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TajinKuo
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Wed, May 23, 2012 11:39:42 AM
Ramanujan
From:
Tajin Kuo <tajin-kuo@att.net>
To: Tajin Kuo <tajin-kuo@att.net>
上承#31.
1968, 我到PSU的数学系去作论文答辩. (这是外系) 到得早, 我就在会议室隔壁的书架上取下了厚厚的一本 Hardy 所著的"Ramanujan" (1940), 因此得窥其全貌. 饶是Hardy使尽全力, 还是扭不过 Ramanujan, 说不动他去遵循数学的规范, 用一步步的逻辑推理来取得数学界认可的证明. 这句话看来无非是很一般的常识, 可说是卑之无甚高论. 细看却又不然. 请看下列所引
"Strictly speaking, all our knowledge outside mathematics and demonstrative logic (which is, in fact, a branch of mathematics) consists of conjectures. ..... There are highly respectable and reliable conjectures as those expressed in certain general laws of physical science. There are other conjectures, neither reliable nor respectable .....
We secure our mathematical knowledge by demonstrative reasoning, but we support our conjectures by plausible reasoning. A mathematical proof is demonstrative reasoning, but the inductive evidence of the physicist, the circumstantial evidence of the lawyer, the documentary evidence of the historian, and the statistical evidence of the economist belong to plausible reasoning." (Preface, in G Polya's "Induction and Analogy in Mathematics", 1954)
这就牵涉到数学证明和物理实验/观察"验证"的关系和区分的根本问题, 其实, 也和理性力学/热力学与非理性力学/热力学的取舍之辩有些关联. 一年半之前, 我向曹广福教授请教这方面的看法, 叫它"数学证明"和"物理(化学, 生物 ....)征信", 见刊于[18], [19], [9]:
http://blog.sciencenet.cn/blog-40247-377797.html
又, 更深一层看, 什么叫充分完备的数学证明? Poincare说, Klein在提出一个拓朴命题时, 引用了电磁学的直观, 便算是证明了事(Poincare並无贬义). Hermite和Picard 屡屡抱怨Poincare论文中欠缺完备的证明程序,不让他教数学, 只准他教数学物理和天文学.(p. 195, Bulletin, American Mathematical Society, April, 1994) Perelman提出了Poincare的根本拓朴猜想解法(及Thurston的几何化落实). 他自認为已经证明完毕. 数学界却以为是高度可信的纲要而已. 这又折腾了一阵子, 才颁了奖給他, 又没的挑起了几宗学界顶尖人士之间的官司.
然而, Perelman尽管高傲, 却还有个高度可信的纲要可循; Ramanujan连个程序纲要都没有. 这有相无相之间, 思维风格的差别还是根本的.
*____________________________________________________________________________________ *
又, Ramanujan 的j该念"硬"j, 如英语念法, 不念"软"j, 如德语念法. 所以中译本的译音该修订. 正音请循Wikipedia "Ramanujan"条.
2012-05-24 02:41
Ramanujan
From:
Tajin Kuo <tajin-kuo@att.net>
To: Tajin Kuo <tajin-kuo@att.net>
上承#31.
1968, 我到PSU的数学系去作论文答辩. (这是外系) 到得早, 我就在会议室隔壁的书架上取下了厚厚的一本 Hardy 所著的"Ramanujan" (1940), 因此得窥其全貌. 饶是Hardy使尽全力, 还是扭不过 Ramanujan, 说不动他去遵循数学的规范, 用一步步的逻辑推理来取得数学界认可的证明. 这句话看来无非是很一般的常识, 可说是卑之无甚高论. 细看却又不然. 请看下列所引
"Strictly speaking, all our knowledge outside mathematics and demonstrative logic (which is, in fact, a branch of mathematics) consists of conjectures. ..... There are highly respectable and reliable conjectures as those expressed in certain general laws of physical science. There are other conjectures, neither reliable nor respectable .....
We secure our mathematical knowledge by demonstrative reasoning, but we support our conjectures by plausible reasoning. A mathematical proof is demonstrative reasoning, but the inductive evidence of the physicist, the circumstantial evidence of the lawyer, the documentary evidence of the historian, and the statistical evidence of the economist belong to plausible reasoning." (Preface, in G Polya's "Induction and Analogy in Mathematics", 1954)
这就牵涉到数学证明和物理实验/观察"验证"的关系和区分的根本问题, 其实, 也和理性力学/热力学与非理性力学/热力学的取舍之辩有些关联. 一年半之前, 我向曹广福教授请教这方面的看法, 叫它"数学证明"和"物理(化学, 生物 ....)征信", 见刊于[18], [19], [9]:
http://blog.sciencenet.cn/blog-40247-377797.html
又, 更深一层看, 什么叫充分完备的数学证明? Poincare说, Klein在提出一个拓朴命题时, 引用了电磁学的直观, 便算是证明了事(Poincare並无贬义). Hermite和Picard 屡屡抱怨Poincare论文中欠缺完备的证明程序,不让他教数学, 只准他教数学物理和天文学.(p. 195, Bulletin, American Mathematical Society, April, 1994) Perelman提出了Poincare的根本拓朴猜想解法(及Thurston的几何化落实). 他自認为已经证明完毕. 数学界却以为是高度可信的纲要而已. 这又折腾了一阵子, 才颁了奖給他, 又没的挑起了几宗学界顶尖人士之间的官司.
然而, Perelman尽管高傲, 却还有个高度可信的纲要可循; Ramanujan连个程序纲要都没有. 这有相无相之间, 思维风格的差别还是根本的.
*____________________________________________________________________________________ *
又, Ramanujan 的j该念"硬"j, 如英语念法, 不念"软"j, 如德语念法. 所以中译本的译音该修订. 正音请循Wikipedia "Ramanujan"条.
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