在1930年证明了哥德尔不完备定理。
第三题 两四面体有相同体积之证明法 已解决。希尔伯特的学生马克斯·德恩以一反例证明了是不可以的了。
第四题 建立所有度量空间使得所有线段为测地线 太隐晦。希尔伯特对于这个问题的定义过于含糊。
第五题 所有连续群是否皆为可微群 已解决。1953年日本数学家山迈英彦已得到完全肯定的结果。
第六题 公理化物理 非数学。对于物理学能否全盘公理化,有很多人质疑。
第七题 若 b 是无理数、a 是非0、1代数数,那么 ab 是否超越数 已解决。分别于1934年、1935年由Gelfond与Schneider独立地解决。
第八题 黎曼猜想及哥德巴赫猜想 部分解决。1966年中国数学家陈景润部分解答了哥德巴赫猜想。
第九题 任意代数数域的一般互反律 部分解决。1921年日本的高木贞治,1927年德国的埃米尔·阿廷(E.Artin)各有部份解答。
第十题 不定方程可解性 已解决。1970年苏联数学家马蒂塞维奇证明:在一般情况答案是否定的。
第十一题 代数系数之二次形式 已解决。有理数的部分由哈塞于1923年解决,实数的部分则由希格尔于1930年解决。
第十二题 扩展代数数 已解决。1920年高木贞治开创了阿贝尔类域理论。
第十三题 以二元函数解任意七次方程 已解决。1957年柯尔莫哥洛夫和阿诺德证明其不可能性。
第十四题 证明一些函数完全系统(Complete system of functions)之有限性 已解决。1962年日本人永田雅宜提出反例。
第十五题 舒伯特列举微积分(Schubert's enumerative calculus)之严格基础 部分解决。一部分在1938年由范德瓦登得到严谨的证明。
第十六题 代数曲线及表面之拓扑结构 未解决
第十七题 把有理函数写成平方和分式 已解决。1927年埃米尔·阿廷(Emil Artin)已解决实封闭域。
第十八题 非正多面体能否密铺空间、球体最紧密的排列 已解决。1910年比伯巴赫做出“n维空间由有限多个群嵌成”
第十九题 拉格朗日系统(Lagrangian)之解是否皆可解析(Analytic) 已解决。1904年由伯恩斯坦(Serge Bernstein)解决。
第二十题 所有有界限条件的变量问题(Variational problem)是否都有解 已解决
第二十一题 证明有线性微分方程有给定的单值群(monodromy group) 已解决
第二十二题 以自守函数(Automorphic functions)一致化可解析关系 已解决。1904年由科比和庞加莱取得解决。
第二十三题 变分法的长远发展 已解决
- Re: 希尔伯特:数学世界的亚历山大posted on 01/12/2009
这个希尔伯特,也不知道是海德堡人,还是尼安德特人的后代。
据说他在大学念书进展特慢,咱大学时也想学他,可惜我那个鸟大学
个个都是催命鬼似的,催出一个个夹生饭。。。
咱中学时象他,学得不亦乐呼。现在,云在青天,子曰诗云。
一叹! - Re: 希尔伯特:数学世界的亚历山大posted on 01/12/2009
xw majored in math? what a surprise! - Re: 希尔伯特:数学世界的亚历山大posted on 01/12/2009
July wrote:
希尔伯特的23个问题
第八题 黎曼猜想及哥德巴赫猜想 部分解决。1966年中国数学家陈景润部分解答了哥德巴赫猜想。
黎曼猜想没有解决(虽然近来一直有人,包括中国人,说他们解决了这个问题),陈景润的成果也不能说是“部分解答”了哥德巴赫猜想。 - Re: 希尔伯特:数学世界的亚历山大posted on 01/12/2009
还是尼安德特人的后代???
xw wrote:
这个希尔伯特,也不知道是海德堡人,还是尼安德特人的后代。
据说他在大学念书进展特慢,咱大学时也想学他,可惜我那个鸟大学
个个都是催命鬼似的,催出一个个夹生饭。。。
咱中学时象他,学得不亦乐呼。现在,云在青天,子曰诗云。
一叹! - posted on 01/13/2009
海德堡人:Homo heidelbergensis
尼安德特人:Homo neanderthalensis
都是德国发现的早先人类啦。一个是德国的北京人,一个是山顶洞人
。说到山顶洞人,上回默克引魏敦瑞的话,说那位老者象克罗马侬人
,说到克罗马侬人,就想到普波战争后德法之间的人类学战争:
In 1868 the world recognized the existence of another type of human with the discovery of five skeletons at the back of the rock shelter of Cro-Magnon, very close to the village of Les Eyzies in the Dordogne region of south-western France. According to Louis Lartet(1840-99), son of the paleontologist Edouard Lartet, who was entrusted with the excavation of Cro-Magnon by the Ministry of Public Instruction, the position of the skeletons and the distribution of the remains accompanying them provided evidence that they had been deliberately buried. Six years after their discovery, in 1874, these people with anatomically modern traints were considered the prototypes of a new race, the so-called Cro-Magnon “race”. Although they were associated with Paleolithic tools—the Paleolithic stretched from two million to 10,000 years ago –the antiquity of these Cro-Magnon people—who succeeded Neanderthals in Europe—was called into question, in particular by the French prehistorian Gabriel de Mortillet(1821-98). He could not bring himself to admit that Paleolithic people were already burying their dead. Some French people who harbored ill feelings towards Germany after the Franco-Prussian War of 1870-1 were delighted that a human type with a well-developed physiognomy and fine and elegant features had been discovered in their country, while the uncivilized Neanderthals, who still gave the impression of being ape-men, had been found in the country next door.
http://www.mayacafe.com/forum/topic1sp.php3?tkey=1071867956
不过我读贾兰坡的英文版“北京人的故事”,魏敦瑞说的不是克罗马
侬人,但不知原话是德语还是英语,故而不好转录。感觉到中文世界
是被翻译误会了。说到人类学战争还有一桩英国丑闻,不能不提:
In 1911 amateur archaeologist Charles Dawson discovered a skull and a lower jaw of a fossil human in a gravel pit near Piltdown, in southern England; it became the subject of several hundred publications. Many scholars were convinced that the earliest-known human from western Europe had been found. In 1953 the truth was revealed: “Piltdown Man” was a fraud—the jaw of an ape had been combined with a human skill. If Charles Dawson was the instigator of this hoax, it was likely, given his lack of expertise, that he would have had an accomplice. Suspicion has fallen on different people. In Piltdown: A Scientific Forgery(1990), Frank Spencer names the famous English anatomist Sir Arthur Keith as the chief suspect.
人类学本有贵族气,这一军备,一丑闻,便成了狗屁。题走远了,但
希尔伯特还是很有族气的:)
tar wrote:
还是尼安德特人的后代???
xw wrote:
这个希尔伯特,也不知道是海德堡人,还是尼安德特人的后代。
据说他在大学念书进展特慢,咱大学时也想学他,可惜我那个鸟大学
个个都是催命鬼似的,催出一个个夹生饭。。。
咱中学时象他,学得不亦乐呼。现在,云在青天,子曰诗云。
一叹! - posted on 01/13/2009
The Germans are not the descendants of the Neanderthals, nor are we Peking man's. They are distinct end nodes on Homo genus family tree, as our own ancestor the Homo sapiens from Africa. The fact that we all came from Africa is indisputable. National Geographic has sponsored a testing program where anybody can submit some DNA samples to them, and in a couple of weeks a result will mail back to you to show the migration route your own ancestors took.
There are suggestions that some interbreeding might have taken place between the modern humans and Neanderthals. But most recent DNA studies cast doubts on such hypothesis. One study concluded that there are about 5% of the genes in West Africans and Europeans may have differed from a direct modern human line. But it's origin is still unknown. An effort to map the entire Neanderthal genome is underway, and may provide the answer to the interbreeding question once and for all.
xw wrote:
海德堡人:Homo heidelbergensis
尼安德特人:Homo neanderthalensis
都是德国发现的早先人类啦。一个是德国的北京人,一个是山顶洞人
。说到山顶洞人,上回默克引魏敦瑞的话,说那位老者象克罗马侬人
,说到克罗马侬人,就想到普波战争后德法之间的人类学战争:
In 1868 the world recognized the existence of another type of human with the discovery of five skeletons at the back of the rock shelter of Cro-Magnon, very close to the village of Les Eyzies in the Dordogne region of south-western France. According to Louis Lartet(1840-99), son of the paleontologist Edouard Lartet, who was entrusted with the excavation of Cro-Magnon by the Ministry of Public Instruction, the position of the skeletons and the distribution of the remains accompanying them provided evidence that they had been deliberately buried. Six years after their discovery, in 1874, these people with anatomically modern traints were considered the prototypes of a new race, the so-called Cro-Magnon “race”. Although they were associated with Paleolithic tools—the Paleolithic stretched from two million to 10,000 years ago –the antiquity of these Cro-Magnon people—who succeeded Neanderthals in Europe—was called into question, in particular by the French prehistorian Gabriel de Mortillet(1821-98). He could not bring himself to admit that Paleolithic people were already burying their dead. Some French people who harbored ill feelings towards Germany after the Franco-Prussian War of 1870-1 were delighted that a human type with a well-developed physiognomy and fine and elegant features had been discovered in their country, while the uncivilized Neanderthals, who still gave the impression of being ape-men, had been found in the country next door.
http://www.mayacafe.com/forum/topic1sp.php3?tkey=1071867956 不过我读贾兰坡的英文版“北京人的故事”,魏敦瑞说的不是克罗马
侬人,但不知原话是德语还是英语,故而不好转录。感觉到中文世界
是被翻译误会了。说到人类学战争还有一桩英国丑闻,不能不提:
In 1911 amateur archaeologist Charles Dawson discovered a skull and a lower jaw of a fossil human in a gravel pit near Piltdown, in southern England; it became the subject of several hundred publications. Many scholars were convinced that the earliest-known human from western Europe had been found. In 1953 the truth was revealed: “Piltdown Man” was a fraud—the jaw of an ape had been combined with a human skill. If Charles Dawson was the instigator of this hoax, it was likely, given his lack of expertise, that he would have had an accomplice. Suspicion has fallen on different people. In Piltdown: A Scientific Forgery(1990), Frank Spencer names the famous English anatomist Sir Arthur Keith as the chief suspect.
人类学本有贵族气,这一军备,一丑闻,便成了狗屁。题走远了,但
希尔伯特还是很有族气的:)
tar wrote:
还是尼安德特人的后代???
xw wrote:
这个希尔伯特,也不知道是海德堡人,还是尼安德特人的后代。
据说他在大学念书进展特慢,咱大学时也想学他,可惜我那个鸟大学
个个都是催命鬼似的,催出一个个夹生饭。。。
咱中学时象他,学得不亦乐呼。现在,云在青天,子曰诗云。
一叹! - posted on 01/13/2009
tar wrote:
The Germans are not the descendants of the Neanderthals, nor are we Peking man's. They are distinct end nodes on Homo genus family tree, as our own ancestor the Homo sapiens from Africa. The fact that we all came from Africa is indisputable.
This is an interesting topic and there have been similar discussions here. The conclusion that all people in the world have common ancestors in Africa is still not quite accepted by all people, including many in China, who still believe 北京人 (or 蓝田人,etc.) the ancestors of Chinese people.
On the other hand, it may seem anti-intuitive that ALL of the relatively advanced groups over the world, including 北京人, got distinct, except one in Africa that eventually spread through out the globe. Isn't it more likely for some local groups to survive, compared to some branches of a single group that migrated through vast distances and most harsh environment, say from Africa all the way to Asia?
Also, there does not seem to have any fossil evidence to support the common ancestor theory, which is only due to the DNA evidence. Just read a pretty new book in Chinese history, which mentioned both theories as possibilities.
- posted on 01/13/2009
When it comes down to scientific evidence, there is no more "scientific" then DNA evidence. As far as I can find, there are no reputable SCIENTISTS who don't believe the out-of-Africa theory. And I don't know what you mean by the lack of fossil evidence. I only heard of critics of evolution theory citing the lack of "gaps" in fossil remains to assail evolution. If you look at the reproduction of facial rendering of both Neanderthal and Peking Man fossils, they are clearly more apelike than our common ancestor.
As far as species became extinct, it happened all the time. Not all monkeys and apes evolve to become human beings. Half-ape half-man species like Neanderthal and Peking Man's extinction is just part of the evolution process, totally on par for the course.
I wouldn't give too much credence to some Chinese authors who still cling to the belief that the Chinese race is special. We are just part of the humanity like the rest of the peoples, no more no less.
gz wrote:
tar wrote:This is an interesting topic and there have been similar discussions here. The conclusion that all people in the world have common ancestors in Africa is still not quite accepted by all people, including many in China, who still believe 北京人 (or 蓝田人,etc.) the ancestors of Chinese people.
The Germans are not the descendants of the Neanderthals, nor are we Peking man's. They are distinct end nodes on Homo genus family tree, as our own ancestor the Homo sapiens from Africa. The fact that we all came from Africa is indisputable.
On the other hand, it may seem anti-intuitive that ALL of the relatively advanced groups over the world, including 北京人, got distinct, except one in Africa that eventually spread through out the globe. Isn't it more likely for some local groups to survive, compared to some branches of a single group that migrated through vast distances and most harsh environment, say from Africa all the way to Asia?
Also, there does not seem to have any fossil evidence to support the common ancestor theory, which is only due to the DNA evidence. Just read a pretty new book in Chinese history, which mentioned both theories as possibilities.
- Re: 希尔伯特:数学世界的亚历山大posted on 01/13/2009
NTY Jan 12, 2007: Skull Supports Theory of Human Migration - posted on 01/13/2009
Again, I strongly recommend everybody to try the test from National Geographic to see your own ancetral migration pattern ( https://genographic.nationalgeographic.com/genographic/participate.html). Give an anonymous sample to them, and they will tell you if your fraternal ancestry (if you are a man), or through mtDNA your maternal ancestry and their migration patterns. In China, most people are descendants of two groups: one came from north on land, the other came from south sea faring.
浮生 wrote:
NTY Jan 12, 2007: http://www.nytimes.com/2007/01/12/science/12skull.html">Skull Supports Theory of Human Migration - posted on 01/14/2009
tar wrote:
When it comes down to scientific evidence, there is no more "scientific" then DNA evidence. As far as I can find, there are no reputable SCIENTISTS who don't believe the out-of-Africa theory. And I don't know what you mean by the lack of fossil evidence. I only heard of critics of evolution theory citing the lack of "gaps" in fossil remains to assail evolution. If you look at the reproduction of facial rendering of both Neanderthal and Peking Man fossils, they are clearly more apelike than our common ancestor.
Are there fossil evidences somewhere along the migration routes from Africa to Europe or Asia? Of course the DNA evidence is scientific, but it will be more convincing if some fossil evidences of those migrating groups can be found along the path of these long journeys, just like those for the Peking man.
As far as species became extinct, it happened all the time. Not all monkeys and apes evolve to become human beings. Half-ape half-man species like Neanderthal and Peking Man's extinction is just part of the evolution process, totally on par for the course.
Of course it is natural that some species get extinct. But it is interesting to ask the question why ALL of the large amount of fossil evidences discovered over the last century or two are from those human-to-be hopeful groups which all got extinct. I would expect some fossil evidences of the only surviving group outside Africa along their journeys, which should have taken thousands of years. And finding such fossil evidences of some relatively recent events 50,000 years ago should be more probable than finding those much older, such as Peking man over 500,000 years ago.
I wouldn't give too much credence to some Chinese authors who still cling to the belief that the Chinese race is special. We are just part of the humanity like the rest of the peoples, no more no less.
No need. I wouldn't either. And no need to convince me regarding the out-of-Africa theory, as I know it is scientifically convincing. I just wonder why it is not more intuitively convincing, in terms of common sense.
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